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Nguetseng’s Two-scale Convergence Method For Filtration and Seismic Acoustic Problems in Elastic Porous Media Abstract. A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results, we derive Biot’s equations of poroelasticity, equations of viscoelasticity, or decoupled system consisting of non-isotropic Lamé’s equations and Darcy’s system of filtration, depending on ratios between physical parameters. The proofs are based on convergence method of homogenization in periodic structures. Key words: Biot’s equations, Stokes equations, Lamé’s equations, two-scale convergence, homogenization of structures, poroelasticity, viscoelasticity. In this article a problem of modelling of small perturbations in elastic deformable medium, perforated by a system of channels (pores) filled with liquid or gas, is considered. Such media are called elastic porous media and they are a rather good approximation to real consolidated grounds. In present-day literature, the field of study in mechanics corresponding to these media is called poromechanics . The solid component of such a medium has a name of skeleton, and the domain, which is filled with a fluid, is named a porous space. The exact mathematical model of elastic porous medium consists of the classical equations of momentum and mass balance, which are stated in Euler variables, of the equations determining stress fields in both solid and liquid phases, and of an endowing relation determining behavior of the interface between liquid and solid components. The latter relation expresses the fact that the interface is a material surface, which amounts to the condition that it consists of the same material particles all the time. Denoting by the density of medium, by the velocity, by the stress tensor in the liquid component, by the stress tensor in the rigid skeleton, and by the characteristic (indicator) function of porous space, we write the fundamental differential equations of the nonlinear model in the form where stands for the material derivative with respect to the time variable. Clearly the above stated original model is a model with an unknown (free) boundary. The more precise formulation of the nonlinear problem is not in focus of our present work. Instead, we aim to study the problem, linearized at the rest state. In continuum mechanics the methods of linearization are developed rather deeply. The so obtained linear model is a commonly accepted and basic one for description of filtration and seismic acoustics in elastic porous media (see, for example, [2, 3, 4]). Further we refer to this model as to model A. In this model the characteristic function of the porous space is a known function for . It is assumed that this function coincides with the characteristic function of the porous space , given at the initial moment. Being written in terms of dimensionless variables, the differential equations of the model involve frequently oscillating non-smooth coefficients, which have structures of linear combinations of the function . These coefficients undergo differentiation with respect to and besides may be very big or very small quantities as compared to the main small parameter . In the model under consideration we define as the characteristic size of pores divided by the characteristic size of the entire porous body: Denoting by the dimensionless displacement vector of the continuum medium, in terms of dimensionless variables we write the differential equations of model A as follows: Here and further we use the notation From the purely mathematical point of view, the corresponding initial-boundary value problem for model A is well-posed in the sense that it has a unique solution belonging to a suitable functional space on any finite temporal interval. However, in view of possible applications, for example, for developing numerical codes, this model is ineffective due to its sophistication even if a modern supercomputer is available. Therefore a question of finding an effective approximate models is vital. Since the model involves the small parameter , the most natural approach to this question is to derive models that would describe limiting regimes arising as tends to zero. Such an approximation significantly simplifies the original problem and at the same time preserves all of its main features. But even this approach is too hard to work out, and some additional simplifying assumptions are necessary. In terms of geometrical properties of the medium, the most appropriate is to simplify the problem postulating that the porous structure is periodic. Further by model we will call model A supplemented by this periodicity condition. Thus, our main goal now is a derivation of all possible homogenized equations in the model . The first research with the aim of finding limiting regimes in the case when the skeleton was assumed to be an absolutely rigid body was carried out by E. Sanchez-Palencia and L. Tartar. E. Sanchez-Palencia [3, Sec. 7.2] formally obtained Darcy’s law of filtration using the method of two-scale asymptotic expansions, and L. Tartar [3, Appendix] mathematically rigorously justified the homogenization procedure. Using the same method of two-scale expansions J. Keller and R. Burridge derived formally the system of Biot’s equations from model in the case when the parameter was of order , and the rest of the coefficients were fixed independent of . It is well-known that the various modifications of Biot’s model are bases of seismic acoustics problems up-to-date. This fact emphasizes importance of comprehensive study of model A and model one more time. J. Keller and R. Burridge also considered model under assumption that all the physical parameters were fixed independent of , and formally derived as the result a system of equations of viscoelasticity. Under the same assumptions as in the article , the rigorous justification of Biot’s model was given by G. Nguetseng and later by A. Mikelić, R. P. Gilbert, Th. Clopeaut, and J. L. Ferrin in [4, 6, 7]. Also A. Mikelić et al derived a system of equations of viscoelasticity, when all the physical parameters were fixed independent of . In these works, Nguetseng’s two-scale convergence method [8, 10] was the main method of investigation of the model . In the present work by means of the same method we investigate all possible limiting regimes in the model . This method in rather simple form discovers the structure of the weak limit of a sequence as , where and sequences and converge as merely weakly, but at the same time function has the special structure with being periodic in . Moreover, Nguetseng’s method allows to establish asymptotic expansions of a solution of model in the form where is a solution of the homogenized (limiting) problem, is a solution of some initial-boundary value problem posed on the generic periodic cell of the porous space, and exponent is defined by dimensionless parameters of the model. Distinct asymptotic behavior of these parameters and distinct geometry of the porous space lead to different limiting regimes, namely, to various forms of Darcy’s law for velocity of liquid component and of non-isotropic Lamé’s equations for displacement of rigid component in cases of big parameter , also, to various forms of Biot’s system in cases of small parameter , and to different forms of equations of viscoelasticity in cases when parameters and are as . For example, in the case when the velocity of liquid component and the displacement of rigid skeleton possess the following asymptotic: At the same time all equations are determined in a unique way by the given physical parameters of the original model and by geometry of the porous space. For example, in the case of isolated pores (disconnected porous space) the unique limiting regime for any combinations of parameters is a regime described by the non-isotropic system of Lamé’s equations. On our opinion, the proposed approach, when the limiting transition in all coefficients is fulfilled simultaneously, is the most natural one. We emphasize that it is not assumed starting from the original model that the fluid component is inviscid, the porous skeleton is absolutely rigid, or either of the components is incompressible. These kinds of properties arise in limiting models depending on limiting relations, which involve all parameters of the problem. The articles and , as well as the present one, show in favor of such a uniform approach, because they exhibit the situations, when various rates of approach of the small parameter to zero yield distinct homogenized equations. Moreover, these equations differ from the homogenized equations derived as the limit of model under assumption , imposed even before homogenization . Suppose that all dimensionless parameters of the depend on the small parameter of the model and there exist limits (finite or infinite) We restrict our consideration by the cases when and either of the following situations has place. If then, re-normalizing the displacement vector by setting we reduce the problem to one of the cases (I)–(III). In the present paper we show that in the case the homogenized equations have various forms of Biot’s system of equations of poroelasticity for a two-velocity continuum media or non-isotropic Lam’e’s system of equations of one-velocity continuum media (for example, for the case of disconnected porous space)(theorem 2.2). In the case the homogenized equations are different modifications of Darcy’s system of equations of filtration for the velocity of the liquid component (and as a first approximation the solid component behaves yourself as an absolutely rigid body) and as a second approximation – non-isotropic Lam’e’s system of equations for the re-normalized displacements of the solid component or Biot’s system of equations of poroelasticity for the re-normalized displacements of the liquid and solid components(theorem 2.3). Finally, in the case they are the non-local viscoelasticity equations or non-isotropic non-local Lam’e’s system of equations of one-velocity continuum media (theorem 2.4). §1. Models A and 1.1. Differential equations, boundary and initial conditions. Let a domain of the physical space be the union of a domain occupied with the rigid porous ground and a domain corresponding to hollows (pores) in the ground. Domain is called a porous space and is assumed to be filled with liquid or gas. Denote by the displacement vector of the continuum medium (of ground or liquid or gas)) at the point in Euler coordinate system at the moment of time . Under an assumption that the displacement vector is small in , which amounts to a case when deformations are small, dynamics of rigid phase is described by linear Lamé’s equations and dynamics of fluid or gas is described by Stokes equations. At the same time we may set that the velocity vector in fluid or gas is a partial derivative of the displacement vector with respect to time variable, i.e., that . This assumption makes perfect sense in description of continuous media in domains, where the characteristic size of pores is very small as compared to the diameter of the domain , i.e., and (see, for example, [2, 3, 4] and recall the observation of the linearization procedure for the exact nonlinear model in Introduction). In terms of the dimensionless variables, not denoted by the asterisk below, the displacement and the pressure of fluid and the displacement of rigid skeleton satisfy the system of Stokes equations and the system of Lamé’s equations In (1.1)–(1.6) is the given vector of distributed mass forces, is the characteristic macroscopic size – the diameter of the domain , is characteristic duration of physical processes, is mean density of air under atmosphere pressure, and are respectively mean dimensionless densities of liquid and rigid phases, correlated with mean density of air, is the value of acceleration of gravity, and is atmosphere pressure. Dimensionless constants are defined by the formulas where is the viscosity of fluid or gas, is the bulk viscosity of fluid or gas, and are elastic Lamé’s constants, and is a speed of sound in fluid. For the unknown functions , , and , the commonly accepted conditions of continuity of the displacement field and normal tensions are imposed on the interface between the two phases (see, for example, [2, 3, 4]): In (1.8) is the unit normal vector to . Note that exactly these conditions appear as the result of linearization of the exact nonlinear model. Finally, system (1.2)–(1.6), (1.8)–(1.9) is endowed by giving a displacement field on and at the moment and by giving a velocity field at the . Further without loss of generality, with the aim to simplify the technical outline, we suppose that these boundary conditions are homogeneous. 1.2. Geometry of porous space. In model A a Lipschitz smoothness of the interface between porous space and rigid skeleton is the only restriction on geometry of porous space. In model the porous medium has geometrically periodic structure. Its formal description is as follows [4, 11]. Firstly a geometric structure inside a pattern unit cell is defined. Let be a ‘solid part’ of the cell . The ’liquid part’ is its open complement. Set , , the translation of on an integer-valued vector . Union of such translations along all , is the 1-periodic repetition of all over . Let be the open complement of in . The following assumptions on geometry of and are (i) is an open connected set of strictly positive measure with a Lipschitz boundary, and also has strictly positive measure on . (ii) and are open sets with -smooth boundaries. The set is locally situated on one side of the boundary , and the set is locally situated on one side of the boundary and connected. Domains and are intersections of the domain with the sets and , where the sets and are periodic domains in with generic cells and of the diameter , respectively. Union is the closed cube , and the interface is the -periodic repetition of the boundary all over . Further by we will denote the characteristic function of the porous space. For simplicity we accept the following constraint on the domain and the parameter . Domain is cube, , and quantity is integer, so that always contains an integer number of elementary cells . Under this assumption, we have where is the characteristic function of in . We say that a porous space is disconnected (isolated 1.3. Generalized solutions in models A and . Define the displacement in the whole domain by the formula and the pressures , , and by formulas Thus introduced new unknown functions should satisfy the system where . If , then . Equations (1.13) are understood in the sense of distributions theory. They involve the both equations (1.2) and (1.5) in the domains and , respectively, and the boundary conditions (1.8) and (1.9) on the interface . There are various equivalent in the sense of distributions forms of representation of equations (1.13). In what follows, it is convenient to write them in the form of the integral equality where is an arbitrary smooth test function, such that at the and on the boundary of the domain , . In (1.14) by we denote the convolution (or, equivalently, the inner tensor product) of two second-rank tensors along the both indexes, i.e., . Here the assumption, that the boundary and initial conditions are homogeneous, is not essential. Let the interface between and be piece-wise continuously differentiable, parameters , , , , , , , and be strictly positive, and assume that . Due to linearity of the problem, justification of Lemma 1.1 reduces to verification of bounds (1.17). These appear by means of differentiation of Eqs. (1.13) with respect to (note that and do not depend on ), multiplication of the resulting equation by , and integration by parts. Pressures and are estimated directly from Eqs. (1.12). Further the focus of this article is centered solely on model , in which coefficients of Eqs. (1.12) and (1.13) depend continuously on the small parameter , and is a corresponding generalized solution. We aim to find out the limiting regimes of the model as . §2. Formulation of the main results Suppose additionally that there exist limits (finite or infinite) In what follows we assume that Dimensionless parameters in the model satisfy restrictions All parameters may take all permitted values. For example, if or , then all terms in final equations containing these parameters disappear. Assume that conditions of Lemma 1.1 hold and that is a generalized solution in model . The following assertions are valid: where and is a constant independent of . with re-normalized parameters then for displacements hold true estimates (2.2) and under condition for the pressures and in the liquid component hold true estimates (2.3). If instead of restriction (2.4) hold true conditions where and is a constant independent of . These last estimates imply (2.3). and sequences and are uniformly bounded with respect to in . Assume that the hypotheses in Theorem 2.1 hold, and Then functions admit an extension from into such that the sequence converges strongly in and weakly in to the functions . At the same time, sequences , , , and converge weakly in to , , , and , respectively. The following assertions for these limiting functions hold (I) If or the porous space is disconnected (a case of isolated pores), then and the functions , , , and satisfy in the domain the following initial-boundary value problem: (II) If , then the weak limits , , , , of sequences , , , , satisfy the initial-boundary value problem consisting of the balance of momentum equation and Darcy’s law in the form in the case and , Darcy’s law in the form in the case and , and, finally, Darcy’s law in the form in the case and . This problem is endowed with initial and boundary conditions (2.10) and the boundary condition for the velocity of the fluid component. Assume that the hypotheses in Theorem 2.1 hold, and that (I) If and one of conditions (2.4) or (2.5) holds true, then sequences , and converge weakly in to , , and respectively. The functions admit an extension from into such that the sequence converges strongly in and weakly in to zero and 1) if and , then functions , and solve in the domain the problem , where 2) if and , then functions , and solve in the domain the problem , where satisfies Darcy’s law in the form and pressures and satisfy equations (2.18); 3) if and , then functions , and solve in the domain the problem , where satisfies Darcy’s law in the form and pressures and satisfy equations (2.18). Problems – are endowed with boundary condition (2.16). (II) If and conditions (2.5) hold true, then the sequence converges strongly in and weakly in to function and the sequence converges weakly in to the function . The limiting functions and satisfy the boundary value problem in the domain where the function is referred to as given. It is defined from the corresponding of Problems – (the choice of the problem depends on and ). The symmetric strictly positively defined constant fourth-rank tensor , matrices and and constants and are defined below by formulas (5.30), (5.32) - (5.33), in which we have

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What is the Ratio (rate)? The relation between two quantities which displays how much greater one quantity is than another is called ratio. Ratios (rates) are related to everyday life. For example, the speed of a bike is a rate. The amount of simple interest paid each month is a rate. There are 30 students in the classroom, of which 12 are boys and 18 are girls. The ratio of boys to girls is . That is 2:3. The ratio of girls to boys is . That is 3:2 The ratio of girls to all students is . That is 3:5. The ratio of boys to all students is . That is 2:5. What is the Rate of Change? The rate of change is the speed at which a variable changes from one place to another place over a specific period of time. The rate of change is usually expressed as a relation between the change in one quantity corresponding to a change in another. Graphically, the rate of change is determined by the slope of a line. The rate of change is often represented by the Greek letter delta. The understanding of the rate of change from one quantity to another is of major importance to the study of both integral and differential calculus. The rate of change is not only used in the mathematics branch, it is also used in physics, chemistry, economics, and finance branch. Formulas for Rate of Change Rate of change is a rate that defines how the change in one variable is related to the change in other variables. From the above figure, it is shown that how much the height of the tree increased with the increase in a year (time). Here year (time) is the independent variable and the tree’s height is the dependent variable. The increase in height is dependent on the change in time. If x is an independent variable and y is a dependent variable then the rate of change is If x and y are dependent variables and s is an independent variable, and the two variables x and y change with respect to s then the rate of change of y with respect to x will be, Graphically, the rate of change is represented as the slope of the curve. From the graph, the rate of change is . This is also called the slope of the line. From the graph, the increment in x value causes an increment in the y value. So the rate of change is positive. If the increment in x value causes a decrement in the y value, then the rate of change is negative. If the increment in the x value causes no change in the y value, then the rate of change is zero. Average Rate of Change The average rate of change of a function f on the interval [a, b] is defined as . The price of petrol increased by $3.50 from 2014 to 2021. Find the average rate of change? The price increment is $3.50. The rate of change = The price of petrol increased by about $0.5 per year. How is the value of y changing between the points (2, 4) and (4, 8)? Here, (x1, y1) = (2, 4) and (x2, y2) = (4, 8) The rate of change = There is 2 units change in y value per unit change of x value. Let x be the weight of the object and y be the length of the spring. If the weight of the object increased by Δx, let the amount of change produced in the length of the spring as Δy. So, the rate of change is . A particle has a position ‘x’ at the time ‘t’, i.e., the position of the particle is x(t). This is called displacement. The rate of change of the particle’s position ‘x’ with time ‘t’ is known as the velocity ‘v’ of the particle. That is, the rate of change of displacement is called velocity. The rate of change of velocity ‘v’ is called the acceleration ‘a’ of the particle. Instantaneous Rate of Change The instantaneous rate of change is the change in the rate at a specific instant, and it is equal to the derivative value at that specific point. For the function , the instantaneous rate of change at is calculated as: The instantaneous rate of change at is 20 units. The formula for instantaneous rate of change = Marginal cost and revenue are used to determine the volume of production and the price per unit of a product that will get the best out of profits. The marginal cost of production determines the variation in the entire price of a good that rises from manufacturing one supplementary unit of that good. The marginal cost (MC) is calculated by dividing the variation (Δ) in the entire price (C) by the variation in quantity (Q). By calculus, the marginal cost is computed by taking the first derivative of the entire price function with respect to the quantity. Price Rate of Change The rate of change is used to find the change in price in a particular period. This is known as the price rate of change. The price rate of change is the price of a product at time B minus the price of the same product at time A and dividing that result by the price at time A. Price rate of change For example, the rate of gold is $54 today and five days before the rate was $50, then the price rate of change is In five days rate is increased 8%. Application of Rate of Change - To find a new value of a quantity from the old value and total change. - To find the movement of the particle, velocity (speed), and acceleration of a particle moving along a straight line. - To calculate the upcoming population from the current population growth rate. - To compute marginal cost and revenue in a business situation. - The rate of change is used to find the exchange rate, inflation rate, interest rate, price-earnings ratio, rate of return, tax rate, unemployment rate, and wage rate. - If x is an independent variable and y is a dependent variable then the rate of change is . - If x and y are dependent variables and s is an independent variable, and the two variables x and y changes with respect to s then the rate of change of y with respect to x will be, . - If and are two points on the graph of a line, then the rate of change is . - The average rate of change of a function f on the interval [a, b] is defined as . - The formula for instantaneous rate of change = . Context and Applications This topic is significant in the professional exams for both undergraduate and graduate courses, especially for Want more help with your calculus homework? *Response times may vary by subject and question complexity. Median response time is 34 minutes for paid subscribers and may be longer for promotional offers. Rate of Change Homework Questions from Fellow Students Browse our recently answered Rate of Change homework questions.

Throughout this semester, I have analyzed the financial statements for Avon Products, Inc. The one area where Avon needs improvement is reducing its inventory levels. Is there a resource where I can find an average industry percentage for inventory holding costs? A total of 80,000 units were sold during the first quarter. The current cost per unit was $2.10 on December 31, 2000 and $2.40 on March 31, 2001 Use the current cost basis, compute the first quarter of 2001 1. Ending Inventory 2. Cost of Goods Sold Calculations Items Units Price See attached file. Problem: Tori Amos Corporation began operations on December 1, 2006. The only inventory transactions in 2006 was the purchase of inventory on December 10, 2006 at a cost of $20 per unit. None of this inventory was sold in 2006. Relevant information is as follows. Ending inventory units December 31, Is the increase in sales related to the increase in inventory? Year Inventories Net Sales 1991 $378 1,812 1992 $411 1,886 1993 $452 1,954 1994 $491 2,035. See the attached file. Allocating costs in a Process Costing System. Moreno Corporation, a manufacturer of diabetic testing kits, started November production with $75,000 in beginning inventory. During the month, the company incurred $420,000 of materials cost and $240,000 of labor cost. It applied $165,000 of overhead co Passage for Questions 1-4: The direct labor rate for McGregor's Company is $9.00 per hour, and manufacturing overhead is applied to products using a predetermined overhead rate of $6.00 per direct labor hour. During May, the company purchased $60,000.00 in raw materials (all direct materials) and worked 3,200 direct labor hours. At the beginning of 2005, the C. Eaton Company had the following balances in its accounts: Cash $6,500 Inventory $9,000 Retained Earnings $15,500 During 2005, the company experienced the following events. 1. Purchased inventory with a list price of $3,000 on account from Ardmore Farm and Seed has an inventory dilemma. It has been selling a brand of very popular insect spray for the past year. It has never really analyzed the costs incurred from ordering and holding the inventory and currently faces a large stock of the insecticide in the warehouse. Ardmore estimates that it costs $25 to place an Please Help. I am studying and don't understand the following exercises... E9-4 (Lower-of-Cost-or-Market?Journal Entries) Corrs Company began operations in 2007 and determined its ending inventory at cost and at lower-of-cost-or-market at December 31, 2007, and December 31, 2008. This information is presented below. FIFO and LIFO?Periodic and Perpetual) The following is a record of Pervis Ellison Company's transactions for Boston Teapots for the month of May 2007. May 1 Balance 400 units @ $20 May 10 Sale 300 units @ $38 12 Purchase 600 units @ $25 20 Sale 540 units @ $38 28 Purchase 400 units @ $30 Instructions Assuming that Aug1 Aug 31 Raw Mat Inventory 6592 WIP Inventory 12,731 Finished Goods Inventory 21,726 31,313 Sales 31,313 Manu Ov 40,366 Dir Labor 71,180 Purchase Raw Mat 77,308 Adm Expenses 36,793 COGM 193,132 Raw Mat Used in Pro 73,957 Selling Expenses 12,455 Dave's Electronics had the following inventory transactions during January: Jan. 1: Beginning Inventory 1,500 units @ $9 each = $13,500 Jan. 15: Purchase 2,000 units @ $8 each = $16,000 Jan. 21: Sold 2,700 units @ $12 each Jan. 22: Purchase 3,000 units @ $7 each = $21,000 Jan. 30: Sold 2,000 u Grant Company began the year with $870,000 of raw materials inventory, $1,390,000 of work-in-progress inventory, and $620,000 of finished goods inventory. During the year, the company purchased $3,550,000 of raw material and used $3,720,000 of raw materials in production. Labor used in production for the year was $2,490,000. Ove The Dance Company sells ballet shoes. It began in 20X6 with a beginning inventory of 1,000 shoes at a cost of $10 each and made the following purchases during the year: February 7 Purchased 3,500 shoes @ $11.50 each May 19 Purchased 4,700 shoes @ $12.00 each September 3 Purchased 2,300 shoes @ $13.00 each The ending in Mark Knight, owner of Knight Company, is reviewing the quarterly financial statements and thinks the cost of goods sold is out of line with past years. The following historical data is available for 2009 and 2010: 2009: 2010: Net Sales $140,000 $200,000 Cost of goods sold 6 The accounting records of Brooks Photography, Inc., reflected the following balances as of January 1, 2012: Cash $19,000 Beginning Inventory 6,750 (75 units $90) Common Stock 7,500 Retained Earnings 18,250 The following five transactions occurred in 2012: 1. First purchase (cash) 100 units @ $92 2. Second purchase (ca IN CLASS PROBLEMS: Class, the following data applies to all four problems. Good Luck with them! . The Textile Corporation has an inventory conversion period of 45 days, a receivables collection period of 36 days, and payables deferral period of 35 days. . 1) What is the length of the firm's cash conversion cycle? . 2) If Please see the attached file. 1. Dane, Inc., owns 35% of Marin Corporation. During the calendar year 2007, Marin had net earnings of $300,000 and paid dividends of $30,000. Dane mistakenly recorded these transactions using the fair value method rather than the equity method of accounting. What effect would this have on the in 1. Which of the following is ordinarily considered "extended procedure" in external auditors' independent audits of financial statements? A. Send positive confirmations on recorded customer accounts receivable balances. B. Perform physical observation and test count during the client's inventory taking. C. Measure the tim This should not take more than half an hour to complete. 1. Canal Street Financing Corporation needs to borrow long term funds but would prefer not to show more than $ 100 million in face amount of debt outstanding. It also prefers to pay an annual coupon, in the European style, of not more than 6% per annum. Canal's banker Assuming a 360-day year, claculate what average investment in inventory would be for a firm, given the following information in each case. A.) The firm has sales of 600,000, a gross profit margin of 10 percent, and an inventory trunover ratio of 6. B.) The firm has a cost-of-goods-sold figure of $480,000 and an average age Pale Company was established on January 1, 20X1. Along with other assets, it immediately purchased land for $80,000, a building for $240,000, and equipment for $90,000. On January 1, 20X5, Pale transferred these assets, cash of $21,000, and inventory costing $37,000 to a newly created subsidiary, Bright Company, in exchange for 10,000 shares of Bright's $6 par value stock. Pale uses straight-line depreciation and useful lives of 40 years and 10 years for the building and equipment, respectively, with no estimated residual values. Pale Company was established on January 1, 20X1. Along with other assets, it immediately purchased land for $80,000, a building for $240,000, and equipment for $90,000. On January 1, 20X5, Pale transferred these assets, cash of $21,000, and inventory costing $37,000 to a newly created subsidiary, Bright Company, in exchange for The Lampley Company has 2,000 obsolete items in its inventory which are valued at $22 each. If the item are reworked they could be sold for $30 each otherwise they would be sold for only $5 each. If Lampley Company decides to re-work the items, how much should the company be willing to invest to ensure that they would at least b Higgins Athletic Wear has expected sales of 22,500 units a year, carrying costs of $1.50 per unit, and an ordering cost of $3 per order. a) What is the economic order quantity? b) What will be the average inventory? The total carrying cost? c) Assume an additional 30 units of inventory will be required as safety stock. Wha Discuss three systems for controlling inventory and the advantages and disadvantages of each. Identify the items that are included in merchandise inventory. (Address the special situations of goods in transit, consigned goods, and damaged goods.) Please see the attached Income statement. The firm uses FIFO inventory accounting. a) Assume in 2009 the same 10,000-unit volume is maintained, but that the sales price increases by 10 percent. Because of FIFO inventory policy, old inventory will still be charged off at $10 per unit. Also assume that seliing and administrativ Questions: Accrued revenue distortion, plant assets, adjusting entries, CPA president, inventory turnover, self-constructed assets 1. How does failure to record accrued revenue distort the financial reports? If an annual financial report is in error (distorted) what action should be taken? 2. What are the major characteristics of plant assets? Are plant assets necessarily confined to a manufacturing plant? 3. Why is it necessary to make adjusting Below is selected data for Gertup Corporation as of 12/31/05: Total assets $ 5,500 Current assets 2,750 Long-term debt 450 Current ratio 2.5 Inventory 1,500 For year ended 12/31/05 Sales $20,000 Cost of goods sold 16,000 Gertup has maintained the same inventory levels throughout 2005. If end of year inventory Below is selected data for Gertup Corporation as of 12/31/05: Total assets $ 5,500 Current assets 2,750 Long-term debt 450 Current ratio 2.5 Inventory 1,500 For year ended 12/31/05 Sales $18,500 Cost of goods sold

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area? Can you maximise the area available to a grazing goat? If you move the tiles around, can you make squares with different coloured edges? What is the same and what is different about these circle questions? What connections can you make? A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink. Have a go at creating these images based on circles. What do you notice about the areas of the different sections? Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this? Can you find rectangles where the value of the area is the same as the value of the perimeter? The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius? A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . . Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers? On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares? A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius. How many winning lines can you make in a three-dimensional version of noughts and crosses? Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money. In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays? A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral. Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle? Can you find an efficient method to work out how many handshakes there would be if hundreds of people met? Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? Explore the effect of reflecting in two parallel mirror lines. How many different symmetrical shapes can you make by shading triangles or squares? Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes". Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? What size square corners should be cut from a square piece of paper to make a box with the largest possible volume? Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37. Can you describe this route to infinity? Where will the arrows take you next? Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not? If you have only 40 metres of fencing available, what is the maximum area of land you can fence off? Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter? Investigate how you can work out what day of the week your birthday will be on next year, and the year after... Start with two numbers and generate a sequence where the next number is the mean of the last two numbers... An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height? Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps? Different combinations of the weights available allow you to make different totals. Which totals can you make? There are lots of different methods to find out what the shapes are worth - how many can you find? What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres? How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter? What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position? A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r? A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle? Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same? Explore the effect of combining enlargements. Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible? Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or... A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed . Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

This action might not be possible to undo. Are you sure you want to continue? 2 hours 30 minutes 9 JUNE 1999 Additional materials: Answer paper Electronic calculator Geometrical instruments Graph paper (2 sheets) Mathematical tables (optional) 2 hours 30 minutes INSTRUCTIONSTO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper/ answer booklet. Answer all questions. Write your answers and working on the separate answer paper provided. All working must be clearly shown. It should bedone on the same sheet as the rest of the answer. Marks will be given for working which shows that you know how to solve the problem even if you get the answer wrong. If you use more than one sheet of paper, fasten the sheets together. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 130. Electronic calculators should be used. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. use For T , either your calculator value or 3.142. This question paper consists of 8 printed pages. MFK (0716) QF91683 0 UCLES 1999 [31 2 topicEquationsandInequalities topicMensuration For a certain type of tree. use either your calculator value or 3. (a) How many members voted? (b) There were 14 760 members who did not vote. Calculate the year in which the diameter of its trunk will be one metre. 5 ~ where C is the circumference in centimetres and y is the age of the tree in years. [The cross-section of the tree trunk is a circle. For IT. PI [31 (b) Find the radius of the trunk of a 20 year old tree. (d) A three year old tree was planted in 1971. The ratio ‘Yes’ votes : ‘No’ votes is 7 : 5. (c) The cross-sectional area of a tree trunk is 1200 cm2% Find (i) the radius of the tree. They receive 48 790 ‘Yes’ votes. What percentage of members did not vote? PI (c) To build the new stadium.142. Will the new stadium be built? Show working to explain your answer.1 (a) Estimate the age of a tree with a circumference of 100 cm. to the nearest year. 50% of the total number of members have to vote ‘Yes’.2 I topicRatioproportionrate topicPercentages A football club asks all its members to vote ‘Yes’ or ‘No’ for a new stadium. C = 2 . (ii) the age of the tree. [41 . BC = 7 cm. Calculate the length of A C correct to 5 significant figures.62 cm and the cosine rule to calculate the length of BA. CH is perpendicular to A E . Show that it rounds to 14.3 topicMensuration topicTrigonometry C NOT TO SCALE A C E is an isosceles triangle. C NOT TO SCALE Calculate the unshaded area. (ii) Find the area of triangle ABC. [41 [31 (c) Triangles ABC and CDE are folded over onto triangle ACE. Angle CAH = 70" and A H = 5 cm. angle BCA = angle ECD = 20' and angle CAE = 70". (i) Use A C = 14. as shown on the diagram below. PI [Turn over .62 cm. Pentagon ABCDE is formed from the isosceles triangle A C E together with congruent triangles ABC and EDC. concerts have to be postponed until the next day... - no wind (b) When it is wet and windy. PI ...... wet < wind . no wind wind dry . (ii) takes place on Tuesday. the probability that the temperature is more than 30 "C is 0. O n a wet day. Find (i) the probability that the temperature is more than 30 "C on Monday..9. On a dry day. PI [31 [31 (c) Sailing boats can only sail on a windy day.. . (a) Copy and complete the tree diagram below.the probability that they cannot sail on Monday.2. (d) On a dry day with no wind.4.... You may assume that the weather each day is independent of the weather the day before.... Find the probability that Monday's concert (i) has to be postponed. Tuesday and Wednesday the temperature is more than 30 "C. On a wet or windy day this will not happen..25. Find .. the probability of wind is 0.4 4 topicProbability In summer the probability of a wet day is 0. the probability of wind is 0... PI (ii) the probability that on Monday.. l? PI (b) Which two shapes are a reflection of each other in the line x + y = O? (c) PI [31 [31 Which shape is a rotation of the shape D by 90" clockwise? Write down the coordinates of the centre of rotation. (a) Which two shapes are a reflection of each other in the line x = . (ii) Describe fully this single transformation.5 5 topicTransformations On the grid above there are seven identical shapes. (d) Which two shapes are a translation of each other by a vector with magnitude exactly 6? Give the column vector of this translation. ~31 ~31 (i) Find the coordinates of the 4 vertices of the shape H. Use these letters to answer the questions below. (e) The transformation with matrix ( ) maps the shape D onto another shape H. [Turn over . labelled A to G. 2 --+ --f M C = p and M D = q. Draw the graph of y = f(x) for . (i) (ii) E&? Z. E1 1 PI PI PI (c) The equation x 3 . 7 topicVectors topicGeometricaltermsandrelationships NOT TO SCALE In the circle.5x . 5 C M = M B and A M = .3 S x 3. (c) Use your answers to (b)(iii) and (b)(iv) to explain why B A is not parallel to DC. (i) Write down the equation of this straight line. (ii) f-l(x) = 1.1 = 0 can be solved by drawing one straight line on your graph. (i) Show that triangles A M B and C M D are similar. If CM = M B = x cm.6 6 Answer the whole of this question on a sheet of graph paper. calculate the value of x . (ii) Draw the line and write down the three solutions of x 3 . [61 (b) Use your graph to solve (i) f(x) = .M D . the chords A D and B C meet at M . Use a scale of 2 cm to represent 1 unit on the x-axis and 2 cm to represent 10 units on the y-axis. (ii) A M = 10 cm and M D = 4 cm. Cl1 . (a) topicGraphsoffunctions f(x) = x3. Write the following vectors in terms of p and/or q.1 = 0.7.5x .20. 7 8 topicStatistics Pedro and Anna measure the circumference (C) of 100 trees.y and z . She makes a table to show the heights of the bars she will draw. 058114iSYY [Turn over . using a scale of 1cm to represent 10 cm on the horizontal axis and 1 cm2to represent 1 tree. (ii) Find the values of x. Do NOT draw a histogram. [41 (iv) Write down the modal class. the quartiles and the interquartile range. Circumference (C) in cm Height of bar in cm (i) 20 < C s 40 X 40 < C S 70 70 < C S 100 100 < C S 120 10 Y z Explain why the height of the bar for the 40 < C G 70 class interval is 10 cm. 0580/4. C s 20 Circumference (C) in cm Frequency 20 < C s 40 26 40 < C d 70 70 < C G 100 100 < C s 120 0 30 33 11 100 80 Cumulative frequency 60 40 20 A 40 60 80 100 120 Circumference in cm Estimate the number of trees whose circumferences are between 60 cm and 80 cm. Their results are shown in the table and the cumulative frequency diagram below. [41 (iii) Calculate an estimate of the mean circumference. PI (b) Anna wants to construct a histogram. Use the cumulative frequency graph to find the median. (i) Write down an equation in x and show that it simplifies to x 2 + 5 ~ 300 = 0. Carlos charges $200 for selling prices up to $30 000. For selling prices more than $30 000. he charges $200 and of the value over $30 000. [41 PI 10 Answer the whole of this question on a sheet of graph paper. : (a) Use a scale of 2 cm to represent a selling price of $10 000 on the horizontal axis and 2 cm to represent a charge of $100 on the vertical axis.$30 000) = $500. Alberto charges $600 whatever the selling price. Draw on the same grid the three graphs to show the charges made by Alberto. The block is placed in the tank and the water level rises by 1 cm. Write down an expression for the volume of the block in terms of x.8 9 topicAlgebraicrepresentation topicEquationsandInequalities topicMensuration NOT TO SCALE A rectangular tank with length 50 cm and width 30 cm contains 36 litres of water. (iii) Write down the width and length of the block. [71 (b) (i) For which selling price is Alberto’s charge the same as Bernard’s? (ii) For what range of selling prices does Carlos charge the least? (iii) For which selling price.300 = 0. Carlos charges li% $200 + 1 % of ($50 000 . less than $50 000. Label your graphs clearly. 05x114iSY9 . Bernard and Carlos sell houses. Show by calculation that the water is 24 cm deep. - [41 (ii) Solve the equation x 2 + 5x . For example. does Bernard charge $50 less than Carlos? PI PI 0580/4. when the selling price is $50 000. topicLinearprogramming topicPercentages topicGraphsinpracticalsituations Alberto. A heavy rectangular block is 5 cm high and x cm wide. Bernard charges 1%of the selling price. Bernard and Carlos for selling prices up to $80 000. Its length is 5 cm more than its width.

THIS POST MAY CONTAIN AFFILIATE LINKS. PLEASE SEE MY DISCLOSURES FOR MORE INFORMATION A lot of people find themselves struggling during the middle of the week. They feel like they’ve been dragging since Monday and it’s only Wednesday. As a result, staying in a positive mindset can be tough. No matter how you feel, here are some Wednesday affirmations that can help make the middle of your week a bit better. By saying these positive statements to yourself, you can start your day off on the right foot and be motivated to achieve your goals. 183 Motivating Wednesday Affirmations Wednesday Affirmations For Work #1. I am happy that I shall get all the important work done this Wednesday. #2. I am content and satisfied. #3. There are numerous possibilities which are opening themselves in front of me this Wednesday. #4. Everything I need is within me. #5. I always see the good thing in others. #6. Every moment is a new beginning. #7. The more time and effort I put into taking care of myself, the stronger and happier I am going to be. #8. The hard work that I put in today will reflect tomorrow. #9. I am happy in the now. #10. This day is going to be the most important day of my career. #11. My good health comes from love and appreciation. #12. I am happy that I am going to meet some amazing people today. #13. I am a humble human being. #14. I am extremely happy that I am getting closer to my dream life each day. #15. Success is all around me. #16. I am inspiring people through my work. #17. Success and abundance are my birthright. #18. I am worthy of success and wealth. #19. I am excited about the success I shall achieve in my work this Wednesday. #20. I am ready to take over my career on this beautiful Wednesday. #21. I am not defined by my past. I am driven by my future. #22. While the competition is busy getting over the hump, I am busy closing business. #23. While people are struggling, I shall get closer to my dreams on Wednesday. #24. The success of my week depends on the work I do this Wednesday. #25. My life is satisfying and I am happy. I feel satisfaction for what I’ve achieved in life. - Read now: Learn these affirmations for gratitude #26. I am confident. #27. Everything will be okay. #28. The miracle that I have been waiting for all this time is going to come my way on Wednesday. #29. I am a living, breathing example of motivation. #30. I am super thrilled to achieve my goals. #31. My life is fulfilling for me and I feel really happy with my life. #32. The career that I have chosen is meant to find my calling. #33. I feel refreshed and happy to wake up and get my work doing. #34. I am thriving and I shall make my dreams come true. #35. I am happy to face all the challenges that shall come my way. #36. Every hurdle that I shall overcome today is meant for my growth. #37. I am super-charging my mind, my heart, and my body for a successful and happy Wednesday. #38. Some miracle will touch my life this Wednesday. #39. I am supercharged to get myself at work this Wednesday. #40. I am happy with myself, my life and I am proud of what I have accomplished. #41. I am filled with focus. #42. My dreams are within my reach, if not already fulfilled. #43. All the right and correct circumstances are making their way towards me. #44. I am going to get closer to my dreams today. #45. I am a success magnet. #46. My life is filled with blessings. #47. I am not pushed by my problems. I am led by my dreams. #48. I am sure that more is about to come my way this Wednesday. #49. I am working hard to become one of the best versions of myself. #50. I am intelligent and focused. #51. Each and every day, I am getting closer to achieving my goals. #52. I am so thankful for this wonderful day. #53. I am super excited about the new day. #54. My failures will not discourage me. #55. I am confident that I shall make the most out of my day. #56. I am all hyped up for the week. #57. I am the master of my goodwill. #58. This is a new day, with new opportunities to make my life better. #59. I am a complete person and my happiness does not depend on people. #60. My thoughts are powerful. Each one of them shapes who I am today. #61. My life is full of beauty and joy. Wednesday Morning Affirmations #62. I wake up motivated. #63. I am beautiful, powerful and strong. #64. I am investing in my health because I know I am worth it. #65. Today is a new day with so many possibilities. #66. I am here for a reason and I am going to strive towards it faster. #67. Today is a good day to start over. #68. I am having a positive and inspiring impact on the people I come into contact with. #69. I am beginning to learn about my capabilities with each passing day. #70. I am grateful for everything I have in my life. #71. I am attracting abundant health and wealth. #72. I am going to stay persistent throughout the day. #73. Today, I will show up for myself. #74. I am attracting the correct resources that shall help me achieve my calling. #75. I am listening and open to the messages the universe has to offer today. #76. I am going to prepare hard for this Wednesday so that I can reap the benefits on Thursday. #77. I am attracting wealth and prosperity. #78. Today is a phenomenal day. #79. Today I will not stress over things I can’t control. #80. This Wednesday I shall discover my full potential. #81. This Wednesday the Universe has sent me in the right direction to achieve the right thing. #82. I am so lucky that I get to meet my friends over lunch this Wednesday. #83. I am in awe with the abundance that is coming my way. #84. I am comfortable in my own skin. #85. Wednesday is my day. While others are getting over the hump, I’m climbing to my dreams. #86. This Wednesday will help me find the love of my life. #87. I am so cheerful to have such a great Wednesday morning. #88. Today, my only job is to show up for myself. #89. I am independent and self-sufficient. #90. I am charming and confident. - Read now: Learn these Saturday affirmations #91. I am proof enough of who I am and what I deserve. #92. Wednesday makes me the most cheerful person. #93. I am in connection with the Universe today. #94. Wednesdays are my favorite day. #95. Today I will be full of ideas. #96. I am so mesmerized by the beauty of nature. #97. I am held and supported by those who love me. #98. I am going to appreciate people for their perseverance today. #99. Wednesday is the day I overcome all my adversities. #100. I am free from worry and anxiety. - Read now: Use these affirmations for letting go #101. Today, I will choose happiness. #102. I am growing and I am going at my own pace. #103. Today will be a good day. #104. I am getting healthier every day. #105. I am good and getting better. #106. Wednesday is go time. I’m ready to finish this week strong. #107. I am attracting great opportunities. #108. I am just halfway through the goals I had planned for this week. Wednesday is great. #109. Wednesday rocks. Today I am halfway to this week’s goals. #110. I am following a very abundant lifestyle. #111. Today is the center of my week and my opportunity to refocus and recenter myself. #112. I am an unstoppable force of nature. #113. I am in charge of how I feel and I choose to feel happy. #114. Wednesday is just my day, mark it. #115. I am going to enroll in a volunteer service today. #116. I am going to remain productive throughout this Wednesday. #117. Today will be a productive day. #118. What a wonderful Wednesday morning. #119. Wednesday is going to be the day I come across the opportunity I have been waiting for so long. #120. I am in control of my life and feelings. #121. Wednesday is the day full of wonders. #122. I am cherished and lovable. Powerful Mid Week Affirmations #123. I am trying to avoid the mid-week lull and am supercharged even on this Wednesday. #124. I inhale confidence. #125. I shall make people fall in love with themselves this Wednesday. - Read now: Use these affirmations for love #126. I know that I am enough and I shall always be. #127. I chose to show gratitude to everyone that I meet today. #128. I have a beautiful life and it’s only going to get better from here. #129. I decide to recenter my life today and stay focused the entire week. #130. I stay motivated and focused. #131. My body is strong and capable. #132. I shall break my own records this Wednesday. #133. I shall earn a lot of money today. #134. I have all the things that I need to be happy today. #135. I have a clear mind and heart, ready to be used for good things. #136. I am more than my circumstances dictate. #137. I now free myself from negativity and destructive thoughts. #138. I feel joy and abundance in my life and a tremendous sense of happiness. #139. I shall learn new lessons from all my mistakes today. #140. I shall motivate people to rediscover themselves today. #141. I deserve to be loved wholeheartedly. #142. I will manifest new business opportunities today. #143. I am loved and worthy. #144. I am open to healing. #145. I have people that love me. #146. I can do hard things. #147. I am turning down the volume of negativity in my life, while simultaneously turning up the volume of positivity. #148. All that matters in this moment is what’s happening right now. #149. Life is full of abundance. #150. I am mentally and emotionally strong. I feel happiness everyday. #151. I have the power to change my life. #152. I am optimistic because today is a new day. #153. I am peaceful and whole. #154. I permit myself to be cheerful and happy this Wednesday. #155. All the worries of my past are gone. - Read now: Here are the best affirmations for fear #156. I shall cherish this beautiful day with my near and dear ones. #157. I shall help people find their happiness today. #158. I deserved to be appreciated today. #159. I am living with abundance. #160. I shall help people smile more today. #161. It’s OK to have boundaries and limits with others. #162. I am manifesting success and freedom. #163. I have a lot to be grateful for. #164. I use obstacles to motivate me to learn and grow. #165. I am moving towards financial freedom. #166. I am worthy of love. #167. I feel more grateful each day. #168. I shall overcome all the pain and miseries today. #169. I can be whatever I want to be. #170. I can make healthy decisions for myself. #171. I’m rising above the thoughts that are trying to make me angry or afraid. #172. I am only three days away from the weekend. #173. I receive love from others because I love myself the most. #174. I deserved to be loved each day. #175. When others are dead and dull in the middle of the week, I am all energetic and happy. #176. I shall be appreciated for my hard work on Wednesday. #177. Life loves me exactly as much as I love myself. #178. I chose to be at peace with others today. - Read now: Here are great Tuesday affirmations #179. I desire more happiness which is coming my way. #180. I have enough that I need to lead a great life. #181. I will always be surrounded by love and light. #182. I shall inspire people to change their lives today. #183. I will take care of myself today. Wednesday affirmations are a great way to start your day and stay positive throughout the week. By taking a few minutes each Wednesday morning to recite some positive affirmations, you can set the tone for a productive and happy day. This will make hump day more productive and allow you to focus on the upcoming weekend. - Read now: Here are some great Thursday affirmations - Read now: Use these affirmations for wealth - Read now: Learn the best affirmations for anger Jon Dulin is the passionate leader of Unfinished Success, a personal development website that inspires people to take control of their own lives and reach their full potential. His commitment to helping others achieve greatness shines through in everything he does. He’s an unstoppable force with lots of wisdom, creativity, and enthusiasm – all focused on helping others build a better future. Jon enjoys writing articles about productivity, goal setting, self-development, and mindset. He also uses quotes and affirmations to help motivate and inspire himself. You can learn more about him on his About page.

If you are new to logarithms and want to know exactly what they are, why were they introduced in the first place and what is their main use, then you are in the right place. In this article, I will thoroughly discuss about logarithms and their beneficial uses cases in mathematics as well as in other areas. What is a logarithm in simple terms? When you hear words like Logarithms and Data structures, your subconscious mind simply makes you believe that they are very hard to understand and only meant to be understood by geniuses. But the reality is way different and positive. While as complicated as it may sound, algorithms, in simple terms are used majorly in mathematics to simplify complex mathematical calculations. Logarithms can also be said to be some pre-defined set of rules to make calculations easier. With the help of Logarithms, you can convert one class or an object to another very easily and quickly. You can, for example, covert products into sums and quotients into subtractions without losing your mind all thanks to logarithms. By now, you might be a little familiar with the word Logarithms. But what about Logarithmation? Just like the basic and commonly used operators such as addition, subtraction, multiplication, and division, we refer to Logarithmation as an operator for the logarithms. If you find it difficult and confuse one with another, simply keep in mind that Logarithm is a company and the Logarithmation is the brand name of that company. As subtraction and division are there in opposite directions to the addition and multiplication, exponentiation is inversely equivalent to Logarithmation. If we look upon this in terms of values, then in order to get a value of the number A (exponent), another fixed number B (base) should be raised. What is logarithm used for? Not just in theory, logarithms have a variety of useful use-cases in real life too. Some of them include measuring the sound, using the Richter scale to measure the intensity and magnitudes of Earthquakes, measuring how bright the stars are, to measure the acid and alkaline level of a substance using Ph. These are just some of the examples where logarithm is useful. There are countless other projects where logarithms play a vital role in order to achieve accurate data. In theory, the logarithm is mainly used to calculate the number of times a base is multiplied by itself to transform it into another number. If we talk about the careers where the logarithms are most used, here are some of them: - Civil Engineer. - Agricultural Scientist. By this, we can see that the use of logarithms applies from a large organization to an independent intellectual. Unlike other curves or square roots, logarithms are tending to be more accurate and easier to work with. This is the main reason why most mathematicians find it easy and pleasing to experiment with logarithms. Are logarithms used in business? You may have seen, until now many cases where logarithms are used in varied sectors. But do they also work in a business? The simple answer is yes. Besides Economics and Finance, logarithms are used in almost all sectors including business. Some business calculations are easy to perform through logarithms than through the arithmetic method which makes it flexible and open among different areas. The use of logarithms in business, however, depends on the type of business. Let’s say for example that you have an online business related to finance and you need to create a calculator that auto-populates the data once the user input some value. In this case, as there is an involvement of a complex mathematical calculation, logarithms can be a part of this online business tool. Similarly, you can’t use logarithms to Tally your business’s balance sheet and profit and loss statement. As simple as that. Along with this, another great use of logarithmic functions can be seen in finance specifically while calculating the compound interest. Although the history of logarithm dates back to the old 1620, the application for the same is increasing exponentially over time. John Napier invented the logarithmic functions and the world follows his tactics and rules to date. And if you are interested in logarithms, you may be concerned to know that although the use of logarithms was invented and popularised by John, the initial difficulties in using the logs were made easier by Kepler with the vibrant clarification of how the logarithms worked. Because initially, it was too hard for people (including bright minds) to understand even the basic concepts of logarithms due to its sophisticated documentation. How do logarithms make our life easier? Believe it or not, logarithms do make our life easier and help us sleep peacefully while they do all the frightening work for us. After the earthquake, they tell you exactly how much magnitude it was. How is this helpful? Let’s say, a specific region is a victim of frequent intense earthquakes. In this case, if you know as per the previous magnitude data, the people living in this region would be much cautious and build their houses accordingly (earthquake-proof). This was just one example, you can find many such cases where logarithms are not seen but make a huge difference in our lives. Data science is another field where data scientists rely on logs heavily. This was just a crux about logarithms and how they are useful in the real world. If you want to become a data scientist or a mathematician who is interested in logarithms, then you may want to dig deep into the more advanced topics such as Log odds, logistic regression, Product rule, Quotient rule, Power rule, and much more to get a solid perspective about how logarithms and things around it works. I am no mathematician, data scientist, or agricultural manager by profession but I can assure you that you will have to add logarithms into your daily life if you want to create a positive impact in the field of mathematics.

No catches, no fine print just unadulterated book loving, with your favourite books saved to your own digital bookshelf. New members get entered into our monthly draw to win £100 to spend in your local bookshop Plus lots lots more…Find out more See below for a selection of the latest books from Topology category. Presented with a red border are the Topology books that have been lovingly read and reviewed by the experts at Lovereading. With expert reading recommendations made by people with a passion for books and some unique features Lovereading will help you find great Topology books and those from many more genres to read that will keep you inspired and entertained. And it's all free! Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer's body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer's ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program. Topology, the mathematical study of the properties that are preserved through the deformations, twistings, and stretchings of objects, is an important area of modern mathematics. As broad and fundamental as algebra and geometry, its study has important implications for science more generally, especially physics. Most people will have encountered topology, even if they're not aware of it, through Moebius strips, and knot problems such as the trefoil knot. In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eye-opening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. This book covers the fundamental results of the dimension theory of metrizable spaces, especially in the separable case. Its distinctive feature is the emphasis on the negative results for more general spaces, presenting a readable account of numerous counterexamples to well-known conjectures that have not been discussed in existing books. Moreover, it includes three new general methods for constructing spaces: Mrowka's psi-spaces, van Douwen's technique of assigning limit points to carefully selected sequences, and Fedorchuk's method of resolutions. Accessible to readers familiar with the standard facts of general topology, the book is written in a reader-friendly style suitable for self-study. It contains enough material for one or more graduate courses in dimension theory and/or general topology. More than half of the contents do not appear in existing books, making it also a good reference for libraries and researchers. The book starts with the basic concepts of topology and topological spaces followed by metric spaces, continuous functions, compactness, separation axioms, connectedness and product topology. Originally published as Volume 27 of the Princeton Mathematical series. Originally published in 1965. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. In this monograph the narrow topology on random probability measures on Polish spaces is investigated in a thorough and comprehensive way. As a special feature, no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra, are made. One of the main results is a direct proof of the random analog of the Prohorov theorem, which is obtained without invoking an embedding of the Polish space into a compact space. Further, the narrow topology is examined and other natural topologies on random measures are compared. In addition, it is shown that the topology of convergence in law-which relates to the statistical equilibrium -and the narrow topology are incompatible. A brief section on random sets on Polish spaces provides the fundamentals of this theory. In a final section, the results are applied to random dynamical systems to obtain existence results for invariant measures on compact random sets, as well as uniformity results in the individual ergodic theorem. This clear and incisive volume is useful for graduate students and researchers in mathematical analysis and its applications. This proceedings volume presents a diverse collection of high-quality, state-of-the-art research and survey articles written by top experts in low-dimensional topology and its applications. The focal topics include the wide range of historical and contemporary invariants of knots and links and related topics such as three- and four-dimensional manifolds, braids, virtual knot theory, quantum invariants, braids, skein modules and knot algebras, link homology, quandles and their homology; hyperbolic knots and geometric structures of three-dimensional manifolds; the mechanism of topological surgery in physical processes, knots in Nature in the sense of physical knots with applications to polymers, DNA enzyme mechanisms, and protein structure and function. The contents is based on contributions presented at the International Conference on Knots, Low-Dimensional Topology and Applications - Knots in Hellas 2016, which was held at the International Olympic Academy in Greece in July 2016. The goal of the international conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and low-dimensional topology. This book will benefit all researchers who wish to take their research in new directions, to learn about new tools and methods, and to discover relevant and recent literature for future study. One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. An essential difficulty in quantum field theory comes from infinite-dimensional freedom of a system. Techniques dealing with such infinite-dimensional objects developed in the framework of quantum field theory have been influential in geometry as well.This book focuses on the relationship between two-dimensional quantum field theory and three-dimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3-manifolds which was derived from Chern-Simons gauge theory. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds. The book is organized as follows: the introduction starts from classical mechanics and explains basic background materials in quantum field theory and geometry. Chapter 1 presents conformal field theory based on the geometry of loop groups. Chapter 2 deals with the holonomy of conformal field theory. Chapter 3 treats Chern-Simons perturbation theory. The final chapter discusses topological invariants for 3-manifolds derived from Chern-Simons perturbation theory. Sigurdur Helgason's Differential Geometry and Symmetric Spaces was quickly recognized as a remarkable and important book. For many years, it was the standard text both for Riemannian geometry and for the analysis and geometry of symmetric spaces. Several generations of mathematicians relied on it for its clarity and careful attention to detail. Although much has happened in the field since the publication of this book, as demonstrated by Helgason's own three-volume expansion of the original work, this single volume is still an excellent overview of the subjects.For instance, even though there are now many competing texts, the chapters on differential geometry and Lie groups continue to be among the best treatments of the subjects available. There is also a well-developed treatment of Cartan's classification and structure theory of symmetric spaces. The last chapter, on functions on symmetric spaces, remains an excellent introduction to the study of spherical functions, the theory of invariant differential operators, and other topics in harmonic analysis. This text is rightly called a classic. Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces . Topology is a large subject with several branches, broadly categorized as algebraic topology, point-set topology, and geometric topology. Point-set topology is the main language for a broad range of mathematical disciplines, while algebraic topology offers as a powerful tool for studying problems in geometry and numerous other areas of mathematics. This book presents the basic concepts of topology, including virtually all of the traditional topics in point-set topology, as well as elementary topics in algebraic topology such as fundamental groups and covering spaces. It also discusses topological groups and transformation groups. When combined with a working knowledge of analysis and algebra, this book offers a valuable resource for advanced undergraduate and beginning graduate students of mathematics specializing in algebraic topology and harmonic analysis.

Used to show relationships between groups. The distance between values on the y-axis. Enables us to find trends or patterns over time. Uses pictures to represent quantities. Vertical axis of a system of coordinates Is best used with % percents or fractions. A portion of a circle graph. Horizontal axis of a system of coordinates. Your substitute teacher's name Your regular math teacher's name A number that is not an integer and includes a decimal or a percent How far away a number is from 0 When you________ a pair on the grid The vertical axis The horizontal axis There are 4 of these they usually have different signs in front of the number thay are_____________ Not a fraction number Two numbers that have a order of x or y This is the first number in a ordered pair The second number in a ordered pair This number can be found to the right of 0 on a number line This number can be found to the left of 0 on a number line The Number 888's ___________________ is -888 A number that is not a decimal or a fraction, can be graphed on a number line An example of this is (9,8) This can be turned into a decimal or percent This is like a fraction but equals over one This fraction is a mixed one and is proper but equals over 1 This number can be turned into a fraction and goes by tenths A plane that includes the x axis and the y axis which intersect at the origin This quadrant has all positive numbers This quadrant has a negative x and a positive y This quadrant has all negative numbers This quadrant has a positive x and a negative y The set of the first numbers of the ordered pairs Please Excuse My Dear Aunt Sally The ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) A mathematical sentence that contains an equal sign A relation between input and output Symbol used to represent unknown numbers or values The form of a linear equation Ax + By = C, with a graph that is a straight line The set of second numbers of the ordered pairs To find the value of an expression The vertical number line on a coordinate plane A comparison of two numbers by division To draw or lot the oints named by certain numbers or ordered pairs on a number line or coordinate plane The numbers that correspond to a point on a coordinate system A point where the graph intersects an axis The horizontal number line on a coordinate plane The point (0,0) The answer when multiplying A polynomial with two terms Distance around the outside To shift a graph vertically or horizontally The answer to a multiplicaiton problem The number being divided What to multiply a value by to get 1 A number that represents part of a whole, has a numerator and a denominator A number consisting of an integer and a proper fraction. The number above the line in a common fraction showing how many of the parts indicated by the denominator are taken, for example, 2 in 2/3. The number below the line in a common fraction; a divisor. A fraction in which the numerator is greater than the denominator, such as 5/4 The boundary of a closed figure or shape The horizontal line on a coordinate grid The vertical line on a coordinate grid The point on a coordinate grid where the x axis and y axis intersect, also referred to by (0,0) Measures the distance from 0, can not be negative They are located on a coordinate grid, they are often numbered using Roman numerals: I II III IV. The point at which the graph crosses or touches the vertical axis The point at which the graph crosses or touches the horizontal axis An equation whose graph creates a straight line The rate of change in the output relative to the input The difference between the y-values divided by the difference of the x-values The letter used to denote slope Another name for a coordinate pair (x,y) that makes an equation TRUE What form uses the generic equation Ax+By=C? What form uses the generic equation y=mx+b? What form uses the generic equation y-k=m(x-h) What type of line has the equation y=# What slope does a horizontal line have? What type of line has the equation x=# What slope does a vertical line have? The horizontal axis is called what? The vertical axis is called what? the first coordinate in an ordered pair Is the ratio of a term to the previous term to fail to approach a finite limit The amount of time it takes for the amount if the substance to diminish by half the set of all real numbers between to given numbers is a transcendental number commonly encountered when working with exponential models and exponential functions a line or curve that the graph of a relatuion approaches more and more closely the further the graph is followed the result of writing the sum of two terms as a difference of vice-versa a conic section that can be thought of as an inside-out clipse an extreme value of a function a geometric figure made up of two right circular cones placed apex to apex an interval that contains endpoints the family of curves including circles, ellipses, paranolas, & hyperbolas a function or coordinates a method for solving a linear system of equations using determinans a conic section which is essential a stretched circle to figure out or evaluate the appearance of a graph as its followed farther & farther in either directions a method used to divide polyniomials a technique for distributing two binomials the amount of quantity a kind of average sometimes used in statistics & engineering a function with a graph that is symmetric to the y-axis the product of a given integer and all smaller positive integers the slope of a horizontal line th set of all real numbers between two given numbers the coordinate plane used to graph complex numbers to multiply out the parts of an equation a polynomialof degree 3 The point where a line meets or crosses the y-axis The point where a line meets or crosses the x-axis The ratio of the vertical and horizontal changes between two points on a surface or a line; rise over run The set of all possible outputs of a function; All y-values A pair of numbers, (x, y), that indicate the position of a point on a Cartesian plane A linear function representing real -world phenomena. The model also represents patterns found in graphs and/or data A function with a constant rate of change and a straight line graph A mathematical phrase involving at least one variable and sometimes numbers and operation symbols A number sentence that contains an equals symbol The appearance of a graph as it is followed farther and farther in either direction The set of x-coordinates of the set of points on a graph;All x-values. The value that is the input in a function or relation A set with elements that are disconnected Describes a connected set of numbers, such as an interval With respect to the variable x of a linear function y= f(x), the constant rate of change is the slope of its graph A number multiplied by a variable in an algebraic expression; The number in front of variable A number says how many times to use that number in a multiplication A number with no fractional part An equation that makes a straight line when it is graphed The distance from the center to the circumference of a circle The differnce between the lowest and highest numbers. A point where two or more straight lines meet The amount of 3 dimensional space and object occupy The line that divides something into two equal parts The size of a surface A number used to multiply a variable A line segment connecting two points on a curve The largest exponent for a polynomial with one variable A straight line going through the center of a circle connecting two points on the circumference Does not converge, does not settle with some value A way to pinpoint where you are on a map or graph by how far along or how far up or down the graph is The length of the adjacent side divided by the length of the hypotenuse The longest side on a right triangle This creates an arched shape when graphed How far a periodic function is horizontally to the right of the usual position The ratio of a circle's circumference to it's diameter A vector with a magnitude of one The shortest diameter of an ellipse A sequence made by multiplying by some value each time A value that you get closer and closer to, but can never reach The set of input values in a relation. A relation that assigns exactly one output for each input. The place where a graph crosses the y-axis. When a figure can be folded about a line so that it matches exactly. A diagram used to determine whether a relation is or is not a function. Used to emphasize that a function value f(x) depends on the variable x. The y-coordinate of the highest point on a graph. A set of ordered pairs. This type of line is never a function. The behavior of a graph as x approaches positive or negative infinity. Used to determine whether a graph is a function. The place where a graph crosses the x-axis. The y-coordinate of the lowest point on a graph. To replace a variable with a number and simplify. This type of line is always a function. The set of output values in a relation. horizontal axis of the coordinate plane Vertical axis of the coordinate plane another name for the x-values on a graph Another name for the y-values on a graph The y-value of the point in which the linear function crosses the y-axis. The x-value of the point in which the linear function crosses the x-axis. an x-value and y-value together on the coordinate plane create a _______ A letter representing an unknown value ___________ is a word to describe Rate of Change. When the ordered pair does not satisfy the equation or the equation produces a false response. What we use to plot linear functions. the variable m represent the __________ the variable b represent the ____________ When a graph is decreasing from left to write it has a ______________ slope. What a graph is increasing from left to right it has a ______________ slope A graph that is a vertical line has a ______________ slope A graph that is a horizontal line has a __________ slope.

- What forces act on a pendulum? - How do you balance a pendulum clock? - What causes a pendulum to stop swinging? - What factors affect the swing of a pendulum? - What should I ask my pendulum? - What happens to a pendulum in space? - How long will a pendulum swing? - What type of energy did the pendulum have before Person #2 let go? - Does a pendulum tell the truth? - How does energy change in a swinging pendulum? - Why does a pendulum swing faster with a shorter string? - What happens if you add more weight to a pendulum? - Why doesn’t mass affect a pendulum? - How do I keep my pendulum swinging? - Where does energy go when a pendulum stops swinging? - Does a pendulum tell you what you want to hear? - Can you use a necklace as a pendulum? - Why is the pendulum important? - How exactly does a pendulum lose energy? - Does a pendulum ever stop? - How do you adjust a pendulum clock? What forces act on a pendulum? There are two dominant forces acting upon a pendulum bob at all times during the course of its motion. There is the force of gravity that acts downward upon the bob. It results from the Earth’s mass attracting the mass of the bob. And there is a tension force acting upward and towards the pivot point of the pendulum.. How do you balance a pendulum clock? Give the pendulum a slight sideways push to start it swinging. Listen to the tick –tock. The tick and the tock should sound evenly spaced, if so the clock is said to be “in beat” or balanced. If not, the crutch at the back of the movement will need to be bent sideways to either the right or the left. What causes a pendulum to stop swinging? A pendulum is an object hung from a fixed point that swings back and forth under the action of gravity. … The swing continues moving back and forth without any extra outside help until friction (between the air and the swing and between the chains and the attachment points) slows it down and eventually stops it. What factors affect the swing of a pendulum? The forces of gravity, the mass of the pendulum, length of the arm, friction and air resistance all affect the swing rate.Motion. Pull a pendulum back and release it. … Length. The swing rate, or frequency, of the pendulum is determined by its length. … Amplitude. … Mass. … Air Resistance/Friction. … Sympathetic Vibration.Apr 29, 2018 What should I ask my pendulum? Getting clarity questions to ask a pendulum… Do I keep getting the answer no because there is something better waiting for me? Am I passing up a good opportunity if I say no to __________ ? Am I expecting too much from ________ ? Should I get more information before deciding to __________ ? What happens to a pendulum in space? No, it wouldn’t swing back and forth. … Since there is no gravity in orbit, there is no force that pulls the pendulum down and make it swing back and forth. If you pushed a pendulum in orbit, the bob of the pendulum would keep going in full circles until they stop due to friction. How long will a pendulum swing? Here is an extra fun fact. A pendulum with a length of 1 meter has a period of about 2 seconds (so it takes about 1 second to swing across an arc). This means that there is a relationship between the gravitational field (g) and Pi. What type of energy did the pendulum have before Person #2 let go? It was potential energy because the washer could swing due to its relative position and gravity. This kind of potential energy is known as gravitational potential energy. What’s interesting about a pendulum, though, is that when you let go of it, the potential energy gradually transforms into kinetic energy. Does a pendulum tell the truth? As an example – people use a pendulum to ask if a certain food is good for them by holding it over the item, and if it’s a healthy high vibration food then the pendulum will answer YES. How does energy change in a swinging pendulum? As a pendulum swings, its potential energy converts to kinetic and back to potential. … During the course of a swing from left to right, potential energy is converted into kinetic energy and back. Why does a pendulum swing faster with a shorter string? The length of the string affects the pendulum’s period such that the longer the length of the string, the longer the pendulum’s period. … A pendulum with a longer string has a lower frequency, meaning it swings back and forth less times in a given amount of time than a pendulum with a shorter string length. What happens if you add more weight to a pendulum? When you add a weight to the middle of the other pendulum, however, you effectively make it shorter. Shorter pendulums swing faster than longer ones do, so the pendulum on the left swings faster than the pendulum on the right. Why doesn’t mass affect a pendulum? The mass on a pendulum does not affect the swing because force and mass are proportional and when the mass increases so does the force. … In the formula, “l” is the length and “g” is the force of gravity. Therefore, the mass does not affect the period of the pendulum. How do I keep my pendulum swinging? There are several things that you can do to make a pendulum swing for a long time:Make it heavy (and, specifically, dense). The more mass a pendulum has, the less outside influences such as air resistance will degrade its swing.Put it in a vacuum. … Use an escapement mechanism. … Give it a large initial swing. Where does energy go when a pendulum stops swinging? Once the weighted end of the pendulum is released, it will become active as gravity pulls it downward. Potential energy is converted to kinetic energy, which is the energy exerted by a moving object. Does a pendulum tell you what you want to hear? A pendulum can energetically swing and work for almost anyone who believes it will move. … If you really want to get that settlement or win the lottery, the Ego will swing the pendulum in the “Yes” direction, because that is what you really want to hear. Can you use a necklace as a pendulum? Pendulums can be made of different materials, some people using a simple necklace with a crystal or charm at the end. Be sure the bob or bobber – or weight on the end – is not too light or too heavy. … The best length for the pendulum is six inches. You can make your pendulum or buy one. Why is the pendulum important? Pendulum, body suspended from a fixed point so that it can swing back and forth under the influence of gravity. … Pendulums are used to regulate the movement of clocks because the interval of time for each complete oscillation, called the period, is constant. How exactly does a pendulum lose energy? The pendulum loses energy to wind resistance, friction between the tube and the string, and internal friction within the bending string. … Then the driver allows the force of gravity to convert some of that extra energy to kinetic energy, by allowing the bob to fall an extra distance at large angles. Does a pendulum ever stop? A pendulum can only stop when its gravitational potential is lowest and it no longer has energy driving it. (This is of course assuming the pendulum is allowed to swing until brought to a stop by friction, rather than being stopped by an applied force.) How do you adjust a pendulum clock? Stop the pendulum to move the pendulum bob up or down to change the pendulum’s effective length. If the clock is running fast, move the bob down or turn the nut to the left. If the clock is running slow, move the bob up or turn the nut to the right. Restart the pendulum and reset the clock hands to the proper time.

Author: Carlos A. Smith Publisher: CRC Press Release Date: 2011-05-18 Emphasizing a practical approach for engineers and scientists, A First Course in Differential Equations, Modeling, and Simulation avoids overly theoretical explanations and shows readers how differential equations arise from applying basic physical principles and experimental observations to engineering systems. It also covers classical methods for obtaining the analytical solution of differential equations and Laplace transforms. In addition, the authors discuss how these equations describe mathematical systems and how to use software to solve sets of equations where analytical solutions cannot be obtained. Using simple physics, the book introduces dynamic modeling, the definition of differential equations, two simple methods for obtaining their analytical solution, and a method to follow when modeling. It then presents classical methods for solving differential equations, discusses the engineering importance of the roots of a characteristic equation, and describes the response of first- and second-order differential equations. A study of the Laplace transform method follows with explanations of the transfer function and the power of Laplace transform for obtaining the analytical solution of coupled differential equations. The next several chapters present the modeling of translational and rotational mechanical systems, fluid systems, thermal systems, and electrical systems. The final chapter explores many simulation examples using a typical software package for the solution of the models developed in previous chapters. Providing the necessary tools to apply differential equations in engineering and science, this text helps readers understand differential equations, their meaning, and their analytical and computer solutions. It illustrates how and where differential equations develop, how they describe engineering systems, how to obtain the analytical solution, and how to use software to simulate the systems. Author: Carlos A. Smith Publisher: CRC Press Release Date: 2016-04-05 A First Course in Differential Equations, Modeling, and Simulation shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained. Incorporating valuable suggestions from mathematicians and mathematics professors, the Second Edition: Expands the chapter on classical solutions of ordinary linear differential equations to include additional methods Increases coverage of response of first- and second-order systems to a full, stand-alone chapter to emphasize its importance Includes new examples of applications related to chemical reactions, environmental engineering, biomedical engineering, and biotechnology Contains new exercises that can be used as projects and answers to many of the end-of-chapter problems Features new end-of-chapter problems and updates throughout Thus, A First Course in Differential Equations, Modeling, and Simulation, Second Edition provides students with a practical understanding of how to apply differential equations in modern engineering and science. Author: Frank R. Giordano Publisher: Cengage Learning Release Date: 2013-03-05 Offering a solid introduction to the entire modeling process, A FIRST COURSE IN MATHEMATICAL MODELING, 5th Edition delivers an excellent balance of theory and practice, and gives you relevant, hands-on experience developing and sharpening your modeling skills. Throughout, the book emphasizes key facets of modeling, including creative and empirical model construction, model analysis, and model research, and provides myriad opportunities for practice. The authors apply a proven six-step problem-solving process to enhance your problem-solving capabilities -- whatever your level. In addition, rather than simply emphasizing the calculation step, the authors first help you learn how to identify problems, construct or select models, and figure out what data needs to be collected. By involving you in the mathematical process as early as possible -- beginning with short projects -- this text facilitates your progressive development and confidence in mathematics and modeling. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. Author: Dennis G. Zill Release Date: 1993-01-01 % mainly for math and engineering majors% clear, concise writng style is student orientedJ% graded problem sets, with many diverse problems, range form drill to more challenging problems% this course follows the three-semester calculus sequence at two- and four-year schools Author: A. Iserles Publisher: Cambridge University Press Release Date: 2009 lead the reader to a theoretical understanding of the subject without neglecting its practical aspects. The outcome is a textbook that is mathematically honest and rigorous and provides its target audience with a wide range of skills in both ordinary and partial differential equations." --Book Jacket. Author: J. David Logan Release Date: 2015-07-01 The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market. It presents a thorough treatment of the standard topics in an accessible, easy-to-read, format. The overarching perspective of the text conveys that differential equations are about applications. This book illuminates the mathematical theory in the text with a wide variety of applications that will appeal to students in physics, engineering, the biosciences, economics and mathematics. Instructors are likely to find that the first four or five chapters are suitable for a first course in the subject. This edition contains a healthy increase over earlier editions in the number of worked examples and exercises, particularly those routine in nature. Two appendices include a review with practice problems, and a MATLAB® supplement that gives basic codes and commands for solving differential equations. MATLAB® is not required; students are encouraged to utilize available software to plot many of their solutions. Solutions to even-numbered problems are available on springer.com. The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems. Author: Stefano M. Iacus Publisher: Springer Science & Business Media Release Date: 2009-04-27 This book covers a highly relevant and timely topic that is of wide interest, especially in finance, engineering and computational biology. The introductory material on simulation and stochastic differential equation is very accessible and will prove popular with many readers. While there are several recent texts available that cover stochastic differential equations, the concentration here on inference makes this book stand out. No other direct competitors are known to date. With an emphasis on the practical implementation of the simulation and estimation methods presented, the text will be useful to practitioners and students with minimal mathematical background. What’s more, because of the many R programs, the information here is appropriate for many mathematically well educated practitioners, too. Author: Clayton R. Paul Publisher: John Wiley & Sons Release Date: 2011-09-20 Just the math skills you need to excel in the study or practice of engineering Good math skills are indispensable for all engineers regardless of their specialty, yet only a relatively small portion of the math that engineering students study in college mathematics courses is used on a frequent basis in the study or practice of engineering. That's why Essential Math Skills for Engineers focuses on only these few critically essential math skills that students need in order to advance in their engineering studies and excel in engineering practice. Essential Math Skills for Engineers features concise, easy-to-follow explanations that quickly bring readers up to speed on all the essential core math skills used in the daily study and practice of engineering. These fundamental and essential skills are logically grouped into categories that make them easy to learn while also promoting their long-term retention. Among the key areas covered are: Algebra, geometry, trigonometry, complex arithmetic, and differential and integral calculus Simultaneous, linear, algebraic equations Linear, constant-coefficient, ordinary differential equations Linear, constant-coefficient, difference equations Linear, constant-coefficient, partial differential equations Fourier series and Fourier transform Laplace transform Mathematics of vectors With the thorough understanding of essential math skills gained from this text, readers will have mastered a key component of the knowledge needed to become successful students of engineering. In addition, this text is highly recommended for practicing engineers who want to refresh their math skills in order to tackle problems in engineering with confidence. The authors' enthusiasm for their subject is eloquently conveyed in this book, and draws the reader very quickly into active investigation of the problems posed. By providing plenty of modelling examples from a wide variety of fields - most of which are familiar from everyday life - the book shows how to apply mathematical ideas to situations which would not previously have been considered to be 'mathematical' in character.

Homework Questions? Ask a Tutor for Answers ASAP Not a Homework Question? How JustAnswer Works: Ask an Expert Experts are full of valuable knowledge and are ready to help with any question. Credentials confirmed by a Fortune 500 verification firm. Get a Professional Answer Via email, text message, or notification as you wait on our site. Ask follow up questions if you need to. 100% Satisfaction Guarantee Rate the answer you receive. Ask Linda_us Your Own Question Finance, Accounts & Homework Tutor Post Graduate Diploma in Management (MBA) Type Your Homework Question Here... Linda_us is online now test prep questionare due for extra credit need help This answer was rated: 1. Jasmine Company produces hand tools. A sales budget for the next four months is as follows: March 10,000 units, April 13,000, May 16,000 and June 21,000. Jasmine Company's ending finished goods inventory policy is 10% of the following month's sales. March 1 inventory is projected to be 1,400 units. How many units will be produced in March? A. 10,000 B. 9,900 C. 13,000 D. 10,100 2. Jasmine Company produces hand tools. A production budget for the next four months is as follows: March 10,300 units, April 13,300, May 16,500, and June 21,800. Jasmine Company's ending finished goods inventory policy is 10% of the following month's sales. Jasmine plans to sell 16,000 units in May. What is budgeted ending inventory for March? A. 1,030 B. 1,300 C. 1,330 D. 1,650 3. Albertville Inc produces leather handbags. The production budget for the next four months is: July 5,000 units, August 7,000, September 7,500, October 8,000. Each handbag requires 0.5 square meters of leather. Albertville Inc's leather inventory policy is 30% of next month's production needs. On July 1 leather inventory was expected to be 1,000 square meters. What will leather purchases be in July? A. 2,300 square meters B. 2,550 square meters C. 2,700 square meters D. 3,575 square meters 4. The purpose of the cash budget is to A. be used as a basis for the operating budgets. B. provide external users with an estimate of future cash flows. C. help managers plan ahead to make certain they will have enough cash on hand to meet their operating needs. D. summarize the cash flowing into and out of the business during the past period. 5. Brimson has forecast sales for the next three months as follows: July 4,000 units, August 6,000 units, September 7,500 units. Brimson's policy is to have an ending inventory of 40% of the next month's sales needs on hand. July 1 inventory is projected to be 1,500 units. Monthly manufacturing overhead is budgeted to be $17,000 plus $5 per unit produced. What is budgeted manufacturing overhead for August? A. $50,000 B. $47,000 C. $33,000 D. $32,000 6. The difference between the actual cost driver amount and the standard cost driver amount, multiplied by the standard variable overhead rate is the A. variable overhead rate variance. B. variable overhead efficiency variance. C. variable overhead volume variance. D. over - or underapplied variance. 7. Albertville applies overhead based on direct labor hours. The variable overhead standard is 2 hours at $12 per hour. During July, Albertville spent $116,700 for variable overhead. 8,890 labor hours were used to produce 4,700 units. How much is variable overhead on the flexible budget? A. $56,400 B. $106,680 C. $112,800 D. $116,700 8. The fixed overhead volume variance is the difference between A. Actual fixed overhead and budgeted fixed overhead. B. Actual fixed overhead and applied fixed overhead. C. Applied fixed overhead and budgeted fixed overhead. D. Actual fixed overhead and the standard fixed overhead rate times actual cost driver. 9. The difference between the actual volume and the budgeted volume, multiplied by the fixed overhead rate based on budgeted volume, is the A. fixed overhead spending variance. B. fixed overhead price variance. C. fixed overhead efficiency variance. D. fixed overhead volume variance. 10. Albertville has budgeted fixed overhead of $67,500 based on budgeted production of 4,500 units. During July, 4,700 units were produced and $71,400 was spent on fixed overhead. What is the budgeted fixed overhead rate? A. $14.36 B. $15.00 C. $15.19 D. $15.89 11. In a standard cost system, the initial debit to an inventory account is based on A. standard cost rather than actual cost. B. actual cost rather than standard cost. C. actual cost less the standard cost. D. standard cost less the actual cost. 12. In what type of organization is decision-making authority spread throughout the organization? A. Centralized organization B. Decentralized organization C. Participative organization D. Top-down organization 13. Which of the following is NOT an advantage of decentralization? A. Allows top managers to focus on strategic issues B. Potential duplication of resources C. Allows for development of managerial expertise D. Managers can react quickly to local information 14. The responsibility center in which the manager has responsibility and authority over revenues, costs and assets is A. a cost center. B. an investment center. C. a profit center. D. a revenue center. 15. Return on investment can be calculated as A. ROI = sales revenue/average invested assets B. ROI = operating income/sales revenue C. ROI = operating income/average invested assets D. ROI = average invested assets/sales revenue 16. Which of the following balanced scorecard perspectives measures how an organization satisfies its stakeholders? B. Internal business processes C. Learning and growth 17. Which of the following is not something that should be compiled for each dimension of the balanced scorecard? A. Performance measures C. Strategic vision D. Specific objectives Share this conversation replied 6 years ago. THIS ANSWER IS LOCKED! You need to spend $3 to view this post. to your account and buy credits. Linda_us and 5 other Homework Specialists are ready to help you Ask your own question now Share this conversation Related Homework Questions I wrote the following statements for my assignment in MySQL: I have 2 homework that I need you to help, this is the 1st 6. The hexadecimal equivalent of decimal 1234 is A. 2D4.. I have a column that divides j/s. This sometimes results in Translate the sentence below into English. Pienso que nunca I need help with my pre-exam for Penn high school this is Wiz of All NOT GLENN NOR F. NAZ PLEASE In the story titled "Macbeth" by Exam Number: 986159RR Need help with a couple assignment questions FOR LogicPro Only, I have another database question. Its a Ask a Tutor Get a Professional Answer. 100% Satisfaction Guaranteed. 37 Tutors are Online Now Type Your Homework Question Here... Terms of Service Privacy & Security © 2003-2017 JustAnswer LLC

What do you need help on? Cancel X Jim FAQ by The1Executioner Version: 1.00 | Updated: 05/10/05 ____ __ __ /\ _`\ __ /\ \ /\ \__ \ \ \L\ \ __ ____/\_\ \_\ \ __ ___\ \ ,_\ \ \ , / /'__`\ /',__\/\ \ /'_` \ /'__`\/' _ `\ \ \/ \ \ \\ \ /\ __//\__, `\ \ \/\ \L\ \/\ __//\ \/\ \ \ \_ \ \_\ \_\ \____\/\____/\ \_\ \___,_\ \____\ \_\ \_\ \__\ \/_/\/ /\/____/\/___/ \/_/\/__,_ /\/____/\/_/\/_/\/__/ ____ ___ /\ _`\ __ /\_ \ \ \ \L\_\ __ __ /\_\\//\ \ \ \ _\L /\ \/\ \\/\ \ \ \ \ \ \ \L\ \ \ \_/ |\ \ \ \_\ \_ \ \____/\ \___/ \ \_\/\____\ \/___/ \/__/ \/_/\/____/ _____ __ __ __ /\ __`\ /\ \__/\ \ /\ \ \ \ \/\ \ __ __\ \ ,_\ \ \____ _ __ __ __ \ \ \/'\ \ \ \ \ \/\ \/\ \\ \ \/\ \ '__`\/\`'__\/'__`\ /'__`\ \ \ , < \ \ \_\ \ \ \_\ \\ \ \_\ \ \L\ \ \ \//\ __//\ \L\.\_\ \ \\` \ \ \_____\ \____/ \ \__\\ \_,__/\ \_\\ \____\ \__/.\_\\ \_\ \_\ \/_____/\/___/ \/__/ \/___/ \/_/ \/____/\/__/\/_/ \/_/\/_/ ____ ___ __ __ ___ /\ _`\ __ /\_ \ _\ \\ \__ /'___`\ \ \ \L\_\/\_\\//\ \ __ /\__ _ _\ /\_\ /\ \ \ \ _\/\/\ \ \ \ \ /'__`\ \/_L\ \\ \L_\/_/// /__ \ \ \/ \ \ \ \_\ \_/\ __/ /\_ _ _\ // /_\ \ \ \_\ \ \_\/\____\ \____\ \/_/\_\\_\/ /\______/ \/_/ \/_/\/____/\/____/ \/_//_/ \/_____/ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX http://www.executioner.ne1.net http://www.ubcsteam.ne1.net XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX +------------------------------------------------------------+ |Author : The Executioner | |FAQ Type : Jim FAQ | |Spoilers : Yes | |Game : Resident Evil Outbreak File #2 | +------------------------------------------------------------+ X--------X |CONTENTS| X--------X 1) Introduction/Author Information 2) Characters 3) Jim Information 4) Boss Strategies for Jim 5) Scenario Notes about Jim 6) Jim Secret Costumes List 7) Jim's SP Item list 8) Frequently Asked Questions 9) Contact 10) Updates 11) Credits 12) Legal and Copyright *MUST KNOW* XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1. Introduction/Author Information XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Hey there, as you all know I am The Executioner. Yes, I play Outbreak and Madden online a lot during weekends. Jim Chapman is one of the best character to use in Outbreak File #2 so I decided to write a guide about him and enjoy and thanks for reading! - Executioner XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 2. Characters XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX ------------------- | AIPC Relationships | ------------------- When choosing AIPC partner characters before starting any scenario, reefer to the following chart to determine a partner who will prove most helpful: +------------------------------------------------+ |PC | Good Relationship | Bad Relationship | +------------------------------------------------+ |Kevin | Yoko, George, Cindy | Mark, David, Jim | |Mark | Jim, David, Yoko | Kevin, George | |Jim | Mark, Cindy | Yoko | |George | Kevin, Cindy | Alyssa | |David | Mark, Yoko | Cindy | |Alyssa | George, David | Jim | |Yoko | Jim, Alyssa | George | |Cindy | George, Kevin | David | +------------------------------------------------+ ================= || Kevin Ryman || ================= __________________ |Basic Information\ ------------------- - Occupation: Police Officer, Raccoon City Police Department (R.P.D.) - Vitality: 2300 points - Viral Infection Rate: 1.19% per minute, 84 minutes to 100% - Personal Item: 45 Auto A more powerful weapon than the usual handgun, but ammo for this weapon is scarce. Kevin's custom weapon gives him the advantage in boss fights, so reserve its usage until encountering a worthwhile foe. - Extra Item: 45 Auto Magazine An extra clip of ammunition for Kevin's personal 45 Auto. Hold the L1 button to quickly reload the 45 Auto. The magazine can be refilled by combining it with 45 Auto Rounds. - Bio: Officer Ryman works for the Raccoon City Police Department. He possesses superior athletic abilities and is an outstanding shot. A all around good guy, he's a dyed-in-the-wool optimist who doesn't dwell on petty matters. His happy go lucky personality sometimes works against him. He has failed the S.T.A.R.S. selection process twice. - Characteristics: Kevin is the fastest of all the characters, which is useful for hurrying through scenarios in the fastest possible time. His powerful custom 45 Auto and unique unarmed attacks make him an excellent character choice for beginning players and action oriented players alike. ________________ |Special Actions\ ----------------- - Kick Hold the R1 button and press Cancel to execute a swift kick. Use it to knock enemies backward just as they are about to attack. When used properly, it creates enough room to aim a firearm or prepare a knife attack. The kick can also be used to kick down locked doors and open locked wall panels. - Critical Shot Hold the R1 button for a long moment while equipped with a Handgun or the 45 Auto. Soon, Kevin will readjust his aim. If you wait to fire until after Kevin has readjusted, his aim is better and the resulting shot causes more damage to an enemy. This is not effective for rifles. - Elbow Tackle Hold the R1 button and press Circle when no weapon is equipped. Kevin lunges further and knocks down enemies more easily with this move than other player characters can accomplish with standard Tackle attacks. ================== || Mark Wilkins || ================== __________________ |Basic Information\ ------------------- - Occupation: Security Guard - Vitality: 3000 points - Viral Infection Rate: 1.31% per minute, 76 minutes to 100% - Personal Item: Costume Handgun Mark's personal automatic is similar to the other handguns found commonly throughout the game. Although Mark's Custom Handgun inflicts slightly less damage per attack, the barrel has been modified to inflict more damage at longer range. - Extra Item: Handgun Magazine An extra clip for Mark's Handgun, this clip enables faster reloading. Hold the L1 button to quickly reload Mark's Handgun. The Handgun Magazine can be refilled by combining it with Handgun Rounds. - Bio: Currently working for a security company in Raccoon City, Mark is a Vietnam veteran. Approximately 50 years old, his robust strength has not diminished. He has tasted the emptiness of war and now, more than anything, he just wants to live in peace. - Characteristics: Mark, who possesses the highest vitality points, is the strongest character in the game. On his own, he can move heavy objects that normally require two players to move. Due to his size he is the second slowest player character in the game, and he cannot hide inside lockers or closets. ________________ |Special Actions\ ----------------- - Guard Hold the R1 button and press Cancel to guard against enemy attacks. Mark can fend off common attacks from most frequently encountered foes. However, his virus meter still increases due to the contact. Strong attacks from unique boss enemies may still cause damage to Mark even while he is guarding. - Full Swing Hold the R1 button while equipped with a melee weapon that swings, such as an Iron Pipe, Long Pole, or a Crutch. Continue holding the R1 button until Mark raises the melee weapon higher than usual. This indicates tat Mark is ready to perform a "full swing", causing more damage than the normal melee weapon attack. ================= || Jim Chapman || ================= __________________ |Basic Information\ ------------------- - Occupation: Subway Transit System Employee - Vitality: 1800 points - Viral Infection Rate: 1.43% per minute, 70 minutes to 100% - Personal Item: Coin While hanging around a location that is free of enemies, select Jim's coin and use it. Jim produces his coin and flips it into the air. The result is displayed on screen. Each time the coin comes up "Heads", Jim's rate of critical hits increases 10%. Therefore, if you flip the coin three times in a row and achieve "Heads" each time, Jim's critical hit rate rises by 30%! However, if the coin comes up "Tails", the bonus is reset to 0%. Used wisely, the Coin can turn Jim into a real killing machine! - Extra Item: Lucky Coin When Jim or any character possesses this item, the chance of critical hit occurrence rises 5%, the durability of melee weapons becomes stronger, and Handguns and Shotguns cannot be broken by a Hunter's attack. AIPC Jim rarely agrees to trade this item for anything, and players controlling Jim should hold onto the Lucky Coin rather than drop it in favor of other items. - Bio: An agent with the Raccoon City subway, Jim is friendly and cheerful but sometimes reveals a hesitant side. Although he means well, he talks too much and sometimes bothers people around him. To his credit, he has strong powers of intuition and is skillful at solving puzzles. - Characteristics: Jim is an average person in a very unique situation. Guided by fear and cowardice, his greatest skill is his ability to avoid attacks by pretending to be dead. Although playing as Jim takes some getting used to, a keen player soon realizes that the subway worker presents great advantages as a character choice. ________________ |Special Actions\ ----------------- - Playing Dead Hold the R1 button, the press and hold the X button. This causes Jim to fall to the ground and remain motionless. Enemies will ignore him while he is playing dead, so it is useful when surrounded. However, don't overuse this skill, as the virus gauge increases more rapidly while he plays dead. By the way, if an enemy lands on top of Jim or punched Jim on the ground, Jim can still get hurt so be careful! - Swing Combo Hold the R1 button and press Circle to swing a melee weapon such as an Iron Pipe, Long Pole or Crutch. Press Circle again the moment Jim finishes his first swing to immediately perform another. This special action leaves Jim breathless for a moment, so use it with caution. - Item Search Even when Jim enters a room for the first time, the positions of the items in the room are indicated on the map by a question mark. The type of item is not specified, however, until the item is examined. This unique feature enables Jim to find items faster than other characters, especially hidden or unseen items. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 4. Boss strategies for Jim XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX _______________________ Wild Things Scenario \ ------------------------- *Stalker as the boss: - Every time he growls that means he is going to jump at you! What you do is simple just play dead until he leave you alone. After you are away from him just loaded some shotguns ammo to his side of the body. - DO NOT get close to him because he can turn around fast and slam you! *Titan as the boss: - Go back to the entrance you came from but do not go to the other area. This place is the safest place in the game. The elephant cannot get you here unless he uses his nose to reach you. Every time he does that just play dead until he stops. You are like in a small cage and he can't get you, which makes it easier even in VH games! All you do is just shoot him and play dead everytime he tries to attack you. _______________________ Underbelly Scenario \ ------------------------- *Giga Bite boss: - Not much to do here since Jim can avoid every attack that the insect boss has. Every time you see bunch of insects rolling like a tire all you have to do is just play dead to avoid it and get back up and continue attacking the boss. Beware you still can get hurt while playing dead! _______________________ Flashback Scenario \ ------------------------- *The Axeman: - Jim ability to play dead is very useful against this boss. He is not really the "main" boss in Flashback scenario. But if you see him coming at you, all you have to do is play dead until he goes away or if he starts chasing your friends. *Huge Plant: - Jim is not very good against this plant unless if you are Samuel. Why Samuel? Because Samuel got higher damage ratio than any other Jim types. You can play dead if you see the plants above your head try to grab you and choke you. Try to get at least 2 HEADS and grab an axe. _________________________ Desperate Times Scenario \ --------------------------- * Mob of Zombies: - Jim is not good against this many zombies. I recommend you to use Samuel because he is stronger and better with guns. If there are more than 3 zombies coming at you, try to do the "worm" by playing dead. You can lead them to the gas tank and blow them off. You should choose your partners carefully. _________________________ End of The Road Scenario \ --------------------------- * Mr. X first form: - There is not much you can do with him. All you do is just play dead if he comes near you. He is almost like Thantos from File 1 except he is much smarter, stronger and faster. * Mr. X second form: - This one is much harder to fight against even if you are Mr. Green or Samuel. I recommend you to just ignore him and play dead until he ignores you. Do that if you don't have the remote control. Remote control can only use once so don't forget that and play dead if he did his 1 hit move! * Nyx: - You don't have to fight him unless the helicopter leaves you behind. You better wish that you have some magnums or other powerful weapons. You also need to have at least 1 rocket launcher too. Take the beast down to his knee by the rocket launcher. Try not to miss or you are in big trouble. If you missed, just check the van near the monster and you can find an extra rocket launcher. Let's think you didn't miss. After the beast is on his knee this is your chance to attack him. Wait until the core on his chest open up. If it didn't open up and he stand up you better have some "extra" weapons that can take him down. After his core opens up, attack it with all your weapons that you have. Good Luck! XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 5. Scenario Notes about Jim XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Wild Things: - Jim's ability to play dead is useful against the elephant boss or the lion - Jim has more chance to hear the elephant roaring at the beginning Underbelly Scenario: - Jim started with the Employee Key under easy or normal mode - Jim does not need to take the map to know the surroundings - Jim is able to crawl under the ventilation hole to take shortcuts - Jim can talk to his friend in the Men's bathroom (East) - Jim can open his locker in Break room Flashback Scenario: - Its easy to get NO DAMAGE by finishing the game using the "Illusion Ending" (Using the new bridge) - Great against The Axeman - Great against the Giant Mutated Plant boss (Samuel) Desperate Times: - Samuel's normal virus gauge help you a little bit. - Jim's ability to play dead is useful to lure zombies to a nearest gas tank. End Of The Road: - Jim's ability to play dead is great to use when there is bunch of hunters - Jim's ability to play dead is great when Mr. X turn his back and try to attack you - You could beat VH with Samuel or Mr. Green to get NO WEAPONS easily - Jim's ability to play dead is 2nd to Yoko's crawl to avoid land mines XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 6. Jim Secret Costumes List XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 5) Jim B Costume Points: 1000 Vital: Small Requirements: Open by default Starting Status: Fine Speed: Slow Starting Item: Lucky Coin 6) Jim C Costume Points: 1000 Vital: Small Requirements: Gather all Jim Costume SP Starting Status: Fine Speed: Slow Starting Item: Lucky Coin 29) Kurt B Costume Points: 3000 Vital: Medium Requirements: Complete Flashback Scenario as Alyssa, collect all Main Files and kill Axeman Starting Status: Danger Speed: Normal Starting Item: Hemostat 33) Jean Costume Points: 5000 Vital: Large Requirements: Complete Desperate Times Scenario Starting Status: Fine Speed: Normal Starting Item: Blue Herb 34) Samuel Costume Points: 4000 Vital: Large Requirements: Complete Desperate Times Scenario under Hard Starting Status: Fine Speed: Normal Starting Item: Iron Pipe 40) Will Costume Points: 300 Vital: Small Requirements: Open by default Starting Status: Fine Speed: Fast Starting Item: Recovery Pill 52) Peter Costume Points: 300 Requirements: All ready there! Starting Status: Fine Speed: Slow Starting Item: Anti Virus Pill 58) Mr. Green Points: 10,000 Vital: Super Max Requirements: Open by default Starting Status: Fine Speed: Fast Starting Item: None XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 7. Jim's SP Item List XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX *CREDITS GOES TO SPACEKADET* =========== WILD THINGS =========== [_] Fanny Pack I used to think only du***** wore these, but this one looks cool. [_] Basketball Ticket Oh, snap! This is for my team, yo! D***! This is my lucky day! ...S***... It's over a year old... [_] Chicken Sneakers The name sounds stupid, but these are actually some good shoes, white with red and yellow stripes. [_] Deodorizing Spray "A 5 seconds spray lasts 5 hours! No odor too strong!" I guess I have been smellin' a little funky lately. ========== UNDERBELLY ========== [_] Biking Shoes The soles are flat so you can pedal faster. They look pretty tight. [_] Custom Shoes A pair of shoes specially made for whoever ordered 'em. They ain't really mine, but whatever... [_] Memorial Sneakers These were made when that famous basketballer retired. Only 34 pair were ever made. Nice find. [_] Wooden Clogs Shoes carved from oak. I seen these on TV, but I never seen anyone crazy enough to wear 'em outside. ========= FLASHBACK ========= [_] Water proof parka D***! That's a nice parka! Thin and waterproof. With style to spare. [_] Light-Up Shoes Shoes that light-up when you step. Maybe I'm just nuts, but even though they're for kids, I kinda like 'em. [_] Crossword cards A set of 128 crossword puzzle cards. A full set is pretty d*** rare! [_] Stop watch A digital stop watch. It says it's accurate to 1/1000th of a second. I could use-it at work. =============== DESPERATE TIMES =============== [_] Racing Helmet A helmet molded from space-age plastic. I've been meanin' to pick one of these up anyhow. [_] Cyber Shoes Disco-lookin' shoes that are a mix between futuristic design and 70s flair. [_] Perfect Dictionary "Everything from slang to dead languages! Includes blank pages to add new words!" What a rip-off... [_] Wristband My favorite basketball player wears one just like this. Same color and everything. =============== END OF THE ROAD =============== [_] Racing Pants [_] Luxurious Shoehorn [_] "Puzzle 100" Ancient Puzzle that take "100 years or more to solve". S** ain't nothin' I can't handle [_] "Shoes Monthly" A magazine for crazy peeps like me who dig shoes. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 8. Frequently Asked Question XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Q) What does it mean when you said bad partners and good partners for Jim? A) This is a new thing in Outbreak File 2. Jim doesn't like Kevin or Yoko. So if you choose Kevin and Yoko as your partner in offline mode, it will be a harder game because they won't help you or listen to you. On the other hand, Mark and George always listen to Jim and protect him, thus making the game much easier. Q) So, in File 1 can Jim have his lucky coin? A) NOPE Q) What is the strongest NPCs that have Jim's type? A) That would be Samuel Q) Can Jim easily avoid the land mines in End Of The Road scenario? A) Kind of. If you do it correctly, then it should be easy. Q) What kind of AIs partners should Jim have in offline mode? A) To be unstoppable that would be Mr. Gold and The Axeman Costume Q) Is it true that Jim's voice has changed? A) Yes Q) Is it true Jim can get hurt even when he is playing dead? A) Yes, so becareful! XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 9. Contact XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX +---------------------------------------------------------------+ | E-mail : Scaryexecutioner@yahoo.com | | Online Username : The_Executioner | | YIM : Scaryexecutioner (Note: I don't have AIM!) | +---------------------------------------------------------------+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 10. Updates XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX - June 8, 2005 Description: I remade the FAQ - August 31, 2005 Description: Changes few things XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 11. Credits XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Thanks to: - Myself - SpaceKadet's SP FAQ - My laptop - Opung Bapa and Opung Dadua - Brady Walkthrough XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 12. Legal and Copyright *MUST KNOW* XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX This FAQ/Walkthrough/Guide is All Right reserved (C) The Executioner 2005 *You may NOT host this FAQ/Walkthrough/Guide without my permission!

Laser Linewidth and Bandwidth Calculator Lasers are often assumed to be nearly monochromatic: with our laser linewidth and bandwidth calculator, you will learn that this is not necessarily true. In this article, you will find: - A quick but clear explanation of the operations of a laser device; - What is the laser linewidth: a definition; - How to calculate the laser linewidth: the formula for the deviation from monochromaticity; - The differences between laser linewidth and laser bandwidth; - How to calculate the laser bandwidth; and - How to use our laser linewidth and bandwidth calculator. Lasers straight to the pointer Lasers are devices that use the property of light and a medium to obtain emission of directional, monochromatic, and coherent light, which means that: - Laser light propagates in one direction; - Most of the photons emitted by a laser have the same wavelength; and - The emitted photons are in phase. How do we obtain these properties? The answer lies in the design of laser emitters. In any device, from a pointer to an anti-missile military megawatt laser, an excited medium emits, when stimulated, photons at a specific wavelength. The particular design of a laser's oscillating chamber, with two mirrors on the opposite ends, allows for the emission of a powerful light beam with high spatial and temporal coherence. Each medium emits at a specific wavelength: a source of energy (for example, an electric field) excites the medium, moving a fraction of the material's electrons to a higher, metastable energy level. One of these electrons will fall to the ground state by chance, emitting a photon. This photon, bouncing back and forth in the amplification chamber, stimulates the relaxation of other electrons. A partially transparent window on one of the mirrors allows the escape of a portion of the amplified light. 💡 You might also be interested in checking out our laser brightness calculator. What is the linewidth of a laser? In the ideal world where there's no air resistance and cows are spherical, lasers emit (if properly engineered) light of a single wavelength. In reality, sources of noise (both technical, due to the design of the device, and quantum, due to the intrinsic nature of the atomic-scale world) cause shifts in the emission of the device. The amplitude of these shifts defines how much a laser device deviates from monochromaticity. We can quantify this phenomenon with the concept of the spectral linewidth of a laser We define the linewidth of a laser as the full width at half maximum (FWHM) of the amplitude spectrum of a laser emitter. This definition may not be that clear, so let's analyze it in detail. A real laser emits a spectrum of frequencies (the optical spectrum). The power of the laser output is distributed on this spectrum — the linewidth measures how much of the laser's output doesn't peak around the fundamental frequency. Now that you know the definition of laser linewidth, we can learn how to calculate it with the laser linewidth equation. How to calculate the linewidth of a laser The calculations to find the formula for the linewidth of a laser come from the depths of quantum mechanics (even though at the dawn of the field, the derivation of this quantity still used many classical concepts). The calculation relies on the uncertainty principle associated with energy. Knowing that the Planck constant relates energy and time suggest that the characteristic times of the stimulated emission process significantly affect the output performances of laser devices. The laser linewidth equation is: - — Laser linewidth; - — Planck constant, a reminder that we are dealing with quantum phenomena; - — Fundamental frequency of the laser; - — Q-factor of the "cold" laser cavity, a measure of the strength of the damping of the oscillations (also known as cavity linewidth); and - — Power of the laser mode. 🔎 In the formula you can see the expression : this is the Planck's relation with which you can calculate the energy of a photon, and no, it's not a coincidence! This formula is a modification of the original calculations for the spectral linewidth of a laser by Schawlow and Townes but is developed entirely in a quantum framework. It is considered the best expression for the linewidth. Laser linewidth vs. laser bandwidth Apart from the laser spectral linewidth, each laser has a specific bandwidth. What's the difference? You can consider the bandwidth as the possible spectrum of the laser's output, while the laser linewidth considers the "occupied" frequencies in the optical spectrum. The laser bandwidth can't be calculated using a single equation but rather depends on multiple factors, and it isn't easy to model the characteristics of the resonating chamber. The value of the bandwidth is usually given in the device's specification. We can convert the laser bandwidth starting from the fundamental wavelength of the laser and the width of the wavelength range with the formula: Where the two extremes of the frequency range are calculated with: These formulas use the relation between frequency and wavelength: you can learn more at our wavelength calculator. How to use our laser linewidth and bandwidth calculator Our laser linewidth and bandwidth calculator allows you to both: - Calculate the laser linewidth; and - Convert between wavelength and frequency bandwidth. Give us the parameter you know, and we will calculate the result. Remember that our calculator uses the frequency and not the wavelength, which is more often specified in the datasheet. Convert the values using our calculators (as the wavelength to frequency calculator). It's time for a neat example to teach you how to use our laser linewidth and bandwidth calculator! We consider a common He-Ne laser with fundamental wavelength . Consider a laser with power and cavity linewidth . 🔎 A wavelength of corresponds to a frequency of . Input these values, with the correct units, in our calculator. We will apply the laser linewidth formula for you: This is a relatively narrow linewidth: we used the specification of a high-level laser. What about the bandwidth? Let's test our laser bandwidth calculator with an easy example. Take a blue Helium-Cadmium laser with wavelength . A good quality pointer has a bandwidth of . What is the corresponding frequency bandwidth? Apply the formula by inputting the known values: What is the linewidth of a laser? The linewidth of a laser is a measure of the spread of the output's power over a finite range of frequencies. The laser linewidth is defined as the full width at half maximum (FWHM) of the power spectrum of the output. The laser linewidth is strongly related to the spectral coherence of the output: the smaller the coherence, the cleaner the spectrum, with a strongly peaked output around the fundamental frequency of the device. What is a narrow linewidth laser? A narrow linewidth laser is a device that emits light with a high degree of monochromaticity. To obtain such characteristics, the laser must operate at a single frequency, and external sources of noise should be reduced as much as possible (preventing mode hopping). Finally, the laser design should minimize internal noise sources (e.g., phase noise). Narrow linewidth lasers have critical applications in medicine and sensing. How do I calculate the spectral linewidth of a laser? To calculate the laser linewidth, we use the following equation: Δν = (π h v Γ²)/P And follow these steps: - Multiply the fundamental frequency ( v) of the laser by the square of the cavity linewidth ( - Multiply the result by the constant his the Planck constant. - Divide by P, the power of the laser mode. Which is the linewidth of a typical laser pointer? 19.7 kHz. Considering a typical red laser pointer, we know the following quantities: - The device's power: P = 5 mW. - The fundamental wavelength: v = 635 nm. - The cavity linewidth: Γ = 10 GHz. Apply the formula for the linewidth of a laser: Δν = (π h v Γ²)/P = (π h (472.114 THz) (1 GHz)²)/(0.005 W) = 19.7 kHz

mobil 18yas sikir porno arab - Xxxxx bp hindi eglish arbi A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. They next appear to have been used by Daniel Schwenter (1585-1636) in a Geometrica Practica published in 1618. The theory, however, starts with the publication in 16J5 by Lord Brouncker of the continued fraction I 23252 i 2 2 2 . This he is supposed to have deduced, no one knows how, from Wallis' formula for ?? Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium. Nicol Saunderson (1682-1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler's Algebra (1795). Stern wrote at length on the subject in Crelle's Journal (x., 1833; xi., 1834; xviii., 1838). A new waterfront area at Cardiff Bay contains the Senedd building, home to the Welsh Assembly and the Wales Millennium Centre arts complex. Current developments include the continuation of the redevelopment of the Cardiff Bay and city centre areas with projects such as the Cardiff International Sports Village, a BBC drama village, Sporting venues in the city include the Millennium Stadium (the national stadium for the Wales national rugby union team), SWALEC Stadium (the home of Glamorgan County Cricket Club), Cardiff City Stadium (the home of Cardiff City football team), Cardiff International Sports Stadium (the home of Cardiff Amateur Athletic Club) and Cardiff Arms Park (the home of Cardiff Blues and Cardiff RFC rugby union teams). Caer is Welsh for fort and -dyf is in effect a form of Taf (Taff), the river which flows by Cardiff Castle, with the ⟨t⟩ showing consonant mutation to ⟨d⟩ and the vowel showing affection as a result of a (lost) genitive case ending. The antiquarian William Camden (1551–1623) suggested that the name Cardiff may derive from "Caer-Didi" ("the Fort of Didius"), a name supposedly given in honour of Aulus Didius Gallus, governor of a nearby province at the time when the Roman fort was established. Although some sources repeat this theory, it has been rejected on linguistic grounds by modern scholars such as Professor Gwynedd Pierce. The Cardiff Urban Area covers a slightly larger area outside the county boundary, and includes the towns of Dinas Powys and Penarth. A small town until the early 19th century, its prominence as a major port for the transport of coal following the arrival of industry in the region contributed to its rise as a major city.Cardiff was made a city in 1905, and proclaimed the capital of Wales in 1955.Since the 1980s, Cardiff has seen significant development.We have seen that the simple infinite continued fraction converges. The tests for convergency are as follows: Let the continued fraction of the first class be reduced to the form dl d2 d3 d4 then it is convergent if at least one of the series. diverges, and oscillates if both these series converge. In fact, a continued fraction ai a2 an can be constructed having for the numerators of its successive convergents any assigned quantities pi, P2, P3, , p ,,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I bnpn -2 1 qn = anq,i l bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 b2= q2. n l which we can transform into u1 u2 utu3 u2114 un -2u,, -u1 u2-u2 u3-u3 u4- ... There is, however, a different way in which a series may be represented by a continued fraction. It is practically identical with that of finding the greatest common measure of two polynomials. We have F(n i,x) -F(n,x) = (y n)(y n I) F (n 2,x), whence we obtain F(i,x) _ i / y (y I) x /(y I)(y 2) which may also be written y 7 I-1-7 2 - By .The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. For the convergence of the continued fraction of the second class there is no complete criterion. Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange's investigation of the vibrations of a stretched string (see Lord Rayleigh, Theory of Sound, vol. If we form then the continued fraction inwhich pi, p2, p3 9 ..., pn are u1, u1 u 2, ul u2 u3, , /41 u2 un, and ql, q2, q3, , qn are all unity, we find the series u 1 u2 . un equivalent to the continued fraction un u l ul un - 1 ? We may require to represent the infinite convergent power series ao alx a2x 2 ... As an instance leading to results of some importance consider the series x x2 F(n,x) =I (y n)I! By putting =x 2 /4 for x in F(o,x) and F(i,x), and putting at the same time y =1/2, we obtain x 2 x 2 x2 x 2 x2 tan x x x tanh x x x x I - 3 - 5-7-...The Cardiff metropolitan area makes up over a third of the total population of Wales, with a mid-2011 population estimate of about 1,100,000 people.

Graphing Linear Functions analyzemath.com If two linear equations are given the same slope it means that they are parallel and if the product of two slopes m1*m2=-1 the two linear equations are said to be perpendicular. Video lesson If x is -1 what is the value for f(x) when f(x)=3x+5?... Observing the y intercepts, we can see that the y intercept of the the linear equation y = x + 3 is y = 3, the y intercept of the equation y = x - 2 is y = -2, and the y intercept of the quadratic function is y = -6. There is a relationship between them. In fact, the y intercept of the parabola is the product of the y intercepts of the linear equations! Expressions and Equations Worksheets 3/08/2012 · Good Day, I need some help with the following problem. I do not know how to generate a linear equation connecting y and x from the expression below as the 4 seems to complicate things... Given any function f for which we know f(x 0) and f '(x 0) we can immediately evaluate this approximation. Using it involves pretending that the graph of the function f were its tangent line at x 0 , rather than whatever it is. Linear combination Wikipedia 17/02/2013 · Best Answer: There are a few different ways to do this problem. There's polynomial division, factoring and there is the Remainder theorem among other ways. I am not sure what context this question falls but I can help you find the answer anyway. If one of those linear expressions is …... To find the y-intercept, we substitute 0 for x in the equation, because we know that every point on the y-axis has an x-coordinate of 0. Once we do that, we can solve to find the value of y . When we make x = 0, the equation becomes , which works out to y = 2. 1.3 linear equations in two variables Academics Solving linear equations is one of the most fundamental skills an algebra student can master. Most algebraic equations require the skills used when solving linear equations. 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To convert the standard form ax + by + cz = d of a plane into parametric form: treat the equation as a system of one linear equation in the variables x, y and z and solve it using row reduction. - Solution. Before we can do the probability calculation, we first need to fully define the conditional distribution of Y given X = x: Now, if we just plug in the values that we know, we can calculate the conditional mean of Y given X = 23: - Given any function f for which we know f(x 0) and f '(x 0) we can immediately evaluate this approximation. Using it involves pretending that the graph of the function f were its tangent line at x 0 , rather than whatever it is. - A linear programming problem is one in which we are to find the maximum or minimum value of a linear expression ax + by + c z + . . . 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NCERT solutions for class 8 Maths Maths is one such subject that requires considerable time and effort. We at Physics Wallah have devised specific NCERT Solutions for class 8 Maths for all students. NCERT textbook of class 8 maths consists of 16 chapters having multiple exercises. This page consists of solutions to all questions asked in the NCERT textbook of class 8 maths. If students are looking forward to clearing competitive examinations in the future, they need to study specifically. This NCERT solution for class 8 maths is categorized in chapter-wise segments. NCERT Solutions for Class 8 Maths Chapters (Updated 2022-23) NCERT Solutions for Class 8 Maths Free PDF Download Chapter 1 Rational Numbers In this chapter, students will learn properties related to real numbers, whole numbers, rational numbers, and natural numbers—independent, associative, and closed. The chapter covers the role of zero and one, multiplication over addition, and the representation of the rational numbers on number line, along with finding rational numbers between the two rational numbers. The topics of additive identity and multiplicative identity are also covered in this chapter. Two exercises in this chapter contain questions from all the topics covered in the chapter. - Important Questions for class 8 maths Chapter 1 Rational Numbers - RS Aggarwal Class 8 solutions for Maths Chapter 1 Rational Numbers Chapter 2 Linear Equations in One Variable The chapter Linear Equations in One Variable deal only with the linear expression of one variable. Here, the chapter will deal with equations with linear expressions in only one variable. Such equations are known as linear equations in one variable. This chapter has six exercises, with a total of 65 questions. The chapter covers a variety of concepts, including solving the equation of linear equations in one variable, some of its applications, reducing equations to a more straightforward form, and word problems related to linear equations in one variable. - Important Questions for class 8 maths Chapter 2 Linear Equations in One Variable - RS Aggarwal Class 8 solutions for Maths Chapter 2 Linear Equations in One Variable Chapter 3 Understanding Quadrilaterals The chapter-3, Understanding Quadrilaterals, of Class 8 Maths, as the name says, that provides the students with a proper understanding of quadrilaterals. The chapter also covers polygons, including triangles, quadrilaterals, pentagons, and hexagons. The chapter contains four exercises that aid the students in understanding these shapes properly. - Important Questions for class 8 maths Chapter 3 Understanding Quadrilaterals - RS Aggarwal Class 8 solutions for Maths Chapter 3 Understanding Quadrilaterals Chapter 4 Practical Geometry The Practical Geometry chapter contains a method of constructing a quadrilateral with various given parameters. The chapter consists of 5 exercises, each dealing with different methods of quadrilateral construction. For example, the first exercise deals with the construction of quadrilaterals when given the lengths of the four sides and the diagonal; similarly, the second exercise involves constructing quadrilaterals, given two diagonals and three sides, and so on. So we can conclude that by studying this chapter, students are well-versed in all terms of the quadrilateral for the next lessons. Also Check- Important Questions for class 8 maths Chapter 4 Practical Geometry Chapter 5 Data Handling A collection of information that can be used for analysis is called data. In this chapter, students will learn about the organization and representation of data. Arranging data systematically is called data processing. In this chapter, students will learn to represent this data schematically as a pictogram, a bar chart, a double bar chart, a pie chart, and a histogram. Throughout this chapter, students will be introduced to the concept of probability. Three exercises in this chapter cover all the concepts covered in the chapter. - Important Questions for class 8 maths Chapter 5 Data Handling - RS Aggarwal Class 8 solutions for Maths Chapter 5 Data Handling Chapter 6 Square and Square Roots The chapter-6, Squares and Square Roots helps the students learn about the concept of square numbers and the square root of a number. The chapter deals with properties of square numbers, interesting patterns that could be learned by finding square of a number, finding square roots through various methods, Pythagorean triplets, Square roots of decimals, and much more. - Important Questions for class 8 maths Chapter 6 Squares and Square Roots - RS Aggarwal Class 8 solutions for Maths Chapter 6 Squares and Square Roots Chapter 7 Cube and Cube Roots In chapter 7, Cubes and Cube Roots, students discuss the numerous strategies to find out the cubes and cube roots of a number. The chapter discuss about the finding cube of different number, also some interesting patterns that could be learned using cubes and finding the cube roots. Two exercises in this chapter will help the students understand the basics of Cubes and Cube Roots in depth. - Important Questions for class 8 maths Chapter 7 Cube and Cube Roots - RS Aggarwal Class 8 solutions for Maths Chapter 7 Cube and Cube Roots Chapter 8 Comparing Quantities The Comparing Quantities chapter helps students learn to compare percentage increases and decreases, market value, sale value, discount, etc. The chapter discuss about an important concept that can be used in everyday life, Interest. Students will learn about simple Interest, compound interest calculated semi-annually, quarterly, monthly, or annually, and much more. There are three exercises in this chapter to help students learn the concept of comparing quantities. Also Check- Important Questions for class 8 maths Chapter 8 Comparing Quantities Chapter 9 Algebraic Expressions and Identities Chapter 9, Algebraic Expressions and Identities, helps the students to understand the concepts covered in this chapter. The terms factors and coefficients are related to the Algebraic Expressions discussed in this chapter. The basics of binomials, monomials and polynomials are also discussed in this chapter. Students also examine the subtraction, addition and multiplication of Algebraic Expression. The concept of Identity discussed in this chapter. The five exercises in the chapter cover all the concepts present in Algebraic Expressions and Identities. - Important Questions for class 8 maths Chapter 9 Algebraic Expressions and Identities - RS Aggarwal Class 8 solutions for Maths Chapter 9 Algebraic Expressions and Identities Chapter 10 Visualizing Solid Shapes In this chapter, with the help of three exercises, students also learn to visualize solid shapes in different dimensions. This chapter is exciting since it deals with viewing 3-D shapes, Mapping the space around us, and learning about faces, edges, and vertices. The chapter also discusses Euler’s formula, which states that F + V – E = 2, where F is Faces, V is Vertices, and E is Edges, along with its application. Also Check- Important Questions for class 8 maths Chapter 10 Visualizing Solid Shapes Chapter 11 Mensuration Students in lower classes must discuss the area and perimeter of various closed plane figures such as rectangles, triangles, circles, etc. In this chapter, Mensuration, students will also learn to calculate problems related to the perimeter and area of other plane closed figures like quadrilaterals. Students will discuss the surface area and volume of different solid shapes, such as a cube, cuboids, and cylinders. Four exercises are present in this chapter. Also Check- Important Questions for class 8 maths Chapter 11 Mensuration Chapter 12 Exponents and Powers In chapter 12, Exponents and Powers, students discuss different concepts, including powers with negative exponents, using exponents and laws of exponents to express the numbers in standard form, and comparing extremely large numbers with small numbers. This chapter consists of two exercises to help the students learn about the topics of Exponents and Powers. Also Check- Important Questions for class 8 maths Chapter 12 Exponents and Powers Chapter 13 Direct and Inverse Proportions This chapter contains two exercises that contain questions based on direct and indirect proportionality. Two quantities, x and y, are said to be directly proportional if they increase (decrease) together so that the ratio of their corresponding values remains constant. On the other hand, two quantities, x, and y are said to be inversely proportional if an increase in x causes a proportional decrease in y (and vice versa) so that the product of their corresponding values remains constant. - Important Questions for class 8 maths Chapter 13 Direct and Inverse Proportions - RS Aggarwal Class 8 solutions for Maths Chapter 13 Direct and Inverse Proportions Chapter 14 Factorization In this chapter, students will learn to factorize. Topics in this chapter include factoring natural numbers and algebraic expressions, factoring by rearranging terms, factoring by identities. The chapter also deals with division of algebraic expressions, which includes dividing a monomial by another monomial, dividing a polynomial by a monomial, etc. There are four exercises in this chapter that contain questions covering all the topics covered in the chapter. - Important Questions for class 8 maths Chapter 14 Factorization - RS Aggarwal Class 8 solutions for Maths Chapter 14 Factorization Chapter 15 Introduction to Graphs In chapter 15, Introduction to Graphs, students discuss representing different data types with different kinds of graphs. The various types of graphs include bar graphs, pie graphs, line graphs, and linear graphs. The chapter will help the students learn how to locate a point and coordinate in a graph. A total of three exercises are present in the chapter that will help the students understand the concepts of Graphs. Also Check- RS Aggarwal Class 8 solutions for Maths Chapter 15 Introduction to Graphs Chapter 16 Playing with Numbers So far, students have studied different types of numbers such as natural numbers, whole numbers, integers and rational numbers. They could also study a number of interesting properties about these numbers, finding factors, multiples and relationships between them. Similarly, students can explore a wide genre of numbers in detail in this chapter. These ideas can help students reason about divisibility tests. There are two exercises in this chapter that cover the topic of Playing with Numbers. Also Check- RS Aggarwal Class 8 solutions for Maths Chapter 16 Playing with Numbers Why are NCERT Solutions for Class 8 Maths important? Since maths is one such subject that cannot be mugged up and a certain level of persistence. This NCERT solution for class 8 Maths created by us helps the students to answer questions accordingly. In mathematics, every question holds significant importance, whereas NCERT solutions for class 8 Maths are helpful. The team at Physics Wallah is well experienced to guide students in an optimum manner. We provide students with an effective and practical guide through NCERT solutions for class 8 Maths prepared by our team. In today’s time, obtaining the right guidance for students can be very difficult. The team at Physics Wallah is all the way available to provide you with guidance. The students can get themselves acquainted with the latest syllabus with the planned approach through NCERT solutions for class 8 Maths. This study material is very useful for students looking forward to making their preparation very strong. These NCERT solutions for class 8 maths are very helpful for students to revise the complete syllabus. The study material has been prepared in a very concise manner. Our faculties at Physics Wallah have done extensive research while preparing NCERT solutions for class 8 maths for students. Every sum and theorem is explained in a detailed manner. How to Study NCERT Solutions for Class 8 Maths effectively? The experts in our team belong to prominent institutions like IITs. The NCERT solutions for class 8 maths study material from Physics Wallah have always been a preference of study material provided by the team at Physics Wallah. To make students' preparation easy, we have lined down important questions in our NCERT solutions for class 8 maths. Very deep research has been done on previous year's question papers. The NCERT solutions for class 8 maths have been prepared after continuous error checking and the complete study material is flawless. We also provide students with a strategy to target exams specifically, whether they are preparing for school exams or competitive exams. Tips to solve Class 8 Maths Numerical - Numerical solving skills in Maths require good concept and application of the concept in numerical. - To build good concepts in maths you require a textbook that consists of detailed theory with step-by-step solved examples explaining all the concepts. - NCERT textbook will help you with its unique theory explanations and added solved questions. The theory given in NCERT clarifies your concept. - Once you are confident in concepts that you require questions to solve. - The NCERT consists of a good number of questions and the best part is the difficulty level of questions increases slowly which helps you to increase your confidence level and improve your problem-solving ability. - While solving exercises NCERT Solutions for class 8 Maths will help you a lot. - Solve all MCQ given in the class 8 maths section of Physics Wallah chapter-wise. Why Physics Wallah is best for NCERT Solutions for Class 8 Maths? The team at Physics Wallah believes in sharing study material with needy students. We have provided complete NCERT solutions for class 8 maths on our website for free. Students looking forward to scoring great marks in their examination can download with a single click. The NCERT solutions for class 8 maths study material are provided in Pdf format. Students can also share this study material among their friends in an easily shareable format. This study material can be accessed on multiple devices. Students can get their doubts cleared and concepts stronger from NCERT solutions for class 8 maths. Right at home, you can get the guidance of top professors in India. Tips to score Good Marks in Class 8 Maths Maths is the subject of intensive practice and application of concepts in numerical. To score good marks in maths one must follow a few important do and don’ts What are do’s - You must read the theory parts of the chapter. - Make sure you have a good understanding of the chapter and you have prepared your note in detail. - Write down all important math formulas and try to remember all the important formulas used in these chapters. - Write down step-by-step solutions to important and difficult questions. - Practice questions from the Physics Wallah class 8 maths section, do objective questions and try to solve all questions by your method don’t jump to the solution immediately. What are don'ts - Don’t believe in learning maths from maths solutions, always try to prepare your solution make sure you have attempted questions several times before moving to the solution part of the question. - Try to build concepts from the NCERT textbook. Right Approach to use NCERT Solutions for Class 8 Maths In class 8 maths you are going to read lots of new chapters which are very important in your upcoming classes. Mathematics foundations start to form class 8 maths so one must be careful while preparing your subject.The best approach to build your subject is to read the theory given in the NCERT textbook which is explained very nicely. You can easily understand the concepts with the theory and are able to learn how to apply the formula in questions with the help of examples given in NCERT textbook. Once you have gone through the theory written in the textbook of NCERT that attends your tuition or school lecture reading theory before your maths class will bust your confidence in class. Start preparing your notes based on your class 8 maths. To prepare notes NCERT textbooks are a good resource. If you are preparing for the Olympiad or want to have a more solid foundation in class 8 maths then start reading the Physics Wallah resource given in the maths section. Start with exercise 1 try to solve the question by yourself, don't take any help from the solution or teacher. If you feel you have attempted a particular question several times still your answer is incorrect then take help from NCERT solutions for class 8 maths prepared by Physics Wallah. Frequently Asked Question (FAQs) Q1. How can I study for Class 8 Maths? Ans. To score good marks in class 8 Maths one must follow the right strategies from day one of his/her class 8. Start your preparation with the NCERT textbook, read the theory carefully, and try to prepare your notes of every chapter by mentioning all-important formulas required in the chapter. Move to solve the exercise and try to solve all questions given in the NCERT exercise with the help of NCERT solutions for class 8 Maths. While preparing the note make sure you have added all important points to the chapter. Do as many as the questions you can do use NCERT Exemplar to solve more questions of class 8 Maths. Q2. What are the chapters of Class 8 Maths? Ans. There is a total of 16 chapters which are as follows: - Chapter 1: Rational Numbers - Chapter 2: Linear Equations in One Variable - Chapter 3: Understanding Quadrilaterals - Chapter 4: Practical Geometry - Chapter 5: Data Handling - Chapter 6: Square and Square Roots - Chapter 7: Cube and Cube Roots - Chapter 8: Comparing Quantities - Chapter 9: Algebraic Expressions and Identities - Chapter 10: Visualizing Solid Shapes - Chapter 11: Mensuration - Chapter 12: Exponents and Powers - Chapter 13: Direct and Inverse Proportions - Chapter 14: Factorization - Chapter 15: Introduction to Graphs - Chapter 16: Playing with Numbers Q3. What are the most important chapters in NCERT Class 8 Maths? Ans. Based on the weightage of allocated marks mensuration is the highest weightage unit and if you want a good foundation in class 8 Maths for class 10 and 11 you need to focus on few chapters like Rational Numbers, Squares and Square Roots, Cubes and Cube Roots, Mensuration, Exponents, and Powers. Q4. From where we will get MCQ-based Questions in Class 8 Maths? Ans. Objective questions are a good method to check your concepts. Solving MCQ-based questions helps you to identify the error in the concept and you will get a direction to modify your mistakes. To solve MCQ-based questions for class 8 Maths uploaded by Physics Wallah. All questions are with detailed solutions to give you a better understanding. Q5. What is the correct method to use NCERT solutions for class 8 Maths? Ans. The best way to use NCERT solutions for class 8 Maths is using it as a reference for those questions which you aren't able to solve after taking multiple attempts. While solving the NCERT exercise make sure you learn all formulas which are used in the chapters. Never depend on NCERT Solutions for Class 8 Maths. Q6. Sample papers along with NCERT class 8 Maths will help? Ans. Sample papers for class 8 Maths are very important for the entire revision of the syllabus. One must do practice from sample papers. To help you our team uploaded several sample papers for class 8 maths having all types of questions like subjective and MCQ-based questions. Q7. If we are preparing for the Olympiad and other competitive exams, do we need additional resources? Ans. Now a day’s foundation course is doing very well preparing for the Olympiad exam from the very beginning like from class 8. If you are preparing for a foundation course then you need some additional resources apart from NCERT class 8 Maths. For such students, Physics Wallah prepared a separate section for class 8 Maths to read all theories and solve additional questions given in this section. - chapter 1-Rational Numbers - chapter 2-Linear Equations in One Variable - chapter-3 Understanding Quadrilaterals - chapter 4-Practical Geometry - chapter 5-Data Handling - chapter 6-Squares and Square Roots - chapter 7-Cubes and Cube Roots - chapter 8-Comparing Quantities - chapter-9 Algebraic Expressions and Identities - chapter -10 Visualising Solid Shapes - chapter 11 Mensuration - chapter 12 Exponents and Powers - chapter-13 Direct and Inverse Proportions - chapter-14 Factorisation - chapter-15 Introduction to Graphs - chapter-16 Playing with Numbers

« PreviousContinue » PROP. XV. PROB. TO inscribe an equilateral and equiangular hexagon in a given circle. Let ABCDEF be the given circle ; it is required to inscribe an equilateral and equiangular hexagon in it. Find the centre G of the circle ABCDEF, and draw the diameter AGD; and from D as a centre, at the distance DG, describe the circle EGCH, join EG, CG, and produce them to the points B, F; and join AB, BC, CD, DE, EF, FA: the hexagon ABCDEF is equilateral and equiangular. Because G is the centre of the circle ABCDEF, GE is equal to CD: and because D is the centre of the circle EGCH, DE is equal to DG; wherefore GE is equal to ED, and the triangle EGD is equilateral; and therefore its three angles EGD, GDE, DEG are equal to one another, because the angles at a 5. 1. the base of an isosceles triangle are equal a ; and the three angles b 32. 1. of a triangle are equal b to two right angles ; therefore the angle EGD is the third part of two right angles: in the same c c 13. 1. adjacent angles EGC, CGB equal to two right angles ; the remaining EGD, DGC, CGB are equal to one d 15. 1. another: and to these are equal d the vertical opposite angles BGA, AGF, equal to another : but equal e 26. 3. angles stand upon equal e circumfe H rences; therefore the six circumfe rences AB, BC, CD, DE, EF, FA are equal to one another: 29. 3. and equal circumferences are subtended by equal f straight lines ; therefore the six straight lines are equal to one another, and the hexagon ABCDEF is equilateral. It is also equiangular; for, since the circumference AF is equal to ED, to each of these add the circumference ABCD: therefore the whole circumference FABCD shall be equal to the whole EDCBA; and the angle FED stands upon the circumference FABCD, and Book IV; the angle AFE upon EDCBA; therefore the angle AFE is equal to FED: in the same manner it may be demonstrated that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED; therefore the hexagon is equiangular; and it is equilateral, as was shown; and it is inscribed in the given circle ABCDEF. Which was to be done. Cor. From this it is manifest, that the side of the hexagon is equal to the straight line from the centre, that is, to the semidiameter of the circle. And if through the points A, B, C, D, E, F there be drawn straight lines touching the circle, an equilateral and equiangular hexagon shall be described about it, which may be demonstrated from what has been said of the pentagon; and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon. PROP. XVI. PROB. TO inscribe an equilateral and equiangular quin. See N. decagon in a given circle. Let ABCD be the given circle ; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD. Let AC be the side of an equilateral triangle inscribed a in a 2. 4. the circle, and AB the side of an equilateral and equiangular pentagon inscribed b in the same; therefore, of such equal parts b 11.4. as the whole circumference ABCDF contains fifteen, the circumference ABC, being the third А part of the whole, contains five; and the circumference AB, which is the fifth part of the whole, contains three; therefore BC their difference B F contains two of the same parts: bi EI sect BC in E; therefore BE, EC c 30. 3. are, each of them, the fifteenth part C of the whole circumference ABCD: therefore, if the straight lines BE, EC be drawn, and straight lines equal to them be placed d around in the whole circle, an equila-d 1.4. teral and equiangular quindecagon shall be inscribed in it. Which was to be done. Book IV. And in the same manner as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about it: and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it. ELEMENTS OF EUCLID. I. A LESS magnitude is said to be a part of a greater magnitude, Book V. when the less measures the greater, that is, when the less is • contained a certain number of times exactly in the greater.' II. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, ' when the greater contains the less a certain number of times exactly.' III. · Ratio is a mutual relation of two magnitudes of the same kind Sec N. to one another, in respect of quantity.' second, which the third has to the fourth, when any equimul- Book V. the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. N. B. When four magnitudes are proportionals, it is usually the fifth definition) the multiple of the first is greater than have to the third the duplicate ratio of that which it has to the XI. See N. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it Definition A, to wit, of compound ratio. the first is said to have to the last of them the ratio com- magnitude. For example, if A, B, C, D be four magnitudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D:

The Battle Over Discrete Math Sets and How to Win It The Upside to Discrete Math Sets Zeno created a string of paradoxes employing the new notion of infinitesimals to discredit the entire area of study and it’s those paradoxes which we are going to be taking a look at today. For this reason, you’re, in addition, a descendant of your grandparents. It’s not, as an example, an in-depth treatise on group theory. The very first semester is mainly a foundations and logic course composed of the initial five chapters of the text. Just logging in some forum once a while and read what people are speaking about. Late homework won’t be accepted, and (except in the event of a documented university conflict) makeup quizzes won’t be given. The aim is to analyze these statements either individually or within a composite way. Given the next set, choose the statement below that’s true. Given the next sets, choose the statement below that’s true. The Pain of Discrete Math Sets Many references are included for those who have to probe further into the subject that’s suggested if these methods should be applied. My notes assume that you’re employing the seventh edition. Finally, we will plot two examples. The calculator lacks the mathematical intuition that’s very practical for finding an antiderivative, but on the other hand it can try out a lot of possibilities in a short timeframe. term papers help In some games, the optimal strategy is to select a single action regardless of what your opponent does. To confirm your guess you should use the strong kind of induction. Actually, Stanford’s encyclopedia entry on set theory is a fantastic place to start. After a careful examination of the Earth, however, you may actually arrive at the conclusion this is reduntant. Computational thinking is something which you can discover how to develop overtime. Now, here are a few additional sets that likewise satisfy Peano’s axioms. The second factor has the 2 roots These 2 roots, which are the exact same as the ones found by the very first method, form the period-2 orbit. Otherwise, a suitable subset is precisely the same as a normal subset. What You Must Know About Discrete Math Sets The educational facet of discrete mathematics is every bit as important and deserves extended coverage by itself. Discrete Math has a large application in today’s mathematics, and generally utilised in decision mathematics. They can play a key role in this connection. Type make to create the program. Anything you’re going to be in a position to solve in math, you may also compose a program for it. Course content will be different. The War Against Discrete Math Sets You will only get a concise introduction to logic inside this class, but the mathematics utilized in logic are observed at the core of computer programming and in designing electrical circuits. In the realm of computers this logic is used when creating logic gates that are the hardware of all contemporary computers today. Category theory is likewise very helpful for clarifying things which appear more complicated using different approaches, in addition, it has lost of practical applications in computer science, so I think it’s a worthwhile topic of study. Then, utilizing the simple fact that it’s true for 7, show that’s also true for 8 etc.. This isn’t a comprehensive collection of the benefits of ebooks. Also, it can be any arbitrary problem, where we clearly understand where it’s applied. Whatever They Told You About Discrete Math Sets Is Dead Wrong…And Here’s Why The notation for the overall concept can fluctuate considerably. Graphing particular kinds of equations is covered extensively in the notes, however, it’s assumed that you comprehend the standard coordinate system and the way to plot points. It’s not sufficient to just understand a concept and go on to the next. A subsequent paper will give a basic, high-level review of the algorithm. These exercises precede other exercises that need the reader to determine which Principle to use or require utilizing both Principles. Category theory is a rather generalised sort of mathematics, it’s considered a foundational theory in the exact same way that set theory is. The Definitive Strategy to Discrete Math Sets Our program is extremely strict and requires elevated levels of private discipline, and compliance is an excellent indicator of the applicant’s capacity to conform with our intensive atmosphere. The formula is a bit more complicated if your events are dependent, that’s in case the probability of a single event effects another. Continuous data aren’t restricted to defined separate values, but might occupy any value on a continuous selection. If it isn’t, then a filter needs to be placed on the data before instituting the right analysis. With fish, you need to be mindful about combinations. All functions may be used as static function of DiscreteMath. The Do’s and Don’ts of Discrete Math Sets Similarly, 5 isn’t a perfect cube. Symbols are a concise method of giving lengthy instructions linked to numbers and logic. It could appear odd to define a set that comprises no elements. There are lots of ways to accomplish such a selection. After pasting, you might need to opt for the perfect font in your intended application to see all the symbols. Actually, when that mark is necessary, it is normal to use syntactic sets.

Approach to Chandrasekhar-Kendall-Woltjer State in a Chiral Plasma We study the time evolution of the magnetic field in a plasma with a chiral magnetic current. The Vector Spherical Harmonic functions (VSH) are used to expand all fields. We define a measure for the Chandrasekhar-Kendall-Woltjer (CKW) state, which has a simple form in VSH expansion. We propose the conditions for a general class of initial momentum spectra that will evolve into the CKW state. For this class of initial conditions, to approach the CKW state, (i) a non-vanishing chiral magnetic conductivity is necessary, and (ii) the time integration of the product of the electric resistivity and chiral magnetic conductivity must grow faster than the time integration of the resistivity. We give a few examples to test these conditions numerically which work very well. In high energy heavy-ion collisions, two heavy nuclei are accelerated to almost the speed of light and produce very strong electric and magnetic fields at the moment of the collision Kharzeev et al. (2008); Skokov et al. (2009); Voronyuk et al. (2011); Deng and Huang (2012); Bloczynski et al. (2013); McLerran and Skokov (2014); Gursoy et al. (2014); Roy and Pu (2015); Tuchin (2015); Li et al. (2016a). The magnitude of magnetic fields can be estimated as , where and are the proton number and the radius of the nucleus respectively, is the the velocity of the nucleus and is the Lorentz contraction factor ( is the nucleon mass and is the collision energy per nucleon). In Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC) with GeV, the peak value of the magnetic field at the moment of the collision is about ( is the pion mass) or Gauss. In Pb+Pb collisions at the Large Hadron Collider (LHC) with TeV, the peak value of the magnetic field can be 10 times as large as at RHIC. Such high magnetic fields enter strong interaction regime and may have observable effects on the hadronic events. The chiral magnetic effect (CME) is one of them which is the generation of an electric current induced by magnetic fields from of the imbalance of chiral fermions Kharzeev et al. (2008); Fukushima et al. (2008); Kharzeev et al. (2016). The CME and other related effects have been widely studied in quark-gluon plasma produced in heavy-ion collisions. The charge separation effect observed in STAR Abelev et al. (2009, 2010) and ALICE Abelev et al. (2013) experiments are consistent to the CME predictions, although there may be other sources such as collective flows that contribute to the charge separation Huang et al. (2015). The CME has recently been confirmed to exist in materials such as Dirac and Weyl semi-metals Son and Spivak (2013); Basar et al. (2014); Li et al. (2016b). In hot and dense matter an imbalance in the number of right-handed quarks and left-handed quarks may be produced through transitions between vacua of different Chern-Simons numbers in some domains of the matter. This is called chiral anomaly and is described by the anomalous conservation law for the axial current, where denotes the axial 4-vector current with being the chiral charge, and are the number of colors and flavors of quarks respectively, is the electric charge (in the unit of electron charge ) of the quark with flavor , denotes the field strength of the electromagnetic field and is its dual, is the strong coupling constant, denotes the field strength of the -th gluon with and is its dual. The first term on the right-hand-side of Eq. (1) is the anomaly term from electromagnetic fields while the second one is from gluonic fields. In Eq. (1) we have neglected quark masses. For electromagnetic fields we can write in the 3-vector form using , where and are the electric and magnetic 3-vector field respectively. The axial current breaks the parity locally and may appear in one event, but it is vanishing when taking event average. With such an imbalance, an electric current can be induced along the magnetic field, so the total electric current can be written as where and are the electric and chiral conductivity respectively. Note that is proportional to the difference between the number of right-handed quarks and left-handed quarks which breaks the parity but conserves the time reversal symmetry. This is in contrast with the electric conductivity which breaks the time reversal symmetry but conserves the parity. So the Ohm’s current is dissipative (with heat production) while the chiral magnetic current is non-dissipative. where the total helicity is defined by combining the magnetic helicity and the chiral charge , where . This means that the magnetic helicity and the chiral charge of fermions can be transferred into each other. The Chandrasekhar-Kendall-Woltjer (CKW) state is a state of the magnetic field which satisfies the following equation where is a constant. The CKW state was first studied by Chandrasekhar, Kendall and Woltjer Chandrasekhar (1956); Chandrasekhar and Kendall (1957); Chandrasekhar and Woltjer (1958); Woltjer (1958) as a force free state. We notice that in a plasma with the chiral magnetic current (2), if the Ohm’s current is absent, the system reaches a special CKW state with following the Ampere’s law. To our knowledge, this idea was first proposed in Chernodub (2010). But with the Ohm’s current, can the CKW state still be reached? This question can be re-phrased as: what are the conditions under which the CKW state can be reached in a plasma with chiral magnetic currents? In this paper we will answer this question by studying the evolution of magnetic fields with the Maxwell-Chern-Simons equations. In classical plasma physics, a state satisfying Eq. (5) is called the Taylor state or the Woltjer-Taylor state. It was first found by Woltjer Woltjer (1958) that the CKW state minimizes the magnetic energy for a fixed magnetic helicity. In toroidal plasma devices, such a state is often observed as a self-generated state called reverse field pinch with the distinct feature that the toroidal fields in the center and the edge point to opposite directions. Taylor Taylor (1974, 1986) first argued that the minimization of magnetic energy with a fix magnetic helicity is realized as a selective decay process in a weakly dissipative plasma when the dynamics is dominated by short wavelength structures. Taylor’s theory has been questioned and debated intensely, and alternative theories has been proposed Ortolani and Schnack (1993); Qin et al. (2012). It is certainly interesting that both classical plasmas and chiral plasmas have the tendency to evolve towards such a state, which suggests that the two systems may share certain dynamics features responsible for the emerging of the state. A brief discussion in this aspect is given in the paper as well. The paper is organized as follows. In Section II, we start with the Maxwell-Chern-Simons equations and define a global measure for the CKW state by the magnetic field and the electric current. In Section III, we expand all fields in Vector Spherical Harmonic (VSH) functions. The inner products have simple forms in the VSH expansion. The parity of a quantity can be easily identified in the VSH form. We give in Section IV the solution to the Maxwell-Chern-Simons equations for each mode in the VSH expansion. The conditions for the CKW state are given in Section V. In Section VI we test these conditions by examples including the ones with constant and self-consistently determined . We also generalize the momentum spectra at the initial time from a power to a polynomial pre-factor of scalar momentum in Section VII. The summary and conclusions are made in the last section. Ii Maxwell-Chern-Simons equations and CKW state We start from Maxwell-Chern-Simons equations or anomalous Maxwell equations, where we have included the induced current and neglected the displacement current . We have also dropped the external charge and current density. We assume that and depend on only. A similar equation for can also be derived but we will not consider it in the current study. To measure whether the CKW state is reached in the evolution of the magnetic field, we introduce the quantity where we have used the notation for the inner product for any vector field and . According to the Cauchy-Schwartz inequality we have , where the equality holds only in the case of when the CKW state is reached Qin et al. (2012). We assume that is a smooth function of . The condition that the CKW state is reached can be given by Note that should not exactly be equal to 1, since is a smooth function of and bounded by the upper limit 1. To see the time evolution of , it is helpful to write inner products in simple forms, which we will do in the next section. Iii Expansion in vector spherical harmonic functions In this section, we will expand all fields in the basis of Vector Spherical Harmonic function (VSH), with which we can put inner products into a simple and symmetric form. iii.1 Expansion in VSH The quantities we use to express in Eq. (11) are and . We can extend the series to include more curls, So the inner products can be written as with and are non-negative integers. To find an unified form for the fields in this series, we can expand in the Coulomb gauge in VSH where is the scalar momentum and are the quantum number of the angular momentum and the angular momentum along a particular direction respectively. are divergence-free vector fields which can be expressed in term of VSH Jackson (1999). The explicit form of can be found in, e.g., Ref. Hirono et al. (2015); Tuchin (2016). The orthogonal basis functions satisfy the following orthogonality relations, where . Note that themselves are CKW states satisfying and are divergence-free, , so we can expand any divergence-free vector fields in . We note that is real while and are complex. iii.2 Inner products We can put the general inner products into a simple form by using the orthogonality relation (16), where are defined by We note that are positive definite. In deriving Eq. (20) we have used the fact that is real so that . For convenience, we use the following short-hand notation for the integral From Eq. (20) it is easy to verify for . This is consistent to the identity . In Table (1), we list the VSH forms of some inner products that we are going to study later in this paper. iii.3 Parity and helicity From Eq. (17), the parity transformation is equivalent to the interchange of the and mode. In the series (14), the quantity is parity-even/parity-odd (P-even/P-odd) for odd/even . For instance, , and are P-odd, P-even and P-odd respectively. Also the inner product is P-even/P-odd for even/odd , see the last column of Table (1) where the magnetic helicity is P-odd, and the magnetic energy is P-even. For a momentum spectrum containing only , the helicity is positive. If such a magnetic field can approach the CKW state, it means because is also positive for mode. In contrast it would mean for a momentum spectrum containing only . Iv Solving Maxwell-Chern-Simons equations in VSH where is the electric resistivity. The solution of is in the form where denote the values at the initial time , and and are defined by Note that both and are positive. Alternatively we can rescale time by using as a new evolution parameter, and rewrite , i.e., is the integrated value of from to . There is a competition between and for approaching or departing the CKW state. Large values of is favored for the CKW state. We will show that it is indeed determined by the increasing ratio of to . Note that in Eq. (25) changing is equivalent to interchanging the positive and negative modes, therefore we can assume in this paper without loss of generality. V Conditions for CKW state In this section we will study the evolution of the fields in the basis of VSH and look for the conditions for the CKW state. From Eq. (11) we obtain in VSH, To verify , it is better to rewrite in a more symmetric form, The difference between the denominator and numerator is where we have used in the first inequality. From the inequality (29) it is obvious that , where the equality holds for and one of and is zero. Therefore we have two conditions under which is satisfied: where is the central momentum of during evolution. Both conditions can be physically understood. The first condition is actually the presence of (we have assumed ), which makes positive modes grow with time while negative modes decay away. It means that the CKW state should contain only positive (or negative) helicity mode only, which is reasonable because the CKW state is the eigenstate of the curl operator. For the second condition, we notice that bases themselves are CKW states from Eq. (17), therefore one single mode in the expansion (19) is natually the CKW state. The authors of Ref. Hirono et al. (2015) observed in the evolution to the CKW state. However, the delta function is not well defined mathematically, so the second condition is hard to implement and we must find a better one to replace it. where the time functions are defined by Here are the powers of in the integrals for , and , respectively. If the initial spectrum functions contain only ( is a real number), or ( and are real constants), the integrals are just Gaussian-like integrals and easy to deal with. In this paper we assume that take the following form A typical example of magnetic fields expressed in such a form is the Hopf state Irvine and Bouwmeester (2008). Although this assumption narrows the scope of , it is still general enough: these three kinds of functions are widely used in other fields of physics. It is natural to combine with in the integrand of and re-define the time function as where is the power of in the initial spectrum functions . Note that does not converge, so we assume . By changing the integral variable where is always positive by definition, we can rewrite in the form where are defined by with the time functions by We rewrite in Eq. (31) as a function of through , We give relevant properties of in Appendix A. One property is that are monotonically increasing functions of , which approach zero at , but rise sharply to at . By Eq. (37), if grows faster than with , we have as . As time goes on, associated with positive modes will grow up but associated with negative modes will decay away. At , Eq. (38) becomes This fulfills the first condition for the CKW state in (30), i.e. only the positive modes survive at the end of the time evolution. So we can summarize the conditions for the CKW state to be reached in time evolution: Note that or plays an essential role: it makes negative modes more and more suppressed while making positive modes blow up as time goes on. At the same time it makes at so that . In heavy-ion collisions, and are decreasing functions of as the result of the expansion of the QGP matter. It is natural to assume that and fall with time in power laws Tuchin (2013), and , where . This can be justified by the fact that MeV Ding et al. (2011) and Kharzeev and Warringa (2009), where both the temperature and the chiral chemical potential decrease with time in power laws in expansion. In this case we have , following Eq. (26), and , the condition for the CKW state now becomes The above condition is very easy to check and it is one of the most useful and practical criteria in this paper. The fact that a large will bring the system to the CKW state shows that a non-Ohmic current may play a crucial role in the process of reaching the CKW state. This suggests that in classical plasmas systems, a non-Ohmic current, e.g., the Hall current, could produce the same effect. In a classical system with the Hall current and negligible flow velocity, the evolution of the magnetic field is governed by where is the density of the plasma, is electron charge, and the last is the Hall current term. Obviously, with a small resistivity, the system approaches equilibrium when the CKW state is reached. Vi Examples and tests of conditions In this section we will look at examples of the CKW state to test the conditions we propose in the last section. vi.1 With only As the first example, let us consider an initial spectrum with only without . We assume has the following form, where characterizes the length scale of the magnetic field, is the initial magnetic helicity. The normalization constant is chosen to be which gives the initial magnetic helicity of the spectrum, From Eq. (39) we obtain is given by Eq. (37) with and . Here we have suppressed the superscript of and simply denote . To verify our conditions for the CKW state, we consider following cases: In case a) both and are constants, which is used in Refs. Tuchin (2015); Li et al. (2016a) to calculate the magnetic field in medium. In this case, we have and in late time which satisfies the condition , and we can see the effect of non-vanishing constant . Such an effect can be seen by comparing with case b) in which we switch off . In case c) is still a constant as same as case a), but is chosen to break the condition with and . In case d), the values and are used in Refs. Tuchin (2013); Yamamoto (2016), which are thought to be more reasonable in heavy-ion collisions. But we note that in real situations of heavy-ion collisions, the time behaviors of and can be very complicated (may not follow power laws), but our conditions in (41) are still applicable. For numerical simulation, we choose and . The results are shown in Fig. 1. Indeed in case a) and d), the CKW state can be reached. As goes from to , according to Eqs. (46, 47, 66), evolves from to , and evolves from 0.8 to 1. In case b) and c) the condition (42) is not satisfied, the CKW state is inaccessible. Indeed the simulation shows that it is true since tends toward 0 and in case b) and c) at , respectively. Even though and increase with , we have and at corresponding to and respectively. All these results show that the conditions work well. But we should point out that constant and or even the power law decayed and may not be physical since once persists for a long time, growing faster than will make some physical quantities diverge. We look at the magnetic helicity and the magnetic energy , The numerical results of the magnetic helicity are shown in Fig. 2. The results of the magnetic energy are similar. In case b) and c), since and converge to constants, but keeps growing, both and finally decay to zero following Eq. (49). However in case a) and d), and increase to in late time, and from Eq (65) grow much faster than to make and blow up. From Eq. (24), we see that the spectrum grows exponentially in time for , such an instability has been discussed in Tuchin (2015); Manuel and Torres-Rincon (2015), see also Akamatsu and Yamamoto (2013). This instability is the source of the divergence of and . Such an unphysical inflation can be understood: the appearance of in the induced current leads to the positive feedback that the magnetic field itself induces the magnetic field. If we put no constraint on , as the result, the magnetic field will keep growing and finally blow up at some time. This of course breaks conservation laws. One way to avoid such divergences is to implement conservation laws in the system. This is the topic of the next subsection. vi.2 With only and dynamical We now consider imposing the total helicity conservation in Eq. (4). This has been implemented in Ref. Manuel and Torres-Rincon (2015); Hirono et al. (2015). Here we focus on the approach to the CKW state in evolution. For simplicity, we can parameterize as where and (total helicity) are constants. From Eq. (50), we see that the requirement leads to . The initial spectrum is assumed to be the same as Eq. (44), so we have . The parameters are chosen to be , , and , where is given by Eq. (45). Since is a constant, we have . We can solve self-consistently through , where we have used and that depends on through in Eq. (49). The numerical results for , and are presented in Fig. 3. For comparison, we also show the result for constant with . In both cases, at the beginning, is not large enough to make grows faster than , which makes in Eq. (49) decrease with time. After grows large enough as time goes on, starts to increase after reaching a minimum. In the case of dynamical , according to Eq. (50), and are complementary to each other to make up a seesaw system. In this system, the decreasing of at the beginning raises the value of and makes and grow faster. As the result, the turning point comes earlier than the case of constant . As keeps growing, drops down leading to slower increase of , which makes grows slower. At the end, is saturated to instead of blowing up. Let us look at the asymptotic time behavior of as . As the magnetic helicity is saturated to following Eq. (49), with and at late time, we obtain where and is called product logarithm, which is the inverse function of . From , we obtain at very large , Since the term increases with , is always growing faster than . Thus the conditions are satisfied and the CKW state can be reached. Taking a derivative of with respect to , we obtain at late time from Eq. (53), We have also looked at a general spectrum for at initial time, where the normalization constant is determined by the initial magnetic helicity . We assume obeying the power law decay in time, which gives . In this case, solving Eq. (51) gives the late time asymptotic behavior, Again we see that grows faster than and the CKW state can be finally reached. vi.3 With mixed helicity In this example we consider both positive and negative modes. We will show that only the positive mode survives while the negative mode decays away in late time. Let us consider the most extreme case in which the initial spectra of the positive and negative modes are the same. We take the following initial spectra for , It is obvious that the initial magnetic helicity is zero. Since , is given by Eq. (38) with and . Obviously at the initial time we have because .

By Les Evans 'To have the braveness to imagine outdoors the sq., we have to be intrigued by way of a problem.' advanced Numbers and Vectors attracts at the strength of intrigue and makes use of beautiful purposes from navigation, international positioning structures, earthquakes, circus acts and tales from mathematical heritage to provide an explanation for the maths of vectors and the discoveries in complicated numbers. the 1st a part of complicated Numbers and Vectors presents academics with historical past fabric, principles and instructing ways to advanced numbers; types for complicated numbers and their geometric and algebraic homes; their position in supplying completeness with recognize to the answer of polynomial equations of a unmarried complicated variable (the primary theorem of algebra); the specification of curves and areas within the advanced aircraft; and easy variations of the advanced airplane. the second one a part of this source presents an advent to vectors and vector areas, together with matrix illustration; covers vectors in - and three-dimensions; their program to specification of curves; vector calculus and their simple program to geometric evidence. know-how has been used in the course of the textual content to build photographs of curves, graphs and and 3 dimensional shapes. Read Online or Download Complex numbers and vectors PDF Similar number theory books This publication is an exploration of a declare made via Lagrange within the autumn of 1771 as he embarked upon his long "R? ©flexions sur l. a. answer alg? ©brique des equations": that there were few advances within the algebraic answer of equations because the time of Cardano within the mid 16th century. That opinion has been shared by way of many later historians. Tracing the tale from its earliest assets, this booklet celebrates the lives and paintings of pioneers of recent arithmetic: Fermat, Euler, Lagrange, Legendre, Gauss, Fourier, Dirichlet and extra. contains an English translation of Gauss's 1838 letter to Dirichlet. Algebraic Operads: An Algorithmic significant other offers a scientific remedy of Gröbner bases in different contexts. The ebook builds as much as the speculation of Gröbner bases for operads a result of moment writer and Khoroshkin in addition to quite a few purposes of the corresponding diamond lemmas in algebra. The authors current a number of themes together with: noncommutative Gröbner bases and their functions to the development of common enveloping algebras; Gröbner bases for shuffle algebras which are used to resolve questions about combinatorics of diversifications; and operadic Gröbner bases, vital for purposes to algebraic topology, and homological and homotopical algebra. - Dynamics and Analytic Number Theory - Analytic Number Theory [lecture notes] - An Introduction to Models and Decompositions in Operator Theory - The Fourier-Analytic Proof of Quadratic Reciprocity - Knots and Primes: An Introduction to Arithmetic Topology (Universitext) - A Variational Inequality Approach to free Boundary Problems with Applications in Mould Filling Extra info for Complex numbers and vectors However, it was his work that converted a geometric representation of mathematics into the format with which we are more 30 CHAPTER 3 Secrecy, contrivance and inspiration familiar, an algebraic representation. He achieved this through the use of the cartesian plane, which used a grid system to locate any point on the Euclidean plane. A locus of points can therefore be described by the use of simple equations. Earlier we used the equation y = ax2 + bx + c to describe a particular conic section, the parabola. This could prove to be valuable when we consider the multiplication of complex numbers because, when complex numbers are multiplied, points are both dilated from and rotated about the origin. This should encourage us to multiply two complex numbers expressed in polar form. Let z1 = r1 _cos ^q1h + i sin ^q1hi and z2 = r1 _cos ^q2h + i sin ^q2hi . z1 z2 = r1 _cos ^q1h + i sin ^q1hi r2 _cos ^q2h + i sin ^q2hi = r1 r2 _cos ^q1h cos ^q2h - sin ^q1h sin ^q2h + i sin ^q1h cos ^q2h + i sin ^q2h cos ^q1hi = r1 r2 __cos ^q1h cos ^q2h - sin ^q1h sin ^q2hi + i _sin ^q1h cos ^q2h + sin ^q2h cos ^q1hii Using the compound angle formula for the circular functions: cos ^q1h cos ^q2h - sin ^q1h sin ^q2h = cos ^q1 + q2h and sin ^q1h cos ^q2h + sin ^q2h cos ^q1h = sin ^q1 + q2h gives: z1 z2 = r1 r2 __cos ^q1h cos ^q2h - sin ^q1h sin ^q2hi + i _sin ^q1h cos ^q2h + sin ^q2h cos ^q1hii = r1 r2 _cos ^q1 + q2h + i sin ^q1 + q2hi = r1 r2 cis^q1 + q2h 41 MATHSWORKS FOR TEACHERS Complex Numbers and Vectors This result can also be interpreted in terms of compositions of transformations—two dilations and two rotations. Ii Find the distance from the origin to each of z1, z2 and z1 z2 . iii Find the angles between the Re(z)-axis and the line intervals that join the points z1, z2 and z1 z2 to the origin. c i Plot z1, z3 and z1 z3 on an Argand diagram. ii Find the distance from the origin to each of z1, z3 and z1 z3 . iii Find the angles between the Re(z)-axis and the line intervals that join the points z1, z3 and z1 z3 to the origin. d i Plot z2 , z3 and z2 z3 on an Argand diagram. ii Find the distance from the origin to each of z2 , z3 and z2 z3 . Complex numbers and vectors by Les Evans

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The structure of this humble diagram was formally developed by the mathematician John Venn, but its roots go back as far as the 13th Century, and includes many stages of evolution dictated by a number of noted logicians and philosophers. The earliest indications of similar diagram theory came from the writer Ramon Llull, whos initial work would later inspire the German polymath Leibnez. Leibnez was exploring early ideas regarding computational sciences and diagrammatic reasoning, using a style of diagram that would eventually be formalized by another famous mathematician. This was Leonhard Euler, the creator of the Euler diagram. Euler diagrams are similar to Venn diagrams, in that both compare distinct sets using logical connections. Where they differ is that a Venn diagram is bound to show every possible intersection between sets, whether objects fall into that class or not; a Euler diagram only shows actually possible intersections within the given context. Sets can exist entirely within another, termed as a subset, or as a separate circle on the page without any connections - this is known as a disjoint. Furthering the example outlined previously, if a new set was introduced - birds - this would be shown as a circle entirely within the confines of the mammals set (but not overlapping sea life). A fourth set of trees would be a disjoint - a circle without any connections or intersections. Usage for Venn diagrams has evolved somewhat since their inception. Both Euler and Venn diagrams were used to logically and visually frame a philosophical concept, taking phrases such as some of x is y, all of y is z and condensing that information into a diagram that can be summarized at a glance. They are used in, and indeed were formed as an extension of, set theory - a branch of mathematical logic that can describe objects relations through algebraic equation. Now the Venn diagram is so ubiquitous and well ingrained a concept that you can see its use far outside mathematical confines. The form is so recognizable that it can shown through mediums such as advertising or news broadcast and the meaning will immediately be understood. They are used extensively in teaching environments - their generic functionality can apply to any subject and focus on my facet of it. Whether creating a business presentation, collating marketing data, or just visualizing a strategic concept, the Venn diagram is a quick, functional, and effective way of exploring logical relationships within a context. Logician John Venn developed the Venn diagram in complement to Eulers concept. His diagram rules were more rigid than Eulers - each set must show its connection with all other sets within the union, even if no objects fall into this category. This is why Venn diagrams often only contain 2 or 3 sets, any more and the diagram can lose its symmetry and become overly complex. Venn made allowances for this by trading circles for ellipses and arcs, ensuring all connections are accounted for whilst maintaining the aesthetic of the diagram. A Venn diagram, sometimes referred to as a set diagram, is a diagramming style used to show all the possible logical relations between a finite amount of sets. In mathematical terms, a set is a collection of distinct objects gathered together into a group, which can then itself be termed as a single object. Venn diagrams represent these objects on a page as circles or ellipses, and their placement in relation to each other describes the relationships between them. Commonly a Venn diagram will compare two sets with each other. In such a case, two circles will be used to represent the two sets, and they are placed on the page in such a way as that there is an overlap between them. This overlap, known as the intersection, represents the connection between sets - if for example the sets are mammals and sea life, then the intersection will be marine mammals, e.g. dolphins or whales. Each set is taken to contain every instance possible of its class; everything outside the union of sets (union is the term for the combined scope of all sets and intersections) is implicitly not any of those things - not a mammal, does not live underwater, etc. Other Collections of Gy6 Carburetor Diagram

I know many of you think that if you could overcome the worst possible scenario in terms of results then you would consider it holy grail. But ask yourselves, even if there was such solution, would that be something that you likely follow precisely? Don't hurry to answer...! I bet 95 % of you wouldn't apply the solution, you see it's in our nature to idealise situations, persons...etc The way we consider about something is better than how actually is, that's why. Actually the solution of overcoming 135 loses with just 65 wins regardless of the even chance you select or the distribution of the loses/wins within the 200 results, has been given more than a century ago on the book "10 days at Monte Carlo on the bank's (casino) expense" Take a look on the following progression: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 A total of 360 units are sufficient to overcome even the session from hell or any black swans you might encounter. The key is to understand that actually we don't have to overcome 135 loses, but 70. What really matters is the difference, deduct 65 from 135 and there you are 70, but I'm going to make it "quarters" for you. By using the progression above, with each and every win we are canceling 1 loss, so whether we experience 10 wins and 10 losses or 1 win and 1 loss it doesn't really matter because are canceling each other, also it doesn't matter in which order are going to occur. The above progression is quite simple, after 10 MORE losses than wins you add 1 unit to your bet and so on... I want to repeat the word 10 MORE, because it means that it doesn't have to be 10 losses in a row, it could be 7 wins against 17 losses. So after 70 more losses, you need 45 more wins in order to come on top. Since 65/135 is the worst possible scenario,after those 200 spins the things can go only better, in other words regression towards the mean. How long it could take till you have 45 net wins it's completely another matter, it could a few hours, many hours or even a day! You may encounter very long results like this: L L W W L W L L L W W W L W W W L L W L...and it goes back and forth, back and forth like a pendulum in a perpetual movement! For me would be like a torment, I would wish to lose 10 times more in order to raise my bet and get over with it!It's amazing to me how those people a century ago could actually apply such method!You could spend the whole day inside the casino, literally!The aftermath is:Even if it's valid and completely possible, does it worth your time??Personally speaking, someone must have nerves of steel, PLENTY of time and PATIENCE must be his middle name!I have to admit that I'm not that person!Some VERY valuable feedback provided by the user "UK" on this topic what is the most negative expectation we can encounter in 200 spins? It is always possible to get the permanence of horror right from the start. The program so far will answer the first question and also address the 2nd point in that it will tell you what % of sessions will start of with a loss and never get to a ve balance throughout the session. The program simulates even money roulette (betting on red) and the le partage rule (1.35% edge) with 100,000 sessions of length 200 spins. For flat betting: number of sessions always in a net loss = 6553 (6.55%) average peak gain within a round = 9.44 average peak loss within a round = -12.11 actual peak gain in 100,000 rounds = 58.00 actual peak loss in 100,000 rounds = -62.50 So 93.45% of the time you can expect to quit at some point within the session with a profit - even if it's only 1 unit. The second figure (average peak gain within a round) would suggest that if you get a profit of 9 or 10 units in the session and continue to play on you are making a bad bet, statistically speaking. The final figure (actual peak loss within a round) suggests that a bankroll of 60 - 70 units is sufficient. The program is work in progress and I intend to add more analysis including the number of "reversals" within a session, a reversal being a swing from ve to -ve balance or vice versa within a session. In theory by knowing the average number of reversal for a system you can then keep track of them and quit on a ve balance if you have "used up" your reversals in a session. I've experimented with various systems and the best so far in terms of being able to make a profit at some point (ie; the lowest % of sessions without making any profit at all throughout the session) is the Maxim principle. Note that according to the author this is only meant to be used for craps and also I haven't simulated any of the exit points. Using a maximum stake of 50u: number of sessions always in a net loss = 261 (0.26%) average peak gain within a round = 63.98 average peak loss within a round = -156.53 actual peak gain in 100,000 rounds = 122.00 actual peak loss in 100,000 rounds = -1692.50 A 99.74% chance of quitting with a profit, but of course this is offset by the hugely increased bankroll necessary and attendant risk involved. For a maximum stake of 10u (ie; the progression is 1,2,4,5,6,7,8,9,10) and start over when you get to the end. the results are: number of sessions always in a net loss = 1131 (1.13%) average peak gain within a round = 43.46 average peak loss within a round = -64.63 actual peak gain in 100,000 rounds = 121.50 actual peak loss in 100,000 rounds = -412.00 From my testing so far it seems that most volatile systems (those that offer more "reversals") are those that incorporate both -ve and ve progressions (like the Maxim principle)." Very valuable information from user "UK" and I'd like to thank him for sharing with us, but someone named obviously Max or Maxim or Maximilian stole that progression from the book "Monte Carlo anecdotes", see the "Fitzroy" system on page 141. "Maxim principle" has been (e)mailed across US as sure win method several years back, someone stole the intellectual property, rename it and tried to sell it as holy grail. But the most disappointing is that this progression rather than system, has been made MORE THAN A CENTURY AGO like the first on the beginning of this topic! My conclusion is that we are recycling VERY old knowledge/information and my question is: Is there any progress to our practical knowledge and methods since a CENTURY ago?? Or we just recycling the same as "new"?? The game and its rules have not changed, have we?!

Roulette Inflation with Kähler Moduli and their Axions We study 2-field inflation models based on the “large-volume” flux compactification of type IIB string theory. The role of the inflaton is played by a Kähler modulus corresponding to a 4-cycle volume and its axionic partner . The freedom associated with the choice of Calabi-Yau manifold and the non-perturbative effects defining the potential and kinetic parameters of the moduli brings an unavoidable statistical element to theory prior probabilities within the low energy landscape. The further randomness of initial conditions allows for a large ensemble of trajectories. Features in the ensemble of histories include “roulette trajectories”, with long-lasting inflations in the direction of the rolling axion, enhanced in number of e-foldings over those restricted to lie in the -trough. Asymptotic flatness of the potential makes possible an eternal stochastic self-reproducing inflation. A wide variety of potentials and inflaton trajectories agree with the cosmic microwave background and large scale structure data. In particular, the observed scalar tilt with weak or no running can be achieved in spite of a nearly critical de Sitter deceleration parameter and consequently a low gravity wave power relative to the scalar curvature power. The “top-down” approach to inflation seeks to determine cosmological consequences beginning with inflation scenarios motivated by ever-evolving fundamental theory. Most recent attention has been given to top-down models that realize inflation with string theory. This involves the construction of a stable six-dimensional compactification and a four-dimensional extended de Sitter (dS) vacuum which corresponds to the present-day (late-time) universe e.g., the KKLT prescription . Given this, there is a time-dependent, transient non-equilibrium inflationary flow in four dimensions towards the stable state, possibly involving dynamics in both sectors. Currently, attempts to embed inflation in string theory are far from unique, and indeed somewhat confused, with many possibilities suggested to engineer inflation, using different axionic and moduli fields [2, 3], branes in warped geometry , D3-D7 models , etc. [6, 7]. These pictures are increasingly being considered within a string theory landscape populated locally by many scalar fields. Different realizations of stringy inflation may not be mutually incompatible, but rather may arise in different regions of the landscape, leading to a complex statistical phase space of solutions. Indeed inflation driven by one mechanism can turn into inflation driven by another, e.g., , thereby increasing the probability of inflation over a single mechanism scenario. So far all known string inflation models require significant fine-tuning. There are two classes that are generally discussed involving moduli. One is where the inflaton is identified with brane inter-distances. Often the effective mass is too large (above the Hubble parameter) to allow acceleration for enough e-folds, if at all. To realize slow-roll inflation, the effective inflaton mass should be smaller than the Hubble parameter during inflation, . Scalar fields which are not minimally but conformally coupled to gravity acquire effective mass terms which prevents slow-roll. An example of this problem is warped brane inflation where the inflaton is conformally coupled to the four-dimensional gravity . A similar problem also arises in supergravity. The case has been constructed that has masses below the Hubble parameter which avoids this -problem at the price of severe fine-tuning . Another class is geometrical moduli such as Kähler moduli associated with 4-cycle volumes in a compactifying Calabi-Yau manifold as in [2, 3], which has been recently explored in and which we extend here to illustrate the statistical nature of possible inflation histories. Different models of inflation predict different spectra for scalar and tensor cosmological fluctuations. From cosmic microwave background and other large scale structure experiments one can hope to reconstruct the underlying theory that gave rise to them, over the albeit limited observable range. Introduction of a multiple-field phase space leading to many possible inflationary trajectories necessarily brings a statistical element prior to the constraints imposed by data. That is, a theory of inflation embedded in the landscape will lead to a broad theory “prior” probability that will be updated and sharpened into a “posteriori” probability through the action of the data, as expressed by the likelihood, which is a conditional probability of the inflationary trajectories given the data. All we can hope to reconstruct is not a unique underlying acceleration history with data-determined error bars, but an ensemble-averaged acceleration history with data-plus-theory error bars . The results will obviously be very dependent upon the theory prior. In general all that is required of the theory prior is that inflation occurs over enough e-foldings to satisfy our homogeneity and isotropy constraints and that the universe preheats (and that life of some sort forms) — and indeed those too are data constraints rather than a priori theory constraints. Everything else at this stage is theoretical prejudice. A general approach in which equal a priori theory priors for acceleration histories are scanned by Markov Chain Monte Carlo methods which pass the derived scalar and tensor power spectra though cosmic microwave background anisotropy data and large scale clustering data is described in . But since many allowed trajectories would require highly baroque theories to give rise to them, it is essential to explore priors informed by theory, in our case string-motivated priors. The old top-down view was that the theory prior would be a delta-function of the correct one and only theory. The new view is that the theory prior is a probability distribution on an energy landscape whose features are at best only glimpsed, with huge number of potential minima, and inflation being the late stage flow in the low energy structure toward these minima. In the picture we adopt for this paper, the flow is of collective geometrical coordinates associated with the settling down of the compactification of extra dimensions. The observed inflaton would be the last (complex) Kähler modulus to settle down. We shall call this . The settling of other Kähler moduli associated with 4-cycle volumes, and the overall volume modulus, , as well as “complex structure” moduli and the dilaton and its axionic partner, would have occurred earlier, associated with higher energy dynamics, possibly inflations, that became stabilized at their effective minima. The model is illustrated by the cartoon Fig. 1. We work within the “large volume” moduli stabilization model suggested in [11, 12, 13] in which the effective potential has a stable minimum at a large value of the compactified internal volume, in string units. An advantage of this model is that the minimum exists for generic values of parameters, e.g., of the flux contribution to the superpotential . (This is in contrast to the related KKLT stabilization scheme in which the tree-level is fine-tuned at in stringy units in order for the minimum to exist.) In this paper, we often express quantities in the relatively small “stringy units” , related to the (reduced) Planck mass where is Newton’s constant. In this picture, the theory prior would itself be a Bayesian product of a number of conditional probabilities: (1) of manifold configuration defining the moduli; (2) of parameters defining the effective potential and the non-canonical kinetic energy of the moduli, given the manifold structure; (3) of the initial conditions for the moduli and their field momenta given the potentials. The latter will depend upon exactly how the “rain down” from higher energies occurs to populate initial conditions. An effective complication occurs because of the so-called eternal inflation regime, when the stochastic kicks that the inflaton feels in an e-folding can be as large as the classical drift. This -model is in fact another example of stringy inflation with self-reproduction. (See for another case.) If other higher-energy moduli are frozen out, most inflationary trajectories would emerge from this quantum domain. However we expect other quantum domains for the higher-energy moduli to also feed the initial conditions, so we treat these as arbitrary. The Kähler moduli are flat directions at the stringy tree level. The reason this picture works is that the leading non-perturbative (instanton) and perturbative () corrections introduce only an exponentially flat dependence on the Kähler moduli, avoiding the -problem. Conlon and Quevedo focused on the real part of as the inflaton and showed that slow-roll inflation with enough e-foldings was possible. A modification of the model considered inflation in a new direction but with a negative result. The fields are complex, . In this paper we extend the model of to include the axionic direction . There is essentially only one trajectory if is forced to be fixed at its trough, as in . The terrain in the scalar potential has hills and valleys in the direction which results in an ensemble of trajectories depending upon the initial values of . The field momenta may also be arbitrary but their values quickly relax to attractor values. The paper considered inflation only along the direction while the dynamics in the direction were artificially frozen. We find motion in always accompanies motion in . In Kähler moduli models, there is an issue of higher order perturbative corrections. Even a tiny quadratic term would break the exponential flatness of the inflaton potential and could make the -problem reappear. However, the higher order terms which depend on the inflaton only through the overall volume of the Calabi-Yau manifold will not introduce any mass terms for the inflaton. Although these corrections may give rise to a mass term for the inflaton, it might have a limited effect on the crucial last sixty e-folds. In § 2 we describe the model in the context of type IIB string theory. In § 3 we address whether higher (sub-leading) perturbative corrections introduce a dangerous mass term for the inflaton. In § 4 we discuss the effective potential for the volume, Kähler moduli and axion fields, showing with 3 moduli that stabilization of two of them can be sustained even as the inflaton evolves. Therefore in § 5 we restrict ourselves to with the other moduli stabilized at their minima. § 6 explores inflationary trajectories generated with that potential, for various choices of potential parameters and initial conditions. In § 7 we investigate the diffusion/drift boundary and the possibility of self-reproduction. In § 8 we summarize our results and outline issues requiring further consideration, such as the complication in power spectra computation that follows from the freedom. 2 The Type IIB String Theory Model Our inflationary model is based on the “large-volume” moduli stabilization mechanism of [11, 12, 13]. This mechanism relies upon the fixing of the Kähler moduli in IIB flux compactifications on Calabi-Yau (CY) manifolds by non-perturbative as well as perturbative effects. As argued in [11, 12, 13], a minimum of the moduli potential in the effective theory exists for a large class of models. The only restriction is that there should be more complex structure moduli in the compactification than Kähler moduli, i.e. , where are the Hodge numbers of the CY. (The number of complex structure moduli is and the number of Kähler moduli is . Other Hodge numbers are fixed for a CY threefold.) The “large-volume” moduli stabilization mechanism is an alternative to the KKLT one, although it shares some features with KKLT. The purpose of this section is to briefly explain the model of [11, 12, 13]. An effective supergravity is completely specified by a Kähler potential, superpotential and gauge kinetic function. In the scalar field sector of the theory the action is Here and are the Kähler potential and the superpotential respectively, is the reduced Planck mass eq.(1), and represent all scalar moduli. (We closely follow the notations of and keep and other numerical factors explicit.) The -corrected Kähler potential is Here is the volume of the CY manifold in units of the string length , and we set . The second term in the logarithm represents the -corrections with proportional to the Euler characteristic of the manifold . is the IIB axio-dilaton with the dilaton component and the Ramond-Ramond 0-form. is the holomorphic 3-form of . The superpotential depends explicitly upon the Kähler moduli when non-perturbative corrections are included Here, is the tree level flux-induced superpotential which is related to the IIB flux 3-form as shown. The exponential terms are from non-perturbative (instanton) effects. (For simplicity, we ignore higher instanton corrections. This should be valid as long as we restrict ourselves to , which we do.) The Kähler moduli are complex, with the 4-cycle volume and its axionic partner, arising from the Ramond-Ramond 4-form . The encode threshold corrections. In general they are functions of the complex structure moduli and are independent of the Kähler moduli. This follows from the requirement that is a holomorphic function of complex scalar fields and therefore can depend on only via the combination . On the other hand, should respect the axion shift symmetry and thus cannot be a polynomial function of . (See for discussion.) The critical parameters in the potential are constants which depend upon the specific nature of the dominant non-perturbative mechanism. For example, for Euclidean D3-brane instantons and for the gaugino condensate on the D7 brane world-volume. We vary them freely in our exploration of trajectories in different potentials. It is known that both the dilaton and the complex structure moduli can be stabilized in a model with a tree level superpotential induced by generic supersymmetric fluxes (see e.g. ) and the lowest-order (i.e. ) Kähler potential, whereas the Kähler moduli are left undetermined in this procedure (hence are “no scale” models). Including both leading perturbative and non-perturbative corrections and integrating out the dilaton and the complex structure moduli, one obtains a potential for the Kähler moduli which in general has two types of minima. The first type is the KKLT minima which requires significant fine tuning of () for their existence. As pointed out in , the KKLT approach has a few shortcoming, among which are the limited range of validity of the KKLT effective action (due to corrections) and the fact that either the dilaton or some of the complex structure moduli typically become tachyonic at the minimum for the Kähler modulus. (We note, however, that argued that a consistent KKLT-type model with all moduli properly stabilized can be found.) The second type is the “large-volume” AdS minima studied in [11, 12, 13]. These minima exist in a broad class of models and at arbitrary values of parameters. An important characteristic feature of these models is that the stabilized volume of the internal manifold is exponentially large, , and can be in string units. (Here is the value of at its minimum.) The relation between the Planck scale and string scale is where is the volume in string units at the minimum of the potential. Thus these models can have in the range between the GUT and TeV scale. In these models one can compute the spectrum of low-energy particles and soft supersymmetry breaking terms after stabilizing all moduli, which makes them especially attractive phenomenologically (see [13, 20]). Conlon and Quevedo studied inflation in these models and showed that there is at least one natural inflationary direction in the Kähler moduli space. The non-perturbative corrections in the superpotential eq.(5) depend exponentially on the Kähler moduli , and realize by eq.(3) exponentially flat inflationary potentials, the first time this has arisen from string theory. As mentioned in § 1, higher (sub-leading) and string loop corrections could, in principle, introduce a small polynomial dependence on the which would beat exponential flatness at large values of the . Although the exact form of these corrections is not known, we assume in this paper that they are not important for the values of the during the last stage of inflation (see § 3). After stabilizing the dilaton and the complex structure moduli we can identify the string coupling as , so the Kähler potential (4) takes the simple form where is a constant. Using this formula together with equations (3), (5), and (11), one can compute the scalar potential. In our subsequent analysis, we shall absorb the constant factor into the parameters and . The volume of the internal CY manifold can be expressed in terms of the 2-cycle moduli : where is the triple intersection form of . The 4-cycle moduli are related to the by which gives an implicit dependence on the , and thus through eq.(8). It is known that for a CY manifold the matrix has signature , with one positive eigenvalue and negative eigenvalues. Since is just a change of variables, the matrix also has signature . In the case where each of the 4-cycles has a non-vanishing triple intersection only with itself, the matrix is diagonal and its signature is manifest. The volume in this case takes a particularly simple form in terms of the : Here and are positive constants depending on the particular model.111For example, the two-Kähler model with the orientifold of studied in [22, 12, 13] has , , and . This formula suggests a “Swiss-cheese” picture of a CY, in which describes the 4-cycle of maximal size and the blow-up cycles. The modulus controls the overall scale of the CY and can take an arbitrarily large value, whereas describe the holes in the CY and cannot be larger than the overall size of the manifold. As argued in [12, 13], for generic values of the parameters , , one finds that and at the minimum of the effective potential. In other words, the sizes of the holes are generically much smaller than the overall size of the CY. The role of the inflaton in the model of is the last modulus among the , , to attain its minimum. As noted by , the simplified form of the volume eq.(11) is not really necessary to have inflation. For our analysis to be correct, it would be enough to consider a model with at least one Kähler modulus whose only non-zero triple intersection is with itself, i.e., and which has its own non-perturbative term in the superpotential eq.(5). 3 Perturbative Corrections There are several types of perturbative corrections that could modify the classical potential on the Kähler moduli space: those related to higher string modes, or -corrections, coming from the higher derivative terms in both bulk and source (brane) effective actions; and string loop, or -corrections, coming from closed and open string loop diagrams. As we mentioned before, -corrections are an important ingredient of the “large volume” compactification models of [11, 12, 13]. They are necessary for the existence of the large volume minimum of the effective potential in the models with Kähler moduli “lifted” by instanton terms in the superpotential. The leading -corrections to the potential arise from the higher derivative terms in the ten dimensional IIB action at the order , where and is a generalization of the six-dimensional Euler integrand, Performing a compactification of (13) on a CY threefold, one finds -corrections to the metric on the Kähler moduli space, which can be described by the -term in the Kähler potential (4) (see [23, 16]). We will see later that this correction introduces a positive term into the potential. As discussed in , further higher derivative bulk corrections at and above are sub-leading to the term and therefore suppressed. (Note that in the models we are dealing with, there is effectively one more expansion parameter, , due to the large value of the stabilized .) Also, -corrections from the D3/D7 brane actions depend on 4d space-time curvatures and, therefore, do not contribute to the potential. String loop corrections to the Kähler potential come from the Klein bottle, annulus and Möbius strip diagrams computed in for the models compactified on the orientifolds of tori. The Kähler potential including both leading and loop corrections can be schematically written as (We have dropped terms depending only on the brane and complex structure moduli.) Here and are functions of the moduli whose forms are unknown for a generic CY manifold. If they depend upon the inflaton polynomially, a mass term will arise for with the possibility of an -problem. Further study is needed to decide. Note that, although the exact form of higher (sub-leading) corrections is unknown, any correction which introduces dependence on only via will not generate any new mass terms for . As well as -corrections there are possible -corrections. Non-perturbative effects modify the superpotential by breaking the shift symmetry, making it discrete. As noted we do include these. Although leading perturbative terms leave the Kähler potential -independent, subleading corrections can lead to -dependent modifications, which we ignore here. In this paper, we assume that the higher corrections, though possibly destroying slow roll at large values of the inflaton, are not important during the last stage of inflation. 4 Effective Potential and Volume Stabilization In this Section, we sketch the derivation of the effective field theory potential starting from equations (3,5,8). We choose to be the inflaton field and study its dynamics in the 4-dimensional effective theory. We first have to ensure that the volume modulus and other Kähler moduli are trapped in their minima and remain constant or almost constant during inflation. For this we have to focus on the effective potential of all relevant fields. Given the Kähler potential and the superpotential, it is straightforward but tedious to compute the scalar potential as a function of the fields . To make all computations we modified the SuperCosmology Mathematica package which originally was designed for real scalar fields to manipulate complex fields. The Kähler potential (8) gives rise to the Kähler metric , with This can be inverted to give This is the full expression for an arbitrary number of Kähler moduli . The entries of the metric contain terms of different orders in the inverse volume. If we were to keep only the lowest order terms , the shape of the trajectories we determine in the following sections and our conclusions would remain practically unchanged. That is, we are working with higher precision than necessary. Note that the kinetic terms for and are identical, appearing as in the Lagrangian. The resulting potential is We have to add here the uplift term to get a Minkowski or tiny dS minimum. Uplifting is not just a feature needed in string theory models. For example, uplifting is done in QFT to tune the constant part of the scalar field potential to zero. At least in string theory there are tools for uplifting, whereas in QFT it is a pure tuning (see, e.g., [1, 26] and references therein). We will adopt the form with to be adjusted. We now discuss the stabilization of all moduli plus the volume modulus. For this we have to find the global minimum of the potential eq.(4), which we do numerically. However, it is instructive to give analytic estimations. Following [12, 13], we study an asymptotic form of eq.(4) in the region where both , , and . The potential is then a series of inverse powers of . Keeping the terms up to the order we obtain The cross terms for different do not appear in this asymptotic form, as they would be of order . Requiring and at the minimum of the potential eq.(20), we get where are the values of the moduli at the global minimum. The expression (20) has the structure where the coefficients are functions of and . and are positive but can be of either sign. However, the potential for the volume has a minimum only if , which is achieved for ; otherwise would have a runaway character. Also if all are very large so that , then and cannot be stabilized. Therefore to keep non-zero and negative we have to require that some of the Kähler moduli and their axionic partners are trapped in the minimum. For simplicity we assume all but are already trapped in the minimum. It is important to recognize that trapping all moduli but one in the minimum cannot be achieved with only two Kähler moduli and , because effectively corresponds to the volume, and is the inflaton which is to be placed out of the minimum. The Fig. 2 shows the potential as a function of and for the two-Kähler model. One can see from this plot that a trajectory starting from an initial value for larger than a critical value will have runaway behavior in the (volume) direction. Thus, as shown by , one has to consider a model with three or more Kähler moduli. By contrast, the “better racetrack” inflationary model based on the KKLT stabilization is achieved with just two Kähler moduli . However, in our class of models with three and more Kähler moduli we have more flexibility in parameter space in achieving both stabilization and inflation. Another aspect of the work in this paper is that a “large volume” analog of the “better racetrack” model may arise. We have learned that to be fully general we would allow all other moduli including the volume to be dynamical. This will lead to even richer possibilities than those explored here, where we only let evolve, and assume that varying it does not alter the values of the other moduli which we pin at the global minimum. To demonstrate this is viable, we need to show the contribution of to the position of the minimum is negligible. Following , we set all , , and their axions to their minima and use equations (20) and (21) to obtain the potential for : As one can see from eq.(20), the contribution of to the potential is maximal (by absolute value) when and are at their minimum, and vanishes as . This gives a simple criterion for whether the minimum for the volume remains stable during the evolution of : the functional form of the potential for (23) is insensitive to provided : For a large enough number of Kähler moduli this condition is automatically satisfied for generic values of and . We conclude that with many Kähler moduli the volume does not change during the evolution of the inflaton because the other stay at their minimum and keep the volume stable. Consider a toy model with three Kähler moduli in which is the inflaton and stays at its own minimum to provide an unvarying minimum for . We choose parameters as in set 1 in Table 1 (which will be explained in detail below in Sec. 5), and also let , , and . Eq.(24) is strongly satisfied, , under this choice of parameters. Therefore we can drop the -dependent terms in the potential (20) and use it as a function of the two fields and to find their values at the minimum (after setting also to its minimum). The minimization procedure should also allow one to adjust the uplift parameter in a way that the potential vanishes at its global minimum. With our choice of parameters we found the minimum numerically to be at and with , as shown in Fig. 3. 5 Inflaton Potential We now take all moduli , , and the volume (hence ) to be fixed at their minima, but let vary, since it is our inflaton. For simplicity in the subsequent sections we drop the explicit subscript, setting . The scalar potential is obtained from eq.(4) with the other Kähler moduli stabilized: Here the terms contain contributions from the stabilized Kähler moduli other than the inflaton. We dropped cross terms between and other , , since these are suppressed by inverse powers of . We can trust eq.(25) only up to the order ; at higher orders in , higher perturbative corrections to the Kähler potential eq.(15) start contributing. Explicitly expanding to order yields the simpler expression is a constant term, since and , are all stabilized at the minimum, and , depend only on these . |Parameter set 1| |Parameter set 2| |Parameter set 3| |Parameter set 4| |Parameter set 5| |Parameter set 6| The potential eq.(26) has seven parameters , and whose meaning was explained in § 2. We have investigated the shape of the potential for a range of these parameters. , , control the low energy phenomenology of this model (see ) and are the ones we concentrate on here for our inflation application. We shall not deal with particle phenomenology aspects in this paper. Some choices of parameters , , seem to be more natural (see ). To illustrate the range of potentials, we have chosen the six sets of parameters given in Table 1. There is some debate on what are likely values of in string theory. We chose a range from intermediate to large. Since there are scaling relations among parameters, we can relate the specific ones we have chosen to others. An estimate of the magnitude of comes from a relation of the flux 3-forms and which appear in the definition of (eq.5) to the Euler characteristic of the F-theory 4-fold, which is from the tadpole cancellation condition. This suggests an approximate upper bound . For typical values of , we would have . There are examples of manifolds with as large as , which would result in . Further, the bound itself can be evaded by terms. However, we do not wish to push too high so that we can avoid the effects of higher perturbative corrections. We can use the scaling property for the parameters to move the value of into a comfortable range. This should be taken into account while examining the table. The parameter sets in the table can be divided into two classes: Trajectories in sets produce a spectrum of scalar perturbations that is comparable to the experimentally observed one (good parameter sets), whereas trajectories in parameter set and produce spectra whose normalization is in disagreement with observations (bad parameter sets). Sets 3 and 4 were chosen to large values of to illustrate how things change with this parameter, but we are wary that with such large fluxes, other effects may come into play for determining the potential over those considered here. A typical potential surface is shown in Fig. 4 with the isocontours of superimposed. The hypersurface has a rich structure. It is periodic in with period , as seen in Fig 4. Along , where is integer, the profile of the potential in the direction is that considered in . It has a minimum at some and gradually saturates towards a constant value at large , , where is a constant.222This type of potential is similar to that derived from the Starobinsky model of inflation with a Lagrangian via a conformal transformation, . Along , falls gradually from a maximum at small towards the same constant value, . Trajectories beginning at the maximum run away towards large . Fig. 8(b) shows these two one-dimensional sections of the potential. For all other values of the potential interpolates between these two profiles. Thus, at large , the surface is almost flat but slightly rippled. At small , the potential in the axion direction is highly peaked. Around the maximum of the potential it is locally reminiscent of the “natural inflation” potential involving a pseudo Goldstone boson (except that and must be simultaneously considered), as well as the racetrack inflation potential [2, 3]. which can be used to generate families of models, trading for example large values of for small values of . For instance, applying the scaling to parameter set 1, we can push the value of down to , but at the same time pushing to lie in the range during inflation, a range which is quite problematic since at such small higher order string corrections would become important. More generally, for the supergravity approximation to be valid the parameters have to be adjusted to have at least a few: is the four-cycle volume (in string units) and the supergravity approximation fails when it is of the string scale. However, even if the SUGRA description in terms of the scalar potential is not valid at the minimum, it still can be valid at large , exactly where we wish to realize inflation. The consequence of small is that the end point of inflation, i.e. preheating, would have to be described by string theory degrees of freedom. We will return to this point in the discussion. 5.1 The Canonically-normalized Inflaton If we define a canonically-normalized field by , then It is therefore volume-suppressed. For inflation restricted to the direction, we identify with the inflaton. The field change over the many inflationary e-foldings is given in the last column in Table 1 for a typical radial () trajectory. It is much less than . (Here the scale factor at the end of inflation is so goes in the opposite direction to time.) The variations of the inflaton and the Hubble parameter wrt , suggest we must have a deceleration parameter nearly the de Sitter and the Hubble parameter nearly constant over the bulk of the trajectories. This is shown explicitly in § 6 and Figures 8 and 9. The parameter is the first “slow-roll parameter”, although it only needs to be below unity for inflation. With so small, we are in a very slow-roll situation until near the end of inflation when it rapidly rises from approximately zero to unity and beyond. Equation (30) connects the change in the inflaton field to the tensor to scalar ratio , since to a good approximation . Since we find , we get very small . The following relation [29, 30] gives a lower limit on the field variation in order to make tensor modes detectable We are not close to this bound. If is restricted to be in stringy inflation models, getting observable gravity wave signals is not easy. (A possible way out is to have many fields driving inflation in the spirit of assisted inflation .) When the trajectory is not in the direction, the field identified with the canonically-normalized inflaton becomes trajectory-dependent as we describe in the next section and there is no global transformation. That is why all of our potential contour plots have focused on the Kähler modulus and its axion rather than on the inflaton. 6 Inflationary Trajectories 6.1 The Inflaton Equation of Motion We consider a flat FRW universe with scalar factor and real fields . To find trajectories, we derive their equations of motion in the Hamiltonian form starting from the four dimensional Lagrangian (see and references therein) with canonical momentum , where we used and , stands for . (The usual field momentum is .) Here the non-canonical kinetic term is The Hamiltonian is where . The equations of motion follow from , which reduce to

Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos. What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science. Originally published in 1893, this book was significantly revised and extended by the author (second edition, 1919) to cover the history of mathematics from antiquity to the end of World War I. Since then, three more editions were published, and the current volume is a reproduction of the fifth edition (1991). The book covers the history of ancient mathematics (Babylonian, Egyptian, Roman, Chinese, Japanese, Mayan, Hindu, and Arabic, with a major emphasis on ancient Greek mathematics). The chapters that follow explore European mathematics in the Middle Ages and the mathematics of the sixteenth, seventeenth, and eighteenth centuries (Vieta, Decartes, Newton, Euler, and Lagrange). The last and... This book aims to explain, in clear non-technical language,what it is that mathematicians do, and how that differs from and builds on the mathematics that most people are familiar with from school. It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics. The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. Mathematics education research has blossomed into many different areas, which we can see in the programmes of the ICME conferences, as well as in the various survey articles in the Handbooks. However, all of these lines of research are trying to grapple with the complexity of the same process of learning mathematics. Although our knowledge of the process is through fragmentation of research more extensive and deeper there is a need to overcome this fragmentation and to see learning as one process with different aspects. To overcome this fragmentation, this book identifies six themes: (1) mathematics, culture and society, (2) the structure of mathematics and its influence on the learning process, (3) mathematics learning as a cognitive process, (4) mathematics learning as a social process, (5) affective conditions of the mathematics learning process, (6) new technologies and mathematics learning. This book is addressed to all researchers in mathematic education. It gives an orientation and overview on what is going on and what are the main results and questions what are important books or papers if further information is needed. This second edition contains updates on the career paths of individuals profiled in the first edition, along with many new profiles. The authors of the essays in this volume describe a wide variety of careers for which a background in the mathematical sciences is useful. Each of the jobs presented shows real people in real jobs. Their individual histories demonstrate how the study of mathematics was useful in landing good-paying jobs in predictable places such as IBM, AT&T, and American Airlines, and in surprising places such as FedEx Corporation, L.L. Bean, and Perdue Farms, Inc. You will also learn about job opportunities in the Federal Government as well as exciting careers in the arts, sculpture, music, and television. --back cover. Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained. Contains alphabetically arranged entries that provide definitions and explanations of thousands of mathematical terms and concepts, theories, and principles, and includes biographical sketches of key people in math.

Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group , showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants. Recently, several new Chern-Simons matter theories with a large amount of supersymmetry have been found. In , Gaiotto and Witten found a class of theories with supersymmetry. Closely related are the the theories of ABJM , with SUSY, and BLG , with . All of these theories are superconformal, and arise as the low energy effective actions on certain brane configurations in string theory and -theory. Some of these theories are conjectured to be holographically dual to -theory on orbifold backgrounds, in similarity to super Yang-Mills theory in four dimensions, which was conjectured to be dual to Type IIB string theory on . Recall that, in the latter case, certain supersymmetric Wilson loops are dual to fundamental strings. Thus, as a check of this duality, one can compute the expectation values of these Wilson loops and compare to calculations made in string theory. This is difficult to do perturbatively, as the perturbative region of one theory is the strongly coupled region of its dual. However, supersymmetric operators are often easier to deal with than their less symmetric counterparts, and in it was shown that this is indeed true of supersymmetric Wilson loops in SYM theory. It was demonstrated, using localization, that finding the expectation value of these operators reduces to a calculation in a matrix model. This allows one to compute it much more efficiently at any coupling, and the result provides a non-trivial test of the duality. In this paper we seek to apply the methods of to the supersymmetric Chern-Simons matter theories discussed above. Our main result is that the partition function for a supersymmetric Chern-Simons theory with gauge group and chiral multiplets in a (possibly reducible) representation localizes to the following matrix integral:111The factors of appearing the determinants did not appear in the original version of this paper, but were found later by Drukker, Marino,and Putrov . Here the integration is over the Cartan of the Lie algebra of , is the order of the Weyl group of , and “Tr” defines some invariant inner product on .222We have absorbed the Chern-Simons level into “Tr”, see the next section for more details. We’ve also defined: where the product runs over the weights of the representation (or in the case of the adjoint representation (), over the roots of the Lie algebra). The same technique applies to representations which are not self-conjugate, although the resulting matrix models are more complicated. We also consider the following supersymmetric Wilson loop in a representation : where is an auxiliary scalar in the vector multiplet. We find its expectation value is given by: One application of this result is to a trivial example of a supersymmetric Chern-Simons matter theory: Chern-Simons theory without matter, which can be written in a supersymmetric form . Here we are able to use localization to recover some well known results on the reduction of the Chern-Simons partition function to a matrix model [7, 8], as well as reproducing some very simple knot invariants . Another example is ABJM theory. This is conjectured to be dual to a certain orbifold background in -theory, so it would be interesting to make non-perturbative calculations in this theory. Here we were able to reduce the path integral to a matrix model, although we were not able to compute the resulting matrix integrals exactly. However, evaluating them as a perturbative expansion in the ’t Hooft coupling, we find agreement with a perturbative calculation done in field theory [10, 11, 12], which provides a check of our result. It is possible that the matrix model could be solved exactly in the large limit using a saddle point approximation , as we will briefly mention at the end of the paper. Acknowledgments.This work was supported in part by DOE grant DE-FG02-92ER40701. A.K. would like to thank Lev Rozansky and Alexei Borodin for useful discussions. I.Y. would also like to thank Joseph Marsano, John Schwarz, Ketan Vyas and Ofer Aharony for their input. The class of theories we will be considering, supersymmetric Chern-Simons theory with matter, are described in for Minkowski space. We will work in Euclidean space. In this section we briefly review these theories. We start with the gauge multiplet. This consists of a gauge field , two real auxiliary scalars and , and an auxiliary fermion , which is a -component complex spinor. This is just the dimensional reduction of the vector multiplet in dimensions, with being the reduction of the fourth component of . All fields are valued in the Lie algebra of the gauge group . The kinetic term we will use for the gauge multiplet is a supersymmetric Chern-Simons term. In flat Euclidean space, this is: Here “Tr” denotes some invariant inner product on . For example, for , we will take “Tr” to mean times the trace in the fundamental representation, where is constrained by gauge invariance to be an integer. This action is invariant under the usual (euclideanized) vector multiplet transformations: Here and are -component complex spinors, in the fundamental representation of the spin group . We will take to be the Pauli matrices, which are hermitian, with . Also, is the gauge covariant derivative. Note that, in contrast to the Minkowski space algebra where we would have , in Euclidean space and are independent. Taking would reduce us to the algebra. This is because there is no reality condition on spinors in dimensional Euclidean space, so the least amount of supersymmetry one can have is a single complex spinor. To carry out the localization, we will work on a compact manifold rather than in flat space, as this makes the partition function well-defined. As the above action is conformal, we can transfer it to the unit -sphere, , without changing any of the quantities we are interested in computing. The Lagrangian simply acquires an overall measure factor of . In addition, we must modify the following supersymmetry transformations: where now all derivatives are covariant with respect to both the gauge field and the usual metric on . One can easily check that this leaves the action invariant for arbitrary spinors and . When we add matter, however, it will turn out to be necessary that we take and to be a Killing spinors, which means that they satisfy the following equation: Here is an arbitrary spinor. Note that in dimensions, this is actually equations, one of which determines , with the rest imposing conditions on . We will give the explicit solutions to this equation on below. With a Killing spinor, the above supersymmetries give a representation of the superconformal algebra, anticommuting with each other to conformal transformations. 2.1 The Wilson Loop The operator we will be localizing is the following supersymmetric Wilson loop: This operator has been considered in . Here is the closed world-line of the Wilson loop, and “” denotes the usual path-ordering operator. The variation of this operator under the the supersymmetry (6) is: For this to vanish for all we must have the following two conditions: Note we cannot take here, which is why we need to consider theories with at least supersymmetry. Now we need to impose the condition that and are Killing spinors. This will force us to consider only certain loops, which will turn out to be great circles on . To see this, we will need to determine the Killing spinors on . We start by picking a vielbein. It will be convenient to use the fact that is, as a manifold, the same as , so we can take a local orthonormal basis of left-invariant vector fields . In terms of these, the spin connection is simply: Thus the spinor covariant derivative is: We can immediately see a few solutions to the Killing spinor equation. Namely, take the components of in this basis to be constant, in which case: This gives two of the Killing spinors. There are two more solutions, which can be seen most easily using a right invariant vielbein, and which satisfy: Note that in these cases, is proportional to . This is not true of a general Killing spinor (eg, take a linear combination of the above spinors), although in spaces of constant curvature it is always possible to form a basis of the space of Killing spinors with such special ones [15, 16]. Now let us impose the condition that preserves the Wilson loop. If we pick to be the arc length, we find that must satisfy: This is only possible if is constant, which means must be some fixed linear combination of the , so we may as well pick our loop so that is parallel to one of them, say . The integral curves of these vector fields are great circles, so the Wilson loop must be a great circle to preserve any supersymmetry. Then this equation becomes: So this restricts us to only one of the two left-invariant Killing spinors (there is also a right handed one that preserves it). We could also pick to be one of these Killing spinors. Thus this Wilson loop preserves half of the supersymmetries. Conversely, given a Killing spinor , and taking , there is a family of great circles such that the Wilson loops along these circles are preserved by the corresponding supersymmetry. These are just the integral curves of the vector field , which, since this vector field is left-invariant, form a Hopf fibration. As a result, one could use the localization described here to compute the expectation value of a product of Wilson loops corresponding to a general link consisting of loops from this fibration. We will discuss this more below. Next we would like to add matter to the theory. The matter will come in chiral multiplets, each of which consists of a complex scalar , a fermion , which is a -component complex spinor, and an auxilliary complex scalar . The gauge-coupled action for a chiral multiplet in a representation of the gauge group is described in in the case of flat Minkowski space. It is straightforward to modify this for , giving: where, for example, is a shorthand for: where are indices in , is an index of the Lie algebra, and are the generators of in the representation . Also, is a derivative that is covariant with respect to both the gauge group and the metric on , and we assume the various color indices have been contracted in a gauge invariant way. Note that the second term in (18), which arises from the conformal coupling of scalars to the curvature of , gives the matter scalars a mass. A similar mass term will appear in the localizing term in (56). This action is classically invariant under the following superconformal symmetries: Here and must be a Killing spinors, satisfying (8). In order to perform the localization, it turns out that the theory must be superconformal on the quantum level. This is because the supersymmetry that we will use for localization, together with its conjugate and the Lorentz group, generate the entire superconformal algebra. Thus any hermitian action invariant under is necessarily superconformal. This fact determines which superpotentials are allowed. In the absence of a superpotential, the combined Chern-Simons-matter system is superconformal on the quantum level . This follows from the nonrenormalization of the Chern-Simons couplings (except for finite shifts) together with the standard nonrenormalization theorem for the -terms. An arbitrary quartic superpotential preserves superconformal invariance on the classical level, but in the quantum theory superconformal invariance is destroyed in general. Indeed, if the fields have anomalous dimensions, the scaling dimension of the quartic superpotential is not equal to , and the superpotential perturbation is not marginal. However, for special values of the superpotential couplings it may happen that the theory has enhanced supersymmetry which requires the anomalous dimensions to vanish . This is the case for theories of Gaiotto and Witten, the ABJM theory, and the BLG theory. We will see later that the path-integral localizes to configurations where all matter fields vanish, so the precise choice of the superpotential will not matter, provided it ensures superconformal invariance on the quantum level. One particular example from the class of superconformal Chern-Simons matter theories is the ABJM theory . Here the gauge group is , with the Chern-Simons action for the two factors appearing at levels and . The matter comes in two copies of the bifundamental representation , and two more in . There is also a quartic superpotential which ensures the supersymmetry is enhanced from to . 3.1 Gauge Sector In this section we will closely follow , in which supersymmetric Wilson loops were studied in and super Yang-Mills theory in dimensions, and their expectation values were computed by essentially the same localization method we use here. We will start by considering pure Chern-Simons theory, with no matter. We will discuss how the addition of matter affects the computation in section 3.4. The idea of the localization is as follows. We start by picking a single supersymmetry which preserves the operator we are interested in. We then deform the action by adding a term: Here “’’ is some positive definite inner product on the Lie algebra.333We distinguish it from the trace in the original action, which is not necessarily positive definite (eg, in ABJM, where it has a different sign for the two gauge groups). We assume this term is itself supersymmetric, which amounts to saying that is invariant under the bosonic symmetry . Then the standard argument shows that the addition of this term to the action does not affect the expectation value of any -invariant observable. We pick the term in (21) to deform the action because its bosonic part, , is positive definite. Thus we can take to be very large, and the dominant contribution to the path integral will come from the region of field space where this term vanishes, which is precisely where . In the limit of large the theory becomes free, so we can compute things easily, knowing the results we get are independent of and thus apply at , where they represent the quantities we are interested in. Returning to the case at hand, let us fix a supersymmetric Wilson loop along some great circle on . This will be the operator we want to localize. We start by defining the supersymmetry we will be working with. Let be the unique left-invariant spinor which preserves the Wilson loop, normalized so that , and let . By “” we will mean the infinitesimal supersymmetry variation corresponding to this choice of parameters. It is clear that on the bosonic fields, and therefore on the fermions as well. Thus the -exact term (21) is trivially supersymmetric. As shown in appendix A, for the supersymmetry in question it evaluates to: Note that, unlike the matter scalars, there is no mass term for the scalar . The mass term for will arise not from the conformal coupling to the curvature of but from the supersymmetric Chern-Simons action. Next we need to determine where the theory localizes to. The vanishing of requires: This implies the following two conditions: It is straightforward to show that, on , the only solution to the first equation is to take and constant, and then the second equation implies . Thus the theory localizes to the space of constant , with and all other fields vanishing. One can also see this from the explicit form of above. In the limit of large , the exact result for the path integral becomes equal to the saddle point approximation: Here we integrate the contributions from the saddle points, which are labeled by , together with the determinant factor coming from quadratic fluctuations of the fields about each saddle point. For the partition function, the classical contribution is: where we have used the fact that the volume of is . The Wilson loop gives an additional factor of: Thus we have: This is essentially the same result as was found in for and SYM theory in four dimensions. For , it was shown that , so that the resulting matrix model is Gaussian, while for it was something more complicated, involving the Barnes G-function. In the next section we will find that, in our case, is not , but it is still something relatively tractable. Before moving on, we should mention that, to be precise, we should really be localizing the gauge-fixed theory. That is, we should have started by introducing ghost fields , and a Lagrange multiplier . We would then have the standard BRST transformations , and, continuing to follow , define a new fermionic symmetry: One can check that . We also would modify to: We would then localize with respect to rather than . The variation of the new has four contributions: from and each hitting the two terms in . For the first term, only would contribute, since is a gauge transformation and the term is gauge invariant, so the total contribution would just be the -exact term we found above. For the second term, the variation would give us the usual gauge-fixing term: Then we still need to worry about the remaining term: But if we define , then we are only left with some term multiplying , which can be absorbed into the definition of . In other words, we have shown that we can proceed by starting with the action in (23) and gauge fixing it the usual way, and then computing the -loop determinant. We turn to that calculation now 3.2 1-Loop Determinant In this section we compute the 1-loop determinant coming from quadratic fluctuations of the fields about the saddle points we found in the last section. After introducing ghosts, (23) becomes: We will be interested in the large limit, so we rescale the fields to eliminate the out front: Here represents all fields other than and , and we have treated these fields differently because they have zero modes. Also, represents the non-zero mode part of , and similarly for . Taking to be large then allows us to keep only the quadratic terms in the action: where . The integral over can be performed immediately, and eliminates the squared term. The integral is also easy, giving a delta function constraint which imposes Lorentz gauge. We find: Here is the vector Laplacian. This is a free theory, and we would like to compute its 1-loop determinant. A similar calculation is done in , and we can proceed similarly. First we separate the gauge field into a divergenceless and pure divergence part: where . Then the delta function constraint becomes , and so we can integrate over using the delta function, picking up a jacobian factor of . The integral over gives the same factor, while the integral over the ghosts contributes a factor of . These all cancel (and in any case, are -independent), and we are left with: Now if we go back to (29) for a moment, we see that since the action is gauge invariant, the integrand is invariant under the adjoint action of the group. Thus we can replace the integral over the entire Lie algebra with an integral over some chosen Cartan subalgebra. This introduces a Vandermonde determinant in the measure. There is also the residual gauge symmetry of the Weyl group of , so we should divide by , the order of this group. We’re left with, eg, for the partition function: where runs over the roots of , and runs over the Cartan subalgebra. Thus we only need to know for in the Cartan, and so from now on we will assume is in the Cartan. Now let us decompose as: where are representatives of the root spaces of , normalized so , and runs over the roots of . Here is the component of along the Cartan, but this part of will only contribute a -independent factor to the loop determinant, so we will ignore it. Then we can write: We can do something similar for . Plugging this into the action, we can now write it in terms of ordinary (as opposed to matrix valued) vectors and spinors:444Thanks for F. Benini and A. Yarom for pointing out some errors that appeared in this equation in the original version of this paper. From we know that the eigenvalues of the vector Laplacian acting on divergenceless vector fields are , where , and they occur with degeneracy . Thus the bosonic part of the determinant is: For the gaugino, we note that on , eigenvalues of are with degeneracy , where runs over the positive integers.555This can be seen easily using the results of section 3.4, specifically as a special case of (73) Thus the fermion determinant is: And so the total 1-loop determinant is: We see there is partial cancellation between the numerator and the denominator, and this becomes: Because the eigenvalues of a matrix in the adjoint representation come in positive-negative pairs, only the even part in sigma contributes. We can isolate this by looking at: The infinite constant can be fixed by zeta regularization , and the rest of the product can be done exactly. We find: To summarize, we have shown: where runs over the roots of . Plugging this into (40), we see the denominator cancels against the Vandermonde determinant. Introducing the notation: where is some representation, and the product runs over its weights (which, in the case of the adjoint representation, are just the roots of the algebra), we are left with: 3.3 Chern Simons Theory For a concrete example, we will look at the case where . Then we can take the Cartan as the set of diagonal matrices, setting . The roots of are labeled by integers , and have: Also, as mentioned earlier, we take Tr as times the trace in the fundamental representation, and we also take the Wilson loop in the fundamental representation. Also the Weyl group is , so we should divide by . Then (50) becomes (up to a sign): where all the integrals run over the real line. To interpret this result, note that without matter, we can integrate out the auxiliary fields trivially. The integral over in (5) imposes , and so we see from (9) that, in the case of pure Chern-Simons theory, the supersymmetric Wilson loop we have been considering is just the ordinary Wilson loop operator. Thus the second line of (52) gives a simple way of computing the Wilson loop expectation value in this theory. Both of the integrals above are just sums of Gaussian integrals, and it is straightforward to evaluate them exactly, as shown in appendix B. The result for the Wilson loop expectation value is: reproducing the known result , up to an overall phase. This phase comes from the framing of the loop, which can be seen as follows. As mentioned earlier, the supersymmetry we are using preserves a family of Wilson loops forming a Hopf fibration of the sphere. The framing of a Wilson loop is essentially the choice of a nearby loop so that point-splitting regularization may be performed, and for this procedure to be compatible with supersymmetry, this loop must come from the Hopf fibration, and therefore have linking number with the Wilson loop. One simple extension of this calculation is to a link of Wilson loops. If the loops in such a link come from the Hopf fibration, then they preserve the same supersymmetry . Then it is easy to see that the expectation value of this operator is given simply by inserting more factors of in the matrix model, or equivalently, taking the trace in the product representation. This property of the Chern-Simons invariant of the Hopf link can also be shown by topological means.666Thanks to Lev Rozansky for discussions on this point. 3.4 Matter Sector Next we carry out the localization procedure in the matter sector. Rather than treat the case of multiple chiral multiplets in various representations, we will consider a single chiral multiplet in some possibly reducible representation . The action of the supersymmetry on the matter fields is as follows: with all other variations vanishing. Here we have assumed the rescaling (35) has been done on the gauge multiplet. Since we will be taking to be very large, this means we can ignore all coupling to the gauge sector, except that through . To localize the matter sector, we add a term similar to the one we used in the gauge sector: In Appendix A it is shown that this equals: where . As before, this term is positive definite, and vanishes on the following field configurations: The second equation implies . With a little work, one can check that the first implies must also be zero. Rather than show this directly, we will see in a moment, when we evaluate the 1-loop determinant, that the operator acting on in (56) has no zero modes, which leads to the same conclusion. Thus there is no classical contribution coming from the matter sector, and its only influence is through its effect on the one-loop determinant. Since there is no interaction between the matter and gauge fields at quadratic order, except that throught , the determinant factorizes, and we can write: We will compute this extra factor in the next section. 3.5 1-Loop Determinant - Matter Sector For the scalar field, we see from (56) that the operator we need to diagonalize is: Here “” is actually a matrix representing in the representation . As we did for the gauge multiplet, we can decompose this representation into its weight spaces. Namely, if is a representative of the weight space corresponding to the weight , satisfying (where is some gauge-invariant way of contracting the relevant color indices), then we write: The total -loop determinant will be the product of the one coming from each term in this sum, which are all acting on ordinary (not matrix-valued) scalars. It will be most convenient to use a pair of orthonormal frames, one left-invariant and one right-invariant under the action of (thinking of as and letting it act on itself). We will call these and . Then we can take . It is straightforward to show that the laplacian can be expressed in terms of these fields as: where we think of the vector fields as differential operators on the space of scalar fields. Thus the each term in (60) can be written as: Also, using the fact that the vectors satisfy the algebra: we see that if we define new operators , these satisfy the algebra. In terms of these operators, the operator acting on the scalars becomes: and so computing its eigenvalues reduces to a familiar problem from quantum mechanics. On a spin- representation, the determinant of the operator can be written as: It can be shown that the scalar fields on decompose into the irreps under the action of the left- and right-acting ’s, so the total determinant will be a product of the above expression over all non-negative integers , each raised to the power of the degeneracy, which is owing to the right-acting . Next we consider the fermions. After decomposing them into weights as with the bosons, the operator we need to diagonalize is: If we use the as our vielbein, the covariant derivative acting on spinors can be written as: Then the Dirac operator is: We should be careful to distinguish , which is a differential operator, from , which is just a matrix. Thus the operator acting on the fermions becomes: Or, if we define , which satisfy the algebra, and plug in the : So the problem reduces to computing spin-orbit coupling. Unfortunately, we cannot proceed the standard way since does not commute with the total angular momentum , so we are forced to compute the determinant manually. Let: Note that this operator commutes with both and , so its eigenvectors all have the form: Letting act on these vectors, it is straightforward to compute: Plugging in the relevant values, , we get: We note this is almost equal to the scalar determinant, except for the extra terms out front and the missing the factor in the product. Taking this into account, we can write:

Elementary particles in physics 3 see that these four types of fundamental particle are replicated in two heavier families, (µ−, νµ, c, s) and (τ−, ντ, t, b)the reason for the existence of these. Description elementary number theory, sixth edition, blends classical theory with modern applications and is notable for its outstanding exercise setsa full range of exercises, from basic to challenging, helps students explore key concepts and push their understanding to new heights. This is page 3 printer: opaque this preface this is a textbook about prime numbers, congruences, basic public-key cryptography, quadratic reciprocity, continued fractions, elliptic curves, and. 74 the elementary beam theory in this section, problems involving long and slender beams are addressed as with pressure vessels, the geometry of the beam, and the. 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In conclusion, the theory of elementary waves is no theory at all though some of the problems above are problems of exposition, and others might be deficiencies that could be patched up somehow or other, several are fundamental to the theory: # 1, 2, 5, 6, 9, 12. Qcarl friedrich gauss, born into a poor working class family in brunswick, now lower saxon, germany and died in gottingen, germany he was a child prodigy with genius that did not impress his father who called him a star-gazer his mother, dorothea gauss was the exact opposite of his father as. The elementary proof of the prime number theorem joel spencer and ronald graham p rime numbers are the atoms of our mathematical universe euclid showed that there are infinitely. Elementary number theory, seventh edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students.

So I've been doing some math because I have seen some bad math here. To calculate the voting power of any individual electoral vote, it's a bit more complicated than the Population (P) divided by the total electoral votes (E). This is because each state gets 2 electoral votes automatically. All remaining votes are given to match the number of House of Representatives allocated to that state. So, it's incorrect to say the electoral vote is equal to P/E because 2 of those votes represent the entire population of the state. It is better to say that when someone votes for the president, their votes are used to determine 3 votes, not one. So if we have a unit called Representative Power (R) of a single Electoral vote for a given state, then our formula becomes: R = (((P/(E-2) + (2P))/(E-2))) Thus given the numbers set by @LeeDanielCrocker, let's first see what R of Califorinia would be: Let P = 39,250,00 and E = 55 Thus our equation comes to: R(CA) = (((39,250,00/53) + 2(39,250,000))/53) R(CA) = ((740,566 + 78,500,000)/53) R(CA) = 79,240,566/53 R(CA) = 1,495,105 So, with that in mind, we can now calculate Wyoming using the same numbers: Let P = 585,000 and E = 3 R(WY) = (((585,000/(3-2)) + 2(585,000))/(3-2)) R(WY) = 585,000 + 1,1700,00 R(WY) = 1,755,000 So the R(WY) is definitely greater than the R(CA). In fact, one California electoral vote equals 82% of one Wyoming vote. But this is on a one to one voting weight only. No matter how you slice this though, 3 votes at 100% power is still less than 55 votes that represent 82%. So there is the math behind the idea of the smaller the total electoral votes the more powerful your vote becomes. This would be true of the force behind a Congressional seat as well. This is a result of what is called the Great Compromise. As a Representative Democracy, figuring out how citizens would be represented in Congress needed to be a balancing act. If each state got an equal number of votes, small states could impose their will on large states because their vote is equal to the hundreds of thousands, while the larger state's vote is in the 10s of millions. Conversely, if they were given out by population, then the large states would be more powerful than the small states, because their vote is worth the same, but they have 55 times more votes than the smallest ones. The balance here was to allocate 2 votes to every state and additional vote for each division of population. This makes it that small states have more powerful votes but large states have more numerous votes. To understand the electoral college's design, think of each state as an individual country and the Federal Government as a big treaty organization between all of the member states. The leader of this organization would in effect be the leader of all the states. So it was necessary to pick the leader that showed he had the best interests of the individual states in mind. Hollywood actors and Oklahoma corn farmers have very little in common and very different needs from government. Each state does have its own interests to look out for. Since each state is entitled to pick its representation in the way it sees fit, provided that it is a Republic (read representative democracy which was how the framers understood it), then the balance is that each state is afforded a share of votes equal to their congressional delegations (2 + the House Delegation). They are free to distribute them however they see fit (mostly winner take all; two states delegate the districts to the winner in that district and the remaining two to the overall state winner, and South Caroline historically voted on the full delegation in the legislature; and a number require by law that electors vote for the candidate the state chose). However, the states are only given so many votes in accordance with their congressional power. The most balanced option is to distribute like Maine and Nebraska (congressional districts determine all but 2 of the electoral vote, and the 2 remaining are given to the winner). This would get us closer to the popular vote but we would still run into an occasional popular/electoral split. Now, before you say this is undemocratic, consider this: Switzerland is the world's only Direct Democracy and is, with notable exceptions, modeled after the United States Federal System. Switzerland does give its people the right to directly write and repeal their laws and even amend their constitution (with exception to citizens rights, which cannot be repealed from the constitution). In order to pass in this method, the law must pass with Double Majority. This means that popular vote alone does not get the job done, the law must achieve popular support in a majority of the Cantons (basically the same thing as States) in order to pass. A popular/Canton split means the law is not valid. This check is intended to avoid the tyranny of the majority like the electoral college was in the United States. Double Majority is a little hard to do in the United States Presidential elections because there is no proposal as to what would happen if a Double Majority is not achieved on a leadership position. Swiss got around this by making an Executive council of seven co-equal executive members, with each one rotating into the Council President and Vice President, who would be the de facto leader in a diplomatic settings. This committee is appointed by the legislature, not election of the people, so even in one of the most Democratic nations in the world, the people do not get to directly pick their top leadership by popular vote.

Several volatile organic compounds such as n-butanol, acetone, styrene, toluene, and DMDS are emanated from industrial sources that are unhealthy to humanistic well-being and may motive vomiting, irritability and upset the nervous and respiratory systems . VOCs imperil air aspect and inartificial environs by giving greenhouse consequences and cooperate in the generation of the stratospheric ozone layer . Also, Ketones Lee et al. successfully applied the biofiltration technology for the deterioration of some hydrophilic compounds such as liquors and, oxygen must also disband in the damp layer and spread to the biofilm . In a macro kinetic approach, it is assumed that mass relocates restraint can be ignored . A few results have shown that macro kinetics models fit well with the experimental elimination capacities (EC’s) , though there are several models and expressions available in the literature that corresponds to various phenomena and processes at biofilter. Eshraghi et al. studied the effect of operating temperature on the removal of n-butanol vapor in a biofilter. To the best of our knowledge, there is no rigorous analytical expression available to date for the steady-state concentration. As a result, in this work, we focus on obtaining a feeling for the steady-state concentration of n-butanol in the biofilm phase and gas phase. Further, the expression helps us to analyze the physical response related to the parameters in the biofilter model. 2. Mathematical Modeling of the Boundary Value Problem The modeling was developed by using the following assumption : · The biofilm is formed on the outside surface of the packing materials, and there is no reaction in the pores and the biofilm completely covers the surface of the packaging materials. · Compared to the size of solid particles, the biofilm is very thin; hence planar geometry is used. · n-butanol is the sole reactant that influences the biodegradation rate, and oxygen does not limit the reaction. · Arrhenius equation is used for the temperature dependence of the biodegradation rate constant. · The plug flow model is applied to the gas phase. · The air/biofilm interface concentration of n-butanol meets Henry's rule by assuming the same air/water partition coefficients. · There is no boundary layer at the air/biofilm interface. Thus, gas-phase resistance is assumed negligible. · The biofilm properties ( , and density) are constant all over the bed. · The temperature gradient inside the biofilm is negligible. 1) Mass Balance in the Biofilm Phase The steady-state mass balance equation for Michaelis-Menten kinetics in the biofilm may be written as follows : where S is the concentration of n-butanol, is the maximum of elimination capacity is the Michaelis-Menten constant, D is the diffusion coefficient, E is the activation energy, R is the ideal gas constant and T is the kelvin temperature. The boundary conditions Eshraghi et al. for the above equation at the air/biofilm interface are as follows: 2) Mass Balance in Gas Phase The concentration profile of n-butanol in the gas phase may be written as follows: where u is the superficial velocity of gas flow, is the biofilm specific area, C is the n-butanol concentration in the gas phase and D is diffusion coefficient. The corresponding boundary condition is 3) Dimensionless Mass Balance Equation in the Biofilm Phase The non-linear differential Equation (1) is made dimensionless form by defining the following dimensionless parameters: Using the above dimensionless variables, Equation (1) reduces to the following dimensionless form: The corresponding boundary conditions for the above Equation (7) can be expressed as 4) Dimensionless Mass Balance in the Gas Phase By defining the following dimensionless parameters, the differential Equation (4) is made dimensionless form: Using the variables, Equation (4) can be expressed in the dimensionless form as follows: The respective boundary condition for the above mentioned Equation (11) can be described as 3. Analytical Expression for the Concentrations for Values of Parameter Using HPM HPM couples the homotopy technology and perturbation. The primary deficiencies in applying perturbation methods are that a small parameter is needed in the equations. The HPM was further developed and improved and applied to nonlinear oscillators , nonlinear wave equations , boundary value problem , bifurcation problems , etc. Abukhaled and Khuri obtained a semi-analytical solution of nonlinear equations in amperometric enzymatic reactions. This method was based on constructing a Green’s function and employing a fixed point iterative scheme . In recent years, the application of the homotopy perturbation method in nonlinear problems has been developed by scientists and engineers . Most perturbation methods assume a small parameter exists, but most nonlinear problems have no small parameter at all. Unlike analytical perturbation methods, the HPM and HAM do not depend on a small parameter, which is difficult to find . Using the homotopy perturbation method (Appendix A), the concentration of n-butanol in the biofilm phase is obtained as follows: Solving Equation (11) using boundary condition (12) the concentrations of n-butanol in gas phase can be obtained as follows: 4. Analytical S Expression for the Concentrations Using Hyperbolic Function Method In order to use the new analytical method, the trail solution for Equation (7) is given below: where are constants. Using the boundary conditions (8) and (9), we get the constant Now Equation (15) reduces to where m is constant. This constant can be obtained as follows: Equation (7) can be rewritten as Substituting Equation (18) in Equation (7), we get the following result. When x = 0, the above results becomes Using Equation (20), the value of m becomes as follows:. The concentration of n-butanol can be obtained in the biofilm process by inserting Equation (21) in Equation (17), as follows: Hyperbolic function method is a special case of exponential function method . 5. Results and Discussion Equations (13) & (14) represent the simple and new analytical expressions of the concentration of n-butanol in biofilm-phase ( ) and in the gas-phase ( ) respectively. The concentration of n-butanol in the biofilm-phase and the gas-phase depends upon the parameters and . The variation in the dimensionless variable can be achieved by varying either the thickness ( ) or diffusivity of the biofilm (D). The parameter depends upon the initial concentration ( ) and half-saturation constant ( ). The experimental setup for the biofiltration of this organic compound is given in Figure 1. Figure 2 represents the concentration of n-butanol in the biofilm phase versus dimensionless height for different values of and . From Figure 2(a), it is inferred that the concentration of n-butanol increases when decreases for the fixed value of . Figure 2(b), represents the concentration of n-butanol in the biofilm-phase increases when increases for some fixed values of . Figure 3 exhibits the concentration of n-butanol in the gas phase for different values of and . From Figure 3(a), it is inferred that the concentration of n-butanol in the gas phase increases when decreases From Figure 3(b), it is observed the concentration of n-butanol in the gas phase increases when increases. From Figure 3(c), it is inferred that the concentration of n-butanol in the gas phase increases when A decreases for the fixed value of and . Figure 4 Figure 1. Schematic of the laboratory BF set up . Figure 2. (a) Effect of parameter on the concentration of n-butanol in the biofilm phase using Equation (13); (b) Effect of parameter on the concentration of n-butanol in the biofilm phase using Equation (13). Figure 3. Comparison of concentration n-butanol in the gas phase with simulation results, when (a) for various values of the parameter ; (b) for various values of the parameter and (c) for various values of the parameter A. The key to the graph: Solid lines represented the numerical simulation and dotted lines represent Equation (14). Figure 4. Comparison of concentration of n-butanol Equation (14) with experimental result for the parameters & . represents n-butanol concentration for different values of and compared with analytical method numerical simulation and experimental results (Eshraghi et al. 2016). 6. Differential Sensitivity Analysis of Parameters The sensitivity analysis of the parameter is given in Figure 5 & Figure 6. From the analysis it is inferred that the reaction and diffusion parameter have more impact in the concentration in the biofilm-phase. In contrast, the parameter A has more impact in the concentration in gas-phase. 7. Numerical Simulation In order to investigate the accuracy of the HPM solution with a finite number of terms, the nonlinear differential equation is solved numerically. To show the efficiency of the present method, the analytical expressions of the concentration of n-butanol in biofilm-phase and gas-phase are compared with simulation results in Tables 1-3 for the experimental values of parameters. A satisfactory agreement is Figure 5. Sensitivity analysis of parameters on concentration of n-butanol in the biofilm-phase. Figure 6. Sensitivity analysis of parameters on concentration of n-butanol in gas-phase. Table 1. Comparison of normalized non-steady-state concentration with simulation results when . Table 2. Comparison of normalized non-steady-state concentration with simulation results when and . Table 3. Comparison of normalized non-steady-state concentration with simulation results when and . noted. The detailed Matlab program for numerical simulation is provided in Appendix B and Appendix C. In this paper, the non-linear differential equations in the biofiltration have been solved analytically. Using the homotopy perturbation method and hyperbolic function method, an approximate and closed-form of analytical representation of the concentrations of n-butanol in the biofilm phase is provided. This solution of the concentrations of n-butanol in the biofilm phase and the gas phase is compared with the numerical simulation results. These new analytical results provide a good understanding of the system and the optimization of the parameters in the biofiltration model. This work was supported by consultancy project, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai. The Authors are also thankful to Shri J. Ramachandran, Chancellor, Col. Dr. G. Thiruvasagam, Vice-Chancellor, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai, for their constant encouragement. Supplementary Materials of the Manuscript Appendix A: Analytical Solution of Equation (1) in Gas Phase Using HPM The homotopy perturbation method is used to give the approximate solutions of the non-linear Equation (6). We construct the homotopy for Equation (1) as follows: The analytical solution of Equation (1) is Substituting Equation (A2) into Equation (A1) we get Comparing the coefficients of like powers of p in Equation (A3) we get The boundary conditions for Equation (A1) are as follows Solving Equation (A4) and using the boundary conditions Equation (A6) and (A7), we obtain the following results: By applying the boundary conditions, we get and Substitute & value in (A8) we get, Now Equation (A5) becomes The boundary conditions for the above equation are as follows: Solving Equation (A11) and using the boundary conditions Equations (A12) and (A13), we can get the following result: Considering the two terms, we get which is Equation (15) in text. Appendix B: Matlab Program for the Numerical Solution of Equation (1) m = 0; x = linspace(0,1); sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t); u1 = sol(:,:,1); function [c,f,s] = pdex4pde(x,t,u,DuDx) c = 1; f = 1.*DuDx; function u0 = pdex4ic(x) u0 = ; pl = [ul(1)-1]; ql = ; pr = [ur(1)-0]; qr = ; Appendix C: Matlab Program for the Numerical Solution of Equation (10) function [dx_dt]=Test Function(t,x) Munoz, R., Daugulis, A.J., Hernandez, M. and Quijano, G. (2012) Recent Advances in Two-Phase Partitioning Bioreactors for the Treatment of Volatile Organic Compounds. Biotechnology Advances, 30, 1707-1720. Lee, C.L.S., Heber, A.J., Ni, J. and Huang, H. (2013) Biofiltration of a Mixture of Ethylene, Ammonia, N-Butanol, and Acetone Gases. Bioresource. Technology. 127, 366-377. Delhomenie, M.C., Nikiema, J., Bibeau, L. and Heitz, M. (2008) A New Method to Determine the Microbial Kinetic Parameters in Biological Air Filters. Chemical Engineering Science, 63, 4126-4134. Ramirez, A.V., Benard, S., Giroir-Fendler, A., Jones, J.P. and Heitz, M. (2008) Kinetics of Microbial Growth and Biodegradation of Methanol and Toluene in Biofilters and an Analysis of the Energetic Indicators. Journal of Biotechnology, 138, 88-95. Krailas, S. and Pham, Q.T. (2002) Macrokinetic Determination and Water Movement in a Downward Flow Biofilter for Methanol Removal. Biochemical Engineering Journal, 10, 103-113. Streese, J., Schlegelmilch, M., Heining, K. and Stegmann, R. (2005) A Macrokinetic Model for Dimensioning of Biofilters for VOC and Odour Treatment. Waste Management, 25, 965-974. Eshraghi, M., Parnian, P., Zamir, S.M. and Halladj, R. (2016) Biofiltration of N-Butanol Vapour at Different Operating Temperatures: Experiment Study and Mathematical Modeling. International Biodeterioration & Biodegradation, 119, 361-367. Abukhaled, M. and Khuri, S.A. (2017) A Semi-Analytical Solution of Amperometric Enzymatic Reactions Based on Green’s Functions and Fixed Point Iterative Schemes. Journal of Electroanalytical Chemistry, 792, 66-71. Abukhaled, M. (2017) Green’s Function Iterative Method for Solving a Class of Boundary Value Problems Arising in Heat Transfer. Applied Mathematics & Information Sciences, 11, 229-234. Abukhaled, M. (2017) Green’s Function Iterative Approach for Solving Strongly Nonlinear Oscillators. The Journal of Computational and Nonlinear Dynamics, 12, Article ID: 051021. He, J.H. (2005) Homotopy-Perturbation Method for Bifurcation of Nonlinear Problems. International Journal of Nonlinear Science and Numerical Simulation, 6, 207-208. Liao, S.J. (2009) Notes on the Homotopy Analysis Method: Some Definition and Theorems. Communications in Nonlinear Science and Numerical Simulation, 14, 983-997. He, J.H. (2019) A Simple Approach to One-Dimensional Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes. Journal of Electroanalytical Chemistry, 854, Article ID: 113565.

Presentation on theme: "T-tests, ANOVAs & Regression and their application to the statistical analysis of neuroimaging Carles Falcon & Suz Prejawa."— Presentation transcript: t-tests, ANOVAs & Regression and their application to the statistical analysis of neuroimaging Carles Falcon & Suz Prejawa OVERVIEW Basics, populations and samples T-tests ANOVA Beware! Summary Part 1 Part 2 Basics Hypotheses –H 0 = Null-hypothesis –H 1 = experimental/ research hypothesis Descriptive vs inferential statistics (Gaussian) distributions p-value & alpha-level (probability and significance) Activation in the left occipitotemporal regions, esp the visual word form area, is greatest for written words. Populations and samples Population z-tests and distributions Sample (of a population) t-tests and distributions NOTE: a sample can be 2 sets of scores, eg fMRI data from 2 conditions Comparison between Samples Are these groups different? Comparison between Conditions (fMRI) Reading aloud vs Picture naming Reading aloud (script) vsReading finger spelling (sign) right hemisphereLeft hemisphere lesion site % CI infer comp t-tests Compare the mean between 2 samples/ conditions if 2 samples are taken from the same population, then they should have fairly similar means if 2 means are statistically different, then the samples are likely to be drawn from 2 different populations, ie they really are different Exp. 1 Exp. 2 t-test in VWFA Exp. 1: activation patterns are similar, not significantly different they are similar tasks and recruit the VWFA in a similar way Exp. 2: activation patterns are very (and significantly) different reading aloud recruits the VWFA a lot more than naming Exp. 1 Exp. 2 Formula Reporting convention: t= , df= 9, p< Difference between the means divided by the pooled standard error of the mean Formula cont. Cond. 1Cond. 2 Types of t-tests Independent Samples Related Samples also called dependent means test Interval measures/ parametric Independent samples t-test* Paired samples t-test** Ordinal/ non- parametric Mann-Whitney U-Test Wilcoxon test * 2 experimental conditions and different participants were assigned to each condition ** 2 experimental conditions and the same participants took part in both conditions of the experiments Types of t-tests cont. 2-tailed tests vs one-tailed tests 2 sample t-tests vs 1 sample t-tests 2.5% 5% Mean A known value Comparison of more than 2 samples Tell me the difference between these groups… Thank God I have ANOVA ANOVA in VWFA (2x2) Is activation in VWFA for different for a) naming and reading and b) influenced by age and if so (a + b) how so? H 1 & H 0 H 2 & H 0 H 3 & H 0 reading causes significantly stronger activation in the VWFA but only in the older group so the VWFA is more strongly activated during reading but this seems to be affected by age (related to reading skill?) Naming Reading TASK NamingReading Aloud AGEYoung Old ANOVA ANalysis Of VAriance (ANOVA) –Still compares the differences in means between groups but it uses the variance of data to decide if means are different Terminology (factors and levels) F- statistic –Magnitude of the difference between the different conditions –p-value associated with F is probability that differences between groups could occur by chance if null-hypothesis is correct –need for post-hoc testing (ANOVA can tell you if there is an effect but not where) Reporting convention: F= 65.58, df= 4,45, p<.001 Types of ANOVAs Type2-way ANOVA for independent groups repeated measures ANOVAmixed ANOVA Participants Condition I Condition II Task IParticipan t group A Participant group B Task II Participan t group C Participant group D Condition I Condition II Task IParticipan t group A Task II Participan t group A Condition I Condition II Task IParticipan t group A Participant group B Task II Participan t group A Participant group B NOTE: You may have more than 2 levels in each condition/ task Between- subject design Within-subject design both BEWARE! Errors –Type I: false positives –Type II: false negatives Multiple comparison problem esp prominent in fMRI SUMMARY t-tests compare means between 2 samples and identify if they are significantly/ statistically different may compare two samples to each other OR one sample to a predefined value ANOVAs compare more than two samples, over various conditions (2x2, 2x3 or more) They investigate variances to establish if means are significantly different Common statistical problems (errors, multiple comparison problem) PART 2 Correlation - How much linear is the relationship of two variables? (descriptive) Regression - How good is a linear model to explain my data? (inferential) Correlation: -How much depend the value of one variable on the value of the other one? Y X Y X Y X high positive correlation poor negative correlation no correlation How to describe correlation (1): Covariance -The covariance is a statistic representing the degree to which 2 variables vary together (note that S x 2 = cov(x,x) ) cov(x,y) = mean of products of each point desviation from mean values Geometrical interpretation: mean of signed areas from rectangles defined by points and the mean value lines sign of covariance = sign of correlation Y X Y X Y X Positive correlation: cov > 0Negative correlation: cov < 0 No correlation. cov 0 How to describe correlation (2): Pearson correlation coefficient (r) -r is a kind of normalised (dimensionless) covariance -r takes values fom -1 (perfect negative correlation) to 1 (perfect positive correlation). r=0 means no correlation (S = st dev of sample) Pearson correlation coefficient (r) Problems: -It is sensitive to outlayers -r is an estimate from the sample, but does it represent the population parameter? Linear regression: - Regression: Prediction of one variable from knowledge of one or more other variables - How good is a linear model (y=ax+b) to explain the relationship of two variables? - If there is such a relationship, we can predict the value y for a given x. But, which error could we be doing? (25, 7.498) Preliminars: Lineal dependence between 2 variables Two variables are linearly dependent when the increase of one variable is proportional to the increase of the other one x y Samples: - Energy needed to boil water - Money needed to buy coffeepots The equation y= mx+n that connects both variables has two parameters: -m is the unitary increase/decerease of y (how much increases or decreases y when x increases one unity) - n the value of y when x is zero (usually zero) Samples:m= Energy needed to boil one liter of water, n=0 m = prize of one coffeepot, n= fixed tax/comission to add n m 1 0 Fiting data to a straight line (o viceversa): Here, ŷ = ax + b – ŷ : predicted value of y – a: slope of regression line – b: intercept Residual error (ε i ): Difference between obtained and predicted values of y (i.e. y i - ŷ i ) Best fit line (values of b and a) is the one that minimises the sum of squared errors (SS error ) (y i - ŷ i ) 2 ε iε i ε i = residual = y i, observed = ŷ i, predicted ŷ = ax + b Adjusting the straight line to data: Minimise (y i - ŷ i ) 2, which is (y i -ax i +b) 2 Minimum SS error is at the bottom of the curve where the gradient is zero – and this can found with calculus Take partial derivatives of (y i -ax i -b) 2 respect parametres a and b and solve for 0 as simultaneous equations, giving: This calculus can allways be done, whatever is the data!! How good is the model? We can calculate the regression line for any data, but how well does it fit the data? Total variance = predicted variance + error variance: S y 2 = S ŷ 2 + S er 2 Also, it can be shown that r 2 is the proportion of the variance in y that is explained by our regression model r 2 = S ŷ 2 / S y 2 Insert r 2 S y 2 into S y 2 = S ŷ 2 + S er 2 and rearrange to get: S er 2 = S y 2 (1 – r 2 ) From this we can see that the greater the correlation the smaller the error variance, so the better our prediction Is the model significant? i.e. do we get a significantly better prediction of y from our regression equation than by just predicting the mean? F-statistic: And it follows that: F (df ŷ,df er ) = sŷ2sŷ2 s er 2 r 2 (n - 2) 2 1 – r 2 =......= complicated rearranging t (n-2) = r (n - 2) 1 – r 2 So all we need to know are r and n !!! Generalization to multiple variables Multiple regression is used to determine the effect of a number of independent variables, x 1, x 2, x 3 etc., on a single dependent variable, y The different x variables are combined in a linear way and each has its own regression coefficient: y = x x 2 +…..+ n x n + ε The a parameters reflect the independent contribution of each independent variable, x, to the value of the dependent variable, y i.e. the amount of variance in y that is accounted for by each x variable after all the other x variables have been accounted for Geometric view, 2 variables: ŷ = x x 2 x1x1 x2x2 y ε Plane of regression: Plane nearest all the sample points distributed over a 3D space: y = x x 2 + ε Multiple regression in SPM: y : voxel value x1, x2,… : parameters that are supposed to justify y variation (regressors) GLM: given a set of values yi, (voxel value at a determinated position for a sample of images) and a set of explanatories variables xi (group, factors, age, TIV, … for VBM or condition, movement parameters,…. for fMRI) find the (hiper)plane nearest all the points. The coeficients defining the plane are named 1, 2,…, n equation: y = x x 2 +…..+ n x n + ε Matrix representation and results: Last remarks: - Correlated doesnt mean related. e.g, any two variables increasing or decreasing over time would show a nice correlation: C0 2 air concentration in Antartica and lodging rental cost in London. Beware in longitudinal studies!!! - Relationship between two variables doesnt mean causality (e.g leaves on the forest floor and hours of sun) - Cov(x,y)=0 doesnt mean x,y being independents (yes for linear relationship but it could be quadratic,…) Questions ? Please dont! REFERENCES Field, A. (2005). Discovering Statistics Using SPSS (3rd ed). London: Sage Publications Ltd. Field, A. (2009). Discovering Statistics Using SPSS (2nd ed). London: Sage Publications Ltd. Various stats websites (google yourself happy) Old MfD slides, esp 2008

Two boats set off from 𝑋 at the same time. They travel in straight lines such that boat A remains due north of boat B at all times, as shown in the figure. Boat B sails on a bearing 𝐾 degrees greater than boat A. Part a) Calculate the distance between points 𝑌 and 𝑍. Part b) Which of the following most accurately describes angle 𝑊𝑍𝑉? The options are less than 𝐾 degrees, between 𝐾 degrees and two 𝐾 degrees, exactly 𝐾 degrees, or greater than two 𝐾 degrees. So we’ve been given a lot of information in this question. But actually all of it has been summarised for us in the diagram. The fact that boat B sails on a bearing 𝐾 degrees greater than the bearing boat A sails on just means that the angle between the two lines formed by each boat’s journey is 𝐾 degrees. That’s this angle here in the diagram. Let’s look at what we’ve been asked to do then. Part a says calculate the distance between points 𝑌 and 𝑍. That’s this distance here. And in order to work this out, we need to recognise that there are in fact two similar triangles in this diagram: triangle 𝑋𝑌𝑍 and triangle 𝑋𝑉𝑊. Now let’s look at why these triangles are similar. This angle of 𝐾 degrees is common to both triangles. They both also have a right angle as one of their internal angles. The third angle in these two triangles will also be the same because we know that the sum of the angles in any triangle is 180 degrees. We found then that these two triangles have three angles in common. And therefore, they’re similar triangles. How does this help us with calculating the distance between points 𝑌 and 𝑍? Well, 𝑌𝑍 is a side on triangle 𝑋𝑌𝑍. And in fact, it corresponds with side 𝑉𝑊 on the larger triangle. If we can work out the scale factor for these two similar triangles, we’ll be able to use the length 𝑉𝑊 in order to calculate the length of 𝑌𝑍. In order to find the scale factor, we need to know the lengths of a pair of corresponding sides on the two triangles. We see that, on the smaller triangle, the length of 𝑋𝑌 is 90 miles. And if we add the lengths of 𝑋𝑌 and 𝑌𝑉 together — that’s 90 plus 120 — we see that the length of 𝑋𝑉, which is the corresponding side on the larger triangle, is 210 miles. To find the scale factor, we can divide the length on the larger triangle by the corresponding length on the smaller triangle, giving 210 over 90. Now this scale factor can be simplified because both the numerator and denominator can be divided by 10, leaving us with 21 over nine. But then both of these numbers can be divided by three, leaving us with seven over three. This tells us then that lengths on the larger triangle are seven over three or two and one-third times as big as the corresponding lengths on the smaller triangle. Now if we’re going back the other way, so if we want to work out the length of a side on the smaller triangle, this means that we need to take the corresponding length on the larger triangle, which in our case is the length of 𝑉𝑊. That’s seven miles. And because we’re going from larger to smaller, we need to divide by the scale factor we’ve calculated. So we have that 𝑌𝑍 will be equal to seven divided by seven over three. We then recall that, in order to divide by a fraction, we flip or invert that fraction and we multiply. So seven divided by seven over three is the same as seven multiplied by three over seven. We can think of the integer seven as seven over one if it helps. Before we multiply, we can actually cross-cancel. There’s a seven in the numerator of the first fraction and a seven in the denominator of the second. So we’re left with one multiplied by three, which is three, over one multiplied by one, which is one. And three over one is just equal to three. So this tells us that the length of 𝑌𝑍 is three miles. And we can see straight away that this makes sense. If we recalculate our scale factor using this pair of corresponding sides — so that’s 𝑉𝑊 over 𝑌𝑍 — we get seven over three, which is equal to the scale factor we’ve already calculated. So we’ve got our answer to part a then. The distance between 𝑌 and 𝑍 is three miles. And now let’s consider part b. Part b asked us which of the following most accurately describes angle 𝑊𝑍𝑉. That’s the angle formed when we go from 𝑊 to 𝑍 to 𝑉. That’s this angle here, the one marked with the pink question mark. We don’t want to know how big this angle is, just how big it is in relation to the angle of 𝐾 degrees. Well, let’s start by drawing in this line here, a line which is parallel to the line 𝑌𝑉. And so it meets the line 𝑉𝑊 at a right angle. This line divides angle 𝑊𝑍𝑉 up into two parts, a part labelled with a green dot and a part labelled with an orange dot. So angle 𝑊𝑍𝑉 will be the sum of these two angles. The angle marked with a green dot will actually be equal to 𝐾 degrees, because it is a corresponding angle with the angle of 𝐾 degrees in triangle 𝑋𝑌𝑍. And we know that corresponding angles are equal. So we now know that angle 𝑊𝑍𝑉 is equal to 𝐾 degrees plus some other angle, which means we can actually rule out two of the multiple choice options we were given straight away. Angle 𝑊𝑍𝑉 can’t be less than 𝐾 degrees, and it also can’t be exactly 𝐾 degrees. Now we need to think about this angle marked with an orange dot. And to do so, we need to consider triangle 𝑍𝑉𝑍 prime. To help us visualise what we’re about to do, I’ve drawn this triangle out the other way up. So I’ve flipped it over. The line 𝑍𝑍 prime, which was originally the bottom of this triangle, is now at the top. We can actually work out some of the lengths in this triangle. The vertical side of this triangle will actually be equal to the side 𝑌𝑍, which we’ve already calculated, because they are two sides which are between the parallel lines 𝑌𝑉 and 𝑍𝑍 prime. So we know that the side 𝑍 prime 𝑉 is three miles long. We can also work out the length of the side 𝑍𝑍 prime because it will be the same as the side 𝑌𝑉. So the side 𝑍𝑍 prime is 120 miles long. Now we’re going to compare this triangle with triangle 𝑍𝑍 prime 𝑊. That’s the triangle now marked in orange in the original diagram. We already know that 𝑍𝑍 prime is 120 miles, and we can also work out the length of 𝑍 prime 𝑊. If the total length of 𝑉𝑊 is seven miles and the length of the top portion 𝑉𝑍 prime is three miles, then the length of 𝑍 prime 𝑊 will be seven minus three. That’s four miles. Now if we compare these two triangles, we can see that they have the same base of 120 miles. But we see that the height of the triangle containing the orange angle is less than the height of the triangle containing the green angle, which we already know to be 𝐾 degrees. This tells us that the orange angle must be less than the green angle because the base of the triangles is the same but the height of the triangle with the orange angle is less. We already know that the green angle is 𝐾 degrees. We’ve shown that already. So we found that the orange angle must be less than 𝐾 degrees. That means, for angle 𝑊𝑍𝑉, we’re adding 𝐾 degrees to something less than 𝐾 degrees, which means the answer will be less than two 𝐾 degrees. Angle 𝑊𝑍𝑉 then will be between 𝐾 degrees and two 𝐾 degrees. So we tick the correct option. We’ve now completed the problem. In part a, we found that the distance between the points 𝑌 and 𝑍 was three miles. And in part b, we found that the angle 𝑊𝑍𝑉 is most accurately described as being between 𝐾 degrees and two 𝐾 degrees.

Analytic solutions for Dp branes in SFT This is the follow-up of a previous paper [ArXiv:1105.5926], where we calculated the energy of an analytic lump solution in SFT representing a D24-brane. Here we describe an analytic solution for a D-brane, for any , and compute its energy. This paper is a continuation and extension of . Recently, following an earlier suggestion of , a general method has been proposed, , to obtain new exact analytic solutions in Witten’s cubic open string field theory (OSFT) , and in particular solutions that describe inhomogeneous tachyon condensation. There is a general expectation that an OSFT defined on a particular boundary conformal field theory (BCFT) has classical solutions describing other boundary conformal field theories [7, 8]. Previously analytic solutions were constructed describing the tachyon vacuum [5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and of those describing a general marginal boundary deformations of the initial BCFT [22, 23, 24, 25, 26, 27, 28, 29, 30, 31], see also the reviews [32, 33]. In this panorama an element was missing: the solutions describing inhomogeneous and relevant boundary deformations of the initial BCFT were not known, though their existence was predicted [7, 8, 34]. In [2, 3] such solutions were put forward, in [1, 35] the energy of a D24-brane solution was calculated for the first time. Here we wish to extend the method and the results of to describe analytic SFT solutions corresponding to D(25-)-branes for any . The extension is nontrivial because new aspects and problems arise for . Apart from a greater algebraic complexity, we have a (new) dependence of the solutions on several (gauge) parameters and a different structure of the UV subtractions. But the method remains essentially the same as in . The energy of the various solutions turns out to be the expected ones. The paper is organized as follows. In section 2 we consider a solution for a D23 brane, compute its energy functional, study its UV and IR behaviour and verify that the value of its energy functional depends on the parameter . Next, in section 3, in analogy with , we introduce the -regularized solutions , which represents the tachyon condensation vacuum. Then we verify that the difference , which is a solution to the equation of motion over the vacuum represented by , has the expected energy of a D23-brane. At this point the extension to a generic D(25-)-brane is straightforward and we summarize it in section 5. 2 A D23-brane solution Let us briefly recall the technique to construct lump solutions by incorporating in SFT exact renormalization group flows generated in a 2D CFT by suitable relevant operators. To start with we enlarge the well-known algebra defined by in the sliver frame (obtained by mapping the UHP to an infinite cylinder of circumference 2, by the sliver map ), by adding a state constructed out of a (relevant) matter operator with the properties such that has the following action: One can show that does indeed satisfy the OSFT equation of motion In order to describe the lump solution corresponding to a D24-brane in [3, 1] we used the relevant operator, , defined on , where is a scalar field representing the transverse space dimension, is the coupling inherited from the 2D theory and is a suitable constant. In the case of a –brane solution, we propose, as suggested in , that the relevant operator is given by where and are two coordinate fields corresponding to two different space directions. There is no interaction term between and in the 2D action. Then we require for the following properties under the coordinate rescaling The partition function corresponding to the operator (8) is factorized, [36, 37]: where in (10) we have already made the choice . This choice implies With these properties all the non-triviality requirements of [3, 1] for the solution are satisfied. Therefore we can proceed to compute the energy. To this end we follow the pattern of Appendix D of , with obvious modifications. So, for example, and so on. Going through the usual derivation one gets that the energy functional is given by When writing we mean that we differentiate (only) with respect to the first entry, and when (only) with respect to the second. This can be written also as where now means differentiation with respect to both entries. A further useful form is the following one where and, by definition, . The derivative in acts on both entries. We see that, contrary to , where the dependence was completely absorbed within the integration variable, in (17) there is an explicit dependence on . 2.1 The IR and UV behaviour First of all we have to find out whether is finite and whether it depends on . To start with let us notice that the structure of the dependence is the same as in . Therefore we can use the results already found there, with exactly the same IR () and UV () behaviour. The differences with come from the various factors containing or derivatives thereof. The relevant IR asymptotic behaviour is for large ( is kept fixed to some positive value). The asymptotic behaviour does not change with respect to the D24-brane case (except perhaps for the overall dominant asymptotic coefficient, which is immaterial as far as integrability is concerned), so we can conclude that the integral in (17) is convergent for large , where the overall integrand behaves asymptotically as . Let us come next to the UV behaviour (). To start with let us consider the term not containing . We have The double pole in zero is to be expected. Once we integrate over we obtain a behaviour near . This singularity corresponds to , which can be interpreted as the D25 brane energy density multiplied by the square of the (one-dimensional) volume, see Appendix C of ). In order to extract a finite quantity from the integral (17) we have to subtract this singularity. We proceed as in and find that the function to be subtracted to the LHS of (19) is in the interval and 0 elsewhere. It is important to remark that both the singularity and the subtraction are -dependent. As for the quadratic terms in and the overall UV singularity is and the corresponding function to be subtracted from the overall integrand is in the interval and 0 elsewhere. Also in this case the subtraction is dependent. Finally let us come to the cubic term in and . Altogether the UV singularity due to the cubic terms is The overall function we have to subtract from the corresponding integrand is for and 0 elsewhere. Also in this case the subtraction is dependent. As explained in the result of all these subtractions does not depend on the particular functions we have used, provided the latter satisfy a few very general criteria. After all these subtractions the integral in (17) is finite, but presumably dependent. This is confirmed by a numerical analysis. For instance, for and 2 we get and 0.126457, respectively, where the superscript means UV subtracted. It is clear that this cannot represent a physical energy. This is not surprising. We have already remarked in that the UV subtraction procedure carries with itself a certain amount of arbitrariness. Here we have in addition an explicit dependence that renders this fact even more clear. The way out is the same as in . We will compare the (subtracted) energy of with the (subtracted) energy of a solution representing the tachyon condensation vacuum, and show that the result is independent of the subtraction scheme. 3 The -regularization As we did in section 8 of , we need to introduce the -regularization and the -regularized solution corresponding to (8). We recall the general form of such solution where is an arbitrary small number. In the present case It is convenient to split and associate to the first piece in the RHS of (26) and to the second. We will call the corresponding solution . After the usual manipulations the result is where and . It is worth remarking that the result (3) does not depend on the splitting . The integrand in (3) has the same leading singularity in the UV as the integrand of (17). The subleading singularity on the other hand may depend on . Thus it must undergo an UV subtraction that generically depends on . We will denote the corresponding subtracted integral by . The important remark here is, however, that in the limit both (3) and (17) undergo the same subtraction. The factor of appearing in the integrand of (3) changes completely its IR structure. It is in fact responsible for cutting out the contribution at infinity that characterizes (17) and (modulo the arbitrariness in the UV subtraction) makes up the energy of the D23 brane. In keeping with , we interpret as a tachyon condensation vacuum solution and the energy of such vacuum. This energy is actually - (and possibly )-dependent. We will explain later on how it can be set to 0. 4 The energy of the –brane As explained in , the problem of finding the right energy of the D23 brane consists in constructing a solution over the vacuum represented by (the tachyon condensation vacuum). The equation of motion at such vacuum is One can easily show that is a solution to (30). The action at the tachyon vacuum is Thus the energy is Eq.(31) is the lump solution at the tachyon vacuum, therefore this energy must be the energy of the lump. The two additional terms and are given by Now we insert in (33) the quantities we have just computed together with (17) and (3). We have of course to subtract their UV singularities. As we have already remarked above, such subtractions are the same for all terms in (33) in the limit , therefore they cancel out. So the result we obtain from (33) is subtraction-independent and we expect it to be the physical result. In fact the expression we obtain after the insertion of (17,3,34) and (35) in (33) looks very complicated. But it simplifies drastically in the limit . As was noticed in , in this limit we can drop the factors and in (34) and (35) because of continuity111It is useful to recall that the limit can be taken safely inside the integration only if the integral without the factor or is convergent. This is true for the and integration, but it is not the case for instance for the integral (38) below.. What we cannot drop a priori is the factor . Next it is convenient to introduce and notice that Another useful simplification comes from the fact that (without the or factors) upon integrating over the three terms proportional to , and

The mass will simply be kept constant by having 5g of CaCO3 each time. But the surface area is difficult to keep constant considering that the chips are roughly the same size but irregular shapes and so the surface area may differ. There is no way to make the surface area constant exactly in each case. However, I can reduce the amount of difference between each experiment by using a lot of the chips. To keep the surface area constant in size 10-12mm chips I used 10g. Therefore I deduce that if I use 5g of 2-4mm chips the surface area will still remain constant to a certain extent. I believe this because I am using a lot smaller chips and therefore will need a lower mass to keep the surface area basically equal. Therefore to keep the mass and surface area equal I will use 5g of 2-4mm sized CaCO3 chips. In my preliminary work I established that the reaction was exothermic and so there is a temperature rise this means that I have to keep it constant. I will keep the temperature constant by leaving it alone. One of my preliminary experiments proved two things. Firstly that the temperature rise is so small it can be ignored, but this isn’t a very good way of keeping a constant as it still makes a small difference. The second and more important factor is that the temperature rises from 21i?? C i?? 23i?? C each time and so is already a constant without me having to bother about trying to keep it constant through manual means. Therefore if I leave the experiment alone the temperature will remain constant. The last variable which I will have to make constant is the volume of the HCl. This is very simple I will use 8cm3 each time. I chose 8cm3 because it will produce 96cm3 of CO2 gas while if I had used 10cm3 of HCl it would have produced 120cm3 of CO2. I have proved this below: With 10cm3 1 i?? 10 = 0. 01 = 0. 005 i?? 24000 = 120cm3 1000 2 With 8cm3 1 i?? 8 = 0. 008 = 0. 004 i?? 24000 = 96cm3 1000 2 The experiment will be set out as shown below. Prediction I predict that my graph will look like this: The graph above illustrates that if you have the concentration you will do two things. Firstly you will have the volume of CO2 produced and secondly you will the rate at which the CO2 is produced. The rate will be measured in the first 40 seconds so that we have the steepest and fastest rate to compare. Therefore we will be able to compare the different rates at their optimum before the acid concentration has started to deplete. The gradient can easily be measured by the below equation: Change in the y axis Change in the x axis I think that all this will happen because if you half the amount of H+ ions then the collisions will be reduced by a half as well as the H+ ions will have to find the CaCO3, if you half the amount of collisions then the speed at which the chips are broken down is made half as well. I. e. the rate is halved. The volume is halved as well because you will have half as many H+ ions than before, while keeping the volume of HCl acid the same, and so only a certain amount of CO2 will be able to be given off before the H+ ions have been fully used. FAnalysis As you can see from the graph above there is one anomalous point. This is on the 1M experiment and is highlighted grey for ease of identification. The first experiment is the one which threw the average off. The point, for some reason, is lower than the point before it. This suggests that we had lost gas and then gained more gas in the span of 20 seconds. I do not think that this is likely. Therefore I believe that the anomalous point’s existence has to be because of experimental error. I believe that I must have miss read what the volume of CO2 was after 90 seconds. Apart from that one anomalous point the result are very good. They prove my prediction to be correct. My prediction was: * If you half the concentration then you will half the rate at which the CO2 is given off. * If you half the concentration you will half the volume of CO2 produced. The rate was to be calculated by: – Change in the y axis Change in the x axis Therefore, using the above equation and reading from 40 seconds each time (as stated in my prediction) I get these rates: 1M HCl = 63 = 1. 575 40 0. 75M HCl = 47 = 1. 175 40 0. 5M HCl = 31. 5 = 0. 7875 40 0. 25M HCl = 15. 5 = 0. 3875 40 You can clearly see the relation if you double the gradient of the 0. 5M HCl acid you will equal the gradient of the 0. 5M HCl acid: 2 i?? 0. 25M HCl = 0. 5M HCl 2 i?? 0. 3875 = 0. 775 It is close enough 2 i?? 0. 5M HCl = 1M HCl 2 i?? 0. 7875 = 1. 575 Perfect match 1. 3333333… i?? 0. 75M HCl = 1M HCl 1. 3333333… i?? 1. 175 = 1. 566666… Again it is close enough. The above calculations clearly prove that if you double the concentration you will double the rate of the CO2 loss. This means that if you double the amount of H+ ions in 8cm3 of water you will double the chance of collision. This effectively means that you will have twice as many collisions which in turn means that the CaCO3 will lose its CO2 twice as fast. This is proved by my results and as I stated this in my prediction I have proved my prediction to be correct. In my prediction I also stated that I would be getting the following amounts of CO2 1M HCL – 96cm3 0. 75M HCl – 72cm3 0. 5M HCl – 48cm3 0. 25M HCl – 24cm3 My actual results were: 1M HCl – 95. 8cm3 0. 75M HCl – 72. cm3 0. 5M HCl – 48. cm3 0. 25M HCl – 24. cm3 As you can clearly see again my prediction was accurate on the last three and a mere 0. 2cm3 out on the first. One other thing which is shown by results is that the time it takes to reach constant volume is also being doubled. We can see that: For 1M HCl it took 100seconds to reach constant volume of CO2. For 0. 75M HCl it took 130 seconds to reach constant volume of CO2. For 0. 5M HCl it took 190 seconds to reach constant volume of CO2. For 0. 25M HCl it took 360 seconds to reach constant volume of CO2. This is further proof that if you half the concentration of the acid you will be halving the number of H+ ions and therefore you will double the time taken for the volume of CO2 being produced to remain constant. The above results can be matched with results that are calculated to show my results’ accuracy and reliability. If we take the M HCl’s result to be accurate then we can deduce: In converting 1M HCl to 0. 75M HCl it would be – 100 i?? 1. 333… = 133. 333… cm3 If we half the 1M HCl to 0. 5M HCl it would be – 100 i?? 2 = 200cm3 Therefore if we half the 0. 5M HCl to 0. 25M HCl it would be – 200 i?? 2 = 400cm3 To a certain extent this supports my prediction although out of all off the calculated results these were the most different. I believe that my results aren’t so close to the calculated ones because I didn’t get all the CO2 given off because there was a tiny leak in my equipment or that I didn’t wait long enough for the reaction to have finished. Evaluation I believe that my results are very good, apart from one anomalous point all the others fit comfortably on a line of best fit. Not only this, but they all follow the same trend. That one anomalous point I will disregard due to experimental error. All my rates and initial recordings are accurate and prove my prediction correct. However, the final volume of all the CO2 results aren’t perfect. This is why I repeated the experiments twice each, this makes the data more reliable and accurate as you then average those results to get a final set. The final readings were still out even after I repeated the experiment twice. I believe that this is because the experiment relies too much on the equipment. What I by this is that if there is one leak then the final readings aren’t accurate. And also that if there isn’t a leak you may have to wait a very long time for the final readings to show up. So to get around this problem you could repeat the experiment 3 times and then produce an average. Or you could wait a very long time until the reaction has finished. Unfortunately it is very difficult to tell whether or not the reaction has finished by looking at it. You could add a catalyst to the, this would speed up the reaction and so maybe the final volume of CO2 would come up sooner. But the problem with this is that then the initial rate becomes very difficult to measure and so by human error there would be mistakes in the readings. In my preliminary work I found that some gas escaped before you could place the bung into place. However, I did disregard that that would have made a big difference to the final volume of the CO2, but I could be wrong and maybe my final volumes are wrong because of this very reason. Therefore another improvement in the experiment is to add a sample bottle into the apparatus, this would keep the HCl away from the CaCO3. Therefore you could firmly place the bung into position and so no gas would escape in the time it takes to put the bung into place. Therefore there is another experiment which can be performed alongside this one to prove my prediction correct again or on its own for some more accurate results. In this experiment you weigh out a certain amount of CaCO3, let’s say 10g. Then you add a certain amount of HCl, about 50cm3 and put both of these onto a weighing scale. The constant is the mass of the CaCO3 and the volume of the HCl. You vary the molarity, just like in my original experiment. After having placed the CaCO3 and HCl onto the weighing scale you simply tip the acid into the conical flask containing the CaCO3 and note the weight loss after every 10 seconds or 30 seconds or however many you want.

Описание: This introduction to the theory of Sobolev spaces and Hilbert space methods in partial differential equations is geared toward readers of modest mathematical backgrounds. It offers coherent, accessible demonstrations of the use of these techniques in developing the foundations of the theory of finite element approximations. J. T. Oden is Director of the Institute for Computational Engineering & Sciences (ICES) at the University of Texas at Austin, and J. N. Reddy is a Professor of Engineering at Texas A&M University. They developed this essentially self-contained text from their seminars and courses for students with diverse educational backgrounds. Their effective presentation begins with introductory accounts of the theory of distributions, Sobolev spaces, intermediate spaces and duality, the theory of elliptic equations, and variational boundary value problems. 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In dealing with such topics as number, order, relations, limits and continuity, propositional functions, descriptions, and classes, Russell writes in a clear, accessible manner, requiring neither a knowledge of mathematics nor an aptitude for mathematical symbolism. The result is a thought-provoking excursion into the fascinating realm where mathematics and philosophy meet -- a philosophical classic that will be welcomed by any thinking person interested in this crucial area of modern thought. What are the laws of physics, and how did they develop? This reader-friendly guide offers illustrative examples of the rules of physical science and how they were formulated. It was written by Francis Bitter, a distinguished teacher and inventor who revolutionized the use of resistive magnets with his development of the Bitter plate. Dr. Bitter shares his scientific expertise in direct, nontechnical terminology as he explains methods of fact gathering, analysis, and experimentation. The four-part treatment begins with an introductory section on physical measurement. An overview of the basics of data assembly leads to the path of scientific investigation, which is exemplified by observations on planetary motions such as those of Earth, Venus, and Mercury. The heart of the book explores analytic methods: topics include the role of mathematics as the language of physics; the nature of mechanical vibrations; harmonic motion and shapes; the geometry of the laws of motion; and the geometry of oscillatory motions. A final section surveys experimentation and its procedures, with explanations of magnetic fields, the fields of coils, and variables involved in coil design. Appropriate for anyone with a grasp of high-school-level mathematics, this book is as well suited to classroom use as it is to self-study. Описание: Students must prove all of the theorems in this undergraduate-level text, which features extensive outlines to assist in study and comprehension. Thorough and well-written, the treatment provides sufficient material for a one-year undergraduate course. The logical presentation anticipates students' questions, and complete definitions and expositions of topics relate new concepts to previously discussed subjects. Most of the material focuses on point-set topology with the exception of the last chapter. Topics include sets and functions, infinite sets and transfinite numbers, topological spaces and basic concepts, product spaces, connectivity, and compactness. Additional subjects include separation axioms, complete spaces, and homotopy and the fundamental group. Numerous hints and figures illuminate the text. Описание: Fluid dynamics, the behavior of liquids and gases, is a field of broad impact that encompasses aspects of physics, engineering, oceanography, and meteorology. Full understanding demands fluency in higher mathematics, the only language of fluid dynamics. This introductory text is geared toward advanced undergraduate and graduate students in applied mathematics, engineering, and the physical sciences. It assumes a knowledge of calculus and vector analysis. Author Richard E. Meyer notes, "This core of knowledge concerns the relation between inviscid and viscous fluids, and the bulk of this book is devoted to a discussion of that relation." Dr. Meyer develops basic concepts from a semi-axiomatic foundation, observing that such treatment helps dispel the common impression that the entire subject is built on a quicksand of assorted intuitions. His topics include kinematics, momentum principle and ideal fluid, Newtonian fluid, fluids of small viscosity, some aspects of rotating fluids, and some effects of compressibility. Each chapter concludes with a set of problems. Fluid dynamics, the behavior of liquids and gases, is a field of broad impact -- in physics, engineering, oceanography, and meteorology for example -- yet full understanding demands fluency in higher mathematics, the only language fluid dynamics speaks. Dr. Richard Meyer's work is indeed introductory, while written for advanced undergraduate and graduate students in applied mathematics, engineering, and the physical sciences. A knowledge of calculus and vector analysis is presupposed. The author develops basic concepts from a semi-axiomatic foundation, noting that "for mathematics students such a treatment helps to dispel the all too common impression that the whole subject is built on a quicksand of assorted intuitions." Contents include: Kinematics: Lagrangian and Eulerian descriptions, Circulation and Vorticity. Momentum Principle and Ideal Fluid: Conservation examples, Euler equations, D'Alembert's and Kelvin's theorems. Newtonian Fluid: Constitutive and Kinetic theories, exact solutions. Fluids of Small Viscosity: Singular Perturbation, Boundary Layers. Some Aspects of Rotating Fluids: Rossby number, Ekman layer, Taylor-Proudman Blocking. Some Effects of Compressibility: Thermodynamics, Waves, Shock relations and structure, Navier-Stokes equations. Dr. Meyer writes, "This core of our knowledge concerns the relation between inviscid and viscous fluids, and the bulk of this book is devoted to a discussion of that relation." Автор: Hodel, Richard Название: An Introduction to Mathematical Logic ISBN: 0486497852 ISBN-13(EAN): 9780486497853 Издательство: Dover Рейтинг: Цена: 3443 р. Наличие на складе: Нет в наличии. Описание: Widely praised for its clarity and thorough coverage, this comprehensive overview of mathematical logic is suitable for readers of many different backgrounds. Designed primarily for advanced undergraduates and graduate students of mathematics, the treatment also contains much of interest to advanced students in computer science and philosophy. An introductory section prepares readers for successive chapters on propositional logic and first-order languages and logic. Subsequent chapters shift in emphasis from an approach to logic from a mathematical point of view to the interplay between mathematics and logic. Topics include the theorems of Godel, Church, and Tarski on incompleteness, undecidability, and indefinability; a rigorous treatment of recursive functions and recursive relations; computability theory; and Hilbert's Tenth Problem. Numerous exercises appear throughout the text, and an appendix offers helpful background on number theory. Автор: Rubinow S. I. Название: Introduction to Mathematical Biology ISBN: 0486425320 ISBN-13(EAN): 9780486425320 Издательство: Dover Рейтинг: Цена: 2868 р. Наличие на складе: Нет в наличии. Описание: Designed to explore the applications of mathematical techniques and methods related to biology, this text explores five areas: cell growth, enzymatic reactions, physiological tracers, biological fluid dynamics and diffusion. Topics essentially follow a course in elementary differential equations - some linear algebra and graph theory; requires only a knowledge of elementary calculus. Автор: Goldstein, Marvin Название: Introduction to Abstract Analysis ISBN: 0486789462 ISBN-13(EAN): 9780486789460 Издательство: Dover Рейтинг: Цена: 2293 р. Наличие на складе: Нет в наличии. Описание: Developed from lectures delivered at NASA's Lewis Research Center, this concise text introduces scientists and engineers with backgrounds in applied mathematics to the concepts of abstract analysis. Rather than preparing readers for research in the field, this volume offers background necessary for reading the literature of pure mathematics. Starting with elementary set concepts, the treatment explores real numbers, vector and metric spaces, functions and relations, infinite collections of sets, and limits of sequences. Additional topics include continuity and function algebras, Cauchy completeness of metric space, infinite series, and sequences of functions and function spaces. The relation between convergence and continuity and algebraic operations is discussed in the abstract setting of linear spaces in order to acquaint readers with these important concepts in a fairly simple way. Detailed, easy-to-follow proofs and examples illustrate how the material relates to and serves as a foundation for more advanced subjects. ООО "Логосфера " Тел:+7(495) 980-12-10 www.logobook.ru

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The regression analysis determines the values of the parameters that cause the function to MacMolPlt Despite the name, MacMolPlt is a cross-platform Chemistry visualization tool that can be used to build molecules with an integrated graphical model builder, create input files for the GAMESS computational chemistry program, and then visualize the results o Math Educator Fatih Software Math Educator is a simple tool to teach kids addition, subtraction, multiplication and division. Easy-using and redesignable interface are just some of the ultimate features of Math Educator.What is new in this release:Version 2.0 is a bug f Math Processor Math Processor helps you solve different types of mathematical and statistical problems. It allows you to work on large data sets in an easy and fast environment. It has some plotting capabilities as well. Do not be mistaken by the small size or simple int Basho Unit Converter Basho Unit Converter is the application that converts wide range of units for common measurements. It knows how to convert units of length, area, volume, weight, temperature, pressure, speed, angle and more. Application can convert as well common units as Neural Networks - DCT for Face Identification Neural Networks - DCT for Face Identification. Matlab source code. High information redundancy and correlation in face images result in inefficiencies when such images are used directly for recognition. Discrete cosine transforms are used to reduce image i AnalyticMath Free math/graphing program that will allow you to develop and visually analyze mathematical expressions. Boasts an editor/calculator and a graphing module that permits expressions with up to 8 parameters to be plotted directly, such as y=Asin(kx+b). A tool Solid Geometry Portable Solid Geometry is a tool for fast and accurate calculation of volume, area, and mass for various geometric bodies. Geometrical bodies include cylindrical tank, spherical segment, spherical sector, spherical frustum, pyramid, prism, truncated cone, truncate Tvalx Free Calculator Free Calculator is a general purpose ergonomic calculator which combines use simplicity and calculation power. It handles main arithmetic operations with two operands, addition, subtraction, multiplication, division, and complex formulas with unlimited num Abilities Builder Math Plus Build whole number and fraction computation skills. Number by number problem exercises. Includes whole number math facts, addition, subtraction, multiplication and division of whole numbers and fractions. Also includes English and Metric measurements. SimplEquations Software for teaching and learning of simple equations and inequalities with one unknown. Tutorial with animation for easy understanding. 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In this set of exercises, students find and interpret the standard deviation of a sample if the proportion of female students at union high school is . Standard deviation and high school seniors 1255 words mar 13th, 2013 6 pages part 1 of 1 - | question 1 of 10 | 10 points | consider the following scenario. The sample mean is 128, and the sample standard deviation it is believed that the mean height of high school students who play basketball on the school. The high school-college disconnect in academic expectations is clear- ly reflected by students with scores at one standard deviation above the mean were. First and foremost, we thank the standard-setting panel, 18 educational experts the standard deviation set to 35, based on a calibration group of test takers who students at a community college or high school seniors planning to attend a. Knowledge by a quarter of a standard deviation and led to a 14 percentage point examine a financial literacy course for high school students in the us the. What is the probability that a sample of n = 25 would yield a mean (m) iq of for samples of n = 100, what mean iq scores would comprise the middle in this semester's ps 306 first lab, we collected data from a sample of students, including gpa mean = 0, standard deviation = 1, shape = unchanged ( multimodal and. The standard error of measurement (sem) is the standard deviation of errors of self-test 1 a high school geometry test was administered to 250 students. Banning cellphones in schools reaps the same benefits as extending student test scores improving by 641 percent points of a standard deviation these students are distracted by the presence of phones, and high-ability. Is one (black or white) standard deviation, then when we compare a randomly five other major national surveys of high school seniors conducted since 1965. For example, suppose a researcher wishes to test the hypothesis that a sample of size n = 25 with mean x = 79 and standard deviation s = 10 was drawn at. The scores for all high school seniors taking the verbal section of the test (sat ) in a particular year had a mean of 490 and a standard deviation of 100. Learn high school statistics for free—scatterplots, two-way tables, normal distributions, binomial probability, variance and standard deviation of a population. In a two-tailed test, you will reject the null hypothesis if your sample mean falls in either sacramento county high school seniors have an average sat score of 1,020 and we certainly don't know the true population standard deviation. In the study of 10,912 students, surveyed at 123 schools, the research in general, for students from rural areas or high-poverty schools, as well as bridges on critical thinking skills in terms of standard deviation effect sizes. Given the average height of a group of students and the standard deviation, find the percentage of students who are taller or shorter than the. Students and standard deviation 105 students, the z-score for an absence total of 140 high school a loses funding using the suggested plan would be - . Drawing upon iranian high school teachers' classroom management table 2 demonstrates the mean and standard deviation for the students' scores in the. Chapter 3: are high school counselors preparing students to a low of 75 to a high of 94 the mean across all schools was 353 (standard deviation = 88. But the typical middle school and high school start at 8 am as a the authors multiplied the standard deviation for each naep exam by the change in students with longer sleep times report significantly higher grades than. Ros school a graphic display of shape with a mean µ = 534 inches and standard deviation σ = 18 inches (see figure 221) figure 221 what fraction of all individual students who take the test have scores 21 or higher b suppose we. Sat standard deviation is calculated so that 68% of students score within gre and loves advising students on how to excel in high school. If i wanted to survey 50 cabrini college students about where they prefer to eat for a group of high school seniors is 800 and a standard deviation of 150, find. What is the meaning of s (sample standard deviation) example 4: the mean value the act math scores of 15 high-school seniors: 18 15 25 24 21 17 32 30. Suppose that new york state high school average scores, for students who graduate, standard deviation of 13 a) the “middle” 95% of all nys high school . This study of high school sophomores in 1980 and 1990 compares the experiences of of a standard deviation unit below white students, respectively 28. Semester of each student's final vear of high school) was 1925 years (range 17- 22 years sd = 156) the mean iq for the group was 8291 (sd = 2171. High class achievement might be thought to indicate better teaching, or to “we find that moving up one standard deviation in rank similarly. Notes on understanding, using, and calculating effect sizes for schools section 2 if we have a bunch of data and want to estimate the standard deviation, then the and lowest scores for the middle 95 percent of the students: sd = (highest. The total sat score of high school seniors in recent yearshave mean μ=1026 and standard deviation σ=209 thedistribution of sat scores is roughly normal.Download

Why is the Speed of Light an unbreakable barrier for us? Understand Einstein's theory of relativity: speed limit of light, inertial frame, constant speed of light, and the absence of an absolute frame of reference. Why can't we go faster than the Speed of Light? Einstein's special theory of relativity posits that the speed of light is insurmountable. However, it's crucial to understand that this is based on certain prerequisites. It's important to note that the theory doesn't rule out the possibility of exceeding the speed of light entirely. The special theory of relativity establishes that information propagation speed can't surpass the speed of light within an inertial system, which includes the speed of any object. Therefore, objects belong to the information category. It's worth highlighting that two key elements are relevant here: the inertial frame and the speed of information propagation. One prerequisite of this theory is that the frame must be inertial. What is an inertial system, then? It refers to a reference frame with zero acceleration. If you're motionless or moving uniformly in a straight line relative to a reference frame, that frame will be your inertial frame. The Earth is generally considered to be an inertial system, despite moving in a circular motion. This is because the circular motion of the Earth is executed on a much larger scale and has a minimal effect on us. As a result, in an inertial system, the rate at which information propagates cannot surpass the speed of light. Hence, in a non-inertial system, it is possible to exceed the speed of light. Scientists often use large particle colliders to discover smaller microscopic particles. The primary method is to collide two microscopic particles at a velocity very close to the speed of light. As an example, if the speed of microscopic particles reaches 0.9 times the speed of light, the relative speed between the two is 1.8 times the speed of light. This may seem like it surpasses the speed of light, but it doesn't violate Einstein's special theory of relativity. This is because 1.8 times the speed of light is only the relative motion speed of microscopic particles, not their speed relative to a particular inertial system. In other words, if we consider the inertial system of the Earth, the speed of microscopic particles is impossible to surpass the speed of light. To give another example, suppose a spaceship is traveling at the speed of light C, even though it's only a hypothetical scenario. You are running with a speed of V on the spaceship, and I am on the ground. From my perspective, what is your speed? Is it the speed of the spaceship plus your running speed, i.e., C+V? No, your speed is still the speed of light, not the speed of light + the speed of the spaceship. This is because "I" am an inertial system. In this inertial system, no matter how any two speeds are combined, the final speed cannot exceed the speed of light. This is, in fact, the direct embodiment of Einstein's special theory of relativity regarding "the speed of light is the speed limit." What is the core of special relativity? The principle of constant speed of light and the principle of relativity! The special theory of relativity is also based on these two fundamental postulates. Among them, the principle of relativity emphasizes the inertial system, which is often overlooked by many people. Another crucial point, in addition to the inertial system, is the principle of the constant speed of light. Why is the speed of light constant? There are several popular science articles available that explain this topic, so I will not go into detail here. Essentially, the speed of light is an absolute constant, and it doesn't rely on any frame of reference. The speed of light remains exactly the same in any frame of reference. In mathematical terms, the speed of light remains unchanged, regardless of whether it is added to any other velocity. The "dominant" nature of the speed of light implies that regardless of how quickly you try to catch up to a beam of light, its velocity will always remain constant from your perspective. Even if you approach 99% of the speed of light, you will never be able to keep pace with it. Rather, the light will continue to move away from you at its steady rate. To test your comprehension of the constant speed of light principle, let's consider an example. I turn on a flashlight and while it is shining, you and the light are both in motion. Your speed is half the speed of light, which is 0.5 times the speed of light. Suppose I observe the light from the flashlight traveling two kilometers, then the distance you traveled is one kilometer. Since the speed of light is constant, the light from the flashlight travels two kilometers in your eyes as well. Now, if you observe the light from the flashlight, how far would you see it travel in your reference frame? Would it still be two kilometers? No, the reference frame has changed, and in your reference frame, the light from the flashlight only travels one kilometer. This is because you and the light are both moving, and your relative speed is 0.5 times the speed of light. Since the speed of light is constant, dividing the distance the light traveled by my time is equal to dividing the distance the light traveled by your time, and the result is always the speed of light. What's Your Reaction?

Steady State Error Solved Example Let's view the ramp input response for a step input if we add an integrator and employ a gain K = 1. You should always check the system for stability before performing a steady-state error analysis. Evaluating: Steady-State Error 1. Since this system is type 1, there will be no steady-state error for a step input and an infinite error for a parabolic input. https://www.ee.usyd.edu.au/tutorials_online/matlab/extras/ess/ess.html Steady State Error Solved Example It is your responsibility to check the system for stability before performing a steady-state error analysis. That is, the system type is equal to the value of n when the system is represented as in the following figure: Therefore, a system can be type 0, type 1, This situation is depicted below. Combine feedback system consisting of G(s) and [H(s) -1]. Calculating steady-state errors Before talking about the relationships between steady-state error and system type, we will show how to calculate error regardless of system type or input. Let's zoom in further on this plot and confirm our statement: axis([39.9,40.1,39.9,40.1]) Now let's modify the problem a little bit and say that our system looks as follows: Our G(s) is Input Test signal is step. 4. Steady State Error Solved Problems axis([39.9,40.1,39.9,40.1]) Examination of the above shows that the steady-state error is indeed 0.1 as desired. Example: Static Error Constants for Unity Feedback Department of Mechanical Engineering 16. Steady State Error In Control System Pdf For example, let's say that we have the following system: which is equivalent to the following system: We can calculate the steady state error for this system from either the open Let's say that we have the following system with a disturbance: we can find the steady-state error for a step disturbance input with the following equation: Lastly, we can calculate steady-state Now let's modify the problem a little bit and say that our system has the form shown below. Steady State Error Disturbance s = tf('s'); P = ((s+3)*(s+5))/(s*(s+7)*(s+8)); C = 1/s; sysCL = feedback(C*P,1); t = 0:0.1:250; u = t; [y,t,x] = lsim(sysCL,u,t); plot(t,y,'y',t,u,'m') xlabel('Time (sec)') ylabel('Amplitude') title('Input-purple, Output-yellow') As you can see, Background: Steady-State Error Test Inputs : Department of Mechanical Engineering 7. Steady State Error Solved Example Please try the request again. Steady State Error In Control System Examples From our tables, we know that a system of type 2 gives us zero steady-state error for a ramp input. Since system is Type 1, error stated must apply to ramp function. his comment is here The system returned: (22) Invalid argument The remote host or network may be down. ME 176 Control Systems Engineering Steady-State Errors Department of Mechanical Engineering 2. System is Type 0 3. Steady State Error In Control System Problems The system returned: (22) Invalid argument The remote host or network may be down. Greater the sensitivity, the less desirable. "The ratio of the fractional change in the function to the fractional change in parameter as the fractional change of parameters approaches zero" Department of The system returned: (22) Invalid argument The remote host or network may be down. this contact form Many of the techniques that we present will give an answer even if the system is unstable; obviously this answer is meaningless for an unstable system. The system type is defined as the number of pure integrators in a system. Unity Feedback System Transfer Function Department of Mechanical Engineering 19. You can keep your great finds in clipboards organized around topics. Background: Analysis & Design Objectives "Analysis is the process by which a system's performance is determined." "Design is the process by which a systems performance is created or changed." Transient Response Let's examine this in further detail. The only input that will yield a finite steady-state error in this system is a ramp input. Generated Sun, 30 Oct 2016 10:13:56 GMT by s_sg2 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection How To Reduce Steady State Error Note: Steady-state error analysis is only useful for stable systems. Type 0 system Step Input Ramp Input Parabolic Input Steady-State Error Formula 1/(1+Kp) 1/Kv 1/Ka Static Error Constant Kp = constant Kv = 0 Ka = 0 Error 1/(1+Kp) infinity infinity The steady-state error will depend on the type of input (step, ramp, etc.) as well as the system type (0, I, or II). Manipulating the blocks, we can transform the system into an equivalent unity-feedback structure as shown below. navigate here Knowing the value of these constants, as well as the system type, we can predict if our system is going to have a finite steady-state error. axis([40,41,40,41]) The amplitude = 40 at t = 40 for our input, and time = 40.1 for our output. Name* Description Visibility Others can see my Clipboard Cancel Save Effects Tips TIPS ABOUT Tutorials Contact BASICS MATLAB Simulink HARDWARE Overview RC circuit LRC circuit Pendulum Lightbulb BoostConverter DC motor INDEX The system returned: (22) Invalid argument The remote host or network may be down. Example: Sensitivity Calculate sensitivity of the closed-loop transfer function to changes in parameter K and a, with ramp inputs: Department of Mechanical Engineering Recommended Strategic Planning Fundamentals Solving Business Problems Competitive From our tables, we know that a system of type 2 gives us zero steady-state error for a ramp input. Example: Steady-State Error for Unity Feedback Find the steady-state errors for inputs of 5u(t), 5tu(t), and 5t^2u(t). Your cache administrator is webmaster. Generated Sun, 30 Oct 2016 10:13:56 GMT by s_sg2 (squid/3.5.20) Facebook Twitter LinkedIn Google+ Link Public clipboards featuring this slide × No public clipboards found for this slide × Save the most important slides with Clipping Clipping is a handy Example: Steady-State Error for Disturbances Find the steady-state error component due to a step disturbance. Published with MATLAB 7.14 SYSTEM MODELING ANALYSIS CONTROL PID ROOTLOCUS FREQUENCY STATE-SPACE DIGITAL SIMULINK MODELING CONTROL All contents licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. We know from our problem statement that the steady state error must be 0.1. Please try the request again. Example: Static Error Constants for Unity Feedback Department of Mechanical Engineering 18. Department of Mechanical Engineering 21. Compute resulting G(s) and H(s). Generated Sun, 30 Oct 2016 10:13:56 GMT by s_sg2 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection Now we want to achieve zero steady-state error for a ramp input. Generated Sun, 30 Oct 2016 10:13:56 GMT by s_sg2 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection SlideShare Explore Search You Upload Login Signup Home Technology Education More Topics For Uploaders Get Started Tips & Tricks Tools Lecture 12 ME 176 6 Steady State Error Upcoming SlideShare Loading

Table of contents: - What does real analysis mean? - What is the point of real analysis? - Why is real analysis so difficult? - What comes after real analysis? - How can I be good at statistics? - What type of math is statistics? - Is there a lot of math in statistics? - Is statistics easier than pre calculus? - Is Statistics harder than algebra 2? - What is the easiest math class in college? - Is Precalc harder than calculus? - How quickly can you learn calculus? - Can calculus be self taught? - Can I learn calculus in 3 months? - Can I learn calculus on my own? - Is calculus hard to learn? - What is the hardest math question in the world? - What is the hardest form of calculus? What does real analysis mean? In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. ... Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. What is the point of real analysis? Real Analysis brings intellecture closure to ideas like real numbers. The emphasis on proofs hones analytical skill to a sharp point, which is necessary for further study. Calculus theorems are sometimes only stated. Why is real analysis so difficult? Besides the fact that it's just plain harder, the way you learn real analysis is not by memorizing formulas or algorithms and plugging things in. ... Real analysis is hard. This topic is probably your introduction to proof-based mathemat- ics, which makes it even harder. What comes after real analysis? after real analysis, you can take topology and differential geometry. How can I be good at statistics? Study Tips for the Student of Basic Statistics - Use distributive practice rather than massed practice. ... - Study in triads or quads of students at least once every week. ... - Don't try to memorize formulas (A good instructor will never ask you to do this). ... - Work as many and varied problems and exercises as you possibly can. ... - Look for reoccurring themes in statistics. What type of math is statistics? Statistics is a part of Applied Mathematics that uses probability theory to generalize the collected sample data. It helps to characterize the likelihood where the generalizations of data are accurate. This is known as statistical inference. Is there a lot of math in statistics? Originally Answered: Is statistics a field of math? No. Statistics is its own field separate from mathematics similar to physics. Although the two are closely related because the concept of probability is indeed studied in mathematics, and statistics makes use of many tools of analysis/calculus. Is statistics easier than pre calculus? Statistics (the AP course) in my opinion is a slightly less challenging class than pre-calc. I am enrolled in both of these classes right now at the high school level. ... But as a single-year math course, pre-calculus is bit more challenging. Is Statistics harder than algebra 2? a fundamental course in statistics, then, generally, statistics is more difficult. ... Algebra concepts are much easier to grasp, Stats concepts are harder to grasp but the work itself at an INTRO level stat class will be easier as most of it is just memorizing a bunch of formulas and plugging them in. What is the easiest math class in college? Is Precalc harder than calculus? For me Calc 1 was easier than precalc. Precalc is a lot of memorization and understanding trigonometry (seriously make sure you have trig down!), but calculus is just understand some new concepts that all flow pretty well. The CALCULUS is easy, it's the algebra that can be difficult. How quickly can you learn calculus? These courses are central to understanding the mathematics that aid in learning the basis of physical equations. Learning high-school calculus in a high-school class takes roughly 150 hours + 100 hours of homework/studying. Learning the same in a college class takes roughly 40 hours + 80 hours of homework/studying. Can calculus be self taught? "Calculus made easy" by Thomson. It will be easy to teach it to yourself. Youtube or Khanacademy anything you don't understand and be sure to do some practice problems. Can I learn calculus in 3 months? If you plan on moving onto higher-level calculus and analysis courses, the more time you invest into truly mastering single-variable calculus, the easier those will seem. I was able to independently cover two semesters' worth of calculus in roughly 2-3 months, so it is most definitely possible. Can I learn calculus on my own? You can absolutely learn a full calculus course by yourself, from pretty much any book. Is calculus hard to learn? The math involved in learning calculus is not hard at all, it's basically all just algebra and trig. Sure you can make it hard but for the most part it is not. Learning calculus is hard in that it demands more effort to understand it. What is the hardest math question in the world? Today's mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It's one of the seven Millennium Prize Problems, with a million dollar reward for its solution. What is the hardest form of calculus? - What is a characteristic of fields? - What does it mean when dreams become reality? - Where is a compass used in real life? - What is scientific reality? - Are there real force fields? - What are the four stages of perception? - What NFL fields have real grass? - Where are the confetti fields? - What is the field of view for human beings with two eyes? - Is field hockey a posh sport? You will be interested - What is an example of a natural experiment? - What is meant by real number? - What FOV is real life? - Do electric field lines really exist? - Why do they call it a field trip? - How much are field stones? - How do I become an oilfield inspector? - What are the fields of mathematics? - What is real number in math? - Can civilians get MREs?

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . . A description of some experiments in which you can make discoveries about triangles. Formulate and investigate a simple mathematical model for the design of a table mat. What shapes should Elly cut out to make a witch's hat? How can she make a taller hat? How many different sets of numbers with at least four members can you find in the numbers in this box? There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time? Numbers arranged in a square but some exceptional spatial awareness probably needed. All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . . What do these two triangles have in common? How are they related? What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it? What is the largest cuboid you can wrap in an A3 sheet of paper? Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs. This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly? How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways? Can you find ways of joining cubes together so that 28 faces are visible? How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green? Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not? Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given? In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations. In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest? Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles? Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all? We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought. Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc. In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off. Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table? Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make? An activity making various patterns with 2 x 1 rectangular tiles. Investigate what happens when you add house numbers along a street in different ways. Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles? I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take? Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"? Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot. Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented? Why does the tower look a different size in each of these pictures? Work with numbers big and small to estimate and calculate various quantities in biological contexts. Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here. Explore one of these five pictures. A group of children are discussing the height of a tall tree. How would you go about finding out its height? How many tiles do we need to tile these patios? Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers? This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six? A follow-up activity to Tiles in the Garden. How will you decide which way of flipping over and/or turning the grid will give you the highest total? In how many ways can you stack these rods, following the rules? How many models can you find which obey these rules? Can you create more models that follow these rules? What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes? What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules. What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

# **Why is There a Sewer Smell in My Bathroom?** ## *Uncovering the Reasons and Solutions for Unpleasant Odors* Sewer smells in the bathroom can be incredibly unpleasant and disruptive to our daily routines. These pungent and often nauseating odors can ruin our peaceful relaxation time or make it embarrassing to have guests over. Understanding the underlying causes of these odors is crucial to finding effective solutions. In this article, we will explore the various reasons why there might be a sewer smell in your bathroom and provide practical solutions to eliminate these odors. ## **1. Common Causes of Sewer Smell in the Bathroom** Sewer smells can originate from several sources within your bathroom. Identifying the exact cause will allow you to address the issue more effectively. Here are some common culprits: ### 1.1. Dry P-Trap A dry P-trap is one of the most common causes of sewer smells in the bathroom. The P-trap, a curved pipe beneath your sink or shower drain, is designed to hold water and create a barrier between your living space and the sewer system. When the water in the P-trap evaporates due to infrequent use, a direct connection is made between your bathroom and the sewer, resulting in foul odors. ### 1.2. Cracked or Damaged Pipes Cracked or damaged pipes are another potential cause of sewer smells. These issues can allow sewer gases to escape into your bathroom, leading to unpleasant odors. Aging pipes or those exposed to extreme temperatures are particularly susceptible to cracks and damage. ### 1.3. Blocked Ventilation Pipes Blocked ventilation pipes, also known as vent stacks, can prevent proper airflow in your plumbing system. Without proper ventilation, sewer gases can accumulate and find their way into your bathroom. Blockages in the vent stack can be caused by debris, leaves, or even small animals. ### 1.4. Faulty or Loose Toilet Seal A deteriorating toilet seal can result in sewer smells emanating from your bathroom. The seal is responsible for maintaining a watertight connection between the toilet base and the drain pipe. If the seal becomes loose or damaged, sewer gases can escape and permeate your bathroom, creating an unpleasant environment. ### 1.5. Improperly Installed Drains or Plumbing Fixtures Improper installation of drains or plumbing fixtures can lead to sewer smells in the bathroom. Inadequate sealing or incorrect connections may cause sewer gases to leak into your living space. It is essential to ensure that any installation or repair work is carried out by experienced professionals. ## **2. Practical Solutions to Eliminate Sewer Smells** Now that we have identified some common causes of sewer smells in the bathroom, let’s explore practical solutions to eliminate these odors and restore a fresh and pleasant environment. ### 2.1. Running Water in Infrequently Used Drains To prevent the water in your P-trap from evaporating, remember to run water through infrequently used drains regularly. This simple maintenance step can help create a barrier between your bathroom and the sewer system, eliminating sewer smells caused by dry P-traps. ### 2.2. Checking for Cracks and Damages Inspect your bathroom pipes for any signs of cracks or damages. If you come across any issues, it is advisable to seek professional help. Experienced plumbers can repair or replace the damaged sections, ensuring a proper seal to prevent sewer smell infiltration. ### 2.3. Clearing Blocked Ventilation Pipes To clear blockages in your ventilation pipes, it is recommended to hire a professional plumber. They have the necessary tools and expertise to safely remove debris and restore proper airflow. Regular maintenance of the vent stack can prevent future issues and maintain a fresh bathroom environment. ### 2.4. Repairing or Replacing Faulty Toilet Seals If you suspect a faulty toilet seal, it is best to address the issue promptly. Contact a licensed plumber who can assess the situation and replace the seal if necessary. Taking swift action can prevent further damages and eliminate sewer smells in your bathroom. ### 2.5. Seeking Professional Help for Installation and Repairs When it comes to plumbing installation or repair work, it is crucial to rely on professional assistance. Experienced plumbers can ensure proper installation, adequate sealing, and correct connections, minimizing the chances of sewer smells in your bathroom. Don’t hesitate to contact a trusted plumber who can provide reliable and efficient service. ## **3. FAQs about Sewer Smell in the Bathroom** ### 3.1. FAQ #1: Can a sewer smell in the bathroom be harmful to my health? No, sewer smells in the bathroom are usually not harmful to your health. However, they can indicate underlying plumbing issues that should be addressed promptly to prevent health hazards. ### 3.2. FAQ #2: How can I differentiate a sewer smell from other bathroom odors? Sewer smells often have a distinct rotten egg-like odor. If you notice this specific smell in your bathroom, it is likely caused by sewer gases escaping from the plumbing system. ### 3.3. FAQ #3: Why does the sewer smell only occur in my bathroom? Sewer smells in your bathroom may be due to specific issues within the bathroom’s plumbing system, such as dry P-traps, damaged pipes, or faulty seals. These problems can create a direct connection to the sewer system, resulting in unpleasant odors confined to your bathroom. ### 3.4. FAQ #4: Is it safe to use commercial air fresheners to mask sewer smells? While commercial air fresheners may temporarily mask sewer smells, they do not address the underlying issue. It is best to identify and eliminate the cause rather than relying on temporary solutions. ### 3.5. FAQ #5: Can I fix a sewer smell in my bathroom without professional help? Although some DIY methods exist, it is advisable to seek professional help for persistent sewer smells. Plumbers have the expertise and tools to identify and resolve the root cause effectively. The presence of a sewer smell in your bathroom can be frustrating and unpleasant. By understanding the causes and implementing the appropriate solutions, you can eliminate these odors and restore a fresh and pleasant bathroom environment. Remember to regularly maintain your drains, inspect plumbing fixtures, and seek professional assistance when needed. Don’t let sewer smells ruin your bathroom experience – take action today for a fresher tomorrow!

Please read these answers after you work on the problems. Reading the answers without working on the problems is not a useful strategy! Notation eFrog will be called exp(Frog) here, mostly because it is easier to type and read. Also, sqrt(Toad) will denote the square root of Toad (so sqrt(4) is 2). Comment Answers will generally not be "simplified" unnecessarily. It is the philosophy of the management that formulas are delicate, and there is some risk of damaging them any time they are touched, so ... unless you really need to change them, don't bother. Problem 1 B, the amplitude, is half of the difference between the maximum and the minimum, so B=2. The vertical shift A is the average of the maximum and minimum, so A= 7. Substituting x=0 will yield a value for C: algebraically y = A+Bsin(0+C) = 7+2sin(C). But (0,7.6) seems to be the y-intercept of the graph (at least approximately!) so that 7.6= 7+2sin(C). Solve this with your calculator (be sure to use radians!) and .3 will be the approximate value of arcsin(.3). Problem 2 Arctan of a large number is close to Pi/2 (look at the graph of y = arctan x, and always use radians!). Pi is approximately 3.14, so arctan (1,000) is approximately 1.57, and 1,000 arctan (1,000) is 1570 (more or less) and the answer requested is 1600 (being careful to find the nearest 100!). Problem 3 a) The functions ln and exp are inverses. Also, 3 ln (sqrt(x))= ln (x3/2), so exp(3 ln(sqrt(x)) ) = exp (ln b) The standard properties of logs will be used. ln(B2 sqrt(A)) = 2 ln(B) + (1/2) ln A = 2 (5.3) + (1/2) (1.2)= 11.2. For the second part of the problem, we recall (look at the graph of ln!) that ln(Frog) < ln(Toad) exactly when Frog < Toad. Therefore we can compare A9 and B2 by comparing their lns. But ln(A9) = 9 ln(A)= 9 (1.2) =10.8 and ln(B2)=2 ln(B) = 2 (5.3) = 10.6, so that the first number is larger than the second. Problem 4 a) The equation of the circle is (x-2)2 + y2 = 9. b)The equation of the line is y=-2x+6. c) Maybe the easiest way is to substitute "y" from b) into the equation in a). This yields a quadratic in x which can be solved using the quadratic formula (important moral lesson: very few quadratics, even with small integer coefficients, can be "solved" by factoring!). The second coordinates can be gotten by using the equation in b). The two points gotten are ((14+sqrt(41))/5, (2-2sqrt(41))/5) and ((14-sqrt(41))/5, (2+2sqrt(41))/5). These points are approximately (1.52,-2.16) and (4.08,2.96). The picture may help some people understand the answers. Problem 5 a) The graph of y=Z(x) in the accompanying figure is the solid broken line. b) Since 2 is in the interval [0,3], Z(2) should be evaluated using the first expression so Z(2) = 2*2 = 4. Since 4 belongs to the interval [3,infinity), we evaluate Z(4) using the second expression: Z(4) = 4+3=7. c) Z is strictly increasing so it has an inverse N. For x in [0,3], Z(x)=2x, so the values of Z are in [0,6]. That means N(x) = (1/2)x for x in [0,6] (just solve y=2x for x!). Otherwise (for x > 6) N(x)= x-3 (gotten by solving y=x+3 for x). To find N(2) we use the first formula, so N(2)=(1/2)2=1, and N(8) uses the second formula, so N(8)=5. d) The graph of y=N(x) in the accompanying figure is the dashed broken line. Problem 6 The inverse function is given by n(p) = ((p-100)/3)2 and its domain coincides with the range of p(n) which is [103,130]. n(p) represents the number of boxes of detergent which can be produced for p dollars in this model. Problem 7 a) The limit of the top is 0, and the limit of the bottom is 8 which is not 0. Therefore the limit of the expression is 0. b) The bottom is x2 -4 = (x-2)(x+2), so if x is not equal to 2, the whole expression simplifies to 1/(x+2). The limit as x approaches 2 of this is just 1/4. c) The bottom has a non-zero limit as x approaches 4, so we can get the limit just by evaluating the top and bottom at 4. The result is 0. d) Multiply the top and bottom by sqrt(x)+2. Then the bottom becomes (sqrt(x)-2)(sqrt(x)+2) = x-4 which cancels the factor of (x-4) in the top and we're left with just sqrt(x)+2 on top. This has limit 4 as x approaches 4. Problem 8 The derivative of f is f'(x) = 6x and f'(1)=6. So 6 is the slope of the line tangent to this curve when x=1. The tangent line has to pass through the point (1,f(1))=(1,7), so one equation for this line is y-7 = 6(x-1). Problem 9In this problem, dx will denote the expression "delta x". Then f(x+dx)-f(x) = (3/(x+dx))-(3/x)= (3(x)-3(x+dx))/((x+dx)x)= (-3dx)/((x+dx)x). Therefore (f(x+dx)-f(x))/dx = -3/((x+dx)x). The limit of this last expression as dx approaches 0 is just -3/(x2). Problem 10 a) (x3-3x+17)' = 3x2-3 b) (exp(x) sin(x))'= exp(x)' sin(x) + exp(x) (sin(x))' = exp(x) sin(x) + exp(x)cos(x). c) ((2x+3)/(x2+1))' is a complicated fraction. For typographical purposes, think of it as Top/Bottom where Top= 2(x2 +1) -(2x+3)(2x) and Bottom = (x2+1)2. Please note: you are not asked to "simplify" the answer by any additional algebra. d) (3exp(x) + (x/(2+5cos(x)) )' = 3exp(x) + another Top/Bottom where here Top= 1(2+5cos(x))-(0-5sin(x))x and Bottom= (2+5cos(x))2. Again, no additional "simplification" is needed. Problem 11 By the chain rule the derivative of the left-hand side f'(g(x)) g'(x) which when x=2 is f'(g(2))g'(2)= f'(4)g'(2)= 2g'(2). The derivative of the right-hand side is 16x which when x=2 is 32. So we know that 2g'(2) = 32, and we get g'(2)=16. Problem 12 a) tan(?) = 10/15 = 2/3, so ? = arctan(2/3) which is approximately 0.588 radians. b) Here tan(alpha) = 10/A, so alpha = arctan(10/A). Problem 13 a) If f is continuous at 0, the limit from the left should be equal to f(0). But the left-hand limit of f at 0 is equal to the left-hand limit of x2+2 at 0, and this limit is just 2. That means f(0) should be 2. We also know that f(0)=02 +A = A, and we know that the right-hand limit of f at 0 is the right-hand limit of the formula x2+A at 0, which is also A. Now the two limits and the value of f at 0 should agree to make f continuous at 0. This will occur when A=2. b) The left-hand limit of f at 2 uses the formula x2 + 2 Here we substitute 2 for A, since 2 was the value found in the previous part of the problem. Therefore the left-hand limit of f at 2 is 22 +2 =6. The right-hand limit at 2 uses the formula x2+1, which has limit 5 at x=2. Since 5 is not equal to 6, f is not continuous at x=2. Problem 14 At A the function is continuous (there are no jumps or breaks in the graph there) but it not differentiable. The graph has a "corner" at (A,g(A)): more formally, it looks like the right-hand part of the limit defining the derivative at A is 0 while the left-hand part of that limit is infinite (vertical lines have no slope). These don't agree, so the function is not differentiable at The function is differentiable at B, and, since differentiable functions are continuous, it is also continuous at B. The graph has a jump at x=C, so the function is not continuous there: the left- and right-hand limits differ (and neither agrees with the value of g at C. The function is not differentiable at C since if it were, it would have to be continuous there also. Problem 15 Vertical asymptotes are x=0 (the y-axis) and x=3. The accompanying graph satisfies the conditions given in the question. Note the "empty hole" at (5,1) since the function's domain does not include 5 (tricky!). Problem 16 a) Consider the interval [2.5,3]. Enzyme concentration is increasing, since the slope of the tangent line is positive. But the rate of increase is "slowing" as time increases (moving from left to right along the curve) because the tangent line's slope is decreasing (such a curve is concave down). b) Now consider the interval [1,2], for example. Here the enzyme concentration is also increasing because the slope of the tangent line is always positive. But in this interval as time increases the slope of the tangent line is increasing -- so the rate of increase is increasing here. This part of the curve is concave up. In both parts of this problem, the slope of the tangent line must be recognized as the rate of change of enzyme concentration. Also, there are other correct answers to this problem than the ones given here. Problem 17 F(1) = f(1)g(1) = 2(5) = 10 and the product rule for derivatives gives the following: F'(1) = f'(1)g(1) + f(1)g'(1) = -1. G(1) = f(1)/(g(1)+12) = 2/(5+1)= 2/6. The quotient rule shows that G'(x) = (f'(x)(g(x)+x2)-f(x)(g'(x)+2x))/((g(x)+x2)2) and evaluating this when x=1 yields G'(1)=-1. Problem 18 a)(exp(3cos(2x)))'= exp(3cos(2x)) (3 cos(2x))' = exp(3cos(2x)) 3 (cos(2x))' = exp(3cos(2x))(3) (-sin(2x)) (2x)'= b)(ln(x3+5x+9))'= (1/(x3+5x+9))(x3+5x+9)'= (1/(x3+5x+9) (3x2+5) c) (sin((x+1)/(x2+2)))' = cos((x+1)/(x2+2)) ((x+1)/(x2+2))' = cos((x+1)/(x2+2)) (Top/Bottom) where here Top = (1)(x2+2) - 2x (x+1) and Bottom = (x2+2)2.

Cluster analysis divides a data set into several groups using information found only in the data points. Clustering can be used in an exploratory manner to discover meaningful groupings within a data set, or it can serve as the starting point for more advanced analyses. As such, applications of clustering abound in machine learning and data analysis, including, inter alia: genetic expression analysis (Sharan et al., 2002), market segmentation (Chaturvedi et al., 1997), social network analysis (Handcock et al., 2007), image segmentation (Haralick and Shapiro, 1985)2009), collaborative filtering (Ungar and Foster, 1998), and fast approximate learning of non-linear models (Si et al., 2014). -means clustering is a standard and well-regarded approach to cluster analysis that partitions input vectorsinto clusters, in an unsupervised manner, by assigning each vector to the cluster with the nearest centroid. Formally, linear -means clustering seeks to partition the set into disjoint sets by solving Despite its popularity, linear -means clustering is not a universal solution to all clustering problems. In particular, linear -means clustering strongly biases the recovered clusters towards isotropy and sphericity. Applied to the data in Figure 1(a), Lloyd’s algorithm is perfectly capable of partitioning the data into three clusters which fit these assumptions. However, the data in Figure 1(b) do not fit these assumptions: the clusters are ring-shaped and have coincident centers, so minimizing the linear -means objective does not recover these clusters. To extend the scope of -means clustering to include anistotropic, non-spherical clusters such as those depicted in Figure 1(b), Schölkopf et al. (1998) proposed to perform linear -means clustering in a nonlinear feature space instead of the input space. After choosing a feature function to map the input vectors non-linearly into feature vectors, they propose minimizing the objective function where denotes a -partition of . The “kernel trick” enables us to minimize this objective without explicitly computing the potentially high-dimensional features, as inner products in feature space can be computed implicitly by evaluating the kernel function Thus the information required to solve the kernel -means problem (2), is present in the kernel matrix . be the full eigenvalue decomposition (EVD) of the kernel matrix andbe the rows of . It can be shown (see Appendix B.3) that the solution of (2) is identical to the solution of To demonstrate the power of kernel -means clustering, consider the dataset in Figure 1(b). We use the Gaussian RBF kernel with , and form the corresponding kernel matrix of the data in Figure 1(b). Figure 1(c) scatterplots the first two coordinates of the feature vectors . Clearly, the first coordinate of the feature vectors already separates the two classes well, so -means clustering using the non-linear features has a better chance of separating the two classes. Although it is more generally applicable than linear -means clustering, kernel -means clustering is computationally expensive. As a baseline, we consider the cost of optimizing (3). The formation of the kernel matrix given the input vectors costs time. The objective in (3) can then be (approximately) minimized using Lloyd’s algorithm at a cost of time per iteration. This requires the -dimensional non-linear feature vectors obtained from the full EVD of ; computing these feature vectors takes time, because is, in general, full-rank. Thus, approximately solving the kernel -means clustering problem by optimizing (3) costs time, where is the number of iterations of Lloyd’s algorithm. Kernel approximation techniques, including the Nyström method (Nyström, 1930; Williams and Seeger, 2001; Gittens and Mahoney, 2016) and random feature maps (Rahimi and Recht, 2007), have been applied to decrease the cost of solving kernelized machine learning problems: the idea is to replace with a low-rank approximation, which allows for more efficient computations. Chitta et al. (2011, 2012) proposed to apply kernel approximations to efficiently approximate kernel -means clustering. Although kernel approximations mitigate the computational challenges of kernel -means clustering, the aforementioned works do not provide guarantees on the clustering performance: how accurate must the low-rank approximation of be to ensure near optimality of the approximate clustering obtained via this method? We propose a provable approximate solution to the kernel -means problem based on the Nyström approximation. Our method has three steps: first, extract () features using the Nyström method; second, reduce the features to dimensions () using the truncated SVD;111This is why our variant is called the rank-restricted Nyström approximation. The rank-restriction serves two purposes. First, although we do not know whether the rank-restriction is necessary for the bound, we are unable to establish the bound without it. Second, the rank-restriction makes the third step, linear -means clustering, much less costly. For the computational benefit, previous works (Boutsidis et al., 2009, 2010, 2015; Cohen et al., 2015; Feldman et al., 2013) have considered dimensionality reduction for linear -means clustering. third, apply any off-the-shelf linear -means clustering algorithm upon the -dimensional features to obtain the final clusters. The total time complexity of the first two steps is . The time complexity of the third step depends on the specific linear -means algorithm; for example, using Lloyd’s algorithm, the per-iteration complexity is , and the number of iterations may depend on .222Without the rank-restriction, the per-iteration cost would be , and the number of iterations may depend on . Our method comes with a strong approximation ratio guarantee. Suppose we set and for any , where is the coherence parameter of the dominant -dimensional singular space of the kernel matrix . Also suppose the standard kernel -means and our approximate method use the same linear -means clustering algorithm, e.g., Lloyd’s algorithm or some other algorithm that comes with different provable approximation guarantees. As guaranteed by Theorem 1, when the quality of the clustering is measured by the cost function defined in (2 ), with probability at least, our algorithm returns a clustering that is at most times worse than the standard kernel -means clustering. Our theory makes explicit the trade-off between accuracy and computation: larger and lead to high accuracy and also high computational cost. Spectral clustering (Shi and Malik, 2000; Ng et al., 2002) is a popular alternative to kernel -means clustering that can also partition non-linearly separable data such as those in Figure 1(b). Unfortunately, because it requires computing an affinity matrix and the top eigenvectors of the corresponding graph Laplacian, spectral clustering is inefficient for large . Fowlkes et al. (2004) applied the Nyström approximation to increase the scalability of spectral clustering. Since then, spectral clustering with Nyström approximation has been used in many works, e.g., (Arikan, 2006; Berg et al., 2004; Chen et al., 2011; Wang et al., 2016b; Weiss et al., 2009; Zhang and Kwok, 2010). Despite its popularity in practice, this approach does not come with guarantees on the approximation ratio for the obtained clustering. Our algorithm, which combines kernel -means with Nyström approximation, is an equally computationally efficient alternative that comes with strong bounds on the approximation ratio, and can be used wherever spectral clustering is applied. Using tools developed in (Boutsidis et al., 2015; Cohen et al., 2015; Feldman et al., 2013), we rigorously analyze the performance of approximate kernel -means clustering with the Nyström approximation, and show that a rank- Nyström approximation delivers a approximation ratio guarantee, relative to the guarantee provided by the same algorithm without the use of the Nyström method. As part of the analysis of kernel -means with Nyström approximation, we establish the first relative-error bound for rank-restricted Nyström approximation,333Similar relative-error bounds were independently developed by contemporaneous work of Tropp et al. (2017), in service of the analysis of a novel streaming algorithm for fixed-rank approximation of positive semidefinite matrices. Preliminary versions of this work and theirs were simultaneously submitted to arXiv in June 2017. which has independent interest. Kernel -means clustering and spectral clustering are competing solutions to the nonlinear clustering problem, neither of which scales well with . Fowlkes et al. (2004) introduced the use of Nyström approximations to make spectral clustering scalable; this approach has become popular in machine learning. We identify fundamental mathematical problems with this heuristic. These concerns and an empirical comparison establish that our proposed combination of kernel-means with rank-restricted Nyström approximation is a theoretically well-founded and empirically competitive alternative to spectral clustering with Nyström approximation. Finally, we demonstrate the scalability of this approach by measuring the performance of an Apache Spark implementation of a distributed version of our approximate kernel -means clustering algorithm using the MNIST8M data set, which has million instances and classes. 1.2 Relation to Prior Work The key to our analysis of the proposed approximate kernel -means clustering algorithm is a novel relative-error trace norm bound for a rank-restricted Nyström approximation. We restrict the rank of the Nyström approximation in a non-standard manner (see Remark 1). Our relative-error trace norm bound is not a simple consequence of the existing bounds for non-rank-restricted Nyström approximation such as the ones provided by Gittens and Mahoney (2016). The relative-error bound which we provide for the rank-restricted Nyström approximation is potentially useful in other applications involving the Nyström method. The projection-cost preservation (PCP) property (Cohen et al., 2015; Feldman et al., 2013) is an important tool for analyzing approximate linear -means clustering. We apply our novel relative-error trace norm bound as well as existing tools in (Cohen et al., 2015) to prove that the rank-restricted Nyström approximation enjoys the PCP property. We do not rule out the possibility that the non-rank-restricted (rank-) Nyström approximation satisfies the PCP property and/or also enjoys a approximation ratio guarantee when applied to kernel -means clustering. However, the cost of the linear -means clustering step in the algorithm is proportional to the dimensionality of the feature vectors, so the rank-restricted Nyström approximation, which produces -dimensional feature vectors, where , is more computationally desirable. Musco and Musco (2017) similarly establishes a approximation ratio for the kernel -means objective when a Nyström approximation is used in place of the full kernel matrix. Specifically, Musco and Musco (2017) shows that when columns of are sampled using ridge leverage score (RLS) sampling (Alaoui and Mahoney, 2015; Cohen et al., 2017; Musco and Musco, 2017) and are used to form a Nyström approximation, then applying linear -means clustering to the -dimensional Nyström features returns a clustering that has objective value at most times as large as the objective value of the best clustering. Our theory is independent of that in Musco and Musco (2017), and differs in that (1) Musco and Musco (2017) applies specifically to Nyström approximations formed using RLS sampling, whereas our guarantees apply to any sketching method that satisfies the “subspace embedding” and “matrix multiplication” properties (see Lemma A.2 for definitions of these two properties); (2) Musco and Musco (2017) establishes a approximation ratio for the non-rank-restricted RLS-Nyström approximation, whereas we establish a approximation ratio for the (more computationally efficient) rank-restricted Nyström approximation. 1.3 Paper Organization In Section 2, we start with a definition of the notation used throughout this paper as well as a background on matrix sketching methods. Then, in Section 3, we present our main theoretical results: Section 3.1 presents an improved relative-error rank-restricted Nyström approximation; Section 3.2 presents the main theoretical results on kernel -means with Nyström approximation; and Section 3.3 studies kernel -means with kernel principal component analysis. Section4 discusses and evaluates the theoretical and empirical merits of kernel -means clustering versus spectral clustering, when each is approximated using Nyström approximation. Section 5 empirically compares the Nyström method and the random feature maps for the kernel -means clustering on medium-scale data. Section 6 presents a large-scale distributed implementation in Apache Spark and its empirical evaluation on a data set with million points. Section 7 provides a brief conclusion. Proofs are provided in the Appendices. This section defines the notation used throughout this paper. A set of commonly used parameters is summarized in Table 1. |number of samples| |number of features (attributes)| |number of clusters| |target rank of the Nyström approximation| |sketch size of the Nyström approximation| Matrices and vectors. We take to be the identity matrix, to be a vector or matrix of all zeros of the appropriate size, and to be the -dimensional vector of all ones. The set is written as . We call a -partition of if and when . Let denote the cardinality of the set . Singular value decomposition (SVD). . A (compact) singular value decomposition (SVD) is defined by where , , are an column-orthogonal matrix, adiagonal matrix with nonnegative entries, and a column-orthogonal matrix, respectively. If is symmetric positive semi-definite (SPSD), then , and this decomposition is also called the (reduced) eigenvalue decomposition (EVD). By convention, we take . The matrix is a rank- truncated SVD of , and is an optimal rank- approximation to when the approximation error is measured in a unitarily invariant norm. The Moore-Penrose inverse of is defined by . Leverage score and coherence. Let be defined in the above and be the -th row of . The row leverage scores of are for . The row coherence of is . The leverage scores for a matrix can be computed exactly in the time it takes to compute the matrix ; and the leverage scores can be approximated (in theory (Drineas et al., 2012) and in practice (Gittens and Mahoney, 2016)) in roughly the time it takes to apply a random projection matrix to the matrix . We use three matrix norms in this paper: Any square matrix satisfies . If additionally is SPSD, then . Here, we briefly review matrix sketching methods that are commonly used within randomized linear algebra (RLA) (Mahoney, 2011). Given a matrix , we call (typically ) a sketch of and a sketching matrix. Within RLA, sketching has emerged as a powerful primitive, where one is primarily interested in using random projections and random sampling to construct randomzed sketches (Mahoney, 2011; Drineas and Mahoney, 2016). In particular, sketching is useful as it allows large matrices to be replaced with smaller matrices which are more amenable to efficient computation, but provably retain almost optimal accuracy in many computations (Mahoney, 2011; Woodruff, 2014). The columns of typically comprise a rescaled subset of the columns of , or random linear combinations of the columns of ; the former type of sketching is called column selection or random sampling, and the latter is referred to as random projection. Column selection forms using a randomly sampled and rescaled subset of the columns of . Let be the sampling probabilities associated with the columns of (so that, in particular, ). The columns of the sketch are selected identically and independently as follows: each column of is randomly sampled from the columns of according to the sampling probabilities and rescaled by where is the index of the column of that was selected. In our matrix multiplication formulation for sketching, column selection corresponds to a sketching matrix that has exactly one non-zero entry in each column, whose position and magnitude correspond to the index of the column selected from . Uniform sampling is column sampling with , and leverage score sampling takes for , where is the -th leverage score of some matrix (typically , , or an randomized approximation thereto) (Drineas et al., 2012). Gaussian projection is a type of random projection where the sketching matrix is taken to be ; here the entries of are i.i.d. random variables. Gaussian projection is inefficient relative to column sampling: the formation of a Gaussian sketch of a dense matrix requires time. The Subsampled Randomized Hadamard Transform (SRHT) is a more efficient alternative that enjoys similar properties to the Gaussian projection (Drineas et al., 2011; Lu et al., 2013; Tropp, 2011), and can be applied to a dense matrix in only time. The CountSketch is even more efficient: it can be applied to any matrix in time (Clarkson and Woodruff, 2013; Meng and Mahoney, 2013; Nelson and Nguyên, 2013), where denotes the number of nonzero entries in a matrix. 3 Our Main Results: Improved SPSD Matrix Approximation and Kernel -means Approximation In this section, we present our main theoretical results. We start, in Section 3.1, by presenting Theorem 3.1, a novel result on SPSD matrix approximation with the rank-restricted Nyström method. This result is of independent interest, and so we present it in detail, but in this paper we will use it to establish our main result. Then, in Section 3.2, we present Theorem 1, which is our main result for approximate kernel -means with the Nyström approximation. In Section 3.3, we establish novel guarantees on kernel -means with dimensionality reduction. 3.1 The Nyström Method The Nyström method (Nyström, 1930) is the most popular kernel approximation method in the machine learning community. Let be an SPSD matrix and be a sketching matrix. The Nyström method approximates with , where and . The Nyström method was introduced to the machine learning community by Williams and Seeger (2001); since then, numerous works have studied its theoretical properties, e.g., (Drineas and Mahoney, 2005; Gittens and Mahoney, 2016; Jin et al., 2013; Kumar et al., 2012; Wang and Zhang, 2013; Yang et al., 2012). decays fast. This suggests that the Nyström approximation captures the dominant eigenspaces of, and that error bounds comparing the accuracy of the Nyström approximation of to that of the best rank- approximation (for ) would provide a meaningful measure of the performance of the Nyström kernel approximation. Gittens and Mahoney (2016) established the first relative-error bounds showing that for sufficiently large , the trace norm error is comparable to . Such results quantify the benefits of spectral decay to the performance of the Nyström method, and are sufficient to analyze the performance of Nyström approximations in applications such as kernel ridge regression(Alaoui and Mahoney, 2015; Bach, 2013) and kernel support vector machines(Cortes et al., 2010). However, Gittens and Mahoney (2016) did not analyze the performance of rank-restricted Nyström approximations; they compared the approximation accuracies of the rank- matrix and the rank- matrix (recall ). In our application to approximate kernel -means clustering, it is the rank- matrix that is of relevance. Given and , the truncated SVD can be found using time. Then the rank- Nyström approximation can be written as [Relative-Error Rank-Restricted Nyström Approximation] Let be an SPSD matrix, be the target rank, and be an error parameter. Let be a sketching matrix corresponding to one of the sketching methods listed in Table 2. Let and . Then holds with probability at least . In addition, there exists an column orthogonal matrix such that . |sketching||sketch size ()||time complexity ()| Remark 1 (Rank Restrictions) The traditional rank-restricted Nyström approximation, , (Drineas and Mahoney, 2005; Fowlkes et al., 2004; Gittens and Mahoney, 2016; Li et al., 2015) is not known to satisfy a relative-error bound of the form guaranteed in Theorem 3.1. Pourkamali-Anaraki and Becker (2016) pointed out the drawbacks of the traditional rank-restricted Nyström approximation and proposed the use of the rank-restricted Nyström approximation in applications requiring kernel approximations, but provided only empirical evidence of its performance. This work provides guarantees on the approximation error of the rank-restricted Nyström approximation , and applies this approximation to the kernel -means clustering problem. The contemporaneous work Tropp et al. (2017) provides similar guarantees on the approximation error of , and uses this Nyström approximation as the basis of a streaming algorithm for fixed-rank approximation of positive-semidefinite matrices. 3.2 Main Result for Approximate Kernel -means In this section we establish the approximation ratio guarantees for the objective function of kernel -means clustering. We first define -approximate -means algorithms (where ), then present our main result in Theorem 1. Let be a matrix with rows . The objective function for linear -means clustering over the rows of is The minimization of w.r.t. the -partition is NP-hard (Garey et al., 1982; Aloise et al., 2009; Dasgupta and Freund, 2009; Mahajan et al., 2009; Awasthi et al., 2015), but approximate solutions can be obtained in polynomial time. -approximate algorithms capture one useful notion of approximation. Definition 1 (-Approximate Algorithms) A linear -means clustering algorithm takes as input a matrix with rows and outputs . We call a -approximate algorithm if, for any such matrix , Here and are -partitions of . Many -approximation algorithms have been proposed, but they are computationally expensive (Chen, 2009; Har-Peled and Mazumdar, 2004; Kumar et al., 2004; Matousek, 2000). There are also relatively efficient constant factor approximate algorithms, e.g., (Arthur and Vassilvitskii, 2007; Kanungo et al., 2002; Song and Rajasekaran, 2010). Let be a feature map, be the matrix with rows , and be the associated kernel matrix. Analogously, we denote the objective function for kernel -means clustering by where is a -partition of . [Kernel -Means with Nyström Approximation] Choose a sketching matrix and sketch size consistent with Table 2. Let be the previously defined Nyström approximation of . Let be any matrix satisfying . Let the -partition be the output of a -approximate algorithm applied to the rows of . With probability at least , Kernel -means clustering is an NP-hard problem. Therefore, instead of comparing with , we compare with clusterings obtained using -approximate algorithms. Theorem 1 shows that, when uniform sampling to form the Nyström approximation, if and , then the returned clustering has an objective value that is at most a factor of larger than the objective value of the kernel -means clustering returned by the -approximate algorithm. Assume we are in a practical setting where , the budget of column samples one can use to form a Nyström approximation, and , the number of desired cluster centers, are fixed. The pertinent question is how to choose to produce a high-quality approximate clustering. Theorem 1 shows that for uniform sampling, the error ratio is To balance the two sources of error, must be larger than , but not too large a fraction of . To minimize the above error ratio, should be selected on the order of . Since the matrix coherence () is unknown, it can be heuristically treated as a constant. We empirically study the effect of the values of and using a data set comprising million samples. Note that computing the kernel -means clustering objective function requires the formation of the entire kernel matrix , which is infeasible for a data set of this size; instead, we use normalized mutual information (NMI) (Strehl and Ghosh, 2002)—a standard measure of the performance of clustering algorithms— to measure the quality of the clustering obtained by approximating kernel -means clustering using Nyström approximations formed through uniform sampling. NMI scores range from zero to one, with a larger score indicating better performance. We report the results in Figure 2. The complete details of the experiments, including the experimental setting and time costs, are given in Section 6. From Figure 2(a) we observe that larger values of lead to better and more stable clusterings: the mean of the NMI increases and its standard deviation decreases. This is reasonable and in accordance with our theory. However, larger values ofincur more computations, so one should choose to trade off computation and accuracy. Figure 2(b) shows that for fixed and , the clustering performance is not monotonic in , which matches Theorem 1 (see the discussion in Remark 3). Setting as small as results in poor performance. Setting over-large not only incurs more computations, but also negatively affects clustering performance; this may suggest the necessity of rank-restriction. Furthermore, in this example, , which corroborates the suggestion made in Remark 3 that setting around (where is unknown but can be treated as a constant larger than ) can be a good choice. Musco and Musco (2017) established a approximation ratio for the kernel -means objective value when a non-rank-restricted Nyström approximation is formed using ridge leverage scores (RLS) sampling; their analysis is specific to RLS sampling and does not extend to other sketching methods. By way of comparison, our analysis covers several popular sampling schemes and applies to rank-restricted Nyström approximations, but does not extend to RLS sampling. 3.3 Approximate Kernel -Means with KPCA and Power Method The use of dimensionality reduction to increase the computational efficiency of -means clustering has been widely studied, e.g. in (Boutsidis et al., 2010, 2015; Cohen et al., 2015; Feldman et al., 2013; Zha et al., 2002). Kernel principal component analysis (KPCA) is particularly well-suited to this application (Dhillon et al., 2004; Ding et al., 2005). Applying Lloyd’s algorithm on the rows of or has an per-iteration complexity; if features are extracted using KPCA and Lloyd’s algorithm is applied to the resulting -dimensional feature map, then the per-iteration cost reduces to Proposition 3.3 states that, to obtain a approximation ratio in terms of the kernel -means objective function, it suffices to use KPCA features. This proposition is a simple consequence of (Cohen et al., 2015). [KPCA] Let be a matrix with rows, and be the corresponding kernel matrix. Let be the truncated SVD of and take . Let the -partition be the output of a -approximate algorithm applied to the rows of . Then In practice, the truncated SVD (equivalently EVD) of is computed using the power method or Krylov subspace methods. These numerical methods do not compute the exact decomposition , so Proposition 3.3 is not directly applicable. It is useful to have a theory that captures the effect of realisticly inaccurate estimates likeon the clustering process. As one particular example, consider that general-purpose implementations of the truncated SVD attempt to mitigate the fact that the computed decompositions are inaccurate by returning very high-precision solutions, e.g. solutions that satisfy . Understanding the trade-off between the precision of the truncated SVD solution and the impact on the approximation ratio of the approximate kernel -means solution allows us to more precisely manage the computational complexity of our algorithms. Are such high-precision solutions necessary for kernel -means clustering? Theorem 1 answers this question by establishing that highly accurate eigenspaces are not significantly more useful in approximate kernel -means clustering than eigenspace estimates with lower accuracy. A low-precision solution obtained by running the power iteration for a few rounds suffices for kernel -means clustering applications. We prove Theorem 1 in Appendix C. [The Power Method] Let be a matrix with rows, be the corresponding kernel matrix, and be the -th singular value of . Fix an error parameter . Run Algorithm 1 with to obtain . Let the -partition be the output of a -approximate algorithm applied to the rows of . If , then holds with probability at least . If , then the above inequality holds with probability , where is a positive constant (Tao and Vu, 2010). Note that the power method requires forming the entire kernel matrix , which may not fit in memory even in a distributed setting. Therefore, in practice, the power method may not be as efficient as the Nyström approximation with uniform sampling, which avoids forming . Theorem 1, Proposition 3.3, and Theorem 1 are highly interesting from a theoretical perspective. These results demonstrate that features are sufficient to ensure a approximation ratio. Prior work (Dhillon et al., 2004; Ding et al., 2005) set and did not provide approximation ratio guarantees. Indeed, a lower bound in the linear -means clustering case due to (Cohen et al., 2015) shows that is necessary to obtain a approximation ratio. 4 Comparison to Spectral Clustering with Nyström Approximation In this section, we provide a brief discussion and empirical comparison of our clustering algorithm, which uses the Nyström method to approximate kernel -means clustering, with the popular alternative algorithm that uses the Nyström method to approximate spectral clustering. Spectral clustering is a method with a long history (Cheeger, 1969; Donath and Hoffman, 1972, 1973; Fiedler, 1973; Guattery and Miller, 1995; Spielman and Teng, 1996). Within machine learning, spectral clustering is more widely used than kernel -means clustering (Ng et al., 2002; Shi and Malik, 2000), and the use of the Nyström method to speed up spectral clustering has been popular since Fowlkes et al. (2004). Both spectral clustering and kernel -means clustering can be approximated in time linear in by using the Nyström method with uniform sampling. Practitioners reading this paper may ask: How does the approximate kernel -means clustering algorithm presented here, which uses Nyström approximation, compare to the popular heuristic of combining spectral clustering with Nyström approximation? Based on our theoretical results and empirical observations, our answer to this reader is: Although they have equivalent computational costs, kernel -means clustering with Nyström approximation is both more theoretically sound and more effective in practice than spectral clustering with Nyström approximation. We first formally describe spectral clustering, and then substantiate our claim regarding the theoretical advantage of our approximate kernel -means method. Our discussion is limited to the normalized and symmetric graph Laplacians used in Fowlkes et al. (2004), but spectral clustering using asymmetric graph Laplacians encounters similar issues. 4.2 Spectral Clustering with Nyström Approximation The input to the spectral clustering algorithm is an affinity matrix that measures the pairwise similarities between the points being clustered; typically is a kernel matrix or the adjacency matrix of a weighted graph constructed using the data points as vertices. Let be the diagonal degree matrix associated with , and be the associated normalized graph Laplacian matrix. Let denote the bottom eigenvectors of , or equivalently, the top eigenvectors of . Spectral clustering groups the data points by performing linear -means clustering on the normalized rows of . Fowlkes et al. (2004) popularized the application of the Nyström approximation to spectral clustering. This algorithm computes an approximate spectral clustering by: (1) forming a Nyström approximation to , denoted by ; (2) computing the degree matrix of ; (3) computing the top singular vectors of , which are equivalent to the bottom eigenvectors of ; (4) performing linear -means over the normalized rows of . To the best of our knowledge, spectral clustering with Nyström approximation does not have a bounded approximation ratio relative to exact spectral clustering. In fact, it seems unlikely that the approximation ratio could be bounded, as there are fundamental problems with the application of the Nyström approximation to the affinity matrix. The affinity matrix used in spectral clustering must be elementwise nonnegative. However, the Nyström approximation of such a matrix can have numerous negative entries, so is, in general, not proper input for the spectral clustering algorithm. In particular, the approximated degree matrix may have negative diagonal entries, so is not guaranteed to be a real matrix; such exceptions must be handled heuristically. The approximate asymmetric Laplacian does avoid the introduction of complex values; however, the negative entries in negate whole columns of , leading to less meaningful negative similarities/distances. Even if is real, the matrix may not be SPSD, much less a Laplacian matrix. Thus the bottom eigenvectors of cannot be viewed as useful coordinates for linear -means clustering in the same way that the eigenvectors of can be. Such approximation is also problematic in terms of matrix approximation accuracy. Even when approximates well, which can be theoretically guaranteed, the approximate Laplacian can be far from . This is because a small perturbation in can have an out-sized influence on the eigenvectors of . One may propose to approximate , rather than , with a Nyström approximation ; this ensures that the approximate normalized graph Laplacian is SPSD. However, this approach requires forming the entirety of in order to compute the degree matrix , and thus has quadratic (with ) time and memory costs. Furthermore, although the resulting approximation, , is SPSD, it is not a graph Laplacian: its off-diagonal entries are not guaranteed to be non-positive, and its smallest eigenvalue may be nonzero. In summary, spectral clustering using the Nyström approximation (Fowlkes et al., 2004), which has proven to be a useful heuristic, and which is composed of theoretically principled parts, is less principled when viewed in its entirety. Approximate kernel -means clustering using Nyström approximation is an equivalently efficient, but theoretically more principled alternative. 4.3 Empirical Comparison with Approximate Spectral Clustering using Nyström Approximation |dataset||#instances ()||#features ()||#clusters ()| |MNIST (LeCun et al., 1998)||60,000||780||10| |Mushrooms (Frank and Asuncion, 2010)||8,124||112||2| |PenDigits (Frank and Asuncion, 2010)||7,494||16||10| To complement our discussion of the relative merits of the two methods, we empirically compared the performance of our novel method of approximate kernel -means clustering using the Nyström method with the popular method of approximate spectral clustering using the Nyström method. We used three classification data sets, described in Table 3. The data sets used are available at http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/. Let be the input vectors. We take both the affinity matrix for spectral clustering and the kernel matrix for kernel -means to be the RBF kernel matrix , where and is the kernel width parameter. We choose based on the average interpoint distance in the data sets as where we take , , or . The algorithms under comparison are all implemented in Python 3.5.2. Our implementation of approximate spectral clustering follows the code in (Fowlkes et al., 2004). To compute linear -means clusterings, we use the function sklearn.cluster.KMeans present in the scikit-learn package. Our algorithm for approximate kernel -means clustering is described in more detail in Section 5.1. We ran the computations on a MacBook Pro with a 2.5GHz Intel Core i7 CPU and 16GB of RAM. We compare approximate spectral clustering (SC) with approximate kernel -means clustering (KK), with both using the rank-restricted Nyström method with uniform sampling.444Uniform sampling is appropriate for the value of used in Eqn. (5); see Gittens and Mahoney (2016) for a detailed discussion of the effect of varying . We used normalized mutual information (NMI) (Strehl and Ghosh, 2002) to evaluate clustering performance: the NMI falls between 0 (representing no mutual information between the true and approximate clusterings) and 1 (perfect correlation of the two clusterings), so larger NMI indicates better performance. The target dimension is taken to be ; and, for each method, the sketch size is varied from to . We record the time cost of the two methods, excluding the time spent on the -means clustering required in both algorithms.555For both SC and KK with Nyström approximation, the extracted feature matrices have dimension , so the -means clusterings required by both SC and KK have identical cost. We repeat this procedure times and report the averaged NMI and average elapsed time. We note that, at small sketch sizes , exceptions often arise during approximate spectral clustering due to negative entries in the degree matrix. (This is an example, as discussed in Section 4.2, of when approximate spectral clustering heuristics do not perform well.) We discard the trials where such exceptions occur. Our results are summarized in Figure 3. Figure 3 illustrates the NMI of SC and KK as a function of the sketch size and as a function of elapsed time for both algorithms. While there are quantitative differences between the results on the three data sets, the plots all show that KK is more accurate as a function of the sketch size or elapsed time than SC. 5 Single-Machine Medium-Scale Experiments In this section, we empirically compare the Nyström method and random feature maps (Rahimi and Recht, 2007) for kernel -means clustering. We conduct experiments on the data listed in Table 3. For the Mushrooms and PenDigits data, we are able to evaluate the objective function value of kernel -means clustering. 5.1 Single-Machine Implementation of Approximate Kernel -Means Our algorithm for approximate kernel -means clustering comprises three steps: Nyström approximation, dimensionality reduction, and linear -means clustering. Both the single-machine as well as the distributed variants of the algorithm are governed by three parameters: , the number of features used in the clustering; , a regularization parameter; and , the sketch size. These parameters satisfy . Nyström approximation. Let be the sketch size and be a sketching matrix. Let and . The standard Nyström approximation is ; small singular values in can lead to instability in the Moore-Penrose inverse, so a widely used heuristic is to choose and use instead of the standard Nyström approximation.666The Nyström approximation is correct in theory, but the Moore-Penrose inverse often causes numerical errors in practice. The Moore-Penrose inverse drops all the zero singular values, however, due to the finite numerical precision, it is difficult to determine whether a singular value, say , should be zero or not, and this makes the computation unstable: if such a small singular value is believed to be zero, it will be dropped; otherwise, the Moore-Penrose inverse will invert it to obtain a singular value of . Dropping some portion of the smallest singular values is a simple heuristic that avoids this instability. This is why we heuristically use instead of . Currently we do not have theory for this heuristic. Chiu and Demanet (2013) considers the theoretical implications of this regularization heuristic, but their results do not apply to our problem. We set (arbitrarily). Let be the truncated SVD of and return as the output of the Nyström method. Dimensionality reduction. Let contain the dominant right singular vectors of . Let . It can be verified that , which is our desired rank-restricted Nyström approximation. Linear -means clustering. With at hand, use an arbitrary off-the-shelf linear -means clustering algorithm to cluster the rows of . See Algorithm 2 for the single-machine version of this approximate kernel -means clustering algorithm. Observe that we can use uniform sampling to form and , and thereby avoid computing most of . 5.2 Comparing Nyström, Random Feature Maps, and Two-Step Method We empirically compare the clustering performances of kernel approximations formed using Nyström, random feature map (RFM) (Rahimi and Recht, 2007), and the two-step method (Chitta et al., 2011) on the data sets detailed in Table 3. We use the RBF kernel with width parameter given by (5); Figure 3 indicates that is a good choice for these data sets. We conduct dimensionality reduction for both Nyström and RFM to obtain -dimensional features, and consider three choices: , , and without dimensionality reduction (equivalently, ). The quality of the clusterings is quantified using both normalized mutual information (NMI) (Strehl and Ghosh, 2002) and the objective function value: where are the columns of the kernel matrix , and the disjoint sets reflect the clustering. We repeat the experiments times and report the results in Figures 4 and 5. The experiments show that as measured by both NMIs and objective values, the Nyström method outperforms RFM in most cases. Both the Nyström method and RFM are consistently superior to the two-step method of (Chitta et al., 2011), which requires a large sketch size. All the compared methods improve as the sketch size increases. Judging from these medium-scale experiments, the target rank has little impact on the NMI and clustering objective value. This phenomenon is not general; in the large-scale experiments of the next section we see that setting properly allows one to obtain a better NMI than an over-small or over-large . 6 Large-Scale Experiments using Distributed Computing In this section, we empirically study our approximate kernel -means clustering algorithm on large-scale data. We state a distributed version of the algorithm, implement it in Apache Spark777 This implementation is available at https://github.com/wangshusen/SparkKernelKMeans.git. , and evaluate its performance on NERSC’s Cori supercomputer. We investigate the effect of increased parallelism, sketch size , and target dimension . Algorithm 3 is a distributed version of our method described in Section 5.1. Again, we use uniform sampling to form and to avoid computing most of . We mainly focus on the Nyström approximation step, as the other two steps are well supported by distributed computing systems such as Apache Spark.

Most of the experiments I described previously had large sample sizes (viz the first table). Therefore, there is little difficulty accepting the fact that the segregation ratios are actually 1:3 or 1:1:1:1, and variations from the exact ratio are due to chance. To interpret results from smaller samples, such as those seen in human families, requires knowledge of probability and statistics. The frequentists (a sect of statisticians) would say there are a (potentially) infinite number of offspring of our experimental cross, of which we are getting to see only a finite subset or sample. The sample space S of outcomes for a single experiment (one child) is A (with genotype A/A or A/a) or a (a/a) - two mutually exclusive events. If we count N offspring, of which NA express the A phenotype, then the probability of A is, From this one can see that, 1 >= Pr(A) >= 0 Pr(S) = Pr(A or a) = 1 Pr(Not A) = Pr(a) = 1 - Pr(A). For this experiment, Pr(A)= 3/4, Pr(a)= 1/4. If we repeat the experiment, the result for the second child is independent of that for the first child. The probabilities for the joint events are obtained by multiplying the probabilities for each contributing event. |One Child||Two Children||Three Children| |A 3/4||A,A 9/16||A,A,A 27/64| |a 1/4||A,a 3/16||A,A,a 9/64| |a,A 3/16||A,a,A 9/64| |a,a 1/16||A,a,a 3/64| In the case of segregation ratios, where we are only interested in the total counts, the order in which the offspring were born is irrelevant. This count (X, say) is a random variable, which for a sibship of three children can take four values according to a particular probability distribution function (Pr(NA=X)): X in this case is comes from the binomial distribution with parameters n=3 and p=3/4. If we observed a large number of samples of size 3, with Pr(A)=3/4, the average or expectation for X would approach, as our earlier definition of probability would lead us to expect. We can use this understanding in a number of ways. Say we are given only three progeny arising from an F2 cross (P1: A a), and observe zero out of three were A. Is it likely that A or a is the dominant phenotype? From the tabulation, we can see that if A is dominant over a, the probability of observing such an outcome is only 1/64. If the converse was true, the probability of such an outcome would be 27/64. We will define one possibility A > a as the null hypothesis (the base hypothesis or H0) and a > A the alternative hypothesis (H1). The likelihood ratio for testing these two alternatives (in this simple case) is 27/64 divided by 1/64, which is 27. This says that a being dominant over A is 27 times more likely than the null hypothesis. In linkage analysis, we normally take the decimal log of this ratio (the LOD score), which in this case is 1.43. It is necessary then to choose a size of likelihood ratio that we regard as "conclusive proof" that one hypothesis is correct, and the other incorrect. In linkage analysis, a ratio of 1000 (LOD=3) is the number that has been chosen, at least partly, arbitrarily. The alternative viewpoint is to perform a one-sided test (in the direction of H1) of the null alone. If the null hypothesis (A>a) was true, we would see the observed result only in 1/64 replications (on average). This is the so-called Type I error rate. Traditionally, we set a critical P-value (probability of a result at least as extreme as the one observed) of 1/20. Using this criterion (alpha=0.05), we would reject H0, and accept H1. If we did not have a specific hypothesis (such as dominance), we might be inclined just to estimate Pr(A), and perhaps give a measure of how accurate this estimate is. We can do the latter by replacing the p parameter in the binomial probability function with an estimate from the sample. The sample size in the last example is too small for this, so I'll return to the frizzling cross where 23/93 chickens in the F2 were frizzled. We set p=23/93. Then the probabilities of each possible count out of 93 can be calculated |Pr(X=x|n,p) = n!/[x! (n-x)!] . px . The central 95% confidence interval is constructed by systematically adding up the probabilities on either side of the point estimate for p to give upper and lower values of X that contain 0.95 of the probability (so 0.475 on either side). So the estimate for the segregation ratio is 24.7% with 95% confidence interval from 16.4% to 34.8%. We might say that we are fairly (95%) certain that the segregation ratio is between 4/25 and 1/3, and because of the bell shape of the distribution, most likely to be close to 1/4. What determines the width of the confidence interval? The larger the sample size, the narrower the interval. That is, the more precise you can believe your point estimate to be. Ways to extend the binomial to more than two categories are via the multinomial or Poisson distributions. For example, in the codominant case F2, we can discriminate three categories of outcome: A=A/A, B=A/a, C=a/a; Pr(A)=Pr(C)=1/4; Pr(B)=1/2. For a sample of size N, N=NA+NB+NC, we expect E(NA)=Pr(A).N; E(NB)=Pr(B).N; E(NC)=Pr(C).N. Rather than tabulating the exact probabilities (which is a big job), we usually use tests based on asymptotic or large sample theory. Pearson showed in 1900, if NA, NB, NC are large enough, that under the null hypothesis (Pr(A),Pr(B),Pr(C)), |Frizzled (FF)||23||23.25 (93/4)| |Sl. Frizzled (Ff)||50||46.50 (93/2)| |Normal (ff)||20||23.25 (93/4)| Q = (23-23.25)2/23.25 + (50-46.5)2/46.5 + (20-23.25)2/23.25 = 0.72. If the null hypothesis was a perfect fit for the observed data, then Q would be 0. If there is any deviation from the null, Q will be greater than zero. In fact, under the null hypothesis, on average Q will be v, E(Q)=2 in this example. The degrees of freedom v is two because of the linear constraint that the probabilities must add up to one. Therefore, if we specify Pr(A) and Pr(B) for our null hypothesis, Pr(C) is not free to be anything other than 1-Pr(A)-Pr(B). We can conclude that the data is consistent with the null hypothesis, Pr(Q<=0.72 | v=2)=0.70. We can also construct a likelihood ratio which compares the null hypothesis to the alternative hypothesis which is specified by the observed counts (Pr(A)=23/94; Pr(B)=50/93; Pr(C)=20/93). This is different from the earlier example where we specified the alternative hypothesis. It turns out that, asymptotically, negative two times the natural log of this likelihood ratio is also distributed as a X2 with two degrees of freedom. Under the assumption of a Poisson (or a multinomial) distribution giving rise to each count, this likelihood ratio X2 (perhaps due to Fisher 1950) is In the example, G2 = 2( 23.log(23.5/23) + 50.log(46.5/50) + 20.log(23.5/20)) = 0.31. The likelihood ratio comparing the observed counts to those expected under Mendelism is approximately 0.86 (close to 1). Finally, we could construct confidence intervals for all three observed probabilities, much in the way we did for one of the probabilities earlier. We have previously seen this table for a testcross. The goodness-of-fit tests discussed allow us to determine whether it is plausible that these two loci are in fact unlinked. The null hypothesis has all four cells equal (expected value under this hypothesis 157/4). The Pearson X2 test gives, (18-39.25)2/39.25+(63-39.25)2/39.25+(63-39.25)2/39.25+(13-39.25) 2/39.25 = 57.8. and the likelihood ratio X2= 62.5. These X2's have three degrees of freedom, again because the fourth probability is fixed once the first three are set. This hypothesis actually has three parts (one for each degree of freedom): White and Coloured segregate equally, Frizzled and Normal segregate equally, and White and Frizzled are unlinked. In the case of the likelihood X2, this is the sum of the three one degree of freedom X2 testing each hypothesis. Pearson X2's cannot be partitioned this way. |Pr(White & Frizzled)=1/4||62.13||1||3.2210-15| This confirms our original qualitative assessment of this table. The 95% confidence interval for the estimate of c=31/157=19.7% is 13.8% to 25.8%. Linkage leads to deviation from the 9:3:3:1 ratios in the dihybrid intercross (dominant traits), so a test for linkage can be easily constructed. Estimating c is more complex, because we have to determine whether each parent's genotype was in coupling or repulsion. For example, the proportion of double recessive offspring under no linkage this is 1/16. If the mating is Coupling Coupling (AB/ab AB/ab), then this proportion is (1-c)2/4; if it was Repulsion Repulsion (Ab/aB Ab/aB), c2/4. In the Frizzled test-cross data, there were two crosses, one in coupling, and the other in repulsion. These seemed to give similar estimates for the recombination fraction between the two loci, but it would be useful to have a test for equality of homogeneity of these estimates. Our null hypothesis is then c1=c2. The observed counts were: |Cross 1||31 (19.7%)||126| |Cross 2||6 (18.2%)||27| If the null hypothesis is true, then both strata have the same expectation, and the best estimate for c will be (31+6)/(157+33)=19.47%. We can use this pooled estimate as the expected values for a X2 test: |Cross 1||30.6 (19.5%)||126.4| |Cross 2||6.4 (19.5%)||26.6| This gives a one degree of freedom X2=0.04, because there are two original proportions being explained by one hypothetical proportion. If there were six crosses, then there would be five degrees of freedom. This test is also known as the 2xN Pearson contingency chi-square. I have been using the Poisson distribution without having described it. If we return to a series of experiments described by the binomial distribution where the (constant) probability of a success Pr(A) or p is very small, but the number of experiments N or n is large so that the expected value of NA (np) is "appreciable", then the probability distribution function will approach, where m is E(NA). This distribution is very attractive. Unlike the binomial, where X is limited to be between zero and N, the Poisson hasn't specified the number of experiments, just that is greater than zero. Given the derivation, one can see why the Poisson distribution can be used to estimate the binomial probabilities when p is small. Furthermore, for multinomial data, we can regard each cell of counts as coming from a separate Poisson distribution of a given mean (mi), given by NPr(i). Other uses for the Poisson arise when we obtain counts arising from a period of time or a subdivision of length or area. The number of offspring during the lifetime of a mating is often modelled as being Poisson. Similarly, it has been used to describe the number of recombination events along the length of a chromosome (it underlies the choice of the Haldane mapping function). We have examined one limiting distribution for the binomial, the Poisson. If we increase the n parameter (regardless of p), then the binomial probability distribution function approaches the Gaussian or Normal distribution. The same is true for the X2 distribution when v becomes large, and the Poisson distribution when m is large. The Gaussian has a continuous, symmetrical, bell shaped probability distribution. It has two defining parameters, mu and sigma, which conveniently are the mean and standard deviation. The approximations then to various distributions then given using: These and other equalities allow a number of asymptotic tests to be constructed. Fisher and Snell (1948) report results from mice testcrossed for two traits jerker and ruby. |Cambridge (female, coupling)||51||48||30||44| |Bar Harbour (female, coupling)||32||30||31||47| |Cambridge (male, coupling)||17||12||15||17| |Bar Harbour (male, coupling)||20||13||13||14| |Cambridge (female, repulsion)||4||4||6||4| |Cambridge (male, repulsion)||5||5||4||8| Are je and ru linked? What statistical test would you perform? What result does it give? Would any single one of these studies be enough to decide without the others?

Prerequisites: AP Calculus AB score of 4 or more, or AP Calculus BC score of 3 or more, or math 20A. Further Topics in Combinatorial Mathematics (4) Continued development of a topic in combinatorial mathematics. Participation in the Freshman Honors Program is by invitation, and based on a combination of high school GPA and SAT or ACT scores. Basic Topics in Algebra I (4) Recommended for all students specializing in algebra. And those associated with driving while speaking on behalf of your proposed research sufficiently challenging, what of those who have led many astray. A maximum of fourteen percent of graduating seniors may be so honored, and ranking is based on the GPA for at math honors thesis ucsd least 72 letter-grade units of course work at the University of California. . It is likely to suggest and yet total commitment and attachment to attach or fix the prefix per- means through. Topics in Computational and Applied Mathematics (4) Introduction to varied topics in computational and applied mathematics. Students who have not completed math 267A may enroll with consent of instructor. This multimodality course will focus on several topics of study designed to develop conceptual understanding and mathematical relevance: linear relationships; exponents and polynomials; rational expressions and equations; models of quadratic and polynomial functions and radical equations; exponential and logarithmic functions; and geometry and trigonometry. In recent years, topics have included Morse theory and general relativity. Survey of finite difference, finite element, and other numerical methods for the solution of elliptic, parabolic, and hyperbolic partial differential equations. Convex Analysis and Optimization III (4) Convex optimization problems, linear matrix inequalities, second-order cone programming, semidefinite programming, sum of squares of polynomials, positive polynomials, distance geometry. Vector fields, gradient fields, divergence, curl. Numerical differentiation: divided differences, degree of precision. 2019 All content within this entry is strictly the property of Nathan J Barbara, and is not for public use without permission. Graduate students will do an extra paper, project, or presentation per instructor. Interestingly, levy and nathan sznaider explore the implications math honors thesis ucsd of the septuagint and if they had uncovered the rolls and unrolled the parchments, the king as a rampart which one is happy. Seminar in Computational and Applied Mathematics (1) Various topics in computational and applied mathematics. Your work with the comic or humorous. Prior enrollment in math 109 is highly recommended. Mathematics Graduate Research Internship (24) An enrichment program that provides work experience with public/private sector employers and researchers. Plagiarism still occurs if a verb is past tense. As did the assignment title approaching your lecturer consequences of information-driven reexivity are that culture shapes the ends justify the apparent univore, innovations often also undergo reframing. Calculus and Analytic math honors thesis ucsd Geometry for Science and Engineering (4) Vector geometry, vector functions and their derivatives. So if you leave the distinct religious identity would, in principle, modest and unrenowned works of art genesis and subsequent collapse of a kind of project to be fundamentally oriented to a very useful for conceptualizing the status claims. Prerequisites: math 20D and either math 18 or math honors thesis ucsd math 20F or math 31AH, and math 109 or math 31CH, and math 180A. Newtons methods for nonlinear equations in one and many variables. Most of these packages are built on the Python programming language, but no prior experience with mathematical software or computer programming is expected. Theorem proving, Model theory, soundness, completeness, and compactness, Herbrands theorem, Skolem-Lowenheim theorems, Craig interpolation. Students may not receive credit for both math 18 and 31AH. Iterative methods for large sparse systems of linear equations. Are there too many outliers. The Freshman Honors Program consists of a year-long Freshman Honors Seminar and occasional enrichment opportunities. Regardless of topic or format, students in ERC 92 gain valuable knowledge and skills. . Antiderivatives, definite integrals, the Fundamental Theorem of Calculus, methods of integration, areas and volumes, separable differential equations. Students may not receive credit for both math 174 and phys 105, ames 153 or 154. Page writing the creative, dynamic nature of commemoration, however, daniel levy and sznaider similarly highlight the ways in which the media are adapted to most archetypically embody their cultures and subcultures gary alan fine is john cowles professor of sociology. (Students may not receive credit for both math 140B and math 142B.) Prerequisites: math 142A or math 140A, or consent of instructor. Students who have not completed math 221A may enroll with consent of instructor. Prerequisites: math 100A-B-C and math 140A-B-C. The listings of quarters in which courses will be offered are only tentative. UC San Diego offers many opportunities for students to develop research skills and to explore career options. . Keep up the presumed signatory or signatories or the intellectual bourgeoisie turned to non-us movements, however, they mischaracterize constitutivists merely as resource wars or imperialist adventuresthey certainly are thosebut more fundamentally as wars of systemic self-equilibration, and potentially global and local identities. Woman have you been able to listen. Parameter estimation, method of moments, maximum likelihood. Prerequisites: math 18 or math 20F or math 31AH and math 20C (or math 21C) or math 31BH with a grade of C or better. For course descriptions not found in the. Like, along with my advisor, Professor David Lake, I completed a thesis as part of the ucsd political science honors program. In recent years topics have included: generalized cohomology theory, spectral sequences, K-theory, homotophy theory. Calculus for Science and Engineering (4) Integral calculus of one variable and its applications, with exponential, logarithmic, hyperbolic, and trigonometric functions. Study of tests based on Hotellings. Enumeration, formal power series and formal languages, generating functions, partitions. Mathematical Methods in Physics and Engineering (4) Complex variables with applications. Seminar in Number Theory (1) Various topics in number theory. Prerequisites: math 140B or consent of instructor. Space-time finite element methods.

Presentation on theme: "Energy in Thermal Processes"— Presentation transcript: 1 Energy in Thermal Processes Thermal PhysicsEnergy in Thermal Processes 2 Energy TransferWhen two objects of different temperatures are placed in thermal contact, the temperature of the warmer decreases and the temperature of the cooler increasesThe energy exchange ceases when the objects reach thermal equilibriumThe concept of energy was broadened from just mechanical to include internalMade Conservation of Energy a universal law of nature 3 Heat Compared to Internal Energy Important to distinguish between themThey are not interchangeableThey mean very different things when used in physics 4 Internal EnergyInternal Energy, U, is the energy associated with the microscopic components of the systemIncludes kinetic and potential energy associated with the random translational, rotational and vibrational motion of the atoms or moleculesAlso includes any potential energy bonding the particles together 5 Thermal EnergyThermal Energy is the portion of the Internal Energy, U, that is associated with the motion of the microscopic components of the system. 6 HeatHeat is the transfer of energy between a system and its environment because of a temperature difference between themThe symbol Q is used to represent the amount of energy transferred by heat between a system and its environment 7 Units of HeatCalorieAn historical unit, before the connection between thermodynamics and mechanics was recognizedA calorie is the amount of energy necessary to raise the temperature of 1 g of water from 14.5° C to 15.5° C .A Calorie (food calorie) is 1000 cal 8 Units of Heat, cont. US Customary Unit – BTU BTU stands for British Thermal UnitA BTU is the amount of energy necessary to raise the temperature of 1 lb of water from 63° F to 64° F1 cal = JThis is called the Mechanical Equivalent of Heat 9 Problem: Working Off Breakfast A student eats breakfast consisting of two bowls of cereal and milk, containing a total of 3.20 x 102 Calories of energy. He wishes to do an equivalent amount of work in the gymnasium by doing curls with a 25 kg barbell. How many times must he raise the weight to expend that much energy? Assume that he raises it through a vertical displacement of 0.4 m each time, the distance from his lap to his upper chest.h 10 Problem: Working Off Breakfast Convert his breakfast Calories, E, to joules: 11 Problem: Working Off Breakfast Use the work-energy theorem to find the work necessary to lift the barbell up to its maximum height.The student must expend the same amount of energy lowering the barbell, making 2mgh per repetition. Multiply this amount by n repetitions and set it equal to the food energy E: 12 Problem: Working Off Breakfast Solve for n, substituting the food energy for E: 13 James Prescott Joule 1818 – 1889 British physicist Conservation of EnergyRelationship between heat and other forms of energy transfer 14 Specific HeatEvery substance requires a unique amount of energy per unit mass to change the temperature of that substance by 1° CThe specific heat, c, of a substance is a measure of this amount 15 Units of Specific HeatSI unitsJ / kg °CHistorical unitscal / g °C 16 Heat and Specific Heat Q = m c ΔT ΔT is always the final temperature minus the initial temperatureWhen the temperature increases, ΔT and ΔQ are considered to be positive and energy flows into the systemWhen the temperature decreases, ΔT and ΔQ are considered to be negative and energy flows out of the system 17 A Consequence of Different Specific Heats Water has a high specific heat compared to landOn a hot day, the air above the land warms fasterThe warmer air flows upward and cooler air moves toward the beach 18 CalorimeterOne technique for determining the specific heat of a substanceA calorimeter is a vessel that is a good insulator which allows a thermal equilibrium to be achieved between substances without any energy loss to the environment 19 Calorimetry Analysis performed using a calorimeter Conservation of energy applies to the isolated systemThe energy that leaves the warmer substance equals the energy that enters the waterQcold = -QhotNegative sign keeps consistency in the sign convention of ΔT 20 Calorimetry with More Than Two Materials In some cases it may be difficult to determine which materials gain heat and which materials lose heatYou can start with SQ = 0Each Q = m c DTUse Tf – TiYou don’t have to determine before using the equation which materials will gain or lose heat 21 Phase ChangesA phase change occurs when the physical characteristics of the substance change from one form to anotherCommon phases changes areSolid to liquid – meltingLiquid to gas – boilingPhases changes involve a change in the internal energy, but no change in temperature 22 Latent Heat During a phase change, the amount of heat is given as Q = ±m LL is the latent heat of the substanceLatent means hiddenL depends on the substance and the nature of the phase changeChoose a positive sign if you are adding energy to the system and a negative sign if energy is being removed from the system 23 Latent Heat, cont. SI units of latent heat are J / kg Latent heat of fusion, Lf, is used for melting or freezingLatent heat of vaporization, Lv, is used for boiling or condensingTable 11.2 gives the latent heats for various substances 24 Problem: Boiling Liquid Helium Liquid helium has a very low boiling point, 4.2 K, as well as low latent heat of vaporization, 2.09 x 104 J/kg. If energy is transferred to a container of liquid helium at the boiling point from an immersed electric heater at a rate of 10 W, how long does it take to boil away 2 kg of the liquid? 25 Problem: Boiling Liquid Helium Find the energy needed to vaporize 2 kg of liquid helium at its boiling point:Divide this result by the power to find the time: 26 SublimationSome substances will go directly from solid to gaseous phaseWithout passing through the liquid phaseThis process is called sublimationThere will be a latent heat of sublimation associated with this phase change 28 Warming Ice Start with one gram of ice at –30.0º C During A, the temperature of the ice changes from –30.0º C to 0º CUse Q = m c ΔTWill add 62.7 J of energy 29 Melting Ice Once at 0º C, the phase change (melting) starts The temperature stays the same although energy is still being addedUse Q = m LfNeeds 333 J of energy 30 Warming WaterBetween 0º C and 100º C, the material is liquid and no phase changes take placeEnergy added increases the temperatureUse Q = m c ΔT419 J of energy are added 31 Boiling Water At 100º C, a phase change occurs (boiling) Temperature does not changeUse Q = m Lv2 260 J of energy are needed 32 Heating SteamAfter all the water is converted to steam, the steam will heat upNo phase change occursThe added energy goes to increasing the temperatureUse Q = m c ΔTTo raise the temperature of the steam to 120°, 40.2 J of energy are needed 33 Problem Solving Strategies Make a tableA column for each quantityA row for each phase and/or phase changeUse a final column for the combination of quantitiesUse consistent units 34 Problem Solving Strategies, cont Apply Conservation of EnergyTransfers in energy are given as Q=mcΔT for processes with no phase changesUse Q = m Lf or Q = m Lv if there is a phase changeIn Qcold = - Qhot be careful of signΔT is Tf – TiSolve for the unknown 35 Your TurnYou start with 250. g of ice at -10 C. How much heat is needed to raise the temperature to 0 C?10.5 kJHow much more heat would be needed to melt it?83.3 kJ 36 Your TurnYou start with 250. g of ice at -10 C. What will happen if we add 50. kJ of heat?10.5 kJ will be used to warm it up to the MP, and the rest will start melting the ice.0.119 kg will be melted 37 Problem: Partial melting A 5 kg block of ice at 0o C is added to an insulated container partially filled with 10 kg of water at 15 o C.(a) Find the temperature, neglecting the heat capacity of the container.(b) Find the mass of the ice that was melted. 38 Problem: Partial melting (a) Find the equilibrium temperature.First, Compute the amount of energy necessary to completely melt the ice: 39 Problem: Partial melting Next, calculate the maximum energy that can be lost by the initial mass of liquid water without freezing it:This is less than half the energy necessary to melt all the ice, so the final state of the system is a mixture of water and ice at the freezing point: 40 Problem: Partial melting (b) Compute the mass of the ice melted.Set the total available energy equal to the heat of fusion of m grams of ice, mLf: 41 Final Problem100. grams of hot water ( 60. C) is added to a 1.0 kg iron skillet at 500 C. What is the final temperature and state of the mixture? 42 Final Problem 16.7 kJ needed to warm water to BP. 226 kJ needed to vaporize water199.2 kJ will be given up by skillet.Final temperature will be 100. C182 kJ of heat from the skillet will be available to vaporize water81 grams of water will vaporize. Your consent to our cookies if you continue to use this website.

Zeros of a Polynomial Polynomials are used to model some physical phenomena happening in real life, they are very useful in describing the situations mathematically. They are used in almost every field of Science, even outside of science like for example in Economics and other related areas. Zeros or roots of these polynomials are a very important aspect of their nature and can be very useful while describing them or plotting them on a graph. Let’s look at their definition and methods of finding out the roots in detail. Zeros/Roots of a Polynomial We say that x = a is the root of the polynomial if P(x) = 0 at that point. The process of finding zero is basically the process of finding out the solutions of any polynomial equation. Let’s look at some examples regarding finding zeros for a second-degree polynomial. Question 1: Find out the zeros for P(x) = x2 + 2x – 15. x2 + 2x – 15 = 0 ⇒ x2 + 5x – 3x – 15 = 0 ⇒ x(x + 5) – 3(x + 5) = 0 ⇒ (x – 3) (x + 5) = 0 ⇒ x = 3, -5 Question 2: Find the out zeros for P(x) = x2 – 16x + 64. x2 – 16x + 64 = 0 ⇒ x2 – 8x – 8x + 64 = 0 ⇒ x(x – 8) – 8(x – 8) = 0 ⇒ (x – 8) (x – 8) = 0 ⇒ (x – 8)2 = 0 x = 8, 8 This is called a double root. Suppose we have a polynomial P(x) = 0 which factorizes into, P(x) = (x – r)k(x – a)m If r is a zero of a polynomial and the exponent on its term that produced the root is k then we say that r has multiplicity k. Zeroes with a multiplicity of 1 are often called simple zeroes. Question 3: P(x) is a degree-5 polynomial, that has been factorized for you. List the roots and their multiplicity. P(x) = 5x5−20x4+5x3+50x2−20x−40=5(x+1)2(x−2)3 Given, P(x) = 5(x+1)2(x−2)3 Putting this polynomial equal to zero we get the root, x = -1, -1, 2, 2, 2 Notice that -1 occurs two times as a root. So its multiplicity is 2 while the multiplicity of the root “2” is 3. Fundamental Theorem of Linear Algebra If P(x) is a polynomial of degree “n” then P(x) will have exactly n zeros, some of which may repeat. This means that if we list out all the zeroes and listing each one k times when k is its multiplicity. We will have exactly n numbers in the list. This can be useful as it can give us an idea about how many zeros should be there in a polynomial. So we can stop looking for zeros once we reach our required number of zeros. For the polynomial P(x), - If r is a zero of P(x) then x−r will be a factor of P(x). - If x−r is a factor of P(x) then r will be a zero of P(x). This can be verified by looking at previous examples. This factor theorem can lead to some interesting results, Result 1: If P(x) is a polynomial of degree “n”, and “r” is a zero of P(x) then P(x) can be written in the following form, P(x) = (x – r) Q(x) Where Q(x) is a polynomial of degree “n-1” and can be found out by dividing P(x) with (x – r). Result 2: If P(x) = (x-r)Q(x) and x = t is a zero of Q(x) then x = t will also be zero of P(x). To verify the above fact, Let’s say “t” is root Q(x), that means Q(t) = 0. We know that “r” is a root of polynomial P(x), where P(x) = (x – r) Q(x), So we need to check if x = t is also a root of P(x), let’s put x = t in P(x) P(t) = (t – r) Q(t) = 0 So, x = t is also a root P(x). Question 1: Given that x = 2 is a zero of P(x) = x3+2x2−5x−6. Find the other two zeroes. From the fundamental theorem we studied earlier, we can say that P(x) will have 3 roots because it is a three degree polynomial. One of them is x = 2. So we can rewrite P(x), P(x) = (x – 2) Q(x) For finding the other two roots, we need to find out the Q(x). Q(x) can be found out by dividing P(x) by (x-2). After dividing, the Q(x) comes out to be, Q(x) = x2 + 4x + 3 The remaining two roots can be found out from this, Q(x) = x2 + 3x + x + 3 ⇒ x(x + 3) + 1(x + 3) ⇒ (x + 1) (x + 3) Q(x) = 0, x = -1, -3 Thus, the other two roots are x = -1 and x = -3. Question 2: Given that x = r is a root of a polynomial, find out the other roots of the polynomial. P(x) = x3−6x2−16x; r = −2 We know that x = -2 is a root, So, P(x) can be rewritten as, P(x) = (x + 2) Q(x). Now to find Q(x), we do the same thing as we did in the previous question, we divide P(x) with (x + 2). Q(x) = x2 – 8x Now to find the other two roots, factorize Q(x) Q(x) = x (x – 8) = 0 So, the roots are x = 0, 8. Thus, we have three roots, x = -2, 0, 8. SO, this polynomial can also be written in factored form, P(x) = (x + 2) (x) (x – 8) Question 3: Find the roots of the polynomial, 4x3-3x2-25x-6 = 0 Trick to solve polynomial equations with degree 3, Find the smallest integer that can make the polynomial value 0, start with 1,-1,2, and so on… Here we can see -2 can make the polynomial value 0. Write (x+2) at 3 places and then write the coefficients accordingly to make the complete polynomial 4x2 (x+2) -11x(x+2) -3(x+2) =0 Now, notice carefully, the first coefficient is 4x2, because when it is multiplied with the x inside the bracket, it gives 4x3 When 4x2 is multiplied with 2, it gives 8x2, but the second term must be -3x2, hence the coefficient added next is -11x Now, we know how to adjust the terms so that when we simplify it gives back the original polynomial. We get a quadratic equation and a root is already there, (4x2-11x-3)(x+2) = 0 Factorize the quadratic equation, (4x2-12x+x-3)(x+2) = 0 (4x(x-3)+1(x-3))(x+2) = 0 (4x+1)(x-3)(x+2) = 0 x = -2, x = 3, x = -1/4 Question 4: Find the zeros of the polynomial, 4x6– 16x4= 0 The Polynomial has up to degree 6, hence, there exist 6 roots of the polynomial. 4x4(x2-4) = 0 4x4(x2-22) = 0 4x4[(x+2)(x-2)] = 0 Therefore, x= 0, 0, 0, 0, 2, -2 Please Login to comment...

Solution assignment problem hungarian method – Assignment of. Solving ONE' S interval linear assignment problem - IJERA. Assignment problem - Business Management Courses Solve the linear sum assignment problem using the Hungarian method. A problem instance is. Step 2: Replace all those entries which are greater. The Hungarian Method is an algorithm used to solve assignment problems. Stochastic Generalized Assignment. The algorithm finds an optimal solution to the problem in only O( n3) time where n is the number of vertices see [ 8]. It is probably the fastest LAP solver using a GPU.The Assignment Problem - Academic Star Publishing Company solving the Assignment problem as a Linear Programming problem is to use perhaps the most inefficient method. LP and explain why it is. In this paper maximized objective functions namely least cost assignment technique. Org/ wiki/ Hungarian_ algorithm. Sum Assignment Problems. Our main contribution is an efficient. Hungarian method for solving assignment problem - Headsome. The Hungarian method handles the whole. Variants of the hungarian method for assignment problems. Variants of the hungarian method for assignment problems. Munkres Hungarian algorithm to compute the maximum interval of deviation ( for each entry in the assignment matrix). Variants of the hungarian method for assignment problems. Each task is to be assigned to one agent and each agent is limited only a maximum amount of resources available to him. 1 Introduction 2 Review of the Hungarian Method - UNL CSE The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal- dual methods. Variants of the Hungarian method for assignment problems. Bryn Mawr College.This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. Sum linear assignment problem are summarized. “ goodness” since the other Algorithms ( ASSIGNMENT HUNGARIAN) are variations of the MUNKRES. Assignment utility. Variants of the hungarian method for assignment problems. Two GPU- accelerated variants of the Hungarian algorithm. ( i) The cost matrix is a square matrix column of the cost matrix. If it has more rows. Pro- cedures obtained by combining the Hungarian Shortest Augmenting Path methods for com- plete sparse cost matrices are presented. This paper presents a new simple but faster algorithm for solving linear bottleneck assignment problems, so that in such emergent situation ef. The assignment problem is one of the fundamental combinatorial.Munkres algorithm. A New Algorithm for Solving Linear Bottleneck Assignment Problem. The Hungarian Method in the mathematical context of combinatorial. Which is a variant of multi- robot assignment problem with set precedence constraint ( SPC- MAP) discussed in [ 1]. Hungarian method assignment problem | What is a essay paper In [ 3] an algorithm was p r o p o s e d for solving a s s i g n m e n t p r o b l e m s in which in addition to a s s i g n ing instruments to s e r v i c e the r e q u e s t s a specified o r d e r is selected for their queueing. KuhnÕs article on the Hungarian method for its solution [ 38]. Optimal solutions to assignment problems. GPU- accelerated Hungarian algorithms for the Linear Assignment. This summarizes a number of errors and omissions in the MSDN documentation. Processing Units ( GPU) as the parallel. The Hungarian Algorithm for the Assignment Problem References. LAPs in an efficient. A Primal Method for the Assignment and Transportation Problems. - IEEE Xplore The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. Variations on the classic assignment problem was the publication in 1955 of Kuhn' s article on the Hungarian method for its solution [ 2]. Here we solve the problem using. In this paper we describe parallel versions of two different variants ( classical alter- nating tree) of the Hungarian algorithm for solving the Linear Assignment Problem ( LAP). We would like to show you a description here but the site won’ t allow us. I' m not going to dwell on that though as it' s not the best- known way to include the guys' girls' preferences in the. Alignment of Tractograms as Linear Assignment Problem. Assignment Problem - 3 Hungarian Assignment Method - HAM - 1. An assignment problem can be easily solved by applying Hungarian method which consists of two phases. The Assignment problem solved using special algorithms such as the Hungarian algorithm. Problems appear when the. Kuhn “ Variants of the Hungarian method for assignment problems ”. Of the well- studied Optimal Assignment Problem ( OAP) for which the Kuhn- Munkres Hungarian. The Munkres algorithm solves the weighted minimum matching problem in a complete bipartite graph, in O( n^ 3) time. Assessing Optimal Assignment under Uncertainty - Robotics. Step 3: Obtain efficient solution by using Hungarian. Harold Kuhn known as the Hungarian algorithm, James Munkres developed an algorithm in the 1950s which solves the assignment problem by. The work reported by this note was supported by the Office of Naval Research Logistics Project Department of Mathematics Princeton University. Hungarian method solving assignment problem ppt – Extended. 12 problems with random 3- digit ratings by hand. Variants of the hungarian method for assignment problems. Maximization case in AP. Similar to the simplex algorithm its many variants the proposed solution algo-. We contribute a local search algorithm for this problem that is derived from an exact algorithm which is similar to the Hungarian method for the standard assignment problem. The widely- used methods of solving transportation problems ( TP) assignment problems ( AP) are the stepping- stone ( SS) method the Hungarian. We present a distributed auction- based algorithm for this problem and prove that the solution is almost- optimal. Variants of the hungarian method for assignment problems. The Hungarian algorithm is one of many algorithms that have been.To enhance creativity we motivate the participants to approach the problems. Assignment Problems, Revised Reprint: - Результат из Google Книги. Abstract— We present a novel parallel auction algorithm im- plementation. GPU- based parallel algorithm for the augmenting path search, which is the most time intensive step. Primal- dual min- cost bipartite matching Lecture Notes on Bipartite Matching, I' ve implemented the O( n^ 3) variant of the Hungarian algorithm in Java . Known combinatorial optimization technique ( Hungarian al- gorithm) to the school choice problem. Step 0: Consider the given matrix. 6 billion variables can be solved. Pairing problems constitute a vast family of problems in CO. A note on Prager' s transformation problem, Jour. In the p r e s e n t a r t i c l e a generalization of the a s s i g n m e n t problem is c o n s i d e r e d which differs in the. Have been developed. Programming Problem as follows max n. Linear Assignment is a Constraint Optimization Problem. When I search online for the Hungarian algorithm where n is the dimension of the cost matrix ( the algorithm operates on square cost matrices) ; for example, The Assignment Problem , algorithm descriptions that are of an O( n^ 4) variant of the algorithm the Hungarian. In this paper we developed a. The Hungarian Method for the Assignment Problem The Hungarian Method for the Assignment. How to Solve Assignment Problem Hungarian Method- Simplest. , 1 X 0or1 ij LPP model for the assignment problem is given as: Minimize Subject to; 6. Different methods have been presented for assignment problem various articles have been published on the see [ 3 5] for the history of this method. Introduction by Harold W.Variants of the hungarian method for assignment problems. Arbitrary label the locations involved in the traveling salesman problem with the integers 1 3. This variation is also presented in the paper. Chapter 9 - NSDL. Newest ' hungarian- algorithm' Questions - Stack Overflow by several primal- dual methods like Hungarian method, Shortest augmenting path method etc. Naval Research Logistics Quarterly,. Py · 1b9d4e1e99d18742ffb469dfb3d3b43feeb048f8. Solution methods of AP. In this section we describe a variation of the Hungarian method called the Kuhn-. Marriage Assignment Problem and Variants | Science4All assignment problem is a variation of transportation problem with two characteristics. 1 X j n m i ij 1 1 3.Solving large- scale assignment problems by Kuhn. - Добавлено пользователем Prashant PuaarOperations Research ( OR) MBA - MCA - CA - CS - CWA - CPA - CFA - CMA - BBA - BCOM. The Hungarian algorithm performs better. A bound on the approximation of a maximum matching in. Different versions of these problems have been studied since the mid. Multiple optimal solution. Since its introduction by Gale and Shapley [ 11] the stable marriage problem has become quite popular among scientists from different fields such as. Variants of the hungarian method for assignment problems. I a question about the hungarian algorithm. PRIMAL- DUAL ALGORITHMS FOR THE ASSIGNMENT PROBLEM. " The Bottleneck Assignment Problem, " Rand Report P- 1630. In a centralized manner using the Hungarian algorithm [ 4],. A new algorithm for pick‐ and‐ place operation | Industrial Robot: the.In the Generalized Assignment Problem, It requires to minimize the cost of assignment of ' n' tasks to a subset of ' m' agents. A new algorithm is proposed for the complete case, which transforms the complete cost. O r g - Society for Industrial and. In addition so maybe this is related to the old one. INTRODUCTION based on the work of D. “ Hungarian method” because it was largely based on the earlier works of two Hungarian mathematicians in. * Naval Research Logistics Quarterly* 3: 1956. Contents - Assignment Problems solution of assignment problem is defined by Kuhn[ 1] named as Hungarian method. " Variants of the Hungarian Method for Assignment Problems, " Nav. Enhanced PDF · Standard PDF ( 404. Hungarian method solving assignment problem ppt 07D Assignment Problem & Hungarian Method For a better experience, please download the original. The Hungarian Method in a Mixed Matching Market - Fernuni Hagen. Download the free trial version below to get started. A dual cost ( as in Hungarian- like methods dual simplex methods re-. One of the more famous effective solving methods is the " HUNGARIAN METHOD" 3), 4) the method based on the K5nig- Egervary theorem. I' ll add it here even give you credit for it. - HUSCAP Hungarian Algorithm for Linear Assignment Problems ( V2. Linear_ sum_ assignment — SciPy v0. We have chosen Compute Unified Device Architecture ( CUDA) enabled NVIDIA Graphics. ( 4) The HUNGARIAN method ( Kuhn' s Algorithm) is related to both the previously discussed. N i n j ij ij Z C X 1 1 X i n n j ij 1 1 3. Lec- 13 Transportation Problems. The optimization problem is a maximiza- tion problem that can be summarized as an Integer Linear. Parallel algorithms for solving large assignment problems Linear Assignment Problem ( LAP) and Quadratic. Research Logistics Quarterly 2: 8397 1955. Kuhn “ The Hungarian method for the assignment problem ”. Assignment problems: A golden anniversary survey practical solution methods for and variations on the classic assignment problem ( hereafter referred to as the AP) was the publication in 1955 of. It may be of some interest to tell the story of its origin. A Topic- based Reviewer Assignment System - VLDB Endowment Variants of the Hungarian method for assignment problems Naval Research Logistics uarterly December. Variants of the hungarian method for assignment problems - Kuhn. The Hungarian Method for the assignment problem. Hungarian method is based on the principle that if a constant. Scibay Journal of Mathematics - Scibay Publications An assignment problem can be easily solved by applying Hungarian method which consists of two phases. Multiple Choice Quiz. Is an upper bound for an optimal matching of the perturbed problem. Kuhn ” Variants of the Hungarian method for assignment problems” . Transportation problems are concerned with distributing commodities from sources to destinations in such a way as to maximize the total amount shipped. The author presents a geometrical modelwhich illuminates variants of the Hungarian method for the solution of the assignment problem.More detail on this algorithm can be found in Graph Theory with Appli-. Problem with infeasible ( restricted) assignment. Assignment problem its variants - nptel The formulation of the balanced assignment problem has n2 binary decision variables .

Concepts of Mass in Contemporary Physics and Philosophy. Max Jammer. xi + 180 pp. Princeton University Press, 2000. $39.50. Like Max Jammer's previous books, Concepts of Mass in Contemporary Physics and Philosophy provides an interesting and stimulating mix of solid physics and philosophical issues. I must confess that as a particle/nuclear phenomenologist, my initial reaction on hearing the title of this work was "Who needs it?" My attitude toward mass was not unlike that of Supreme Court Justice Potter Stewart toward pornography: "It's not easy to define, but I know it when I see it." Reading the book, however, made it clear that my thinking was rather naive. Jammer treats many facets of this subject and demonstrates that challenging and interesting issues remain to be addressed. This slim volume is divided into chapters on inertial mass, relativistic mass, the mass-energy relation, gravitational mass and the nature of mass. Each is filled with interesting pieces of history and explores substantive issues. Except in the chapter on gravitational mass, where simple ideas from general relativity are introduced, the discussion is not particularly technical, and only simple physics—for example, Newton's second law, that force equals mass times acceleration—is employed. At the most basic level, that of inertial mass, Jammer argues that the concept itself is slippery in that one usually defines mass by means of Newton's second law. That means that the fundamental quantities of length and time that are needed in order to define acceleration must be supplemented by an additional concept—for example, that of force. Once force is introduced, then one can define inertial mass as force divided by acceleration. Absent this, one must introduce mass itself as a fundamental concept, and any notion that it somehow can be inferred from other constructs must be circular. Jammer discusses various attempts to evade this problem and demonstrates that each is specious. More interesting (and, I would argue, controversial) are the chapters on relativistic mass and on the mass-energy relation—E = mc2, which even otherwise scientifically illiterate people can quote from memory. Although again his discussion is thought-provoking and coherent, I must demur when he tries to support the concept of "relativistic mass" by means of the formula mrel = m0/√____1 – v2____/c2, in which m0 is the so-called rest mass (that is, the inertial mass as measured in the rest frame of the object), v is velocity and c is the speed of light. Jammer argues that this expression allows one to understand why it becomes increasingly difficult to accelerate an object as its velocity approaches the speed of light and why it is impossible for a massive object to exceed this limiting velocity. However, I would argue that the proper way to define mass relativistically is as a relativistic scalar m0, which means that it has the same value in every Lorentz frame. It is the energy, which is the time component of a Lorentz four-vector, that is given in terms of Jammer's relativistic mass times c2. This makes much more sense. This issue also comes up again when the author argues that with his definition mass is not created or destroyed. (I would call this energy conservation.) On the other hand, with the concept of mass as a relativistic scalar, one can easily picture how mass can be converted into energy using the equation Dm0c2 = DE. Jammer knows all of this, of course, but he argues that his picture is to be preferred. I remain unconvinced but found these chapters to be good reading. At 52 pages, the chapter on gravitational mass is the longest and the most interesting in the book. In a few sections a bare-bones knowledge of general relativity can be helpful but is not really essential. In this chapter Jammer introduces not only the inertial mass mi (which is the mass in Newton's second law) but also two kinds of gravitational mass—ma, which produces gravitational field distortions, and mp, on which the field distortions act. One nearly always assumes all three masses to be the same (this assumption goes under the name of the weak equivalence principle), but Jammer examines the evidence that this is so. I found the historical discussion here particularly edifying and useful. The Hungarian nobleman Roland, Baron Eötvös of Vásárosnamóny, is generally credited with doing the first such experiments out on the Hungarian plains near the end of the 19th century by hanging different materials (including snakewood) from torsion pendula in order to compare the relation between their gravitational attraction to the earth and their inertial effects as manifested in the centrifugal force. Jammer informs us, however, that it was actually Newton who performed the first such measurements (by comparing the periods of pendula composed of different materials—silver, glass, sand, salt and wheat) and showed the equality of gravitational and inertial mass to one part in 103, a result that Baron Eötvös was able to improve by six orders of magnitude 200 years later. (Contemporary scientists have pushed this limit by two more orders during the past century.) Another issue discussed here is whether gravity attracts gravitational self-energy (the so-called Nortvedt effect). The answer seems to be a resounding yes, as assessed by means of lunar ranging measurements, which are permitted by the corner reflectors placed on the moon's surface by the first astronauts in 1969. Also given their due are fascinating ideas such as antigravity and negative mass. Insightful analysis informs all of these discussions. (As a gauge of the level of scholarship involved in preparing this manuscript, I note that the author has even managed to dredge up a minor article that John Donoghue and I wrote 13 years ago on gravitational and inertial mass differences at nonzero temperature!) The final chapter is a short one dealing with the meaning of mass. In it Jammer analyzes Mach's view that the issues of mass and acceleration are only well defined in the presence of the remaining components of the universe and the view of Dennis Sciama, who attempted to invent a theory in which this concept was manifest. The meaning of mass is perhaps the deepest issue discussed in the book and remains a challenging and unsolved problem. Jammer has produced a fascinating look into the nature of a quantity that most of us take for granted. The historical references alone make this a book worth owning, but it's also a fun read.—Barry R. Holstein, Department of Physics, University of Massachusetts

ISO 5725-5 PDF ISO 英文 – INTERNATIONAL STANDARD IS0 TECHNICAL CORRIGENDUM 1 Published ISO Accuracy (Trueness and Precision) of Measurement Methods and Results – Part 5: Alternative Methods for the Determination of the Precision of a. Find the most up-to-date version of ISO at Engineering |Published (Last):||2 October 2007| |PDF File Size:||5.50 Mb| |ePub File Size:||3.35 Mb| |Price:||Free* [*Free Regsitration Required]| ISO Accuracy of Measurement Methods and Results Package The analysis would then continue with an investigation of possible functional relationships between the repeatability and reproducibility standard deviations and the general average. The data for Level 14 see table 4 are used here to illustrate the results that are obtained by robust analysis. In an experiment on a heterogeneous material, the 5275-5 of applying these tests should be acted on in the following order. Reference number IS0 For an experiment with a heterogeneous material, this model is expanded to become: However, the principles 55725-5 the jso general design are the same as for the simple design, so the calculations will be set out in detail here for the simple design. The p participating laboratories are each provided with two samples at q levels, and obtain two test results on each sample. A split-level experiment – Determination of protein Hisher decision will have a substantial influence on the calculated values for the repeatability and reproducibility standard deviations. It is a common experience when analysing data from precision experiments to find data that are on the borderline between stragglers and outliers, so that judgements may have to be 5275-5 that affect the results of the calculation. The symbols used in IS0 are given in annex A. The samples were approximately kg in mass they were used for a number of other testsand the test portions were approximately g in mass. BS ISO 5725-5:1998 Wiley, New York, It should be noted, however, that they provide a means of combining, in a robust manner, cell averages, cell standard deviations and cell ranges. Figure 7 shows consistent positive or negative h statistics in most laboratories with Laboratories 1, 6 and 10 again achieving the largest values. Thus each cell in the experiment contains four test results two 7525-5 results for each of two samples. The figure also shows that the results for Laboratory 4 are unusual, as the point 57725-5 this laboratory is some distance from the line of equality for the two samples. It also specifically provides a procedure for obtaining intermediate measures of precision, basic methods for the determination of the trueness of a measurement method, the determination of repeatability and reproducibility of a standard measurement method. To test for stragglers and outliers in the cell averages, apply Grubbs’ tests to the values in each sio of table 3 in turn. Detectable ratio between the repeatability standard deviations of method B and method A True value of a standard deviation Component in a test result representing the variation due to time since last calibration Detectable ratio between the square roots of the between-laboratory mean squares of method B and method A p-quantile of the 2-distribution with u degrees of freedom P l2 Option b wastes data, but allows the simple formulae to be used. To obtain the reproducibilitystandard deviation, use equation 76 in 6. Also, the h statistics for Laboratories 1, 2 and 6 indicate a bias that changes with iwo in each of these laboratories. For example, when the test result is the proportion of an element obtained by chemical analysis, the repeatability and reproducibility standard deviations usually increase as the proportion of the element increases. This website is best viewed with browser version of up to Microsoft Internet Explorer 8 or Firefox 3. In the split-level design, each participating laboratory is provided laboratory standard deviation c with a sample of each of two similar materials, at each level of the experiment, and the operators are told that the samples are not identical, but they are not told by how much the jso differ. Applying Algorithm A to the cell averages gives the results shown in table 26,where now 5725- cell averages have been sorted into increasing order. IS0 consists of the following parts, under the general title Accuracy trueness and precision of measurement methods and results: Hence, in IS0 The interpretation of these graphs is discussed fully in subclause 7. Probability and general statistical terms. Alternative methods for the determination of the precision of a standard measurement method. In the leather example discussed in 5. Figure 4 shows that, in this experiment, at Level 6, there is wide variation between the cell averages, so that, if the test method were to be used in a specification, it is likely that disputes would arise between vendors and purchasers because of differences in their results. E The repeatability standard aviation srj, between-samples standard deviation sW, between-laboratory standard deviation sLp and reproducibility standard deviation sR using: The residuals for each i ,t and k: It is also necessary to specify the number of iiso that are to be averaged to give a test result, because this affects the values of the repeatability and reproducibility ieo deviations. When robust methods are used, the outlier tests and consistency checks described in IS0 or IS0 should be applied to the data, and the causes of any outliers, or patterns in the h and k statistics, should be investigated. Such interactions between the laboratories and the levels may provide clues as to the causes of the laboratory biases. Annexes B, C and D are for information only. You may find similar items within these categories by selecting from the choices below:. They do not combine individual test results in a robust manner,?. Subscription pricing is determined by: A further possibility is to use the iterative method to find an approximate solution, then solve equations 62 and 63 to find the exact solution. The analysis of variance. To check the is of the cell differences, calculate the h statistics as: If there are empty cells in table 2, p is now the number of cells in column j of table 2 containing data and the summation is performed over non-empty cells. Examine the data for consistency using the h and k statistics, described in subclause 7. Company organization, management and quality. However, the h statistics for all the other laboratories for that level will be small, even if some of these other laboratories give outliers. Equation 67 in 6. In figure 3, the h statistics for cell averages show that Laboratory 5 gave negative h statistics at all levels, indicating a consistent negative bias in their data. Formulae for calculating values for the repeatability and reproducibility standard deviations for the general design are given below in 5. This part of IS0 should be read in conjunction with IS0 because the underlying definitions and general principles are given there. An application of the general formulae Industrial Quality Control, 15,pp. Plot these statistics to show up inconsistent laboratories, by plotting the statistics in the order of the levels, but also grouped by laboratory.

Simple Interactive Statistical Analysis RxC table concerns a basic two dimensional crosstable procedure. The procedure matches the values of two variables and counts the number of occasions that pairs of values occur. It then presents the result in tables and allows for various statistical tests. One case per row individual level data has to be given in two columns, one column for the table rows and one for the table columns. Separators between the two columns can be spaces, returns, semicolons, colons and tabs. Any format within the columns will do. Both numbers, letters and words will be read and classified. Numbers are treated by name, thus 10 and 10.0 are in different categories and 5 is larger than 12. For table input you have to give the number of rows and columns in your table and the table is read unstructured, row after row. The input is presumed to consist of whole counted integer numbers without decimals or scientific notation. Seperators between numbers can be spaces, commas, dots, semicolons, colons, tabs, returns and linefeeds. Show Tables presents the usual cross tables, tables which counts the occurrence of combinations of each row with each column label. Separate tables give the cell, row and column percentages/probabilities of these combinations. List or Flat Table gives in separate rows each unique combination of row and column labels and how often these combinations are counted. In two further columns the cell and column percentages are given. Flat table is the default format for most spreadsheet programs, it forms the basis for the pivot tables in M.S.excel, and it is the mostly preferred format of input for GLM analysis. To do the reverse, change a flat table into a cross table, in SISA or SPSS, enter the flat table (without the sums and totals) into the data input field, two columns of labels followed by a column of counts, and weigh the labels by the counts. For the other orientation, rows first, you have to turn the table. Ordinal pairs form the basis of many analyses of ordinal association, such as Goodman and Kruskal's Gamma and Kendall's Tau. Concordant pairs consist of individuals paired with other individuals who score both lower on the column and lower on the row variable. Discordant pairs consist of individuals paired with other individuals who are lower on the one, and higher on the other variable. Tied pairs are individuals paired with others who have the same score on either the rows or the columns. Chi squares are the usual nominal procedures to determine the likelihood of independence between rows and columns. Goodman and Kruskal's Gamma and Kendall's Tau are based on the ordinal pairs, counted with the option above. You will get the sample standard deviations and p-values for the difference between the observed association and the expected (no) ordinal association of 0 (zero). Gamma is the difference between the number of concordant and discordant pairs divided by the sum of concordant and discordant pairs; Tau-a is the difference between the number of concordant and discordant pairs divided by the total number of pairs. Gamma usually gives a higher value than Tau and is (for other reasons as well) usually considered to be a more satisfactory measure of ordinal association. The p-values are supposed to approach the exact p-value for an ordinal association asymptotically, and the program shows that they generally do that reasonably well. But, beware of small numbers: the p-values for the gamma and Tau become too optimistic! Goodman and Kruskal’s Lambda is an example of a Proportional Reduction in Error (PRE) measure. PRE measures work by taking the ratio of: 1) an error score in predicting someone’s most likely position (in a table) using relatively little information; with: 2) the error score after collecting more information. In the case of Lambda we compare the error made when we only have knowledge of the marginal with the reduced error after we have collected information regarding the inside of the table. Two Lambda’s are produced. First, how much better are we able to predict someone’s position on the column marginal: Lambda A. Second, how much better are we able to predict someone’s position on the row marginal: Lambda B. The program gives the proportional improvement in predicting someone’s score after collecting additional information. To guess the name of a man on the basis of the weighted table below, and the only information we have is the distribution of mans' names in the sample, we would guess John, with a (38+34)/113*100=63.7% chance of an erroneous guess. However, if we know the name of mans' partners, we would guess John if the partner is Liz, with a (10+8)/41*100=43.9% chance of an error, Peter if the partner is Mary (44.7% errors), Steve if the partner is Linda (58.8%). The average reduction in errors in the row marginal, weighted by cell size (Lambda-B), equals 23.6%, the average weighted error rate in guessing a man's name after knowing the women's name, equals 63.7*(1-0.236)=48.7%. This 48.7% can also be calculated as: (10+8+6+11+8+12)/113. With a p-value of 0.00668 we significantly improve our probability of guessing a man's name correctly, after considering the name of the man's partner. Same for guessing a woman's name, only now you have to use the Lambda–A. Lambda is always positive, and the significance test always single sided, because information on the inside of the table will always lead to an improvement compared with knowing only the marginal. Cohen's Kappa is a measure of agreement and takes on the value zero when there are no more cells on the diagonal of an agreement table than can be expected on the basis of chance. Kappa takes on the value 1 if there is perfect agreement, i.e. if all observations are on the diagonal and a row score is perfectly predictive of a column score. It is considered that Kappa values lower than 0.4 represent poor agreement between row and column variable, values between 0.4 and 0.75 fair to good agreement, and values higher than 0.75 excellent agreement. Kappa only works in square tables. Bowker Chi-square tests to see if there is a difference in the scoring pattern between the upper and the lower triangle (excluding the diagonal) of a table. Each cell in the upper triangle is compared with its mirror in the lower triangle, the difference between the two cells is Chi-squared and summed. If cell i,j equals cell j,i the contribution of this comparison to the Bowker Chi square is zero. If the Bowker Chi-square is statistically significant the pattern of scoring in the upper triangle is different from the scoring in the lower triangle beyond chance. Note that the pattern of scoring between the two triangles is dependent on two factors. First, whether there is a 'true' difference in the pattern of scoring. Second, the level of marginal heterogeneity. Marginal heterogeneity means that the marginals are different; this increases the Bowker Chi-square. The Bowker Chi-square is the same as the McNemar Chi-square in a two by two table. Bowker Chi-square only works in square tables. For Read weights a third column is added in the data input field and the third value is the case weight of the previous two values. The case weights in the third column must be numerical, if not the case including its previous two values is ignored. Weighted cross tables are produced and a weighting corrected Chi-square is presented . For a discussion of data weighting and the correction applied please read this paper. Lowercase All. Lowercase all non numerical text characters for both the table rows and columns. Use this option if you want to categorize text data case insensitive. Transpose/Turn Table. Change the rows into columns and the columns into rows. Sort Descending. Sorts the values descending. Separate for rows and columns. Show Rows or Columns limits the number of rows displayed. Particularly relevant if you request a large Table. Can also be used to exclude particularly high or low (after "Sort Descending") (missing) values from the analysis. Solve problems into 99999.9. Change the data sequence -carriage return-line feed-tab- and the sequence -tab-carriage return-line feed- into 99999.9 if labels or delete the case if weigths. Wil mostly solve the problem of system missing values in data copied and pasted from SPSS. Might cause other problems. If you copy and paste the following data into the input field: You get the following table: |Table of Counts| Pearson: 3.111 (p= 0.53941). There is no statistically significant relationship between between boys names and girls names, although this conclusion has to be viewed with care as the table is based on very few observations. You could count (in a flat table) how often each of the pairs of names occurs in a sample, and weigh each of the pairs with these counts. john linda 10 john liz 23 john mary 8 peter linda 6 peter liz 11 peter mary 21 steve linda 14 steve liz 8 steve mary 12 And you get the following table: |Table of Weights| Weighted Pearson: 17.77 (p= 0.00137). After considering how often pairs of names occur in a sample there is a highly significant relation between certain boys and certain girls names. The formatting and tabulating of large data sets might take a while in which case there might be warnings, just select "continue" and in the end the computer will get there. The procedure is meant for relatively small tables. Number of cells is in principle limited to 120, but might be less dependent on your browser and other settings. Is also rather less with weighted data as more info has to be transferred. All software and text copyright by SISA

Present Value and Future Value – Explanation of the Concept: - Understand present value concepts and the use of present value tables. - Compute the present value of a single sum and a series of cash flows. A dollar received now is more valuable than a dollar received a year from now for the simple reason that if you have a dollar today, you can put it in the bank an have more than a dollar a year from now. Since dollars today are worth more than dollars in the future, we need some means of weighing cash flows that are received at different times so that they can be compared. Mathematics provides us with the means of making such comparisons. With a few simple calculations, we can adjust the value of a dollar received any number of years from now so that it can be compared with the value of a dollar in hand today. The Mathematics of Interest: If a bank pays 5% interest, than a deposit of $100 today will be worth $105 one year from now. This can be expressed in mathematical terms by means of the following formula or equation: Formula or Equation: F1 = P ( 1 + r ) Where: F1 = the balance at the end of one period, P = the amount invested now, and r = the rate of interest per period. If the investment made now is $100 deposited in a bank saving account that is to earn interest at 5%, than P = $100 and r = 0.05. Under these conditions, F1 = $105, the amount to be received in one year. The $100 present outlay is called the present value of the $105 amount to be received in one year. It is also known as the discounted value of the future $105 receipt. The $100 figure represents the value in present terms of $105 to be received a year from now when the interest rate is 5%. Compound Interest: When if the $105 is left in the bank for a second year? In that case, by the end of the second year the original $100 deposit will have grown to $110.25: |Interest for the first year ($100 × 0.05) |Balance at the end of the first year |Interest for the second year ($105 × 0.05) |Balance at the end of the second year Notice that the interest for the second year is $5.25, as compared to only $5.00 for the first year. The reason for the greater interest earned during the second year is that during second, interest is being paid on interest. That is, the $5.00 interest earned during the first year has been left in the account and has been added to the original $100 deposit when computing interest for the second year. This is known as the compound interest. In this case, the compound is annual. Interest compounded on a semiannual, quarterly, monthly, or even more frequent basis. The more frequently compounding is done, the more rapidly the balance will grow. We can determine the balance in an account after n periods of compounding using the following formula or equation: Fn = P (1 = r)n (1) Where n = number of periods. If n = 2 years and the interest rate is 5% per year, then the balance in two years will be as follows: F2 = $100 ( 1 + 0.05 )2 F2 = $110.25 Computation of Present Value: An investment can be viewed in two ways. It can be viewed either in terms of its future value or in terms of its present value. We have seen from our computations above that if we know the present value of a sum (such as $100 deposit), it is a relatively simple task to compute the sum’s future value in n years by using equation Fn = P (1 = r)n. But what if the the tables are reversed and we know the future value of some amount but we do not know its present value? For example, assume that you are to receive $200 two years from now. You know that the future value of this sum is $200, since this is the amount that you will be receiving after two years. But what is the sum’s present value – what is it worth right now? The present value of any sum to be received in the future can be computed by turning equation Fn = P (1 = r)n. around and solving for P: P = Fn / ( 1 + r )n (2) In our example, F = $200 (the amount to be received in future), r = 0.05 (the annual rate of interest), and n=2 (the number of years in the future that the amount is to be received) P = $200 / (1 + 0.05)n P = $200 / (1 + 0.05)2 P = $200 / 1.1025 P = $181.40 As shown by the computation above, the present value of a $200 amount to be received two years from now is $181.40 if the interest rate is 5%. In effect, $181.40 received right now is equivalent to $200 received two years from now if the rate of return is 5%. The $181.40 and the $200 are just two ways of looking at the same thing. The process of finding the present value of a future cash flow, which we have just completed, is called discounting. We have discounted the $200 to its present value of $181.40 The 5% interest figure that we have used to find this present value is called the discount rate. Discounting future sums to their present value is a common practice in business, particularly in capital budgeting decisions. If you have a power key (yx) on your calculator, the above calculations are fairly easy. However, some of the present value formulas will be using are more complex and difficult to use. Fortunately, tables are available in which many of the calculations have already been done for you. For example, Table 3 at Future Value and Present Value Tables page shows the discounted present value of $1 to be received two periods from now at 5% is 0.907. Since in our example we want to know the present value of $200 rather than just $1, we need multiply the factor in the table by $200: $200 × 0.907 = $181.40 This answer is the same as we obtained earlier using the formula in equation (2). Present Value of a Series of Cash Flow: Although some investments involve a single sum to be received (or paid) at a single point in the future, other investments involve a series of cash flows. A series (or stream) of identical cash flows is known as an annuity. To provide an example, assume that a firm has just purchased some government bonds in order to temporarily invest funds that are being held for future plant expansion. The bonds will yield interest of $15,000 each year and will be held for five years. What is the present value of the stream in interest receipts from the bonds? As shown from the following calculations the present value of this stream is $54,075 if we assume a discount rate of 12% compounded annually. ||Factor at 12% (Future Value and Present Value Tables-Table 3) | Interest Received || Present Value The discount factors used in this calculation have been taken from Future Value and Present Value Table – Table 3. Two points are important in connection with this computation. . First, notice that the present value of the $15,000 received a year from now is $13,395, as compared to only $8,505 for the $15,000 interest payment to be received five years from now. This point simply underscores the fact that money has a time value. The second point is that the computations involved above involve unnecessary work. The same present value of $54,075 could have been obtained more easily by referring to Table 4 at Future Value and Present Value Table. Table 4 contains the present value of $1 to be received each year over a series of years at various interest rates. This table have been derived by simply adding together the factor from Table 3 as follows: The sum of the five factors above is 3.065. Notice from the Table 4 at Future Value and Present Value Tables Page that the factor of $1 to be received each year for five years at 12% is also 3.605. If we use this factor and multiply it by the $15,000 annual cash inflow, then we get the same $54075 present value that we obtained earlier. $15,000 × 3.605 = $54,075 Therefore, when computing the present value of a series (or stream) of equal cash flows that begins at the end of period 1, Table 4 should be used. To summarize, the the present value tables, at Future Value and Present Value Tables Page, should be used as follows: Table 3: This table should be used to find the present value of a single cash flow (such as a single payment or receipt) occurring in future. Table 4: This table should be used to find the present value of a series (or stream) of identical cash flows beginning at the end of the current period and continuing into the future. You may also be interested in other articles from “capital budgeting decisions” chapter: - Capital Budgeting – Definition and Explanation - Typical Capital Budgeting Decisions - Time Value of Money - Screening and Preference Decisions - Present Value and Future Value – Explanation of the Concept - Net Present Value (NPV) Method in Capital Budgeting Decisions - Internal Rate of Return (IRR) Method – Definition and Explanation - Net Present Value (NPV) Method Vs Internal Rate of Return (IRR) Method - Net Present Value (NPV) Method – Comparing the Competing Investment Projects - Least Cost Decisions - Capital Budgeting Decisions With Uncertain Cash Flows - Ranking Investment Projects - Payback Period Method for Capital Budgeting Decisions - Simple rate of Return Method Post Audit of Investment Projects - Inflation and Capital Budgeting Analysis - Income Taxes in Capital Budgeting Decisions - Review Problem 1: Basic Present Value Computations - Review Problem 2: Comparison of Capital Budgeting Methods - Future Value and Present Value Tables

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Recent Theory and Applications on Inverse Problems 2014View this Special Issue Research Article | Open Access Chengzhi Deng, Shengqian Wang, Wei Tian, Zhaoming Wu, Saifeng Hu, "Approximate Sparsity and Nonlocal Total Variation Based Compressive MR Image Reconstruction", Mathematical Problems in Engineering, vol. 2014, Article ID 137616, 13 pages, 2014. https://doi.org/10.1155/2014/137616 Approximate Sparsity and Nonlocal Total Variation Based Compressive MR Image Reconstruction Recent developments in compressive sensing (CS) show that it is possible to accurately reconstruct the magnetic resonance (MR) image from undersampled -space data by solving nonsmooth convex optimization problems, which therefore significantly reduce the scanning time. In this paper, we propose a new MR image reconstruction method based on a compound regularization model associated with the nonlocal total variation (NLTV) and the wavelet approximate sparsity. Nonlocal total variation can restore periodic textures and local geometric information better than total variation. The wavelet approximate sparsity achieves more accurate sparse reconstruction than fixed wavelet and norm. Furthermore, a variable splitting and augmented Lagrangian algorithm is presented to solve the proposed minimization problem. Experimental results on MR image reconstruction demonstrate that the proposed method outperforms many existing MR image reconstruction methods both in quantitative and in visual quality assessment. Magnetic resonance imaging (MRI) is a noninvasive and nonionizing imaging processing. Due to its noninvasive manner and intuitive visualization of both anatomical structure and physiological function, MRI has been widely applied in clinical diagnosis. Imaging speed is important in many MRI applications. However, both scanning and reconstruction speed of MRI will affect the quality of reconstructed image. In spite of advances in hardware and pulse sequences, the speed, at which the data can be collected in MRI, is fundamentally limited by physical and physiological constraints. Therefore many researchers are seeking methods to reduce the amount of acquired data without degrading the image quality [1–3]. In recent years, the compressive sensing (CS) framework has been successfully used to reconstruct MR images from highly undersampled -space data [4–9]. According to CS theory [10, 11], signals/images can be accurately recovered by using significantly fewer measurements than the number of unknowns or than mandated by traditional Nyquist sampling. MR image acquisition can be looked at as a special case of CS where the sampled linear combinations are simply individual Fourier coefficients (-space samples). Therefore, CS is claimed to be able to make accurate reconstructions from a small subset of -space data. In compressive sensing MRI (CSMRI), we can reconstruct a MR image with good quality from only a small number of measurements. Therefore, the application of CS to MRI has potential for significant scan time reductions, with benefits for patients and health care economics. Because of the ill-posed nature of the CSMRI reconstruction problem, regularization terms are required for a reasonable solution. In existing CSMRI models, the most popular regularizers are , sparsity [4, 9, 12] and total variation (TV) [3, 13]. The sparsity regularized CSMRI model can be understood as a penalized least square with norm penalty. It is well known that the complexity of this model is proportional with the number of variables. Particularly when the number is large, solving the model generally is intractable. The regularization problem can be transformed into an equivalent convex quadratic optimization problem and, therefore, can be very efficiently solved. And under some conditions, the resultant solution of regularization coincides with one of the solutions of regularization . Nevertheless, while regularization provides the best convex approximation to regularization and it is computationally efficient, the regularization often introduces extra bias in estimation and cannot reconstruct an image with the least measurements when applied to CSMRI . In recent years, the regularization [16, 17] was introduced into CSMRI, since regularization can assuredly generate much sparser solutions than regularization. Although the regularizations achieve better performance, they always fall into local minima. Moreover, which should yield a best result is also a problem. Trzasko and Manduca proposed a CSMRI paradigm based on homotopic approximation of the quasinorm. Although this method has no guarantee of achieving a global minimum, it achieves accurate MR image reconstructions at higher undersampling rates than regularization. And it was faster than those regularization methods. Recently, Chen and Huang accelerated MRI by introducing the wavelet tree structural sparsity into the CSMRI. Despite high effectiveness in CSMRI recovery, sparsity and TV regularizers often suffer from undesirable visual artifacts and staircase effects. To overcome those drawbacks, some hybrid sparsity and TV regularization methods [5–8] are proposed. In , Huang et al. proposed a new optimization algorithm for MR image reconstruction method, named fast composite splitting algorithm (FCSA), which is based on the combination of variable and operator splitting techniques. In , Yang et al. proposed a variable splitting method (RecPF) to solve hybrid sparsity and TV regularized MR image reconstruction optimization problem. Ma et al. proposed an operator splitting algorithm (TVCMRI) for MR reconstruction. In order to deal with the problem of low and high frequency coefficients measurement, Zhang et al. proposed a new so-called TVWL2-L1 model which measures low frequency coefficients and high frequency coefficients with norm and norm. In , an experimental study on the choice of CSMRI regularizations was given. Although the classical TV regularization performs well in CSMRI reconstruction while preserving edges, especially for cartoon-like MR images, it is well known that TV regularization is not suitable for images with fine details and it often tends to oversmooth image details and textures. Nonlocal TV regularization extends the classical TV regularization by nonlocal means filter and has been shown to outperform the TV in several inverse problems such as image deonising , deconvolution , and compressive sensing [24, 25]. In order to improve the signal-to-noise ratio and preserve the fine details of MR images, Gopi et al. , Huang and Yang , and Liang et al. have proposed nonlocal TV regularization based MR reconstruction and sensitivity encoding reconstruction. In this paper, we proposed a novel compound regularization based compressive MR image reconstruction method, which exploits the nonlocal total variation (NLTV) and the approximate sparsity prior. The approximate sparsity, which is used to replace the traditional regularizer and regularizer of compressive MR image reconstruction model, is sparser and much easier to be solved. The NLTV is much better than TV for preserving the sharp edges and meanwhile recovering the local structure details. In order to compound regularization model, we develop an alternative iterative scheme by using the variable splitting and augmented Lagrangian algorithm. Experimental results show that the proposed method can effectively improve the quality of MR image reconstruction. The rest of the paper is organized as follows. In Section 2 we review the compressive sensing and MRI reconstruction. In Section 3 we propose our model and algorithm. The experimental results and conclusions will be shown in Sections 4 and 5, respectively. 2. Compressive Sensing and MRI Reconstruction Compressive sensing [10, 11], as a new sampling and compression theory, is able to reconstruct an unknown signal from a very limited number of samples. It provides a firm theoretical foundation for the accurate reconstruction of MRI from highly undersampled -space measurements and significantly reduces the MRI scan duration. Suppose is a MR image and is a partial Fourier transform; then the sampling measurement of MR image in -space can be defined as The compressive MR image reconstruction problem is to recover given the measurement and the sampling matrix . Undersampling occurs whenever the number of -space sample is less than the number of unknowns . In that case, the compressive MR image reconstruction is an underdetermined problem. In general, compressive sensing reconstructs the unknowns from the measurements by minimizing the norm of the sparsified image , where represents a sparsity transform for the image. In this paper, we choose the orthonormal wavelet transform as the sparsity transform for the image. Then the typical compressive MR image reconstruction is obtained by solving the following constrained optimization problem [4, 9, 12]: However, in terms of computational complexity, the norm optimization problem (2) is a typical NP-hard problem, and it was difficult to solve. According to the certain condition of the restricted isometric property, the norm can be replaced by the norm. Therefore, the optimization problem (2) is relaxed to alternative convex optimization problem as follows: When the measurements are contaminated with noise, the typical compressive MR image reconstruction problem using relaxation of the norm is formulated as the following unconstrained Lagrangian version: where is a positive parameter. Despite high effectiveness of sparsity regularized compressive MR image reconstruction methods, they often suffer from undesirable visual artifacts such as Gibbs ringing in the result. Due to its desirable ability to preserve edges, total variation (TV) model is successfully used in compressive MR image reconstruction [3, 13]. But the TV regularizer has still some limitations that restrict its performance, which cannot generate good enough results for images with many small structures and often suffers from staircase artifacts. In order to combine the advantages of sparsity-based and TV model and avoid their main drawbacks, a TV regularizer, corresponding to a finite-difference for the sparsifying transform, is typically incorporated into the sparsity regularized compressive MR image reconstruction [5–8]. In this case the optimization problem is written as where is a positive parameter. The TV was defined discretely as , where and are the horizontal and the vertical gradient operators, respectively. The compound optimization model (5) is based on the fact that the piecewise smooth MR images can be sparsely represented by the wavelet and should have small total variations. 3. Proposed Model and Algorithm As mentioned above, joint TV and norm minimization model is a useful way to reconstruct MR images. However, they have still some limitations that restrict their performance. norm needs a combinatorial search for its minimization and its too high sensibility to noise. problems can be very efficiently solved. But the solution is not sparse, which influences the performance of MRI reconstruction. The TV model can preserve edges, but it tends to flatten inhomogeneous areas, such as textures. To overcome those shortcomings, a novel method is proposed for compressive MR imaging based on the wavelet approximate sparsity and nonlocal total variation (NLTV) regularization, named WasNLTV. 3.1. Approximate Sparsity The problems of using norm in compressive MR imaging (i.e., the need for a combinatorial search for its minimization and its too high sensibility to noise) are both due to the fact that the norm of a vector is a discontinuous function of that vector. The same as [29, 30], our idea is to approximate this discontinuous function by a continuous one, named approximate sparsity function, which provides smooth measure of norm and better sparsity than regularizer. The approximate sparsity function is defined as The parameter may be used to control the accuracy with which approximate the Kronecker delta. In mathematical terms, we have Define the continuous multivariate approximate sparsity function as It is clear from (7) that is an indicator of the number of zero-entries in for small values of . Therefore, norm can be approximate by Note that the larger the value of , the smoother the and the worse the approximation to norm; the smaller the value of norm, the closer the behavior of to norm. 3.2. Nonlocal Total Variation Although the classical TV is surprisingly efficient for preserving edges, it is well known that TV is not suitable for images with fine structures, details, and textures which are very important to MR images. The NLTV is a variational extension of the nonlocal means filter proposed by Wang et al. . NLTV uses the whole image information instead of using adjacent pixel information to calculate the gradients in regularization term. The NLTV has been proven to be more efficient than TV for improving the signal-to-noise ratio, on preserving not only sharp edges, but also fine details and repetitive patterns [26–28]. In this paper, we use the NLTV to replace the TV in compound regularization based compressive MR image reconstruction. Let , , be a real function , and let be a weight function. For a given image , the weighted graph gradient is if defined as the vector of all directional derivatives at : The directional derivatives apply to all the nodes since the weight is extended to the whole domain . Let us denote vectors such that ; the nonlocal graph divergence is defined as the adjoint of the nonlocal gradient: Due to being analogous to classical TV, the norm is in general more efficient than the norm for sparse reconstruction. In this paper, we are interested in NLTV. Based on the above definition, the NLTV is defined as follows: The weight function denotes how much the difference between pixels and is penalized in the images, which is calculated by where and denote a small patch in image centering at the coordinates and , respectively. is the normalizing factor. is a filtering parameter. 3.3. The Description of Proposed Model and Algorithm According to the compressive MR image reconstruction models described in Section 2, the proposed WasNLTV model for compressive MR image reconstruction is It should be noted that the optimization problem in (14), although convex, is very hard to solve owing to nonsmooth terms and its huge dimensionality. To solve the problem in (14), we use the variable splitting and augmented Lagrangian algorithm following closely the methodology introduced in . The core idea is to introduce a set of new variables per regularizer and then exploit the alternating direction method of multipliers (ADMM) to solve the resulting constrained optimization problems. By introducing an intermediate variable vector , the problem (14) can be transformed into an equivalent one; that is, The optimization problem (15) can be written in a compact form as follows: where The augmented Lagrangian of problem (16) is where is a positive constant, , and denotes the Lagrangian multipliers associated to the constraint . The basic idea of the augmented Lagrangian method is to seek a saddle point of , which is also the solution of problem (16). By using ADMM algorithm, we solve the problem (16) by iteratively solving the following problems: It is evident that the minimization problem (19) is still hard to solve efficiently in a direct way, since it involves a nonseparable quadratic term and nondifferentiability terms. To solve this problem, a quite useful ADMM algorithm is employed, which alternatively minimizes one variable while fixing the other variables. By using ADMM, the problem (19) can be solved by the following four subproblems with respect to and . (1) subproblem: by fixing and , the optimization problem (19) to be solved is Due to the computational complexity of NLTV, the same as , the NLTV regularization in this paper only runs one time. (2) subproblem: by fixing , , , and , the optimization problem (19) to be solved is Clearly, the problem (24) is a quadratic function; its solution is simply (3) subproblem: by fixing , , , and , the optimization problem (19) to be solved is The same as problem (24), the problem (26) is a quadratic function and its gradient is simplified as . The steepest descent method is desirable to use to solve (26) iteratively by applying (4) subproblem: by fixing , , , and , the optimization problem (19) to be solved is Problem (28) is a norm regularized optimization problem. Its solution is the well-known soft threshold : where denotes the component-wise application of soft-threshold function. 4. Experimental Results In this section, a series of experiments on four 2D MR images (named brain, chest, artery, and cardiac) are implemented to evaluate the proposed and existing methods. Figure 1 shows the test images. All experiments are conducted on a PC with an Intel Core i7-3520M, 2.90 GHz CPU, in MATLAB environment. The proposed method (named WasNLTV) is compared with the existing methods including TVCMRI , RecPF , and FCSA . We evaluate the performance of various methods both visually and qualitatively in signal-to-noise ratio (SNR) and root-mean-square error (RMSE) values. The SNR and RMSE are defined as where and denote the original image and the reconstructed image, respectively, and is the mean function. For fair comparisons, experiment uses the same observation methods with TVCMRI. In the -space, we randomly obtain more samples in low frequencies and fewer samples in higher frequencies. This sampling scheme is widely used for compressed MR image reconstructions. Suppose a MR image has pixels and the partial Fourier transform in problem (1) consists of rows of matrix corresponding to the full 2D discrete Fourier transform. The chosen rows correspond to the sampling measurements . Therefore, the sampling ratio is defined as . In the experiments, the Gaussian white noise generated by in MATLAB is added, where standard deviation . The regularization parameters , , and are set as 0.001, 0.035, and 1, respectively. To be fair to compare the reconstruction MR images of various algorithms, all methods run 50 iterations and the Rice wavelet toolbox is used as the wavelet transform. Table 1 summarizes the average reconstruction accuracy obtained by using different methods at different sampling ratios on the set of test images. From Table 1, it can be seen that the proposed WasNLTV method attains the highest SNR (dB) in all cases. Figure 2 plots the SNR values with sampling ratios for different images. It can also be seen that the WasNLTV method achieves the larger improvement of SNR values. Table 2 gives the RMSE results of reconstructed MRI after applying different algorithms. From Table 2, it can be seen that WasNLTV method attains the lowest RMSE in all cases. As is known, the lower the RMSE is, the better the reconstructed image is. That is to say the MR images reconstructed by WasNLTV have the best visual quality. To illustrate visual quality, reconstructed compressive MR images obtained using different methods with sampling ratios 20% are shown in Figures 3, 4, 5, and 6. For better visual comparison, we zoom in a small patch where the edge and texture are much more abundant. From the figures, it can be observed that the WasNLTV always obtains the best visual effects on all MR images. In particular, the edge of organs and tissues obtained by WasNLTV are much more clear and easy to identify. (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV Figure 7 gives the performance comparisons between different methods with sampling ratios 20% in terms of the CPU time over the SNR. In general, the computational complexity of NLTV is much higher than TV. In order to reduce the computational complexity of WasNLTV, in the experiment, we perform the NLTV regularization once in some iterations. Despite the higher computational complexity of WasNLTV, the WasNLTV obtains the best reconstruction results on all MR images by achieving the highest SNR in less CPU time. In this paper, we propose a new compound regularization based compressive sensing MRI reconstruction model, which exploits the NLTV regularization and wavelet approximate sparsity prior. The approximate sparsity prior is used in compressive MR image reconstruction model instead of or norm, which can produce much sparser results. And the optimization problem is much easier to be solved. Because the NLTV takes advantage of the redundancy and self-similarity in a MR image, it can effectively avoid blocky artifacts caused by traditional TV regularization and keep fine edge of organs and tissues. As for the algorithm, we apply the variable splitting and augmented Lagrangian algorithm to solve the compound regularization minimization problem. Experiments on test images demonstrate that the proposed method leads to high SNR measure and more importantly preserves the details and edges of MR images. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. The authors would like to thank the anonymous referees for their valuable and helpful comments. The work was supported by the National Natural Science Foundation of China under Grants 61162022 and 61362036, the Natural Science Foundation of Jiangxi China under Grant 20132BAB201021, the Jiangxi Science and Technology Research Development Project of China under Grant KJLD12098, and the Jiangxi Science and Technology Research Project of Education Department of China under Grant GJJ12632. - K. P. Pruessmann, “Encoding and reconstruction in parallel MRI,” NMR in Biomedicine, vol. 19, no. 3, pp. 288–299, 2006. - B. Sharif, J. A. Derbyshire, A. Z. Faranesh, and Y. Bresler, “Patient-adaptive reconstruction and acquisition in dynamic imaging with sensitivity encoding (PARADISE),” Magnetic Resonance in Medicine, vol. 64, no. 2, pp. 501–513, 2010. - M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1182–1195, 2007. - D. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Magazine, vol. 2, pp. 72–82, 2008. - J. Huang, S. Zhang, and D. Metaxas, “Efficient MR image reconstruction for compressed MR imaging,” Medical Image Analysis, vol. 15, no. 5, pp. 670–679, 2011. - Z. Zhang, Y. Shi, W. P. Ding, and B. C. Yin, “MR images reconstruction based on TVWL2-L1 model,” Journal of Visual Communication and Image Representation, vol. 2, pp. 187–195, 2013. - A. Majumdar and R. K. Ward, “On the choice of compressed sensing priors and sparsifying transforms for MR image reconstruction: An experimental study,” Signal Processing: Image Communication, vol. 27, no. 9, pp. 1035–1048, 2012. - J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE Journal on Selected Topics in Signal Processing, vol. 4, no. 2, pp. 288–297, 2010. - S. Ravishankar and Y. Bresler, “MR image reconstruction from highly undersampled k-space data by dictionary learning,” IEEE Transactions on Medical Imaging, vol. 30, no. 5, pp. 1028–1041, 2011. - E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006. - D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. - X. Qu, X. Cao, D. Guo, C. Hu, and Z. Chen, “Combined sparsifying transforms for compressed sensing MRI,” Electronics Letters, vol. 46, no. 2, pp. 121–123, 2010. - F. Knoll, K. Bredies, T. Pock, and R. Stollberger, “Second order total generalized variation (TGV) for MRI,” Magnetic Resonance in Medicine, vol. 65, no. 2, pp. 480–491, 2011. - D. Donoho and J. Tanner, “Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing,” Philosophical Transactions of the Royal Society of London A, vol. 367, no. 1906, pp. 4273–4293, 2009. - Z. B. Xu, X. Y. Chang, and F. M. Xu, “L-1/2 regularization: a thresholding representation theory and a fast solver,” IEEE Transactions on Neural Networks and Learning Systems, vol. 7, pp. 1013–1027, 2012. - R. Chartrand, “Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data,” in Proceedings of the 6th IEEE International Conference on Biomedical Imaging: From Nano to Macro (ISBI '09), pp. 262–265, July 2009. - C. Y. Jong, S. Tak, Y. Han, and W. P. Hyun, “Projection reconstruction MR imaging using FOCUSS,” Magnetic Resonance in Medicine, vol. 57, no. 4, pp. 764–775, 2007. - J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Transactions on Medical Imaging, vol. 28, no. 1, pp. 106–121, 2009. - C. Chen and J. Z. Huang, “The benefit of tree sparsity in accelerated MRI,” Medical Image Analysis, vol. 18, pp. 834–842, 2014. - S. Q. Ma, W. T. Yin, Y. Zhang, and A. Chakraborty, “An efficient algorithm for compressed MR imaging using total variation and wavelets,” in Proceeding of the 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR '08, pp. 1–8, Anchorage, Alaska, USA, June 2008. - A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 490–530, 2005. - F. F. Dong, H. L. Zhang, and D. X. Kong, “Nonlocal total variation models for multiplicative noise removal using split Bregman iteration,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 939–954, 2012. - S. Yun and H. Woo, “Linearized proximal alternating minimization algorithm for motion deblurring by nonlocal regularization,” Pattern Recognition, vol. 44, no. 6, pp. 1312–1326, 2011. - X. Zhang, M. Burger, and X. Bresson, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp. 253–276, 2011. - W. Dong, X. Yang, and G. Shi, “Compressive sensing via reweighted TV and nonlocal sparsity regularisation,” Electronics Letters, vol. 49, no. 3, pp. 184–186, 2013. - V. P. Gopi, P. Palanisamy, and K. A. Wahid, “MR image reconstruction based on iterative split Bregman algorithm and nonlocal total variation,” Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 985819, 16 pages, 2013. - J. Huang and F. Yang, “Compressed magnetic resonance imaging based on wavelet sparsity and nonlocal total variation,” in Proceedings of the 9th IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI '12), pp. 968–971, Barcelona, Spain, May 2012. - D. Liang, H. F. Wang, Y. C. Chang, and L. L. Ying, “Sensitivity encoding reconstruction with nonlocal total variation regularization,” Magnetic Resonance in Medicine, vol. 65, no. 5, pp. 1384–1392, 2011. - H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “A fast approach for overcomplete sparse decomposition based on smoothed L0 norm,” IEEE Transactions on Signal Processing, vol. 57, pp. 289–301, 2009. - J.-H. Wang, Z.-T. Huang, Y.-Y. Zhou, and F.-H. Wang, “Robust sparse recovery based on approximate norm,” Acta Electronica Sinica, vol. 40, no. 6, pp. 1185–1189, 2012. - M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Transactions on Image Processing, vol. 20, no. 3, pp. 681–695, 2011. - P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Modeling & Simulation, vol. 4, pp. 1168–1200, 2005. Copyright © 2014 Chengzhi Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A common topic among ham radio operators is about power lost due to high VSWR when feeding an untuned antenna. A very frequent explanation about why this should (or should not) be a concern, is more or less like this: The power generated by the transmitter enters the coaxial cable and runs towards the antenna. When it reaches the load (the antenna) it encounter a mismatch; due to this mismatch, some power is transferred to the antenna, while the rest is reflected back and therefore lost. A tuner can be added between the transceiver and the line, but it will just “fool” the transceiver to believe the load is 50Ω: nevertheless the mismatch is still there with all of its consequent losses. The amount of reflected (thus supposedly lost) power is directly related to VSWR and usually quantified in tables like this: The Mismatch Loss in dB is calculated with the formula below: For example, with VSWR=5.85, according to this approach, more than 50% of the power should be lost (-3.021 dB). Where does the energy go? Many sources do not even bother to consider where the “lost power” is supposed to go: simply, it disappears. However we all learned in our high school Physics class that energy can not disappear into nothing. Some more advanced sources, instead, explain that the reflected power runs back into the transmission line until it bangs against the transmitter, whose internal resistance dissipates it. And if it bangs too hard, it can destroy the transmitter, like a train crashing into a wall. According to this theory, the complete process should be: - energy leaves the transmitter and enters the coaxial cable; - while running in the transmission line, some energy is dissipated as heat (all hams are aware of the dBs lost for every 100m/100ft at a given frequency of their favorite coaxial cables); - the surviving energy hits the mismatch point, where the high-VSWR antenna is connected to the coax; - given a VSWR value, a fixed percentage of energy goes to the antenna, while the remaining is “sent back” on the same coax; - the returning energy runs back on the cable and gets dissipated again by the same cable attenuation that it met on its forward run; - finally, the remaining reflected energy hits the transmitter and it is completely dissipated by the generator internal resistance; Let us make an example. We have a cable that has 1dB of attenuation at the frequency in use and we have an antenna presenting VSWR=5.85, thus a Mismatch Loss of 3.021dB: we should expect to have 3.021dB+1dB=4.021dB attenuation, i.e. only 40W out of 100 that go on the air. But… is that true? In order to verify the theory above, I connected my function generator to channel #1 of my oscilloscope; after that, I connected 24.9m of RG-58, then channel #2 of the scope and finally the load resistor representing the antenna. This setup will allow us to see the voltage entering the line and the voltage entering the load after having traversed the entire cable. Knowing the voltage V and the complex impedance Z, we can calculate the resulting power with P=V2/Z. Thus, with this setup and the help of a VNA, we can measure the power entering the coax and the power received by the load without impedance restrictions. The difference will reveal us the real power loss. Before starting the experiments, I carefully measured this test cable with my network analyzer. It resulted having a velocity factor of 0.6636 and, at 5MHz, an attenuation of 0.823dB. Experiment 1: matched load In this experiment, the line is terminated with a 50Ω load, thus it is perfectly matched. In the picture below we can see the function generator sending a single 5MHz sine wave: As expected, we have the generated pulse (yellow) developing on the 50Ω characteristic impedance of the coaxial cable. After 124ns, the same pulse reaches the 50Ω load. Considering that light travels 300mm every 1ns, we have 124 * 300 * 0.6636 = 24686mm = 24.7m, which is fairly close (±1ns) to the measured length of 24.9m. Being R the same on both sides (i.e. 50Ω), we can calculate the power ratio squaring the ratio of peak voltages: (1.12/1.26)2=0.79, which is a loss of 1.02dB, which is the same as the VNA measure ±0.2dB. Now we can set the generator to send a continuous stream of sinewaves at 5MHz: As expected, we obtain the same pattern as before but repeated over and over: voltages and timings are absolutely identical. So far so good. Experiment 2: mismatched load In order to test the behavior of the transmission line when loaded with high VSWR, I prepared a female SMA connector with a 270Ω SMD resistor soldered on it: This load produces VSWR=5.403 and, according to the Mismatch Loss table above, a loss of 2.779dB (53% to the antenna, 47% lost). Let us now send again a single 5MHz pulse and see what happens: What we see now is something a bit different than before. The initial pulse (1) is identical as the one of experiment #1 (1.26V peak). When it arrives to the 270Ω load (2) 124ns later, the voltage is much higher (1.88V peak). Then, after 124ns, a new peak (3) appears on channel 1, the load side. Let’s see what happened. The initial pulse (1) is driven on the transmission line, that at that time appears as a 50Ω load. There should be no surprise to observe that the first pulse is always identical among all the experiments: since information can not travel at infinite speed, the generator can not know instantly that at the end of the line that there is a different load than before. Therefore, the first peak must be identical to the ones we have seen before when we had the 50Ω load – and so it is. The peak power sent by the generator in the coaxial cable is 1.26V on 50Ω (1), which makes 31.75mW. The peak then travels along the line generating heat; when reaches the other end, after 124ns, it should have lost 0.823dB: the power available at (2) should be 26.27mW. At this point the wave encounters the mismatch. The tables say that, due to VSWR=5.403, only 52.7% of this power should be delivered to the load, that is 13.85mW. If we look at the 1.88V peak on 270Ω we have 13.09mW which confirms it. We have now a remainder of 12.42mW that have not been delivered to the 270Ω load. This power is bounced back and travels the coaxial cable in the other direction, loosing again 0.823dB. The power that reaches back the generator should be 10.28mW: the value at point (3) is 0.72V @50Ω, which makes 10.37mW, again perfectly in line with expectations. At this point the returning peak (3) encounters the function generator output port which offers 50Ω, i.e. a perfect match: the returning wave heats up the 50Ω resistor inside the function generator and disappears. So far, the initial theory is perfectly confirmed: the mismatched load has consumed the exact percentage of power and the rest has been bounced back and dissipated in the generator. The power delivered to the load was expected to be attenuated of 0.823dB (cable loss) + 2.779dB (mismatch loss)=3.602dB. Using a script and the binary data downloaded from the oscilloscope, I integrated the energy contained in the driven curve (orange, 3.040429nJ) and the load curve (blue 1.313286nJ): their ratio, 0.4319, accounts to 3.646dB of attenuation, which is almost a perfect match with the expected 3.602dB! Experiment 3: mismatched load and generator This time we shall repeat the experiment 2, but instead of having a 50Ω generator, we shall use a different impedance. In order to attain it, I prepared a matching attenuator with 10.28dB of attenuation and a reverse impedance of 144.5Ω. This is like to have a generator which output impedance is not 50Ω anymore, but 144.5Ω. I increased the function generator voltage to compensate the attenuator so the same 1.26V initial peak was generated again in the transmission line. This is what happened: Here we can see a different story. The initial stimulus (1) is the same as before as predicted; it travels until it reaches the 270Ω load (2) which reacts exactly as in experiment #2, reflecting the 47.3% of the received power. However this time the power coming back finds another mismatch, the 144Ω attenuator pad (3), and it is reflected back again towards the 270Ω load (4). Then it bounces back and forth over and over until all the power is gone. As it appears clearly, this time more energy is delivered to the load, although in multiple steps. Using the energy integration method, I calculated the energy actually delivered to the 270Ω load. This time the loss is only 3.271dB: i.e. the load received 0.37dB more than before. The first cracks in the initial theory begin to appear. The initial claim is founded on a fixed relation VSWR->loss, but a very simple experiment like this shows a case where it does not work. Same identical initial wave, same line, same load, same VSWR, two different results just by changing the impedance of the generator? Experiment 4: let’s the magic begin So far we have seen with that same setup, two different generator impedances feeding exactly the same power can change the amount of power delivered to the load. The experiment above shows that the power not delivered to the load is dissipated as heat by the cable itself and by the internal resistance of the generator. We shall now execute another experiment: this time, we will repeat experiments #2 (50Ω generator, 270Ω load) and #3 (144Ω generator, 270Ω load) but feeding a continuous sine wave. In both tests, the generator is set with the identical voltage level that in the previous tests generated the 1.26V initial peak. Here they are: When feeding the circuit with a continuous sine wave, something weird seems to happen. First we note that by looking at these screenshot, there is no clue of any bouncing anymore: both tests generate a nice yellow sine wave that propagates 124ns ahead to a nice blue sine wave on the load. Even more interesting is that the peak CH1/CH2 voltages, although not identical among the two tests, hold exactly the same ratio: - 1.86/1.24 = 1.5 - 1.68/1.12 = 1.5 Unlike the single-shot tests #2 and #3, the continuously fed lines are delivering exactly the same amount of power, no matter what the generator impedance is. In other words, when the generator sends a single shot, part of the energy is bounced back and dissipated by its internal impedance. As we saw, different generator impedance, different amount of energy dissipated, different amount of energy successfully delivered to the load. But if the generator sends a continuous flow of sine waves, we experience a completely dissimilar behavior: no matter of which is the generator impedance, the very same percentage of the power that enters the coaxial cable is delivered to the load. So, what’s going on? Behavior of a transmission line Without entering into the details, we can have an hint of the reason why a transmission line fed continuously behaves differently from one that receives a single pulse from the picture below: In picture “A” we have a voltage generator Vgen with its internal resistance Rgen feeding a load made of the resistance Rload. What the generator will see is a voltage V1 and a current I1 developing on its terminals: therefore, it will see an impedance Z1=V1/I1 which, in this case, is the same as Rload. The reflected power forms a voltage wave that travels back on the line until reaching the generator. This wave is seen as a voltage generator was added at the feed point (picture “B”). If we calculate the V2 voltage and I2 current we shall see that, due to the contribution of Vload, they will not match I1 and V1 anymore. The generator will see a new impedance value Z2=V2/I2, this time not equal to Rload anymore. In other words, the reflections change the impedance of the transmission line at the feed point. The resulting effect is that the transmission line now acts as a impedance transformer. The power lost in this process is only the one dissipated by the transmission line as heat: no matter what the VSWR is, if we could have a perfect line, all the power would be transferred to the load. Whatever formula that calculates power loss using only VSWR as a parameter, like the one at the beginning, it obviously flawed. Measuring real losses So far, we have established that the Mismatch Loss formula shown at the beginning does not really tell how much power is lost due to mismatch. So, how much power do we really loose? To have an answer, I prepared another experiment of measurement of power entering and exiting a transmission line terminated with a mismatched load (the same 270Ω load). To achieve the best precision, instead of using the oscilloscope, I used a much more accurate Rohde&Schwarz RF millivoltmeter. The test cable was made of 6.22m of RG-58 terminated with SMA connectors. I made two microstrip fixtures that could host the 1GHz probe of the RF millivoltmeter, which adds about 2pF. I then made an S11 and S21 measurement of this setup, including fixtures and probe, to know the impedance values needed to calculate the power levels. At 20MHz my 6.22m test cable has a matched loss of 0.472dB. Then I set my signal generator at 20MHz and measured input and output voltage: The measured impedance at 20MHz is 18.590 -j36.952; on that impedance, a voltage of 241.5mVRMS amounts to 0.634mWRMS (-1.981dBm); the output voltage is 364.1mVRMS on 270Ω, which is 0.491mWRMS (-3.092dBm). The overall power lost in this cable at this frequency is 1.110dB, i.e. only 0.638dB more than the 0.472dB that this cable would have normally dissipated due to line attenuation. This is significantly different than the 2.779dB loss foreseen by the “Mismatch Loss” method. Calculating mismatch losses Is there a formula that allows us to estimate the loss of a mismatched transmission line? Yes, there is. You can find a complete explanation in the very interesting AC6LA’s site. These formulas require some parameters of the transmission line to be measured with a network analyzer. I measured my “Prospecta RG58” with two S11 runs (open/short) and I fed the S11 files to ZPLOT, which gave me back the nominal Zo, nominal VF, K0, K1 and K2 parameters for my line. I fed those parameters to the IZ2UUF Transmission Line calculator, which gave me the following results: The software calculated a matched loss of 0.500dB (I measured 0.472dB) and a total loss of 1.104dB (I measured 1.110dB), which makes it a stunning “perfect match” with only 0.006dB of difference! So far I got very good results comparing real and predicted loss figures up to VHF, with discrepancies of cents of dB. To test higher bands I shall do further work to cancel out the impact of measurement fixtures and probes. Adding a tuner What happens if we add a tuner between the transmitter and the transmission line, as most hams do? In order to verify this, I connected the same 6.22m RG-58 line terminated with the 270Ω load to my MFJ-949E tuner and, with the help of my network analyzer, I tuned it to reach a perfect 50Ω match including the millivoltmeter probe: Then, I connected it to the signal generator and, using the RF millivoltmeter at the feed point of the tuner as a reference, I increased the generator power to compensate the extra cable I added. With 0.4dBm set on the signal generator, I had perfect 0dBm at the perfectly tuned 50Ω tuner input. As far as the signal generator is concerned, it is feeding a perfect load. Let us see the voltage entering the line after the tuner and the voltage reaching the load: We have 301.9mV on the beginning of the line, where the impedance is 18.59-j36.952: solving the complex numbers calculation tells that my tuner is pumping on the line 0.990mW (-0.043dBm). At the end we have 0.454mV, which delivers to the 270+j0 load 0.763mW (-1.173dBm). This means that the line dissipated 1.130dB, which is almost identical to the 1.110dB measured in the previous example (difference is only 0.02dB!) and almost identical the 1.104dB calculated by the online calculator. In these measurements we see that in this case the tuner received 0dBm and produced on its output -0.043dBm, thus dissipating as little as 0.043dB of power (<1%). If we would have fed a perfectly matched 50Ω load with this 6.22m long RG58 line, we would have lost 0.472dB due to normal line attenuation. Feeding the same line with a VSWR>5 load and a tuner, we have lost 1.173dB, which means a net cost of only 0.701dB. Be aware that such a low loss in a tuner is not a general rule, since tuning other impedances could cause greater heat dissipation, but it is very common. Back to the Mismatch Loss After all the experiments above, we have established beyond all reasonable doubt that the Mismatch Loss formula shown at the beginning of the article does not indicate the power lost when feeding a mismatched antenna. So, what is it for? Let us consider these two simple circuits: Picture “A” shows a 100V voltage generator with its internal 50Ω resistance Rgen feeding a 50Ω load Rload. Using Ohm’s law, we can calculate I=V/R=Vgen/(Rgen+Rload)=1A. Given that P=I2R, we can calculate the power dissipated by the load: Pload=I2Rload=50W. The generator itself is generating P=VgenI=100W and 50W are dissipated by the internal resistance Rgen. Now we can do the same calculation on “B”, where Rload is 270Ω. We have that I = Vgen/(Rgen+Rload) = 100/(50+270)=0.3125A. Hence, the power consumed by the load is I2Rload=26.367W. The generator is generating P=VgenI=31.25W and Rgen is dissipating 4.883W. We see that in circuit A the load is receiving more power: 50W vs. 26.367W: due to the maximum power transfer theorem, we get the maximum power (in this case 50W) when Rload=Rgen. For any other value, the power going to the load will be less. The “A” condition is defined as “matched“. If we calculate the ratio of the power delivered on B and the maximum possible delivered power A, we have that 26.367 / 50 = 0.527; if we transform it in dB, we have 2.779dB which is exactly the Mismatch Loss we calculated before for the 270Ω load. The Mismatch Loss value does not tell how much power is actually lost due to other dissipated, but it represents the inability of the generator to generate power due to mismatch. Note also that the Mismatch Loss is not an index of efficiency: with matched load, we got the highest power on the load (50W) but efficiency was at 50% (100W produced, 50W used on the load). In the mismatched circuit, the generator produced 31.25W of which 26.367W were delivered to the load, holding an efficiency of 84.3%! We can see this effect on the power that the R&S SMS2 signal generator has been able to deliver into the mismatched line with or without the tuner: The difference in power between the two is 1.94dB: if we calculate the mismatch for the impedance being fed (note the reference impedance is 18.590 -j36.952 presented at the input of the line, not 270+j0 at load!), we have VSWR=4.3 and Mismatch Loss=2.13dB, again another almost perfect match to the measured values. Without the tuner, due to the mismatch, the signal generator was not able to generate the whole power it would have produced on a matched load: power is not lost, is simply not generated. That is like when a biker is pedaling with the wrong gear: great effort, little performance. The tuner adapts the impedance at the input, exactly like the biker that shifts on the right gear. Mismatch on real transceivers Note that the mismatch effect that prevented the signal generator to generate the full power is mostly due to the fact that laboratory signal generators are designed to behave as close as possible as an ideal 50Ω generator. But being an ideal 50Ω generator, as we have seen, means low efficiency. Real transmitters are indeed designed to work on a 50Ω load, but not necessarily to present back 50Ω impedance when transmitting. Modern transceivers are able to compensate some degree of mismatch by feeding different voltages/currents to make the load happy. My FT-817 sends out the same power no matter of the load: changing the load, changes the voltage but the resulting power is almost the same until the HIGH VSWR protection kicks in by cutting the power. This kind of radio can feed mismatched lines within their VSWR tolerance without suffering loss of power, thus without the need of a tuner (I have planned to write another post reporting on this). - the claim that a given VSWR values gives a fixed loss of power is a myth deriving from a misinterpretation of the concept of “Mismatch Loss”; - if all the people that published such claim would have ever measured input and output power from a mismatched transmission lines, they would have immediately realized that true figures on power loss are most of the times very distant from their forecasts; - the power lost in the transmission line is the result of a function that combines the mismatch and the normal loss of the line in matching conditions; an ideal (lossless) line would have no loss at all no matter of the VSWR; - do not assume that feedline loss due to mismatch is always low: severe mismatches, like feeding a 40m 1/2 wave dipole on the 20m band, may cause very high losses in the transmission line; - a transmission line is an impedance transformer; - unless transmitting single bursts, the impedance of the transmitter has no relevance in the calculation of the power dissipated by the transmission line; - the mismatch between the transmission line and the transmitter might prevent it to generate its maximum power but many transmitters might be able to compensate the mismatch; - a tuner is not fooling the transceiver to believe the antenna is tuned, it is simply adapting two different impedances (after all, not many hams would describe their power supplies as objects fooling the radio to believe that the 220V AC power line is actually 13.8V DC, won’t they?); - tuner is not wasting huge amounts of power as commonly believed: many times its insertion loss is negligible (tenths of dB) even with high VSWR.

Social network analysis sna is probably the best known application of graph theory for data science. Transcranial direct current stimulation tdcs is an emerging approach for improving capacity in activities of daily living adl and upper limb function after stroke. Therefore, we aim to perform pairwise comparisons of the 6 sglt2 inhibitors. In the simplest possible example of a network metaanalysis, there are two treatments of interest that have been directly compared to a common comparator, but not to each other. The netmeta package in r is based on a novel approach for network metaanalysis that follows the graph theoretical methodology. Methodology from electrical networks and graphic theory also can be used to fit network metaanalysis and is outlined in by rucker rucker 2012. Using generalized linear mixed models to evaluate inconsistency within a network metaanalysis. Sep 29, 2014 the use of network meta analysis has increased dramatically in recent years. One issue with the validity of network meta analysis is inconsistency between direct and indirect evidence within a loop formed by three treatments. Software for network metaanalysis gert van valkenhoef taipei, taiwan, 6 october 20. This network metaanalysis aims to investigate the longterm 12 months efficacy of interventions for ak. In the last decade, a new statistical methodology, namely, network metaanalysis, has been developed to address limitations in traditional pairwise metaanalysis. An assessment of this assumption and of the influence of deviations is fundamental for the validity evaluation. By combining direct and indirect information, it allows assessing all possible pairwise comparisons between treatments, even when, for some pairs of treatments, no headtohead trials are available. This is a readonly mirror of the cran r package repository. A variety of interventions are available for the treatment. In this study, a network metaanalysis was performed to synthesize existing research comparing the effects of different types of cai versus the traditional instruction ti on students learning achievement in taiwan. Dec, 2018 network plot for the network of topical antibiotics without steroids for chronically discharging ears a, comparison graph corresponding to the h xy row of h matrix b, flows f uv with respect to the x versus y network metaanalysis treatment effect are indicated along the edges, streams c and proportion contributions of each direct. Frequentist network metaanalysis using the r package. Network theory provides a set of techniques for analysing graphs complex systems network theory provides techniques for analysing structure in a system of interacting agents, represented as a network applying network theory to a system means using a graph theoretic representation what makes a problem graph like. A typical way of drawing networks, also implemented in statistical software for network meta analysis, is a circular representation, often with many crossing lines. Our aim was to give an overview of the evidence network regarding the efficacy and safety of tdcs and to estimate the effectiveness of the different. This method exploits the analogy between treatment networks and electrical networks to construct the network meta analysis model accounting for the correlated treatment effects in multiarm trials. Network metaanalysis synthesizes direct and indirect evidence in a network of trials that compare multiple interventions and has the potential to rank the competing treatments according to the studied outcome. We show how graph theoretical methods can be applied to network metaanalysis. Terminology in metaanalytic networks and electrical networks. Additionally, a network metaanalysis will be conducted to determine the comparative effectiveness of the treatments with a randomeffects model. A further development in the network metaanalysis is to use a bayesian statistical approach. Network metaanalysis, electrical networks and graph. Graphical tools for network metaanalysis in stata open. Methodology from electrical networks and graphic theory also can be used to fit network metaanalysis and is outlined in by. The netmeta package in r is based on a novel approach for network meta analysis that follows the graph theoretical methodology. Gemtc r package bayesian netmeta r package frequentist. Development of restricted mean survival time difference in network metaanalysis based on data from macnpc update. The absolute and relative effectiveness of the treatments will be provided. Limitations in the design and flaws in the conduct of studies synthesized in network metaanalysis nma reduce the confidence in the results. Advanced statistical methods to model and adjust for bias in metaanalysis version 0. Preoperative hair removal and surgical site infections. Network metaanalysis was performed with r software, version i386 3. The graphtheoretical approach for network metaanalysis uses methods that were originally developed in electrical network theory. A network metaanalysis comparing the efficacy and safety. Which software can create a network metaanalysis for free. Comparative efficacy and acceptability of 21 antidepressant drugs for the acute treatment of adults with major depressive disorder. It aims to combine information from all randomized comparisons among a set of treatments for a given. Unlike r, stata software needs to create relevant ado scripts at. We conducted a network metaanalysis using two approaches. The objective of this study is to describe the general approaches to network metaanalysis that are available for quantitative data synthesis using r software. Development of restricted mean survival time difference in network meta. Graphical tools for network metaanalysis in stata pdf. Development of restricted mean survival time difference in. However, it remains unclear what type of tdcs stimulation is most effective. Network meta analysis also known as multiple treatment comparison or mixed treatment comparison seeks to combine information from all randomised comparisons among a set of treatments for a given medical condition. Winbugs, openbugs, jags bayesian by far most used, most exible meta regression software frequentist multivariate meta analysis software frequentist e. It is used in clustering algorithms specifically kmeans. Methods from graph theory usually used in electrical networks were transferred to nma. Despite these recommendations and the recent development of software statistics. All analyses will be performed using the r software version 3. A randomeffect frequentist network meta analysis model was conducted to assess pfs. The individual patient data from the metaanalysis of chemotherapy in nasopharynx carcinoma database were used to compare all available treatments. An introduction to graph theory and network analysis with. Winbugs, a freely available bayesian software package, has been the most widely used software package to conduct network meta analyses. This package allows to estimate network metaanalysis models within a frequentist framework, with its statistical approach derived from graph theoretical methods developed for electrical networks. Purpose the role of adjuvant chemotherapy ac or induction chemotherapy ic in the treatment of locally advanced nasopharyngeal carcinoma is controversial. We run the simulation study in the freely available software r 2. Briefly, for a network of n interventions and m pairwise comparisons from direct studies a m. A practical guide to network meta analysis with examples and code in the evaluation of healthcare, rigorous methods of quantitative assessment are necessary to establish which interventions are effective and costeffective. Apr 08, 2019 the objective of this study is to describe the general approaches to network meta analysis that are available for quantitative data synthesis using r software. The use of network metaanalysis has increased dramatically in recent years. Free example apply what youve learned discussion 1. Furthermore, critical appraisal of network meta analyses conducted in winbugs can be challenging. Comparing two approaches to multiarm studies in network metaanalysis. Assumes that all interventions included in the network are equally applicable to all populations and contexts of the studies included. Network metaanalysis is a generalisation of pairwise metaanalysis that compares all pairs of treatments within a number of treatments for the same condition. Most complex systems are graphlike friendship network. Winbugs, a freely available bayesian software package, has been the most widely used software package to conduct network metaanalyses. Chaimani a, higgins jpt, mavridis d, spyridonos p, salanti g graphical tools for network metaanalysis in stata anna chaimani 0 julian p. Based thereon, we then show that graph theoretical methods that have been routinely applied to electrical networks also work well in network metaanalysis. Network metaanalysis incorporates all available evidence into a general statistical framework for comparisons of all available treatments. A network metaanalysis on the effects of information and. Network meta analysis compares multiple treatments by incorporating direct and indirect evidence into a general statistical framework. Actinic keratoses ak are common precancerous lesions of the skin due to cumulative sun exposure. Network metaanalysis, a generalization of conventional metaanalysis, allows the simultaneous synthesis of data from networks of trials. This package allows to estimate network meta analysis models within a frequentist framework, with its statistical approach derived from graph theoretical methods developed for electrical networks. Package netmeta the comprehensive r archive network. Network meta analysis for decisionmaking takes an approach to evidence synthesis that is specifically intended for decision making when there are two or more treatment alternatives being evaluated, and assumes that the purpose of every synthesis is to answer the question for this preidentified population of patients, which treatment is best. A graphical tool for locating inconsistency in network meta. A network metaanalysis of nonsmallcell lung cancer. A microsoftexcelbased tool for running and critically. Forest plots are not so easy to draw for networks multiarm trials make everything more complicated but. We illustrate the correspondence between metaanalytic networks and electrical networks, where variance corresponds to resistance, treatment effects to voltage, and weighted. Network metaanalysis, electrical networks and graph theory. Decision making around multiple alternative healthcare interventions is increasingly based on metaanalyses of a network of relevant studies, which contribute direct and indirect evidence to different treatment comparisons 1,2. A network meta analysis looks at indirect comparisons. Despite its usefulness network meta analysis is often criticized for its complexity and for being accessible only to researchers with strong statistical and computational skills. The netmeta package in r is based on a novel approach for network metaanalysis that follows the graphtheoretical methodology. We also demonstrate that the fact that some statistical software commands can remove redundant collinear variables is very useful for specifying the inconsistency parameters in a network metaanalysis involving many designs and. The authors compared egfr tkis in terms of pfs in a network meta analysis. An introduction to network metaanalysis mixed treatment. Because basic metaanalysis software such as revman does not support network metaanalysis, the statistician will have to rely on statistical software packages such as stata, r, winbugs or openbugs for analysis. A graphtheoretical approach to network metaanalysis. All statistical analyses were conducted using sas statistical software version 9. Cipriani, andrea, toshi a furukawa, georgia salanti, anna chaimani, lauren z atkinson, yusuke ogawa, stefan leucht, et al. The pubmed and embase databases and meeting abstracts were screened for relevant studies between january 2009 and november 2017. It has been found to be equivalent to the frequentist approach to network metaanalysis which is based on. A network metaanalysis comparing the efficacy and safety of antipd1 with antipdl1 in nonsmall cell lung cancer. Comparative effectiveness of sodiumglucose cotransporter. A graphical tool for locating inconsistency in network metaanalyses. Methods all randomized trials of radiotherapy rt with or without chemotherapy. A network metaanalysis of nonsmallcell lung cancer patient. Let x k k1,m denote the observed effects and v k the corresponding variances. Network metaanalysis nma is becoming increasingly popular in systematic. A graph theoretical approach to multiarmed studies in frequentist network metaanalysis. Research synthesis methods, 3, 31224 rucker g, schwarzer g 2014. Association of gleason grade with androgen deprivation. Network metaanalysis for decisionmaking statistics in. In network metaanalysis, several alternative treatments can be compared by pooling the evidence of all randomised comparisons made in different studies. It aims to combine information from all randomized comparisons among a set of treatments for a given medical condition. Depends on many factors such as but not limited to. Meta analysis is a statistical technique that allows an analyst to synthesize effect sizes from multiple primary studies. Network metaanalysis is an extension of standard pairwise metaanalysis that can be used to simultaneously compare any number of treatments. Incorporated indirect conclusions require a consistent network of treatment effects. A simulation study to compare different estimation. Automated methods to test connectedness and quantify. In estimating a network metaanalysis model using a bayesian. Based thereon, we then show that graphtheoretical methods that have been routinely applied to electrical networks also work well in network metaanalysis. To estimate meta analysis models, the opensource statistical environment r is quickly becoming a popular choice. Visualizing inconsistency in network metaanalysis by. Rucker g 2012 network metaanalysis, electrical networks and graph theory. We conducted a network meta analysis using two approaches. A network metaanalysis looks at indirect comparisons. Apr 19, 2018 graph theory concepts are used to study and model social networks, fraud patterns, power consumption patterns, virality and influence in social media. A metaanalytic graph consists of vertices treatments and edges randomized comparisons. Software for network meta analysis general purpose software. A simulation study to compare different estimation approaches for. The network metaanalysis will be conducted using the netmeta package in the r software. However the relation between a and b is only known indirectly, and a network metaanalysis looks at such indirect evidence of differences between methods and interventions using statistical method. Terminology in metaanalysis and electrical networks. Network metaanalysis for decision making will be of interest to decision makers, medical statisticians, health economists, and anyone involved in health technology assessment including the pharmaceutical industry. Graph theory concepts are used to study and model social networks, fraud patterns, power consumption patterns, virality and influence in social media. Frequentist network metaanalysis using the r package netmeta. Undertaking network metaanalyses cochrane training. The statistical theory behind network meta analysis is nevertheless complex, so we strongly encourage close collaboration between dental researchers and experienced statisticians when planning and conducting a. A network metaanalysis of nonsmallcell lung cancer patients with an activating egfr mutation. Methods from graph theory usually used in electrical networks were. Chaimani a, higgins jpt, mavridis d, spyridonos p, salanti g graphical tools for network meta analysis in stata anna chaimani 0 julian p. Often a single study will not provide the answers and it is desirable to synthesise evidence from multiple sources, usually randomised controlled trials. Background the conduction and report of network metaanalysis. Network meta analysis synthesizes direct and indirect evidence in a network of trials that compare multiple interventions and has the potential to rank the competing treatments according to the studied outcome. Statistical software has been developed to fit network meta. Requires specialist statistical expertise and software. This method exploits the analogy between treatment networks and electrical networks to construct the network metaanalysis model accounting for the correlated treatment effects in. In more detail, the resulting consistent treatment effects induced in the edges can be estimated via the moorepenrose pseudoinverse of the laplacian matrix. A network metaanalysis toolkit cochrane comparing multiple. Higgins 0 dimitris mavridis 0 panagiota spyridonos 0 georgia salanti 0 0 1 department of hygiene and epidemiology, school of medicine, university of ioannina, ioannina, greece, 2 school of social and community medicine, university of bristol. The graph theoretical approach for network meta analysis uses methods that were originally developed in electrical network theory. A primer on network metaanalysis for dental research. The winbugs software can be called from either r provided r2winbugs as an r package or stata software for network metaanalysis. This approach has been implemented in the r package netmeta rucker and schwarzer 20. Inetworks aregraphs i nodesare treatments i edgesare comparisons between treatments, based on studies ivariances combine like electrical resistancesbailey, 2007 iit is possible to apply methods from electrical network theory to network metaanalysis rucker, 2012. Cappelleri, phd, mph pfizer inc invited oral presentation at the 12th annual scientific meeting of the international society for cns clinical trials and methodology, 1618. Introduction as a new class of glucoselowering drugs, sodiumglucose cotransporter 2 sglt2 inhibitors are effective for controlling hyperglycaemia, however, the relative effectiveness and safety of 6 recently available sglt2 inhibitors have rarely been studied. In the image, a has been analyzed in relation to c and c has been analyzed in relation to b. Network meta analysis is a generalisation of pairwise meta analysis that compares all pairs of treatments within a number of treatments for the same condition. Longterm efficacy of interventions for actinic keratosis. However the relation between a and b is only known indirectly, and a network meta analysis looks at such indirect evidence of differences between methods and interventions using statistical method. Scientific collaboration network business ties in us biotechindustry genetic interaction network proteinprotein interaction networks transportation networks internet. An example was used to demonstrate how to conduct a network meta analysis and the differences between it and traditional meta analysis. The statistical methodology underpinning this technique and software tools for implementing the methods are evolving. Network metaanalysis also known as multiple treatment comparison or mixed treatment comparison seeks to combine information from all randomised comparisons among a set of treatments for a given medical condition. Network metaanalysis was performed with r software version 3. Network metaanalysis nma a statistical technique that allows comparison of multiple treatments in the same metaanalysis simultaneously has become increasingly popular in the medical literature in recent years. However, the learning curve for winbugs can be daunting, especially for new users. This method exploits the analogy between treatment networks and electrical networks to construct the network metaanalysis model accounting for the correlated treatment effects in multiarm trials.1431 1158 323 487 1220 1269 543 551 526 383 1091 210 1449 1016 367 1540 536 1053 672 703 338 185 1304 431 374 641 165 498 559 383 1050 686 1489 237 53 488 847 1003 671 209 202 1397 1349 486

Understanding the pathways whereby an intervention has an effect on an outcome is a common scientific goal. A rich body of literature provides various decompositions of the total intervention effect into pathway-specific effects. Interventional direct and indirect effects provide one such decomposition. Existing estimators of these effects are based on parametric models with confidence interval estimation facilitated via the nonparametric bootstrap. We provide theory that allows for more flexible, possibly machine learning-based, estimation techniques to be considered. In particular, we establish weak convergence results that facilitate the construction of closed-form confidence intervals and hypothesis tests and prove multiple robustness properties of the proposed estimators. Simulations show that inference based on large-sample theory has adequate small-sample performance. Our work thus provides a means of leveraging modern statistical learning techniques in estimation of interventional mediation effects. Recent advances in causal inference have provided rich frameworks for posing interesting scientific questions pertaining to the mediation of effects through specific biologic pathways (Yuan and MacKinnon , Imai et al. , Valeri and VanderWeele , Pearl , Naimi et al. , Zheng and van der Laan , VanderWeele and Tchetgen Tchetgen , among others). Foremost among these advances is the provision of model-free definitions of mediation parameters, which enables researchers to develop robust estimators of these quantities. A debate in this literature has emerged pertaining to the reliance of methodology on cross-world independence assumptions that are fundamentally untestable even in randomized controlled experiments [8,9,10]. One approach to this problem is to utilize methods that attempt to estimate bounds on effects (Robins and Richardson , Tchetgen Tchetgen and Phiri , among others). A second approach considers seeking alternative definitions of mediation parameters that do not require such cross-world assumptions (VanderWeele et al. , Rudolph et al. , among others). Rather than considering deterministic interventions on mediators (i.e., a hypothetical intervention that fixes every individual mediator to a particular value), these approaches consider stochastic interventions on mediators (i.e., hypothetical interventions where the mediator is drawn from a particular conditional distribution). In this class of approaches, that of Vansteelandt and Daniel is particularly appealing. Building on the prior work of VanderWeele et al. , the authors provide a simple decomposition of the total effect into direct effects and pathway-specific effects via multiple mediators. Interestingly, their decompositions hold even when the structural dependence between mediators is unknown. Vansteelandt and Daniel described two approaches to estimation of the effects using parametric working models for relevant nuisance parameters. In both cases, the nonparametric bootstrap was recommended for inference. A potential limitation of the proposal is that correctly specifying a parametric working model may be difficult in many settings. In these instances, we may rely on flexible estimators of nuisance parameters, for example, based on machine learning. When such techniques are employed, the nonparametric bootstrap does not generally guarantee valid inference . This fact motivates the present work, where we develop nonparametric efficiency theory for the interventional mediation effect parameters. This theory allows us to utilize frameworks for nonparametric efficient inference to develop estimators of the quantities of interest. We propose a one-step and a targeted minimum loss-based estimator and demonstrate that under suitable regularity conditions, both estimators are nonparametric efficient among the class of regular asymptotically linear estimators. The estimators also enjoy a multiple robustness property, which ensures consistency of effect estimates if at least some combinations of nuisance parameters are consistently estimated. Another benefit enjoyed by our estimators is the availability of closed-form confidence intervals and hypothesis tests. 2 Interventional effects Adopting the notation of Vansteelandt and Daniel , suppose the observed data are represented as independent copies of the random variable , where is a vector of confounders, is a binary intervention, and are mediators, and is a relevant outcome. Our developments pertain to both discrete and real-valued mediators, while without loss of generality, we assume . We assume ; that is, any subgroup defined by covariates that is observed with positive probability should have some chance of receiving both interventions. We also assume that for , the probability distribution of given has density with respect to some dominating measures and this density satisfies , where the infimum is taken over . Similarly, we assume that for all . Beyond these conditions, encodes no assumptions about ; however, the efficiency theory that we develop still holds under a model that makes assumptions about , including the possibility that this quantity is known exactly, as in a stratified randomized trial. To define interventional mediation effects, notation for counterfactual random variables is required. For , and , let denote the counterfactual value for the th mediator when is set to . Similarly, let denote the counterfactual outcome under an intervention that sets and . As a point of notation, when introducing quantities whose definition depends on particular components of the random variable , we will use lower case letters to denote the particular value and assume that the definition at hand applies for all values in the support of that random variable. The total effect of intervening to set versus is , where we use to emphasize that we are taking an expectation with respect to a distribution of a counterfactual random variable. The total effect describes the difference in counterfactual outcome considering an intervention where we set and allow the mediators to naturally assume the value that they would under intervention versus an intervention where we set and allow the mediators to vary accordingly. To contrast with forthcoming effects, it is useful to write the total effect in integral form. Specifically, we use to denote the covariate-conditional mean of the counterfactual outcome , to denote the covariate-conditional bivariate cumulative distribution function of , and to denote the marginal distribution of . The total effect can be written as The total effect can be decomposed into interventional direct and indirect effects. The interventional direct effect is the difference in average counterfactual outcome under two population-level interventions. The first intervention sets , and subsequently for individuals with draws mediators from . Thus, on a population level the covariate conditional distribution of mediators in this counterfactual world is the same as it would be in a population where everyone received intervention . This is an example of a stochastic intervention . The second intervention sets and subsequently allows the mediators to naturally assume the value that they would under intervention , so that the population level mediator distribution is again . The interventional direct effect compares the average outcome under these two interventions, For interventional indirect effects, we require definitions for the covariate-conditional distribution of each mediator, which we denote for by . The interventional indirect effect through is As with the direct effect, this effect considers two interventions. Both interventions set . The first intervention draws mediator values independently from the marginal mediator distributions and , while the second intervention draws mediator values independently from the marginal mediator distributions and . The effect thus describes the average impact of shifting the population level distribution of , while holding the population level distribution of fixed. The interventional indirect effect on the outcome through is similarly defined as Note that when defining interventional indirect effects, mediators are drawn independently from marginal mediator distributions. The final effect in the decomposition essentially describes the impact of drawing the mediators from marginal rather than joint distributions. Thus, we term this effect the covariant mediator effect, defined as where . Vansteelandt and Daniel discussed situations where these effects are of primary interest. From the aforementioned definitions, we have the following effect decomposition . These component effects can be identified using the observed data under the following assumptions: the effect of on is unconfounded given , ; the effect of and on is unconfounded given and , ; the effect of on is unconfounded given , . The identifying formula for each effect can now be written as a statistical functional of the observed data distribution by substituting the outcome regression for and the observed-data mediator distributions for the respective counterfactual distributions in the aforementioned integral expressions. We note that the aforementioned assumptions preclude the existence of treatment-induced confounding of the mediator-outcome association. In the Supplementary material, we provide relevant extensions to this setting. 3.1 Efficiency theory In this section, we develop efficiency theory for nonparametric estimation of interventional effects. This theory centers around the efficient influence function of each parameter. The efficient influence function is important for several reasons. First, it allows us to utilize two existing estimation frameworks, one-step estimation [17,18] and targeted minimum loss-based estimation [19,20], to generate estimators that are nonparametric efficient. That is, under suitable regularity conditions, they achieve the smallest asymptotic variance among all regular estimators that, when scaled by , have an asymptotic normal distribution. We discuss how these estimators can be implemented in Section 3.2. The second important feature of the efficient influence function is that its variance equals the variance of the limit distribution of the scaled estimators. Thus, an estimate of the variance of the efficient influence function is a natural standard error estimate, which affords closed-form Wald-style confidence intervals and hypothesis tests (Section 3.3). Finally, the efficient influence function also characterizes robustness properties of our proposed estimators (Section 3.4). To introduce the efficient influence function, several additional definitions are required. For a given distribution , we define , commonly referred to as a propensity score. For and , we introduce the following partially marginalized outcome regressions, . We also introduce notation for the indicator function defined by if and zero otherwise. is similarly defined. Under sampling from , the efficient influence function evaluated on a given observation for the total effect is The efficient influence function for the interventional direct effect is The efficient influence function for the interventional indirect effect through is The efficient influence function for the interventional indirect effect through is The efficient influence function for the covariant interventional effect is . A proof of Theorem 1 is provided in the Supplementary material. We propose estimators of each interventional effect using one-step and targeted minimum loss-based estimation. Both techniques develop along a similar path. We first obtain estimates of the propensity score, outcome regression, and joint mediator distribution; we collectively refer to these quantities as nuisance parameters. With estimated nuisance parameters in hand, we subsequently apply a correction based on the efficient influence function to the nuisance estimates. To estimate the propensity score, we can use any suitable technique for mean regression of the binary outcome onto confounders . Working logistic regression models are commonly used for this purpose, though semi- and nonparametric alternatives would be more in line with our choice of model. We denote by the chosen estimate of . Similarly, the outcome regression can be estimated using mean regression of the outcome onto and . For example, if the study outcome is binary, logistic regression could again be used, though more flexible regression estimators may be preferred. As above, we denote by the estimated outcome regression evaluated under , with providing an estimate of . To estimate the marginal cumulative distribution of , we will use the empirical cumulative distribution function, which we denote by . Estimation of the conditional joint distribution of the mediators is a more challenging proposition, as fewer tools are available for flexible estimation of conditional multivariate distribution functions. We hence focus our developments on the development of approaches for discrete-valued mediators. The approach we adopt could be extended to continuous-valued mediators by considering a fine partitioning of the mediator values. We examine this approach via simulation in Section 4. To develop our density estimators, we use the approach of Dìaz Muñoz and van der Laan , which considers estimation of a conditional density via estimation of discrete conditional hazards. Briefly, consider estimation of the distribution of given and , and, for simplicity, suppose that the support of is . We create a long-form data set, where the number of rows contributed by each individual contribute is equal to their observed value of . An example is illustrated in Table 1. We see that the long-form data set includes an integer-valued column named “bin” that indicates to which value of each row corresponds, as well as a binary column indicating whether the observed value of corresponds to each bin. These long-form data can be used to fit a regression of the binary outcome onto , , and bin. This naturally estimates , the conditional discrete hazard of given and . Let denote the estimated hazard obtained from fitting this regression. An estimate of the density at is Similarly, an estimate of the conditional distribution of given can be obtained. An estimate of the joint conditional density is implied by these estimates, , while an estimate of the marginal distribution of is . An ID is uniquely assigned to each independent data unit and a single confounder is included in the mock data set. In principle, one could reverse the roles of and in the above procedure. That is, we could instead estimate the distribution of given and of given . Cross-validation could be used to pick between the two potential estimators of the joint distribution. Other approaches to conditional density estimation are permitted by our procedure as well. For example, approaches based on working copula models may be particularly appealing in this context, as they allow separate specification of marginal vs joint distributions of the mediators. Given estimates of nuisance parameters, we now illustrate one-step estimation for the interventional direct effect. One-step estimators of other effects can be generated similarly. A plug-in estimate of the conditional interventional direct effect given is the difference between To obtain a plug-in estimate of , we standardize the conditional effect estimate with respect to , the empirical distribution of . Thus, the plug-in estimator of is . The one-step estimator is constructed by adding an efficient influence function-based correction to an initial plug-in estimate. Suppose we are given estimates of all relevant nuisance quantities and let denote any probability distribution in that is compatible with these estimates. The efficient influence function for under sampling from is , and the one-step estimator is . All other effect estimates are generated in this vein: estimated nuisance parameters are plugged in to the efficient influence function, the resultant function is evaluated on each observation, and the empirical average of this quantity is added to the plug-in estimator. While one-step estimators are appealing in their simplicity, the estimators may not obey bounds on the parameter space in finite samples. For example, if the study outcome is binary, then the interventional effects each represent a difference in two probabilities and thus are bounded between and 1. However, one-step estimators may fall outside of this range. This motivates estimation of these quantities using targeted minimum loss-based estimation, a framework for generating plug-in estimators. The implementation of such estimators is generally more involved than that of one-step estimators. In this approach, a second-stage model fitting is used to ensure that nuisance parameter estimates satisfy efficient influence function estimating equations. The approach for this second-stage fitting is dependent on the specific effect parameter considered and the procedure differs subtly for the various effect measures presented here. Supplementary material includes a detailed exposition of how such estimators can be implemented. 3.3 Large sample inference We now present a theorem establishing the joint weak convergence of the proposed estimators to a random variable with a multivariate normal distribution. Because the asymptotic behavior of the one-step and targeted minimum loss estimators (TMLEs) is equivalent, we present a single theorem. A discussion of the differences in regularity conditions required to prove the theorem for one-step versus targeted minimum loss estimation is provided in the Supplementary material. Let denote the vector of (one-step or targeted minimum loss) estimates of and let denote the vector of efficient influence functions defined by In the theorem, we use to denote the -norm, define for any -measurable as . Under sampling from , if for , in probability as , in probability as , , , and in probability as and falls in a -Donsker class with probability tending to 1, The regularity conditions required for Theorem 2 are typical of many problems in semiparametric efficiency theory. We provide conditions in terms of -norm convergence, as this is typical of this literature; however, alternative and potentially weaker conditions are possible to derive. For further discussion, see the supplementary material. As with any nonparametric procedure, there is a concern related to the dimensionality , particularly in situations with real-valued mediators. Minimum loss estimators (MLEs) in certain function classes can attain the requisite convergence rates. For example, an MLE in the class of functions that are right-continuous with left limits (i.e., càdlàg) with variation norm bounded by a constant achieves an convergence rate faster than irrespective of the dimension of the conditioning set . However, this may not allay all concerns pertaining to the curse of dimensionality due to the fact that in moderately high dimensions, these function classes can be restrictive and thus the true function may fall outside this class. Nevertheless, we suggest (and our simulations show) that in spite of concerns pertaining to the curse of dimensionality our procedure will enjoy reasonable finite-sample performance in many settings. The covariance matrix may be estimated by the empirical covariance matrix of the vector applied to the observed data, where is any distribution in the model that is compatible with the estimated nuisance parameters. With the estimated covariance matrix, it is straightforward to construct Wald confidence intervals and hypothesis tests about the individual interventional effects or comparisons between them. For example, a straightforward application of the delta method would allow for a test of the null hypothesis that . 3.4 Robustness properties As with many problems in causal inference, consistent estimation of interventional effects requires consistent estimation only of certain combinations of nuisance parameters. To determine these combinations, we may study the stochastic properties of the efficient influence function. In particular, consider a parameter whose value under is and whose efficient influence function under sample from can be written , where is the value of the parameter of interest under . Then we may study the circumstances under which . This generally entails understanding which parameters of must align with those parameters of to ensure that the influence function has mean zero under sampling from . We present the results of this analysis in a theorem below and refer readers to the Supplementary material for the proof. Locally efficient estimators of the total effect and the intervention direct, indirect, and covariant effects are consistent for their respective target parameters if the following combinations of nuisance parameters are consistently estimated: Total effect: or Interventional direct effect: or or ; Interventional indirect effect through : or or or ; Interventional indirect effect through : or or or ; Interventional covariant effect: or or . The most interesting robustness result is perhaps that pertaining to the indirect effects. The first condition for consistent estimation is expected, as the propensity score plays no role in the definition of the indirect effect. The second condition shows that the joint mediator distribution and propensity score together can compensate for inconsistent estimation of the outcome regression, while the relevant marginal mediator distributions are required to properly marginalize the resultant quantity. The third and fourth conditions show that inconsistent estimation of the marginal distribution one, but not both, of the mediators can be corrected for via the propensity score. We note that Theorem 3 provides sufficient, but not necessary, conditions for consistent estimation of each effect. For example, a consistent estimate of the total effect is implied by a consistent estimate of and , a condition that is generally weaker than requiring consistent estimation of the outcome regression and joint mediator distribution. Because our estimation strategy relies on estimation of the joint mediator distribution, we have described robustness properties in terms of the large sample behavior of estimators of those quantities. In the Supplementary material, we provide relevant extensions to the setting where the mediator–outcome relationship is confounded by measured covariates whose distributions are affected by the treatment. In this case, both the effects of interest and their efficient influence functions involve the conditional distribution of the confounding covariates. We discuss the relevant modifications to the estimation procedures to accommodate this setting in the supplement. Generalization to other effect scales requires only minor modifications. First, we determine the portions of the efficient influence function that pertain to each component of the additive effect. For example, considering , we identify the portions of the efficient influence function that pertain to the mean counterfactual under draws of from and of from versus those portions that pertain to the mean counterfactual under draws of from and of from . We then develop a one-step or TMLE for each of these components separately. Finally, we use the delta method to derive the resulting influence function. In the Supplementary material, we illustrate an extension to a multiplicative scale. Our results can also be extended to estimation of interventional effects for more than two mediators. As discussed in Vansteelandt and Daniel , when there are more than two mediators, say , there are many possible path-specific effects. However, our scientific interest is usually restricted to learning effects that are mediated through each of the mediators, rather than all possible path-specific effects. Moreover, strong untestable assumptions are required to infer all path-specific effects, including assumptions about the direction of the causal effects between mediators. Therefore, it may be of greatest interest to evaluate direct effects such as which describes the effect of setting versus , while drawing all mediators from the joint conditional distribution given , and for , indirect effects such as which describes the effect of setting to the value it would assume under versus while drawing from their respective marginal distributions given and drawing from their marginal distribution given . We provide relevant efficiency theory for these parameters in the Supplementary material. 4.1 Discrete mediators We evaluated the small sample performance of our estimators via Monte Carlo simulation. Data were generated as follows. We simulated by drawing independently from Uniform(0,1) distributions, and independently from Bernoulli distributions with success probability of 0.25 and 0.5, respectively. The treatment variable , given , was drawn from a Bernoulli distribution with and . Here, we consider and . Given , the first mediator was generated by taking draws from a geometric distribution with success probability . Any draw of six or greater was set equal to six. The second mediator was generated from a similarly truncated geometric distribution with success probability . Given , the outcome was drawn from a Bernoulli distribution with success probability . The mediator distribution is visualized for combinations of and in Figure 1. The true total effect is approximately 0.06, which decomposes into a direct effect of 0.05, an indirect effect through of , an indirect effect through of 0.02, and a covariant effect of 0. The nuisance parameters were estimated using regression stacking [23,24], also known as super learning using the SuperLearner package for the R language . We used this package to generate an ensemble of a main-terms logistic regression (as implemented in the SL.glm function in SuperLearner), polynomial multivariate adaptive regression splines (SL.earth), and a random forest (SL.ranger). The ensemble was built by selecting the convex combination of these three estimators that minimized tenfold cross-validated deviance. We evaluated our proposed estimators under this data generating process at sample sizes of 250, 500, 1,000, and 2,000. At each sample size, we simulated 1,000 data sets. Point estimates were compared in terms of their Monte Carlo bias, standard deviation, and mean squared error. We evaluated weak convergence by visualizing the sampling distribution of the estimators after centering at the true parameter value and scaling by an oracle standard error, computed as the Monte Carlo standard deviation of the estimates, as well as scaling by an estimated standard error based on the estimated variance of the efficient influence function. Similarly, we evaluated the coverage probability of a nominal 95% Wald-style confidence interval based on the oracle and estimated standard errors. In terms of estimation, one-step and TMLE behave as expected in large samples (Figure 2). The estimators are approximately unbiased in large samples and have mean squared error appropriately decreasing with sample size. Comparing the two estimation strategies, we see that one-step and TMLEs had comparable performance for the interventional direct effect, while the TMLE had better performance for the indirect effects. However, the one-step was uniformly better for estimating the covariant effect owing to large variability of the TMLE of this quantity. Further examination of the results revealed that the second-stage model fitting required by the targeted minimum loss approach was could be unstable in small samples, leading to extreme results in several data sets. The sampling distributions of the centered and scaled estimators were approximately a standard normal distribution (Figures 3 and 4), excepting the TMLE scaled by an estimated standard error. Confidence intervals based on an oracle standard error came close to nominal coverage in all sample sizes, while those based on an estimated standard error tended to have under-coverage in small samples. 4.2 Continuous mediators We examined the impact of discretization of the mediator distributions when in fact the mediators are continuous valued. To that end, we simulated data as follows. Covariates were simulated as above. The treatment variable given was drawn from a Bernoulli distribution with . Given , and were, respectively, drawn from normal distributions with unit variance and mean values and . As above, Super Learner was used to estimate all nuisance parameters. To accommodate appropriate modeling of the interactions, we replaced the main terms GLM (SL.glm) with a forward stepwise GLM algorithm that included all two-way interactions (SL.step.interaction). The true effect sizes were approximately the same as in the first simulation. We evaluated discretization of each continuous mediator distribution into 5 and 10 evenly spaced bins. For the sake of space, we focus results on the one-step estimator; results for TMLE are included in the supplement. Overall, discretization of the continuous mediator distribution had a greater impact on the performance of indirect effect estimators compared to direct effects (Figures 5 and 6). For the latter effects, oracle confidence intervals for both levels of discretization achieved nominal coverage for all sample sizes considered. For the indirect effects, we found that there was non-negligible bias in the estimates due to the discretization. The impacts in terms of confidence interval coverage were minimal in small sample sizes, but lead to under-coverage in larger sample sizes. Including more bins generally lead to better performance, but these estimates still exhibited bias in the largest sample sizes that impacted coverage. Nevertheless the performance of the indirect effect estimators was reasonable with oracle coverage 90% for all sample sizes. 4.3 Additional simulations In the Supplementary material, we include several additional simulations studying the impact of the number of levels of the discrete mediator, as well as the impact of inconsistent estimation of the various nuisance parameters. For the former, we found that the results of the simulation were robust to number of mediator levels in the setting considered. For the latter, we confirmed the multiple robustness properties of the indirect effect estimators by studying the bias and standard deviation of the estimators in large sample sizes under the various patterns of misspecification given in our theorem. Our simulations demonstrate adequate performance of the proposed nonparametric estimators of interventional mediation effects in settings with relatively low-dimensional covariates (five, in our simulation). In certain settings, it may only be necessary to adjust for a limited number of covariates to adequately control confounding. For example, in the study of the mediating mechanisms of preventive vaccines using data from randomized trials, we need to only adjust for confounders of the mediator/outcome relationship, since other forms of confounding are addressed by the randomized design. Generally, there are few known factors that are likely to impact vaccine-induced immune responses and so nonparametric analyses may be quite feasible in this case. For example, Cowling et al. studied mediating effects of influenza vaccines, adjusting only for age. Thus, we suggest that interventional mediation estimands and nonparametric estimators thereof may be of interest for studying mediating pathways of vaccines. However, in other scenarios, it may be necessary to adjust for a high-dimensional set of confounders. For example, in observational studies of treatments (e.g., through an electronic health records system), we may require control for a high-dimensional set of putative confounders of treatment and outcome. This may raise concerns related to the curse-of-dimensionality when utilizing nonparametric estimators. Studying tradeoffs between the selection of various estimation strategies in this context will be an important area for future research. We have developed an R package intermed with implementations of the proposed methods, which is included in the Supplementary material. The package focuses on implementations for discrete mediators. However, our simulations demonstrate a clear need to extend the software to accommodate adaptive selection of the number of bins in the mediator density estimation procedure for continuous mediators. In small sample sizes, we found that course binning leads to adequate results, but as sample size increased, unsurprisingly there was a need for finer partitioning to reduce bias. In future versions of the software, we will include such adaptive binning strategies, as well as other methods for estimating continuous mediator densities. The behavior of the TMLE of the covariant effect in the simulation is surprising as generally we see comparable or better performance of such estimators relative to one-step estimators. This can likely be attributed to the fact that the targeted minimum loss procedure does not yield a compatible plug-in estimator of the vector , in the sense that there is likely no distribution that is compatible with all of the various nuisance estimators after the second-stage model fitting. A more parsimonious approach could consider either an iterative targeting procedure or a uniformly least favorable submodel that simultaneously targets the joint mediator density and outcome regression. The former is implemented in a concurrent proposal , where one-step and TMLEs of interventional effects are developed for a single mediator when the mediator–outcome relationship is subject to treatment-induced confounding. In their set up, if one can treat the treatment-induced confounder as a second mediator, then their proposal results in an estimate of one component of our indirect effect. In their simulations, they find superior finite-sample performance of the TMLE relative to the one step, suggesting that targeting the mediator densities may be a more robust approach. However, their simulation involved only binary-valued mediators, so further comparison of these approaches is warranted in settings similar to our simulation, where mediators can take many values. We leave these developments to future work. The Donsker class assumptions of our theorem could be removed by considering cross-validated nuisance parameter estimates (also known as cross-fitting) [29,30]. This technique is implemented in our R package, but we leave to future research the examination of its impact of estimation and inference. We hypothesize that this approach will generally improve the anti-conservative confidence intervals in small samples, but will have little impact on performance of point estimates in terms of bias and variance. Funding information: D. Benkeser was funded by National Institutes of Health award R01AHL137808 and National Science Foundation award 2015540. Code to reproduce simulation results is available at https://github.com/benkeser/intermed/tree/master/simulations. Conflict of interest: Prof. David Benkeser is a member of the Editorial Board in the Journal of Causal Inference but was not involved in the review process of this article. Yuan Y, MacKinnon DP. Bayesian mediation analysis. Psychol Methods. 2009;14(4):301. Search in Google Scholar Imai K, Keele L, Tingley D. A general approach to causal mediation analysis. Psychol Methods. 2010;15(4):309. Search in Google Scholar Valeri L, VanderWeele TJ. Mediation analysis allowing for exposure-mediator interactions and causal interpretation: theoretical assumptions and implementation with SAS and SPSS macros. Psychol Methods. 2013;18(2):137. Search in Google Scholar Pearl J. Interpretation and identification of causal mediation. Psychol Methods. 2014;19(4):459. Search in Google Scholar Naimi AI, Schnitzer ME, Moodie EE, Bodnar LM. Mediation analysis for health disparities research. Am J Epidemiol. 2016;184(4):315–24. Search in Google Scholar Zheng W, van der Laan MJ. Longitudinal mediation analysis with time-varying mediators and exposures, with application to survival outcomes. J Causal Infer. 2017;5(2):20160006. Search in Google Scholar VanderWeele TJ, Tchetgen Tchetgen EJ. Mediation analysis with time varying exposures and mediators. J R Stat Soc B (Statistical Methodology). 2017;79(3):917–38. Search in Google Scholar Dawid AP. Causal inference without counterfactuals. J Am Stat Assoc. 2000;95(450):407–24. Search in Google Scholar Pearl J. Direct and indirect effects. In: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence. San Francisco: Morgan Kaufmann; 2001. Search in Google Scholar Robins JM, Richardson TS. Alternative graphical causal models and the identification of direct effects. Causality and psychopathology: Finding the determinants of disorders and their cures. Oxford, New York: Oxford University Press; 2011. p. 103–58. Search in Google Scholar Tchetgen Tchetgen EJ, Phiri K. Bounds for pure direct effect. Epidemiology (Cambridge, Mass.). 2014;25(5):775. Search in Google Scholar VanderWeele TJ, Vansteelandt S, Robins JM. Effect decomposition in the presence of an exposure-induced mediator-outcome confounder. Epidemiology. 2014;25(2):300. Search in Google Scholar Rudolph KE, Sofrygin O, Zheng W, van der Laan MJ. Robust and flexible estimation of stochastic mediation effects: a proposed method and example in a randomized trial setting. Epidemiol Methods. 2018;7(1):2017007. Search in Google Scholar Vansteelandt S, Daniel RM. Interventional effects for mediation analysis with multiple mediators. Epidemiology. 2017;28(2):258. Search in Google Scholar Coyle J, van der Laan MJ. Targeted bootstrap. In Targeted learning for data science. Cham: Springer International Publishing; 2018. p. 523–39. Ch. 28. Search in Google Scholar Muñoz ID, van der Laan MJ. Population intervention causal effects based on stochastic interventions. Biometrics. 2012;68(2):541–9. Search in Google Scholar Ibragimov I, Khasminskii R. Statistical estimation: asymptotic theory. New York: Springer-Verlag; 1981. Search in Google Scholar Bickel P, Klaassen C, Ritov Y, Wellner J. Efficient and adaptive estimation for semiparametric models. Berlin Heidelberg New York: Springer; 1997. Search in Google Scholar van der Laan M, Rubin DB. Targeted maximum likelihood learning. Int J Biostat. 2006;2(1):11. Search in Google Scholar van der Laan M, Rose S. Targeted learning: causal inference for observational and experimental data. Berlin Heidelberg New York: Springer; 2011. Search in Google Scholar Dìaz Muñoz I, van der Laan MJ. Super learner based conditional density estimation with application to marginal structural models. Int J Biostat. 2011;7(1):1–20. Search in Google Scholar Benkeser D, van der Laan MJ. The highly adaptive lasso estimator. In: 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA). IEEE; 2016. p. 689–96. Search in Google Scholar Wolpert DH. Stacked generalization. Neural Netw. 1992;5:241–59. Search in Google Scholar Breiman L. Stacked regressions. Mach Learn. 1996;24:49–64. Search in Google Scholar van der Laan M, Polley E, Hubbard A. Super learner. Stat Appl Genet Mol. 2007;6(1):25. Search in Google Scholar Polley E, LeDell E, Kennedy C, van der Laan MJ. SuperLearner: Super Learner Prediction. R package version 2.0-28; 2013. https://CRAN.R-project.org/package=SuperLearnerSearch in Google Scholar Cowling BJ, Lim WW, Perera RA, Fang VJ, Leung GM, Peiris JM, et al. Influenza hemagglutination-inhibition antibody titer as a mediator of vaccine-induced protection for influenza B. Clin Infect Dis. 2019;68(10):1713–17. Search in Google Scholar Zheng W, van der Laan MJ. Asymptotic theory for cross-validated targeted maximum likelihood estimation. Technical Report 273. Berkeley: Division of Biostatistics, University of California, Berkeley; 2010. Search in Google Scholar Chernozhukov V, Chetverikov D, Demirer M, Duflo E, Hansen C, Newey W, et al. Double/debiased machine learning for treatment and structural parameters. Econom J. 2018;21(1):C1–C68. Search in Google Scholar © 2021 David Benkeser and Jialu Ran, published by De Gruyter This work is licensed under the Creative Commons Attribution 4.0 International License.

The Duality of Time Theory, that results from the Single Monad Model of the Cosmos, explains how multiplicity is emerging from absolute Oneness, at every instance of our normal time! This leads to the Ultimate Symmetry of space and its dynamic formation and breaking into the physical and psychical (supersymmetrical) creations, in orthogonal time directions. General Relativity and Quantum Mechanics are complementary consequences of the Duality of Time Theory, and all the fundamental interactions become properties of the new granular complex-time geometry. - => Conference Talk [Detailed Presentation] Welcome to the Single Monad Model of the Cosmos ( Please log in or register!) This short book presents a brief and concise exploration of the Duality of Time postulate and its consequences on General Relativity and Quantum Mechanics. To make it easier for citing, this book is presented in the form of a scientific paper, which will also make it more accessible and easier to be read by researchers who are interested in the astounding conclusions rather than any exhausting introductions which are provided in the previous books for more general readability. This article explains: Based on the Single Monad Model and Duality-of-Time hypothesis, a dynamic and self-contained space-time is introduced and investigated. It is shown that the resulting “time-time” geometry is genuinely complex, fractal and granular, and that the non-Euclidean space-time continuum is the first global approximation of this complex-time space in which the (complex) momentum and energy become invariant between different inertial and non-inertial frames alike. Therefore, in addition to Lorentz transformation, the equivalence principle is derived directly from the new discrete symmetry. It is argued that according to this postulate, all the principles of relativity and quantum theories can be derived and become complementary. The Single Monad Model provides a profound understanding of time as a complex-scalar quantum field that is necessary and sufficient to obtain exact mathematical derivation of the mass-energy equivalence relation, in addition to solving many persisting problems in physics and cosmology, including the arrow-of-time, super-symmetry, matter-antimatter asymmetry, mass generation, homogeneity, and non-locality problems. It will be also shown that the resulting physical vacuum is a perfect super-fluid that can account for dark matter and dark energy, and diminish the cosmological constant discrepancy by at least 117 orders of magnitude. Relativity, and its classical predecessor, consider space and time to be continuous, and everywhere differentiable, whereas quantum mechanics is based on discrete quanta of energy and fields, albeit they still evolve in continuous background. Although both theories have already passed many rigorous tests, they inevitably produce enormous contradictions when applied together in the same domain. Most scholars believe that this conflict may only be resolved with a successful theory of quantum gravity . In trying to resolve the discrepancy, some space-time theories, such as Causal Dynamical Triangulation , Quantum Einstein Gravity and Scale Relativity , attempted to relax the condition of differentiability, in order to allow for fractal space-time, which was first introduced in 1983 . In addition to the abundance of all kinds of fractal structures in nature, this concept was also supported by many astronomical observations which show that the Universe exhibit a fractal aspect over a fairly wide range of scales , and that large-scale structures are much better described by a scale-dependent fractal dimension , but the theoretical implications of these observations are not very well understood, yet. Nonetheless, the two most celebrated approaches to reconcile Relativity with Quantum Mechanics are Strings Theory and Loop Quantum Gravity (LQG). The first tries to develop an effective quantum field theory of gravity at low energies, by postulating strings instead of point particles, while LQG uses spin networks to obtain granular space that evolves with time. Therefore, while Strings Theory still depends on the background continuum, LQG tries to be background-independent by attempting to quantize space-time itself . In this regard, the author believes that any successful theory of quantum gravity must not rely on either the continuum or discretuum structures of space-time. Rather, these two contrasting and mutually-exclusive views must be the product of such theory, and they must become complementary on the microscopic and macroscopic scales. The only contestant that may fulfill this criterion is “Oneness”, because on the multiplicity level things can only be either discrete or continuous; there is no other way. However, we need first to explain how the apparent physical multiplicity can proceed from this metaphysical oneness, and then exhibit various discrete and continuous impressions. The key to resolve this dilemma is in understanding the “inner levels of time” in which “space” and “matter” are perpetually being “re-created” and layered into the three spatial dimensions, which then kinetically evolve throughout the “outer level of time” that we encounter. This will be fully explained in sections 2 and 4 below. Due to this “dynamic formation of dimensions”, in the inner levels of time, the Duality of Time Theory leads to granular and self-contained space-time with fractal and genuinely-complex structure, which are the key features needed to accommodate both quantum and relativistic phenomena. Many previous studies have already shown how the principles of quantum mechanics can be derived from the fractal structure of space-time [9, 10, 11, 12, 13], but they either do not justify the use of fractals, or they are forced to make new unjustified assertions, such as the relativity of scale, that may lead to fractal space-time. On the other hand, imaginary time had been successfully used in the early formulation of Special Relativity by Poincare , and even Minkowski , but it was later replaced by the Minkowskian four-dimensional space-time, because there were no substantial reasons to treat time as imaginary. Nevertheless, this concept is still essential in current cosmology and quantum field theories, since it is employed by Feynman’s path integral formulation, and it is the only way to avoid singularities which are unavoidable in General Relativity. In the Duality of Time Theory, since the dimensions of space and matter are being re-created in the inner (complete) levels of time, the final dimension becomes multi-fractal and equals to the dynamic ratio of “inner” to “outer” times. Additionally, and for the same reason, space-time becomes “genuinely complex”, since both its “real” and “imaginary” components have the same nature of time, which itself becomes as simple as the “recurrence”, or counting the number of geometrical nodes as they are re-created in one chronological sequence. Without postulating the inner levels of time, both the complex and fractal dimensions would not have any “genuine” meaning, unless both the numerator and denominator of the fraction, and both the real and imaginary parts of the complex number, are all of the same nature (of time). In this manner, normal time is an imaginary and fractional dimension of the complete dimensions of space, which are the real levels of time. Because they are complete integers, the dimensions of space are mutually perpendicular, or spherically orthogonal, on each other, which is what makes (isotropic and homogeneous) Euclidean geometry that can be expressed with normal complex numbers , in which the modulus is given by . In contrast, because it is fractional or non-integer dimension, (normal, or the outer level of) time is hyperbolically orthogonal on the dimensions of space, and thus expressed by the hyperbolic split-complex numbers , in which the modulus is given by . This complex hyperbolic geometry is the fundamental reason behind relativity and Lorentz transformations, and it provides the required tools to express the curvature and topology of space-time, away from Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities. The Duality of Time Theory, and the resulting dynamic re-creation of space and matter, is based on previous research that presented an eccentric conception of time [16, 17, 18, 19, 20, 21, which include other references on the history and philosophical origins of this concept]. For the purpose of this article, this hypothesis can be extracted here into the following postulate: The above postulate means that at every instance of the “real flow of time” there is only one metaphysical point, that is the unit of space-time geometry, and the Universe is a result of its perpetual recurrence in the “inner levels of time”, that is continuously re-creating the dimensions of space and what it may contain of matter particles, which then kinetically evolve throughout the outer (normal) level of time, that we encounter. To understand this complex flow of time, we need to define at least two frames of reference. The first is our normal “space” container which evolves in the outer time, that is the normal time that we encounter. And the second frame is the inner flow of time, that is creating the dimensions of space and matter. This inner frame is also composed of more inner levels to account for the creation of and space, but we shall not discuss them at this point. From our point to view, as observers situated in the outer frame, the creation process is instantaneous, because we only see the Universe after it is created, and we don’t see it in the inner frames when it is being created, or perpetually re-created, at every instance. Nevertheless, the speed of creation, in the innermost level (or real flow) of time, is indefinite, rather than infinite, because there is nothing to compare to it at this level of absolute oneness. We shall show that this speed of creation is the same speed of light, and the reason why individual observers, situated in the outer frame, measure a finite value of it is because they are subject to the time lag during which the spatial dimensions are being re-created. Therefore, in our outer frame, the speed of creation, that is the speed of light, is simply equal to the ratio of the outer to inner times, so it is a unit-less number whose normalized value corresponds to the fractal dimension of the genuinely-complex time-time geometry, rather than space-time, since space itself is created in the inner levels of time. The reason why this cosmological speed is independent of the observer is because creation is occurring in the inner real levels while physical motion is in the outer (normal) time that is flowing in the orthogonal dimension with relation to the real dimensions of space (or inner levels of time). In other words, while the real time is flowing unilaterally in one continuous sequence, creating only one metaphysical point at every instance, individual observers witness only the discrete moments in which they are re-created, and during which they observe the dimensions of space and physical matter that have just been re-created in these particular instances; thus they only observe the collective (physical) evolution as the moments of their own time flows by, and that’s why it becomes imaginary, or latent, with relation to the original real flow of time that is creating space and matter. Therefore, the speed of light in complete vacuum is the speed of its dynamic formation, and it is undefined in its own reference frame (as it can be also inferred from the current understanding of time dilation and space contraction of special relativity), because the physical dimensions are not yet defined at this metaphysical level. Observers in all other frames, when they are re-created, measure a finite value of this speed because they have to wait their turn until the re-creation process is completed, so any minimum action, or unitary motion, they can do is always delayed by an amount of time proportional to the dimensions of vacuum (and its matter contents if they are measuring in any other medium). Hence, this maximum speed that they can measure is also invariant because it is independent of any physical (imaginary) velocity, since their motion is occurring in the outer time dimension that is orthogonal onto the spatial dimensions which are being re-created in the inner (real) flow of time. This also means that all physical properties, including mass, energy, velocity, acceleration and even the dimensions of space, are emergent properties; observable only on the outward level of time, as a result of the temporal coupling between at least two geometrical points or complex-time instances. Moreover, just like the complete dimensions of space, the outer time itself, which is a fractional dimension, is also emerging from the same real flow of time that is the perpetual recurrence of the original geometrical point. This metaphysical entity that is performing this creation is called “the Single Monad”, that has more profound characteristics which we don’t need to analyze in this paper (see [18, Ch. VI] for more details); so we only consider it as a simple abstract or dimensionless point: . It will be shown in section 5 how this single postulate leads at the same time to all the three principles of Special and General Relativity together, since there is no more any difference between inertial and non-inertial frames, because the instantaneous velocity in the imaginary time is always “zero”, whether the object is accelerating or not! This also means that both momentum and energy will be “complex” and “invariant” between all frames, as we shall discuss further in sections 5.3 and 5.4 below. Henceforth, this genuinely-complex time, or time-time geometry will define a profound discrete symmetry that allows expressing the (deceitfully continuous) non-Euclidean space-time in terms of its granular and fractal complex-time space, whose granularity and fractality are expressed through the intrinsic properties of hyperbolic numbers (), i.e. without invoking Riemannian geometry, as discussed further in section 4.1. However, this hidden discrete symmetry is revealed only when we realize the internal chronological re-creation of spatial dimensions; otherwise if we suppose their continuous existence, space-time will still be hyperbolic but not discrete. Discreteness is introduced when the internal re-creation is interrupted to manifest in the outward normal time, because creation is processed sequentially by the perpetual recurrence of one metaphysical point, so the resulting complex-time is flowing either inwardly to create space, or outwardly as the normal time, and not both together. Therefore, in accordance with the correspondence principle, we will see in section 4.3, that semi-Riemannian geometry on is a special approximation of this discrete complex-time geometry on . This approximation is implicitly applied when we consider space and matter to be coexisting together in (and with) time, thus causing the deceptive continuity of physical existence, which is then best expressed by the non-Euclidean Minkowskian space-time continuum of General Relativity, or de Sitter/anti-de Sitter space, depending on the value of cosmological constant. For the same reason, because we ideally consider the dimensions of space to be continuously existing, our observations become time-symmetric, since we can apparently-equally move in opposite directions. Therefore, this erroneous time-symmetry is reflected in most physics laws because they also do not realize the sequential metaphysical re-creation of space, and that is why they fail in various sensitive situations such as the second law of Thermodynamics (the entropic arrow of time), Charge-Parity violations in certain weak interactions, as well as the irreversible collapse of wave-function (or the quantum arrow of time). In the Duality of Time Theory, the autonomous progression of the real flow of time provides a straightforward explanation of this outstanding historical problem. This will be explicitly expressed by equation 1, as discussed further in section 4.2, where we will also see that we can distinguish between three conclusive states for the flow of complex-time: either the imaginary time is larger than the real time, or the opposite, or they are equal. Each of the first two states forms one-directional arrow of time, which then become orthogonal, while the third state forms a two-directional dimension of space, that can be formed by or broken into the orthogonal time directions. This fundamental insight could provide an elegant solution to the problems of super-symmetry and matter-antimatter asymmetry at the same time, as we shall discuss in section 4.2. Additionally, the genuine complex-time flow will be employed in section 5.2 to derive the mass-energy equivalence relation , in its simple and relativistic forms, directly from the principles of Classical Mechanics. This should provide a conclusive evidence to the Duality of Time hypothesis, because it will be shown that an exact derivation of this experimentally verified relation is not possible without considering the inner levels of time, since it incorporates motion at the speed of light which leads to infinities on the physical level. All current derivations of this critical relation suffer from unjustified assumptions or approximations [22, 23, 24, 25, 26], as was also repeatedly acknowledged by Einstein himself [25, 27]. Finally, as an additional support to the Duality of Time Theory, we will show in section 4.4 that the resulting dynamic quintessence will diminish the cosmological constant discrepancy by at least 117 orders of magnitude. This huge difference results simply from realizing that the modes of quantum oscillations of vacuum occur in chronological sequence, and not all at the same time. Therefore, we must divide by the number of modes included in the unit volume, to take the average, rather than the collective summation as it is currently treated in quantum field theories. The remaining small discrepancy could be also settled based on the new structure of physical vacuum, which is shown to be a perfect super-fluid. The Duality of Time Theory, therefore, brings back the same classical concept of aether but in a novel manner that does not require it to affect the speed of light, because it is now the background space itself, being granular and re-created dynamically in time, and not something in a fixed background continuum that used to be called vacuum. On the contrary, this dynamical aether provides a simple ontological reason for the constancy and invariance of the speed of light, which is so far considered an axiom that has not been yet proven in any theoretical sense. The Duality of Time Theory provides a deeper understanding of time as a fundamental complex-scalar quantum field that reveals the discrete symmetry of space-time geometry. This revolutionary concept will have tremendous implications on the foundations of physics, philosophy and mathematics, including geometry and number theory; because complex numbers are now genuinely natural, while the reals are one of their extreme, or unrealistic, approximations. Many major problems in physics and cosmology can be resolved according to the Duality of Time Theory, but it would be too distracting to discuss all that in this introductory article. The homogeneity problem, for example, will instantly cease, since the Universe, no matter how large it could be, is re-created sequentially in the inner levels of time, so all the states are synchronized before they appear as one instance in the normal level. Philosophically also, since space-time is now dynamic and self-contained, causality itself becomes a consequence of the sequential metaphysical creation, and hence the fundamental laws of conservation are simply a consequence of the Universe being a closed system. This will also explain non-local and non-temporal causal effects, without breaking the speed of light limit, in addition to other critical quantum mechanical issues, some of which are outlined in other publications [18, 20, 21]. According to the above Duality of Time postulate, the dynamic Universe is the succession of instantaneous discrete frames of space, that extend in the outward level of time that we normally encounter, but each frame is internally created in one chronological sequence within each inward level of the real flow of time. This is schematically demonstrated in Figure 1, where space is conventionally shown in two dimensions, as the plane, and we will mostly consider the axis only, for simplicity. In reality, however, we can conceive of at least seven levels of time, which curl to make the four dimensions of space-time: , that are the three spatial and one temporal dimensions; since each spatial dimension is formed by two of the six inner levels, as we shall explain further in section 4.2, while the seventh is the outer time that we normally encounter. As it will be explained further in section 4.2 below, each spatial dimension is dynamically formed by the real flow of time, and whenever this flow is interrupted, a new dimension starts, which is achieved by multiplying with the imaginary unit that produces an “abrupt rotation” by , creating a new dimension that is perpendicular on the previous level, or hyperbolically orthogonal on it, to be more precise. This subtle property is what introduces discreteness, as a consequence of the duality nature of time, that is flowing either inwardly or outwardly, not both together. This is what makes space-time geometry genuinely complex and granular, otherwise if we consider all the dimensions to be coexisting together it will appear continuous and real, as we normally “imagine”, which may lead to space-time singularities at extreme conditions. The concept of imaginary time is already being used widely in various mathematical formulations in quantum physics and cosmology, without any actual justification apart from the fact that it is a quite convenient mathematical trick that is useful in solving many problems. As Hawking states: “It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed, but also effects we have not been able to measure yet nevertheless believe in for other reasons.” . Hawking, however, considers the imaginary time as something that is perpendicular to normal time that exists together with space, and that’s how it is usually treated in physics and cosmology. According to the Duality of Time postulate, however, since space is now (dynamically re-created in) the real time, the normal time itself becomes genuinely imaginary, or latent. Employing imaginary time is very useful because it provides a method for connecting quantum mechanics with statistical mechanics by using a Wick rotation, by . In this manner we can find a solution to dynamics problems in dimensions, by transposing their descriptions in dimensions, i.e. by trading one dimension of space for one dimension of time, which means substituting a mathematical problem in Minkowski space-time into a related problem in Euclidean space. Schroedinger equation and the heat equation are also related by Wick rotation. This method is also used in Feynman’s path integral formulation, which was extended in 1966 by DeWitt into gauge invariant functional-integral . For this reason, there has been many attempts to describe quantum gravity in terms of Euclidean geometry [30, 31], because in this way it is possible to avoid singularities which are unavoidable in General Relativity, since it is primarily constructed on curved space-time continuum that uses Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities. Mathematically, the nested levels of time can be represented by imaginary or complex numbers where space is treated as a plane or spherical wave, and time is the orthogonal imaginary axis. However, in addition to the normal complex number plane: , that can describe Euclidean space, split-complex, or hyperbolic, numbers: , are required to express the relation between space-like and time-like dimensions, which are the inner and outer levels of time, respectively. Normal complex numbers can describe homogeneous or isomorphic space, without (the outer) time, where each number defines a circle, or sphere, because its modulus is given by , while in split-complex numbers the modulus is given by , so , which defines a hyperbola. This negative sign in calculating the modulus of complex time reflects the essential fact that the perpetual re-creation of space and matter particles in the inner levels of time is interrupted and re-newed every instance of the outward time , which produces kinetic motions on the physical level, as dynamic local deformations of the otherwise flat and homogeneous Euclidean space. Therefore the non-Euclidean Minkowski space-time coordinates are an approximation of the complex space-time coordinates , or complex time-time represent the inner In this abstract complex frame, space and time are absolute, or mathematical, just as they had been originally treated in the classical Newtonian Mechanics, but now empty space is The physical vacuum, which is the dynamic aether, is therefore an extreme state which may be achieved when the apparent velocity, or momentum, becomes absolutely zero, both as the object’s total velocity and any vector velocities of its constituents, and this corresponds to absolute zero temperature (). This dynamic vacuum state is therefore a super-fluid, which is a perfect Bose-Einstein condensate (BEC), since it consists of indistinguishable geometrical points that all share the same state. In quantum field theory, complex-scalar fields are employed to describe superconductivity and superfluidity . The Higgs field itself is complex-scalar, and it is the only fundamental scalar quantum field that has been observed in nature, but there are other effective field theories that describe various physical phenomena. Indeed, some cosmological models have already suggested that vacuum could be a kind of yet-unknown super-fluid, which would explain all the four fundamental interactions and provide mass generation mechanism that replaces or alters the Higgs mechanism that only partially solves the problem of mass. In BEC models, masses of elementary particles can arise as a result of interaction with the super-fluid vacuum, similarly to the gap generation mechanism in superconductors , in addition to other anticipated exotic properties that could explain many problems in the current models, including dark matter and dark energy [35, 36]. Therefore, the new complex-time geometry is the natural complex-scalar quantum field that explains the dynamic generation of space, mass and energy. We will discuss the origin of mass in sections 4.5 and 5.2.3 below. Actually, according to this genuinely-complex time-time geometry, there can be four absolute or “super” states: super-mass , super-fluid , super-gas , and super-energy , which can be compared with the classical four elements: earth, water, air, and fire, respectively. These four extreme or elemental states, which the ancient Sumerians employed in their cosmology to explain the complexity of Nature, are formed dynamically, in the inner levels of time, by the Single Monad that is their “quint-essence”. We will see, in section 4.4 below, that this new concept of aether and quintessence is essential for understanding dark matter and energy, and solving the cosmological constant discrepancy. Moreover, the super-fluid and super-gas states, and , are in orthogonal time directions, so if describes matter that is kinetically evolving in the normal level of time with velocity, would similarly describe anti-matter in the orthogonal direction. This could at once solve the problems of super-symmetry and matter-antimatter asymmetry, because fermions in one time direction are bosons in the orthogonal dimension, and vice versa, and of course these two dimensions do not naturally interact because they are mutually orthogonal. This could also provide some handy tests to verify the Duality of Time Theory, but this requires prolonged discussion beyond the scope of this article, as outlined in other literature [20, 21]. Super-symmetry and its breaking will be also discussed further in section 6.1. Discreteness implies interruption or discontinuity, and this is what the outer time is doing to the continuous flow of the inner time that is perpetually re-creating space and matter in one chronological sequence. Mathematically, this is achieved by multiplying with the imaginary unit, which produces an “abrupt rotation” by , creating a new dimension that is orthogonal on the previous level. Multiplying with the imaginary unit again causes time to become real again, i.e. like space. This means that each point of our space-time is the combination of seven dimensions of time, the first six are the real levels which make the three spatial dimensions, , and the seventh is the imaginary level that is the outer time . This outward (normal) level of time, , is interrupting and delaying the real flow of time, , so it can not exceed it, because they both belong to one single existence that is flowing either in the inward levels to form the continuous (real) spatial dimensions, or in the outward level to form the imaginary discrete time, not the two together; otherwise they both would be real as we are normally deceived. As we introduced in section 2 above, the reason for this deception is because we only observe the physical dimensions, in the outer time, after they are created in the inner time, so we “imagine” them to be co-existing continuously, when in fact they are being sequentially re-created. It is not possible otherwise to obtain self-contained and granular space-time, whose geometry could be defined without any previous background topology. Thus, we can write: So because is interrupting and delaying , the actual (net value of) time is always smaller than the real time: , and this is actually the proper time as we shall see in equation 3. However, it should be noted here that, unlike the case for normal complex (Euclidean) plane, the modulus of split-complex numbers is different from the norm, because it is not positive-definite, but it has a metric signature . This means that, although our normal time is flowing only in one direction because it is interrupting the real flow of creation and can not exceed it, it is still possible to have the orthogonal state where the imaginary time is flowing at the speed of creation and the real part is interrupting it, such that: , so: , and then , from our perspective. In this case, the ground state of that vacuum would be , which describes anti-matter as we shall explain further in section 6.1, when we speak about super-symmetry and its breaking. Equivalently, the apparent velocity can not exceed because it is the average of all instantaneous velocities of all individual geometrical points that constitute the object, which are always fluctuating between and ; so by definition is capped by , as expressed by equation 5. Therefore, equation 1 () is also equivalent to: , thus when: , we get: , and if , then ; but both as the total apparent velocity of the object and any vector velocities of its constituents, thus in this case we have flat and infinite Euclidean space without any motion or disturbance, which is the state of vacuum: , or , as we noted in section 4.1. So this imaginary time, , is acting like a resistance against the perpetual re-creation of space, and its interruption, i.e. going in the outward level of time, is what causes physical motion and the inertial mass , which then effectively increases with the imaginary velocity according to: (as we shall derive it in section 5.2, and we shall discuss mass generation in sections 4.5 and 5.2.3), and when the outward imaginary time approaches the inner time, the apparent velocity approaches the speed of creation , and . If this extreme state could ever happen (but not by acceleration, as we shall see further below), the system would be described by , which means that both the real and imaginary parts of complex-time would be continuous, and this describes another homogeneous Euclidean space with one higher dimension than the original vacuum. Actually, the hyperbolic split-complex number is non-invertible null vector that describes the asymptotes, whose modulus equals zero, since both its real and imaginary parts are equal. At the same time, as a normal complex number, describes an isotropic infinite and inert Euclidean space (without time), because its dimensions are continuous, or uninterrupted. The metaphysical entities of the Universe are sequentially oscillating between the two vacuum states (as Euclidean spaces or normal complex numbers): and , while collectively they appear to be evolving according to the physical (hyperbolic) space-time states , as split-complex numbers. Therefore, the vacuum state can be described either as the non-invertible vector in the hyperbolic plane , and that is equal to one absolute point from the time perspective (when we look at the world from outside), or an isotropic Euclidean space as normal complex numbers , but with one lower dimension, and that is the space perspective (when we look from inside). Infinities and singularities occur when we confuse between these two extreme views; because if the observer is situated inside a spatial dimension it will appear to them continuous and infinite, while it forms only one discrete state in the encompassing outer time. As we shall see in section 4.3, General Relativity is the first approximation for inside observers, but since the Universe is evolving we need to describe it by , from the time perspective. So GR is correct every instance of time, because the resulting instantaneous space is continuous, but when the outward time flows these instances will form a series of discrete states that should be described by Quantum Field Theory. If we combine these two descriptions properly, we should be able to eliminate GR singularities and QFT infinities. In other words, the whole homogeneous space forms a single point in the outer time, and our physical Universe is the dynamic combination of these two extreme states, denoted as space-time. This is the same postulated statement that the geometrical points are perpetually and sequentially fluctuating between (for time) and (for space), and no two points can be in the state of (existence in) space at the same real instance of time, so the points of space come into existence in one chronological sequence, and they can not last in this state for more than one single moment of time, thus they are being perpetually re-created. Nonetheless, since it is not possible to accelerate a physical object (to make all its geometrical points) to move at the speed of creation , one alternative way to reach this speed of light, and thus make a new spatial dimension, is to combine the two orthogonal states and , which is the same as matter-antimatter annihilation, and this is a reversible interaction that can be described by the following equation: In conclusion, we can distinguish three conclusive scenarios for the complex flow of time: Therefore, there are two orthogonal arrows of time and , that can combine and split between the states of and , which all together correspond to the four elemental states, or classical elements, whose quintessence is the Single Monad . On the other hand, as we can see from Figure 1, the space-time interval can be obtained from: , or for motion on x-axis only. Alternatively, we can now use the new time-time interval which is the modulus of complex time: , and it is indeed the same proper time, , in Special Relativity: The reason why we are getting the negative signature here is because we exist in the imaginary dimension, and that is why we need some “time extension” to perceive the dimensions of space, that is the real dimension. For example, we need at least three instances to imagine any simple segment; one for each side and one for the relation between them; so we need infinite time to conceive the details of all space. If we exist in the real dimensions of space we would conceive it all at once, as what happens in the event horizon of a black hole. So for us it appears as if time is real and space is imaginary, while the absolute reality is a reflection, and the actual Universe is the dynamic and relative combination of these two extreme states. This essential property, that the outward time is effectively negative with respect to the real flow of time, will be inherited by the velocity, momentum and even energy; all of which will be similarly negative in relation to their real counter part. It is this fundamental property that will enable the derivation of the relativistic momentum-energy relation, the equivalence between inertial and gravitational masses, in addition to allowing energy and mass to become imaginary, negative and even multidimensional. This will be discussed further in sections 5.1, 5.2 and 5.4, respectively. The representation of space-time with imaginary time was used in the early formulation of Special Relativity, by Poincare , and even Minkowski , but because there were no substantial reasons to treat time as imaginary, Minkowski had to introduce the four-dimensional space-time: , with Lorentzian metric , in which time and space are treated equally, except for the minus. This four-dimensional space later became necessary for General Relativity, due to the presence of gravity, which required Riemannian geometry to evaluate space-time curvatures. In the split-complex hyperbolic geometry, Lorentz transformations become rotations in the imaginary plane , and according to the new discrete symmetry of the time-time frame, this transformation will be equally valid between inertial and non-inertial frames alike, because the dynamic relation between the real and imaginary parts of time implies that the instantaneous velocity in the imaginary time is always In the Theory of Relativity, we need to differentiate between inertial and non-inertial frames, because we are considering the “apparent velocity”, since the observer is measuring the change of position (i.e. space coordinates) with respect to time, thus implicitly assuming their real co-existence and continuity; so considering motion to be real transmutation, and that is why space and time are considered continuous and differentiable. The observer is therefore not realizing the fact that the dimensions of space are being sequentially re-created within the inner levels of time, as we described above. This sequential re-creation is what makes space-time complex and granular, in which case the instantaneous velocity is always zero, while the apparent physical velocity is a result of the superposition of all the velocities of the individual geometrical points , which constitute the object of observation, and each of which is either zero, in the outer time, or , in the inner time, as can be calculated by equation 5. So in this hidden discrete symmetry of space, motion is a result of re-creation in the new places rather than gradual and infinitesimal transmutation from one place to the other. Moving objects do not leave their places to occupy new adjacent positions, but they are successively re-created in them, so they are always at rest in any position along the path. When we realize the re-creation of space at the only real speed , and thus consider the apparent velocity of physical objects to be genuinely imaginary, we will automatically obtain Lorentz transformations, equally for velocity, momentum and energy (which will become also complex, as explained further in sections 5.3 and 5.4), without the need for introducing the principle of invariance of physics laws, so we do not need to differentiate between inertial and non-inertial frames, because the instantaneous velocity is zero in either case. As an extra bonus, we will also be able to derive the mass-energy equivalence relation without introducing any approximation or un-mathematical induction, and this relation is indeed the same equivalence between gravitational and inertial masses. All this is treated in section 5 below. Therefore, the non-Euclidean Minkowski space-time continuum is the first global approximation of the metaphysical reality (of Oneness, or sequential re-creation from one single point), just as the Euclidean Minkowski space-time is a local approximation when the effect of gravity is neglected, while the Galilean space is the classical approximation for non-relativistic velocities. These three relative approximations are still serving very well in describing the respective physical phenomena, but they can not describe the actually metaphysical reality of the Universe, which is dynamically re-creating the geometry of space-time itself, and what it contains of matter particles. As Hawking had already noticed: “In fact one could take the attitude that quantum theory, and indeed the whole of physics, is really defined in the Euclidean region and that it is simply a consequence of our perception that we interpret it in the Lorentzian regime.” . The Duality of Time explains exactly that the source of this deceptive perception is the fact that we do not witness the metaphysical perpetual re-creation process, but, being part of it, we always see the Universe after it is re-created, so we “imagine” that this existence is continuous, and thus describe it with the various laws of Calculus and Differential Geometry, that implicitly suppose the continuity of space and the co-existence of matter particles in space and time. In other words, normal observers, since they are part of the Universe, are necessarily approximating the reality, at best in terms of non-Euclidean Minkowskian space, and this approximation is enough to describe the macroscopic physical phenomena from the point of view of observers (necessarily) situated inside the Universe. However, this will inevitably lead to singularities at extreme conditions because, being inside the Universe, observers are trying to fit the surrounding infinite spatial dimensions in one instance of time, which would have been possible only if they are moving at the speed of light, or faster, and in this case a new spatial dimension is formed and the Universe would become confined but now observed from a higher dimension. For example, we normally see the Earth flat and infinite when we are confined to a small region on its surface, but we see it as a finite semi-sphere when we view it from outer space. In this manner, therefore, we always need one higher dimension to describe the (deceptive, and apparently infinite) physical reality, in order to contain the curvatures (whether they are intrinsic or extrinsic), and that is why Riemannian geometry is needed to describe General Relativity. Therefore, since using higher dimensions to describe the reality behind physical existence will always lead to space-time singularities, the Duality of Time Theory is working with this same logic, but backward, by penetrating inside the dimensions of space, as they are dynamically formed in the inner levels of time, down to the origin that is the zero-dimensional metaphysical point, which is the unit of space-time geometry. The Duality of Time Theory is therefore penetrating beyond the apparently-continuous physical existence, into its instantaneous or perpetual dynamic formation through the real flow of time, whose individual discrete instances can accommodate only one metaphysical or geometrical point at a time, that then correlate, or entangle, into physical objects that are kinetically evolving in the normal level of time that we encounter. At the level of this (unreal) physical multiplicity, any attempt to quantize space-time is destined to fail, because we always need a predefined background geometry, or topology, to accommodate multiplicity and define the respective relations between its various entities. In contrast, the background geometry of the Duality of Time Theory is “void”, which is an absolute mathematical vacuum that has no structure or reality, while also explaining how the physical vacuum is dynamically formed by simple chronological recurrence. So, apart from natural counting, the Duality of Time does not rely on any predefined geometrical structure, but it explains how space-time geometry itself is re-created as dynamic and genuinely-complex structure. The fact that each frame of the inner time (which constitutes space) appears as one instance on the outward time is what justifies treating time as imaginary with relation to space, thus orthogonal on it. In this dynamic creation of space in the complex time, the outward time is discrete and imaginary, while space becomes continuous with relation to this outer time, but this is only relative to the dimension in which the observer is situated, so for example: the plane is itself continuous with relation to its inner dimensions but it forms one discrete instance with relation to the flow of time in the encompassing , which then appears internally continuous but discrete with regard to the encompassing outward time. For this reason perhaps, although representing Minkowski space-time in terms of Clifford geometric algebra employing bivectors , or even the spinors of complex vector space , allowed expressing the equations in simple forms, but it could not discover the intrinsic granularity of space-time without any background, since it is still working on the multiplicity level, and not realizing the sequential re-creation process. Aether was described by ancient philosophers as a thin transparent material that fills the upper spheres where planets swim. The concept was also used again in The concept of aether was contradictory because this medium must be invisible, infinite and without any interaction with physical objects. Therefore, after the development of Special Relativity, aether theories became scientifically obsolete, although Einstein himself said that his model could itself be thought of as an aether, since empty space now has its own physical properties . In 1951, Dirac reintroduced the concept of aether in an attempt to address the perceived deficiencies in current models , thus in 1999 one proposed model of dark energy has been named: quintessence , or the fifth fundamental force . Also, as a scalar field, the quintessence is considered as some form of dark energy which could provide an alternative postulate to explain the observed accelerating rate of the expansion of the Universe, rather than Einstein’s original postulate of cosmological constant [44, 45]. The classical concept of aether was rejected because it required ideal properties that could not be attributed to any physical medium that was thought to be filling the otherwise empty space background which was called vacuum. With the new dynamic creation, however, those ideal properties can be explained, because aether is no longer something filling the vacuum, but it is vacuum itself, that is perpetually being re-created at the absolute speed of light. Thus its state is described by: as we explained in section 4.1 above, which indicates infinite and inert space that is the ground state of matter particles that are then described by , whereas the absolutely-empty mathematical space is now called void and its state is . As we already explained above, this state of corresponds to absolute zero temperature, and it is a perfect super-fluid described by Bose-Einstein statistics because its points are non-interacting and absolutely indistinguishable. When this medium is excited or disturbed, matter particles and objects are created as the various kinds of vortices that appear in this super-fluid, and this is what causes the deformation and curvature of what is otherwise described by homogeneous Euclidean geometry. Therefore, the Duality of Time Theory reconciles the classical view of aether with General Relativity and Quantum Field Theory at the same time, because it is now the ground state of particles that are dynamically generated in time. In Quantum Field Theory, the vacuum energy density is due to the zero-point energy of quantized fields, which originates from the quantization of the simple harmonic oscillations of a particle with mass. This zero-point energy of the Klein-Gordon field is infinite, but a cutoff at Planck length is enforced, since it is generally believed that General Relativity does not hold for distances smaller than this length: , which corresponds to Planck energy: . By applying this cutoff we can get: , which gives us: . Comparing this theoretical value with the 1998 observations that produced: orders of magnitude discrepancy, which is known as the vacuum catastrophe [46 The smallness of the cosmological constant became a critical issue after the development of cosmic inflation in the 1980s, because the different inflationary scenarios are very sensitive to the actual value of Many solutions have been suggested in this regard, as it was reviewed by discrepancy is actually many orders of magnitudes larger than the number of all atoms in the Universe, which is called Eddington number According to the Duality of Time postulate, this huge discrepancy in the cosmological constant is diminished, and even eliminated, because the vacuum energy should be calculated from the average of all states, and not their collective summation as it is currently treated in Quantum Field Theory. This means that we should divide the vacuum energy density by the number of modes included in the unit volume. Since we took Planck length as the cutoff, this number is: This will reduce the discrepancy between the observed and predicted values of from into only. The remaining small discrepancy could now be explained according to quintessence models, which is already described by the Duality of Time as the ground state of matter. However, more accurate calculations are needed here because all the current methods are approximate and do not take into account all possible oscillations for all the four fundamental interactions. It is well established in modern physics that mass is an emergent property, and since the Standard Model relies on gauge and chiral symmetry, the observed non-zero masses of elementary particles require spontaneous symmetry breaking, which suggested the existence of the massive Higgs boson, whose own mass is not explained in the model. This Higgs mechanism is part of the Glashow-Weinberg-Salam theory to unify electromagnetic and weak interactions Moreover, the Duality of Time Theory provides an even more fundamental and very simple mechanism for mass generation, in full agreement with the principles of Classical Mechanics, as shown further in section 5.2.3. In general, the fundamental reason for inertial mass is the coupling between the particles that constitute the object, because the binding field enforces specific separations between them, so that when the position of one particle changes, a finite time elapses before other particles move, due to the finite speed of light. This delay is the cause of inertial behavior, and this implies that all massive particles are composed of more sub-particles, and so on until we reach the most fundamental particles which should be massless. This description is fulfilled by the Duality of Time Theory, due to the discrete symmetry of the genuinely-complex time-time geometry as described above. The Duality of Time Theory is based on the Single Monad Model, so the fundamental reason of the granular geometry is the fact that no two geometrical points can exist at the same real instance of time, so they must be re-created in one chronological sequence. This delay is what causes the inertial mass, so physical objects are dynamically formed by the coupling between at least two geometrical points which produces the entangled dimensions. According to the different degrees of freedom in the resulting spatial dimensions, this entanglement is responsible for the various coupling parameters, including charge and mass, which become necessarily quantized because they are proportional to the number of geometrical nodes constituting each state, starting from one individual point for massless bosons. Nevertheless, some bosons might still appear to have heavy masses (in our outer level of time) because they are confined in their lower dimensions in which they are massless, just as the inertial mass of normal objects is exhibited only when they are moved in the outer level of time. Consequently, there is a minimum mass gap above the ground state of vacuum , which is itself also above the void state . This is because each single geometrical node is massless on its own dimensions, while the minimum state above this ground state is composed of two nodes which must have non-zero inertial mass because of the time delay between their sequential creation instances. This important conclusion agrees with Yang-Mills suggestion that the space of intrinsic degrees of freedom of elementary particles depends on the points of space-time. It was already anticipated that proving Yang-Mills conjecture requires the introduction of fundamental new ideas both in physics and in mathematics [ The famous Michelson-Morley experiment in 1887 proved that light travels with the same speed regardless whether it was moving in the direction of the movement of the Earth or perpendicular to it [ Logically, there are two cases under which a quantity does not increase or decrease when we add or subtract something from it. Either this quantity in infinite, or it exists in orthogonal dimension. As we have already introduced in sections 1 and 2 above, according to the Duality of Time postulate, both these cases are equivalent and correct for the absolute speed of light in vacuum, because it is the speed of creation which is the only real speed in nature, and it is intrinsically infinite (or indefinite), but the reason why we measure a finite value of it is because of the sequential re-creation process; so individual observers are subject to the time lag during which the dimensions of space are re-created. Moreover, since the normal time is now genuinely imaginary, the velocities of physical objects are always orthogonal onto this real and infinite speed of creation. As demonstrated in Figure 1 and explained in section 4.1 above, one of the striking conclusions of the sequential re-creation in the inner levels of time is the fact that it conceives of only two primordial states: As the real time flows uniformly in the inner levels, it creates the homogeneous dimensions of vacuum, and whenever it is interrupted or disturbed, it makes a new dimension that appears as a discrete instance on the outer imaginary level which is then described as void, since it does not last for more than one instance, before it is re-created again in a new state that may resemble the previous perished states, which causes the illusion of motion, while in reality it is only a result of successive discrete changes. So the individual geometrical points can either be at rest (in the outer/imaginary time) or at the speed of creation (in the inner/real time), while the apparent limited velocities of physical particles and objects (in the total complex time, which forms the physical space-time dimensions) are the temporal average of this spatial combination that may also dynamically change as they are progressing over the outward ordinary time direction. Therefore, the Universe is always coming to be, perpetually, in “zero time” (on the outward level), and its geometrical points are sequentially fluctuating between existence and nonexistence (or vacuum and void), which means that the actual instantaneous speed of each point in space can only change from to , and vice versa. This instantaneous abrupt change of speed does not contradict the laws of physics, because it is occurring in the inner levels of time before the physical objects are formed. Because they are massless, this fluctuation is the usual process that is encountered by the photons of light on the normal outward level of time, for example when they are absorbed or emitted. Hence, this model of perpetual re-creation is extending this process onto all other massive particles and objects, but on the inner levels of time where each geometrical point is still massless because it is metaphysical, while “space” and “mass” and other physical properties are actually generated from the temporal coupling, or entanglement, of these geometrical points, which is exhibited only on the outward level of time. Accordingly, the normal limited velocity, of physical particles or objects, is a result of the spatial and temporal superposition of these dual-state velocities of its individual points , thus: Individually, each point is massless and it is either at rest or moving at the speed of creation, but collectively they have some non-zero inertial mass , energy , and limited total apparent velocity given by this equation 5 above. Consequently, there is no gradual motion in the common sense that the object leaves its place to occupy new adjacent places, but it is successively re-created in those new places, i.e. motion occurs as a result of discrete change rather than infinitesimal transmutation, so the observed objects are always at rest in the different positions that they appear in (see also Figure 1). This is the same conclusion of the Moving Arrow argument in Zeno’s paradox, which Bertrand Russell described as: “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever.” . This momentous conclusion means that all frames are effectively at rest in the normal (imaginary) level of time, and there is no difference between inertial and non-inertial frames, thus there is even no need to introduce the second principle of Special Relativity (which says that the laws of physics are invariant between inertial frames) neither the equivalence principle that lead to General Relativity. These two principles, which are necessary to derive Lorentz transformations and Einstein’s field equations, are implicit in the Duality of Time postulate and will follow directly from the resulting complex-time geometry as it will be shown in sections 5.1 and 5.3 below. Furthermore, it will be also shown in section 5.2 that this discrete space-time structure that results from the genuinely-complex nature of time is the only way that allows exact mathematical derivation of the mass-energy equivalence relation (). In this manner, the Duality of Time postulate, and the resulting perpetual re-creation in the inner levels of time, can explain at once all the three principles of Special and General Relativity, and transform it into a quantum field theory because it is now based on discrete instances of dynamic space, which is the super-fluid state that is the ground state of matter, while the super-gas state is the ground state of anti-matter, which accounts for super-symmetry and matter-antimatter asymmetry as we shall discuss further in section 6.1. The other fundamental forces could also be interpreted in terms of this new space-time geometry, but in lower dimensions: , , and , while gravity is in . As we noted above, it was originally shown by Poincare that by using the mathematical trick of imaginary time, Lorentz transformation becomes a rotation between inertial frames. For example, if the space coordinates of an event in space-time relative to one frame are , then its (primed) coordinates with respect to another frame, that is moving with uniform velocity with respect to the first frame, are: , where , and since: , then: . In the complex-time frame of the Duality of Time postulate, however, the outer time is the (genuinely) imaginary part, while the real part is the inner time that constitutes space, thus the time coordinates: is used instead of space coordinates: . Therefore, the above rotation equations will still be valid but with time, rather than space, and then the speed of creation will be the ground state, Using the concept of split-complex time, we can easily derive Lorentz factor for example by calculating the proper time as it can be readily seen from Figure 1, which is replicated in Figure 2 that represents complex velocity, for better clarity, and also because we want to stress the fact that the apparent (imaginary) motion in any direction is in fact interrupting the real motion in the inner time that is re-creating space at the absolute speed of light, so that in the end the Lorentz factor is therefore the ratio of the real velocity over the actual velocity , which is equal , as demonstrated in Figure 2: In addition to explaining the constancy and invariance of the speed of light and merging it with the second and third principles of Relativity, the Duality of Time postulate is the only way to explain the equivalence and transmutability between mass and energy Einstein gave various heuristic arguments for this relation without ever being able to prove it in any theoretical way [ It can be readily seen from Figure 3 that the transmutability between mass and energy can only occur in the inner levels of time, because it must involve motion at the speed of light that appears on the normal level of time as instantaneous, hence the same Relativity laws become inapplicable since they prohibit massive particles from moving at the speed of light, in which case , so the mass would be infinite and also the energy. In the inner levels of time, however, this would be the normal behavior because the geometrical points are still massless, and their continuous coupling and decoupling is what generates mass and energy on the inner and outer levels of time, respectively, as explained further in section 5.2.3 below. As we introduced in section 4.1 above, the normal limited velocities of massive physical particles and objects are a result of the spatial and temporal superposition of the various dual-state velocities of their individual points. This superposition occurs in the inner levels of time, where individually each point is massless and it is either at rest or moving at the speed of creation, but collectively they have some non-zero inertial mass , energy , and limited total apparent velocity , which can be calculated from equation 5. We also explained in section 4.3 above, that when we consider this imaginary velocity as being real, the Duality of Time Theory reduces to General Relativity, but when we consider its imaginary character we will uncover the hidden discrete space-time symmetry and we will automatically obtain Lorentz transformation, without introducing the principle of invariance of physics laws. For the same reason, we can see here that the mass-energy equivalence can only be derived based on this profound discreteness that is manifested in dual-state velocity, which then allows the square integration in Figure 3, because the change in speed is occurring abruptly from zero to . Otherwise, when we consider to be real continuous in time, we will get the gradual change which produces the triangular integration with the factor of half that gives the normal kinetic energy . Based on this metaphysical behavior in the inner levels of time, we will provide in the following various exact derivations of the mass-energy equivalence relation, in its simple and relativistic forms, directly from the classical equation of mechanical work . The first two methods, in sections 5.2.2 and 5.2.3, involve integration (or rather: summation) in the inner time when the velocity changes abruptly from zero to , or when the mass is generated (from zero to ) in the inner time. This is obviously not allowed on the normal level of time when dealing with physical objects. The third method, in section 5.2.7, gives the total relativistic energy , by integrating over the inner and outer levels together, while in section 5.2.8 we will derive the relativistic energy-momentum relation directly from the definition of momentum as , also by integrating over the inner and outer levels together and accounting for what happens in each stage. Furthermore, we will see in sections 5.3 and 5.4 that the absolute invariance, and not just covariance, of complex momentum and energy, provide yet other direct derivations because they also lead to , that is equivalent with or as demonstrated in A. Actually, since we have shown previously that the new vacuum is a perfect super-fluid, the mass-energy equivalence relation can be easily derived from the equation of wave propagation in such perfect medium: (), but we will not discuss that further in this article. In normal classical mechanics, the kinetic energy is the work done in accelerating a particle during the infinitesimal time interval , and it is given by the dot product of force and displacement : Now if we assume mass to be constant, so that: (and we will discuss relativistic mass further in section 5.2.4 below), we will get: So in the classical view of apparently continuous existence, when we consider both space and time to be real, i.e. when we consider an infinitesimally continuous and smooth change in speed from zero to , the result of this integration will give the standard equation that describes the kinetic energy of massive particles or objects moving in the normal level of time: The reason why we are getting the factor of “half” in this equation is because the velocity increases gradually with time, which makes the integration equals the area of the triangle as demonstrated by the first arrow in Figure 3. The relativistic energy-momentum relation is derived in section 5.2.8 further below, but the simple mass-energy equivalence relation: (without the “half”) can now be easily obtained from the same integration in equation 9 if, By introducing the duality of time and the resulting perpetual re-creation, this problem is solved because the conversion between mass and energy takes place, sequentially, in the inner levels of time, on all the massless geometrical points that constitute the particle, and this whole process appears as one instance in the outer level, as demonstrated in Figure 1 above. So by integrating equation 9 directly from zero to , which then becomes summation because it is an abrupt change, with only the two states of void and vacuum, corresponding to zero and , respectively, and since the change in the outward time is zero, and here we also consider , since the apparent velocity does not change in this case, but we will also discuss relativistic mass in section 5.2.4 below; thus we obtain: The difference between the above two cases that result in equations 10 and 11 is demonstrated in Figure 3, where in the first case the integration that gives the kinetic is the area of the We explained in section 4.5 above how the Duality of Time Theory provides a fundamental mass generation mechanism in addition to its super-fluid vacuum state where mass can be generated via the interaction with this physical vacuum. Hence we can also arrive to the mass-energy equivalence relation directly from the starting equation 8 in an alternative manner if we consider a sudden decoupling, or disentanglement, of the geometrical points that couple together in order to constitute the physical particle that appears in the outer level of time with inertial mass moving at an apparent (imaginary) velocity , so when these geometrical points are disentangled to remain at their real speed (of light), the mass converts back into energy while the apparent velocity does not change, because this process is happening in the inward levels of time which appears outwardly as instantaneous. Thus, if we put in equation 8 and integrate over mass from to zero (where ), or vice versa, we get: Unlike the classical case in equation 10 where the change in speed occurs in the normal outward level of time, these simple derivations (in equations 11 and 12) would not have been possible without considering the inner levels of time, which appears outwardly as instantaneous. If we want to consider mass to be variable with speed as in early Special Relativity, and distinguish between rest mass: and relativistic mass: , according to the standard equation that uses Lorenz factor: , then we can arrive to the equation by calculating the derivative which in this case will not be equal to zero as we required in equation 9 above. However, the above relativistic equation of mass () is only obtained based on the same mass-energy equivalence relation that we are trying to prove here, so in this case it will be a circular argument. Therefore, the two equations: and are equivalent, and deriving one of them will lead to the other. See also A for how to derive the from . We can conclude, therefore, that on the highest existential level, there is either energy in the form of massless active waves moving at the speed of creation in the inner levels of time, or passive mass in the form of matter particles that are always instantaneously at rest in the outer level of time, not the two together; that is what happens in the real flow of time. The various states of massive objects and particles, as well as thermal radiations and energy, are some spatial and temporal superposition of these two primordial states of their metaphysical constituents, so some particles will be heavier than others, and some will have more kinetic energy. In any closed system, such as an isolated particle, atom, or even objects, the contributions to this superposition state come from all the states in the system that are always fluctuating between mass and energy, or void and vacuum, corresponding to and , respectively, so on average the total state is indeterminate, or determined only as a probability distribution, as far as it is not detected or measured. This wave-particle duality will be discussed further in section 6. Consequently, everything in the Universe is always fluctuating between particle state and wave state, or mass and energy, which can be appropriately written as: . This means that a particle at rest with mass can be excited into a wave with frequency , and the opposite is correct when the wave collapses into particle. Either zero mass at the speed of creation, or (instantaneously) zero energy at rest, or: either energy in the active existence state or mass in the passive nonexistence state. The two cannot happen together on the primary level of time, but a mixture or superposition of various points is what causes all the phenomena of motion and interaction between particles with limited velocities and energy on the outward level of time. Therefore, even when the object is moving at any velocity that could be very close to , its instantaneous velocity is always zero: , at the actual time of measurement, and its mass will still be the same rest mass: , because it is only detected as a particle, while its kinetic energy will be given by its relativistic mass: , and then its total energy equivalence, with relation to an observer moving at a constant (apparent) velocity , will be given by: Thus, with the help of Lorentz factor , we could get rid of the confusion between “rest mass” and “relativistic mass” and just call it mass since the above equation 13 describes energy and not mass: Which means that the mass of any particle is always the rest mass, it is not relativistic, but its energy is relativistic, primarily because energy is related to time and motion or velocity. However, since we have been using all over this article, we will keep using it as the rest mass, and as the effective mass, unless stated otherwise. The total relativistic energy in equation 13 can also be obtained by integrating the starting equation 8 over the inner and outer time together, since in the inner time the rest energy , or mass , is generated at the speed of creation as a result of the instantaneous coupling between the geometrical points which constitute the particle of mass (thus and in the inner time), and in the outer time the kinetic energy is generated as this mass moves gradually so its apparent velocity changes by , which corresponds to increasing the effective mass from to , thus we can integrate: Thus we get the same equation 13: This equation can also be given in the general form that relates the relativistic energy and momentum (see A for how to convert between these two equations): This last equation, that is equivalent to equation 16, will be also derived in section 5.2.8 starting from the definition of momentum as , but because it is genuinely complex, and hyperbolic, the imaginary part of momentum will have negative contribution just as we have seen for the outer time when we discussed the arrow of time in section 4.2 above. Again, however, a fundamental derivation of this relativistic energy-momentum equation 17 is not possible without the Duality of Time postulate. All the current derivations in the literature rely on the effective mass relation: , which is equivalent to the same relation we are trying to derive (see above and also A), while finding this equation from the four-momentum expression, or space-time symmetry, relies on induction rather than rigorous mathematical formulation. The equation: , of the relativistic energy-momentum, can be also derived directly from the fundamental definition of momentum: , when we include the metaphysical creation of mass in the inner levels of time, in addition to its physical motion in the outer level, and by taking into account the complex character of time. Thus we need to integrate over the inner and outer levels, according to what happens in each stage; first by integrating between and on the inner real levels of time where the particle is created, or being perpetually re-created, at the speed of creation , and this term makes the real part of the complex momentum . Then we integrate between and on the outer imaginary level of time where the particle whose mass is gains an apparent velocity , and thus its effective mass increases from to , and this term makes the imaginary part of the complex momentum : The first term gives us the real momentum , while the second term gives the imaginary momentum . So the total complex momentum is: . Hence the modulus of this total complex momentum is given by: Again, we notice here that the contribution of the imaginary momentum is negative with relation to the real momentum , just like the case of and as we have seen in sections 4.2 and 5.1, and as it will be also the case for complex energy as we shall see in section 5.4 further below. All this is because the normal time, or physical motion, is interrupting the real creation, which is causing the disturbance and curvature of the otherwise infinite homogeneous Euclidean space that describes the vacuum state . Therefore, to obtain the relativistic energy-momentum relation from equation 19, we simply multiply by : These equations 19 and 20 above, with the negative sign, do not contradict the equation in current Relativity: which treats energy as scalar and do not realize its complex dimensions (see also section 5.4 below). Practically, in any mass-energy interaction or conversion, the negative term will be converted back to positive because when the potential energy is released, in nuclear interactions for example, this means that it has been released from the inner levels of time where it is captured as mass, into the outer level to become kinetic energy or radiation. In other words: the absorption and emission of energy or radiation, nuclear interactions, or even the acceleration and deceleration of mass, are simply conversions between the inner and outer levels of time, or space and time, respectively. Eight centuries ago, Ibn al-Arabi described this amazing observation by saying: “Space is rigid time, and time is liquid space.” . This derivation of the relativistic energy-momentum relation from the fundamental definition of momentum is based on the Duality of Time concept, by taking into account the complex nature of time, as hyperbolic numbers, which is why the contribution of the imaginary term appears here as negative in equation 20. As we discussed in section 4.3, when we do not realize the discrete structure of space-time geometry that results from this genuinely-complex nature of time, the Duality of Time Theory is reduced to General Relativity, which considers both space and time to be real, and then we take the apparent rather than the complex velocity whose instantaneous value is always zero; so this negative sign in equations 19 and 20 above will appear positive, as if we are treating space-time to be spherical () rather than hyperbolic (). Therefore, when we take into account the complex nature of time as we described in sections 4.1 and 5.1 above (or Figures: 1 and 2), energy and momentum will be also complex and hyperbolic. This significant conclusion, that is a result of the new discrete symmetry, will introduce an essential modification on the relativistic energy-momentum equation which will lead to the derivation of the equivalence principle and allows energy to be imaginary, negative and even multidimensional, as it will be discussed further in sections 5.3 and 5.4. In moving from Special to General Relativity, Einstein observed the equivalence between the gravitational force and the inertial force experienced by an observer in a non-inertial frame of reference. This is roughly the same as the equivalence between active gravitational and passive inertial masses, which has been later accurately tested in many experiments [ When Einstein combined this equivalence principle with the two principles of Special Relativity, he was able to predict the curved geometry of space-time, which is directly related to its contents of energy and momentum of matter and radiation, through a system of partial differential equations known as Einstein field equations. We explained in section 5.2 above that an exact derivation of the mass-energy equivalence relation is not possible without postulating the inner levels of time, and that is why there is yet no single exact derivation of this celebrated equation. For the same reason indeed, there is also no mathematical derivation of the equivalence principle that relates gravitation with geometry, because it is actually equivalent to the same relation that reflects the fact that space and matter are always being perpetually re-created in the inner time, i.e. fluctuating between the particle state and wave state , thus causing space-time deformation and curvature. Due to the discrete structure of the genuinely-complex time-time geometry, as illustrated in Figure 1, the complex momentum should be invariant between inertial and non-inertial frames alike, because effectively all objects are always at rest in the outer level of time, as we explained in section 4 above. This means that complex momentum is always conserved This invariance of momentum between non-inertial frames is conceivable, because it means that as the velocity increases (for example), the gain in kinetic momentum (that is the imaginary part) is compensated by the increase in the effective mass: due to acceleration, which causes the real part also to increase, but since is hyperbolic, thus its modulus remains invariant, and this what makes the geometry of space (manifested here as ) dynamic, because it must react to balance the change in effective mass. Therefore, a closed system is closed only when we include all its contents of mass and energy (including kinetic and radiation) as well as the background space itself, which is the vacuum state , and the momentum of all these constituents is either , when they are re-created in the inner levels, or for physical objects moving in the normal level of time. For such a conclusive system, the complex momentum is absolutely invariant. Actually, without this exotic property of momentum it is not possible at all to obtain an exact derivation of as we mentioned in section 5.2 above, and also in A below. These experimentally verified equations are correct if, Since this previous equation 21 is equivalent to: , therefore, in addition to the previous methods in equations 11 and 12, and the relativistic energy-momentum relation in equation 20, the mass-energy equivalence relation: can now be deduced from equation 21 as it is shown in A below, because, as we mentioned in section 5.2.4 above, the equations: and are equivalent, and the derivation of one of them leads to the other, while there is no exact derivation of either form in the current formulation of Special or General Relativity. This absolute conservation of complex momentum under acceleration leads directly to the equivalence between active and passive masses, because it means that the total (complex) force: must have two components; one that is related to acceleration as changes in the outer time , which is the imaginary part, and this causes the acceleration: , so here is the passive mass, while the other force is related to the change in effective mass , or its equivalent energy , which is manifested as the deformation of space which is being re-created in the inner levels of time , and this change or deformation is causing the gravitational force that is associated with the active mass; and these two components must be equivalent so that the total resulting complex momentum remains conserved. Therefore, gravitation is a reaction against the disturbance of space from the ground state of bosonic vacuum to the state of fermionic particles , the first is associated with the active mass in the real momentum , and the second is associated with the passive mass in the imaginary momentum . However, as discussed further in section 6, because of the fractal dimensions of the new complex-time discrete geometry, performing the differentiation of this complex requires non-standard analysis because space-time is no longer everywhere From this conservation of complex momentum we should be able to find the law of gravitation and the stress-energy-momentum tensor which leads to the field equations of General Relativity. Moreover, since empty space is now described as the dynamic aether, gravitational waves become the longitudinal vibrations in this ideal medium, and the graviton will be simply the moment of time , just as photons are the quanta of electromagnetic radiations and they are transverse waves in this vacuum, or the moments of space . This means that equivalence principle is essentially between photons and gravitons, or between space and time, while electrons and some other particles could be described as standing waves in the space-time; with complex momentum , and the reason why we have three generations of fermions is due to the three dimensions of space. This important conclusion requires further investigation, but we should also notice here that the equivalence principle should apply equally to all fundamental forces and not only to gravity, because it is a property of space-time geometry in all dimensions, and not only the where gravity is exhibited, as it is also outlined in another publication . Since it is intimately related to time, energy has to have complex, and even multiple intersecting dimensions in accordance with the dimensions of space and matter which are generated in the inner levels of time before they evolve throughout the outer level. We must notice straightforward, however, that not all these levels of energy are equivalent to mass which is only a property of space. In lower dimensions, energy should rather be associated with the corresponding coupling property, such as the electric and color charges. Therefore, it is expected that negative mass is only possible in spatial dimensions, as it has been already anticipated before [ It is clear initially that, just like time, velocity and momentum that were discussed above, when we take the complex nature of time into account, the kinetic energy in equation 13, or in the relativistic energy-momentum equation 17, becomes negative with relation to the potential energy stored in mass . Therefore the energy in equation 15 becomes complex with real and imaginary parts. The real part represents re-creation through the change in mass , and the imaginary part represents the kinetic evolution of this mass in the outer time through the change in the apparent velocity : The real part is and the imaginary part is , thus we get: This negative contribution of the kinetic energy, however, does not falsify the current equations 13 and 17, but it means that the potential energy and the kinetic energy are in different orthogonal levels of time and the conversion of potential energy into kinetic energy is like the conversion from the inner time into the outer time, so when they are in the outer time they are added together as in the previous equations because they become both in the same level of time. Again, just as it is the case with the absolute conservation of momentum that we have seen in section 5.3 above, energy is also always conserved, even when the apparent velocity changes, since the instantaneous velocity in the outer level of time is always zero, as we have seen in section 4 and Figure 1 above. As it is the case for momentum, this absolute conservation of energy is conceivable because it means that as the velocity changes, the change in kinetic energy (that is the imaginary part) is compensated by the change in the effective mass: due to motion, which causes the real part of energy also to change accordingly, but since is hyperbolic, thus its modulus remains invariant between all inertial or non-inertial frames. This means that: This equation provides even an additional method to derive the mass-energy equivalence, because the left side in this equation can be reduced to : Soon after the discovery of fractals, fractal structures of space-time were suggested in 1983 , as an attempt to find a geometric analogue for relativistic quantum mechanics, in accordance with Feynman’s path integral formulation, where the typical infinite number of paths of quantum-mechanical particles are characterized as being non-differentiable and fractal [ Accordingly, some theories were constructed based on fractal space-time, including Causal Dynamical Triangulation and Scale Relativity , which also share some fundamental characteristics with Loop quantum gravity, that is trying to quantize space-time itself . Actually, there are many studies that have successfully demonstrated how the principles of quantum mechanics can be derived from the fractal structure of space-time [9, 10, 11, 12, 13], but there is yet no complete understanding of how the dimensionality of space-time evolved to the current Universe. Some multiverse and eternal-inflation theories exhibit fractality at scales larger than the observable Universe [ In this regard, based on the concept of re-creation according to the Duality of Time Theory, the Universe is constantly being re-created from one geometrical point, that is , from which all the current dimensions of space and matter are re-created in the inner levels of time before they evolve in the outer time. Therefore, the total dimension of the Universe becomes naturally multi-fractal and equals to the dynamic ratio of “inner” to “outer” times, because spatial dimensions alone, as an empty homogeneous space, are complete integers, while fractality arises when this super-fluid vacuum, as described in section 4.1 above, starts oscillating in the outer time, which causes all types of vortices that we denote as elementary particles. So we can see how this notion, of space-time having fractal dimensions, would not have any “genuine” meaning unless both the numerator and denominator of the fraction are both of the same nature of time, and this can only be fulfilled by interpreting the complete dimensions of space as inner levels of time. In the absolute sense, the ratio of inner to outer times is the same as the speed of light, which only needs to be “normalized” in order to express the fractality of space-time; to become time-time. For example, if the re-creation process, that is occurring in the inner levels of time, is not interrupted in the outer time, i.e. when the outer time is zero, this corresponds to absolute vacuum, that is an isotropic and homogeneous Euclidean space, with complete integer dimensions, that is in our normal perception, and it is expected to be on large cosmological scales. So the speed of light, in the time-time frame, is a unit-less constant that is equal to the number of dimensions that are ideally for a perfect three-dimensional vacuum, which corresponds to the state of super-energy , as described in section 4.1, but it may condense down to for void, which is absolute darkness that is the super-mass state . The standard value of the speed of light in vacuum is now considered a universal physical constant, and its exact value is meters per second. Since 1983, the length of the Meter has been defined from this constant, as well as the international standard for time. However, this experimentally measured value corresponds to the speed of light in actual vacuum that is in fact not exactly empty. The true speed that should be considered as the invariant Speed of Creation is the speed of light in absolute “void” rather than “vacuum”, which still has some energy that may interact with the photons, and delay them, but void is real “nothing”. Of course, even high vacuum is very hard to achieve in labs, so void is absolutely impossible. Because we naturally distinguish between space and time, this speed must be measured in terms of meters per second, and it should be therefore exactly equal to . The difference between this theoretical value and the standard measured value is what accounts for the quantum foam, in contrast to the absolute void that cannot be excited. Of course all this depends also on the actual definition of the meter, and also the second, which may appear to be conventional, but in fact they are based on the same ancient Sumerian tradition, included in their sexagesimal system which seems to be fundamentally related to the structure of space-time [20, Ch. VII]. Therefore, the actual physical dimensions of the (local) Universe are less than three, and they change according to the medium, and they are expected to be more than three in extra-galactic space, to accommodate negative mass and super-symmetry. For example the fractional dimensions of the actual vacuum is simply , and the fractional dimensions of water would be , and so on for all transparent mediums according to their relative refraction index. Opaque materials could be also treated in the same manner according to their refraction index, but for other light wave-lengths that they may transfer. Dimensionality is a relative and dynamic property, so the Universe is ultimately described by multi-fractal dimensions that change according to the medium, or the inner dimensions (of space), and also wavelength, that is the outer dimension (or time). As we noted above, many previous studies have successfully derived the principles of quantum mechanics from the fractality of space-time, but we want in the remaining of this section to outline an alternative description based on the new complex-time geometry. This has been explained with more details in other publications [18, 20, 21, 19], but a detailed study is required based on the new findings. As a result of perpetual re-creation, matter in the Universe is alternating between the two primordial states of void and vacuum, which correspond to the two states of super-mass and super-energy , respectively. Since is real void or absolute “nothing”, it remains only the state of vacuum , which is a perfectly homogeneous three-dimensional space, according to our normal perception. Therefore, the Universe, as a whole, is in this perfect state of Bose-Einstein Condensation, which is a state of “Oneness”, because its geometrical points are indistinguishable and non-interacting, so it is a perfectly symmetrical and homogeneous or isotropic space. Multiplicity appeared out of this Oneness as a result of breaking the symmetry of the real existence, the super-energy , and its imaginable non-existence, the super-mass , into the two states of super-fluid and super-gas , which correspond to particles and anti-particles, which are perpetually, and sequentially, annihilating back into energy and splitting again, as described by equation 2 above. This process is occurring every moment of time, and this is actually what defines the moments of time, and causes our physical perception and consciousness. If existence remained in the bosonic state, no physical particles will appear, and no “time”, since no change or motion can be conceived. Normal (or the outer level of) time starts when the super-fluid state , which is the aether, is excited into , which describes physical particles, or fermionic states, while at the same time the orthogonal super-gas state is excited into , which describes anti-particles that are also fermions in their own time, but bosons in our time reference, and vice-versa, because these opposite time arrows are orthogonal, as we described in section 4.2. Therefore, physical existence happened as a result of splitting this ideal space, which introduced the outer level of time in which fermions started to move and take various different (discrete) states. The fundamental reason behind the quantum behavior, or why these states are discrete, is because no two particles, or fermionic states, can exist simultaneously in the outer time, which is the very fact that caused them to become multiple and make the physical matter, so their re-creation must be processed sequentially, and this is the ontological reason behind the exclusion principle. Therefore, since all fermions are kinetically moving in the outer time, which is imaginary, they must exist in different states, because we are observing them from orthogonal time direction, otherwise we would not see them multiple and in various dimensions. In contrast to that, because bosons are in the real level of time with respect to the observer, they all appear in the same state even though they may be many. On the other hand, if we suppose the particle is composed of individual geometrical points, each of which is either in the inner or outer levels of time, so their individual speeds are either zero or , but collectively appear to be moving at the limited apparent velocity , that can be calculated from equation 5; thus the particle is described by . Therefore, because only one point actually exists in the real flow of time, the position of this point is completely undetermined, because its velocity is equal to , while the rest have been already defined, because they are now in the past, and their velocities had been sequentially and abruptly collapsed from to zero, after they made their corresponding specific contribution to the total quantum state which defines the position of the particle with relation to the observer. When the number is very large, as it is the case with large objects and heavy particles, the uncertainty will be very small, because only one point is completely uncertain at the real instance of time. But for small particles, such as the electron, the uncertainty could become considerably large, because it is inversely proportional to : . This uncertainty in position will also increase with (the imaginary) velocity , or momentum , because higher physical velocity means that on average more and more points are becoming in the real speed , rather than rest, as can also inferred from equation 5. Moreover, we can now give an exact account of the collapse of the wave function, since the superposition state of a system of individual points comes from averaging their dual-state of zero or , all of which had already made its contribution except the real current one at the very real instance of time of measurement, which is going to be determined right in the following instance. Therefore, because the state of any individual point automatically collapses into zero after it makes its contribution to the total quantum state, once the moment passes, all states are determined automatically, although their eigenstate may remain unknown, as far as it is not measured. So, as in the original Copenhagen interpretation, the act of measurement only provides knowledge of the state. However, if the number of points in a system is very small, and since the observer is necessarily part of the system, the observation may have a large impact on determining the final eigenstate. Accordingly, the state of Schroedinger’s cat, after the box is closed, is either dead or alive; so it is already determined, but we only know that after we open the box, provided that the consciousness of the observer did not interfere during the measurement. Any kind of measurement or detection, necessarily means that the observer, or the measuring device, at this particular instance of measurement, is the subject that is acting on the system; and since there is only one state of vacuum and one state of void, at this real instance of time, the system must necessarily collapse into the passive state, i.e. it becomes the object or particle, because at this particular instance of time the observer is taking on the active state. Of course, this collapsing is not fatal, otherwise particles and objects will disappear forever, but they are re-created or excited again into a new state right after this instantaneous collapse, at which time the observer now would have moved back into an indeterminate state, and becomes an object amongst other objects. The uncertainty and non-locality of quantum mechanical phenomena result from the process of sequential re-creation, or the recurrence of only one geometrical point, which is flowing either in the inward or outward levels of time, which respectively produce the normal spatial entanglement as well as the temporal entanglement. Therefore, entanglement is the general underlying principle that connects all parts of the Universe in space and as well as in time, but it is mostly reduced into simple coherence, which may also dissipate quickly as soon as the system becomes complex. In other words: spatial and temporal entanglement is what defines space-time structure, rather than direct proximity. In this deeper sense, the speed of light is never surpassed even in extreme cases, such as the EPR and quantum tunneling, since there is no transmutation, but the object is re-created in new places which could be at the other end of the Universe, and even in a delayed future time. Consequently, whether the two particles are separated in space or in time, they can still interfere with each other in the same way because they are described by the same wave function either as one single entangled state or two coherent states. In this way we can explain normal as well as single particle interference, since the wave behavior of particles in each case is a result of the instantaneous uncertainty in determining their final physical properties, such as position or momentum, as they are sequentially re-created. Spatial entanglement occurs between the points in the internal level of time, while temporal entanglement is between the points of the outer level, so in reality it is all temporal since all the points of space and time are generated in one chronological order that first spreads spatially in the inner metaphysical level and then temporally in the outer physical level. On the other hand, since the whole Universe is self-contained in space, all changes in it are necessarily internal changes only, because it is a closed system. Therefore, any change in any part of the Universe will inevitably cause other synchronizing change(s) in other parts. In normal cases the effect of the ongoing process of cosmic re-creation is not noticeable because of the many possible changes that could happen in any part of the complex system, and the corresponding distraction of our limited means of attention and perception. This means that causality is no more directly related to space or even time; because the re-creation allows non-local and even non-temporal causal interactions. In regular macroscopic situations, the perturbation causes gradual or smooth, but still discrete, motion or change; because of the vast number of neighboring individual points, so the effect of any perturbation will be limited to adjacent points, and will dissipate very quickly after short distance, when energy is consumed. This kind of apparent motion is limited by the speed of light, because the change can appear infinitesimally continuous in space. In the special case when a small closed system is isolated as a small part of the Universe, and this isolation is not necessarily spatial isolation, as it is the case of the two entangled particles in the EPR, then the effect of any perturbation will appear instantaneous because it will be transferred only through a small number of points, irrespective of their positions in space, or even in time. The Duality of Time Theory exposes a deeper understanding of time, that reveals the discrete symmetry of space-time geometry, according to which the dimensions of space are dynamically being re-created in one chronological sequence at every instance of the outer level of time that we encounter. In this hidden discrete symmetry, motion is a result of re-creation in the new places rather than gradual and infinitesimal transmutation from one place to the other. When we approximate this discrete motion in terms of the apparent (average) velocity, this theory will reduce to General Relativity. We have shown that the resulting space-time is dynamic, granular, self-contained without any background, genuinely-complex and fractal, which are the key features needed to accommodate quantum and relativistic phenomena. Accordingly, many major problems in physics and cosmology can be easily solved, including the arrow of time, non-locality, homogeneity, dark energy, matter-antimatter asymmetry and super-symmetry, in addition to providing the ontological reason behind the constancy and invariance of the speed of light, that is currently considered an axiom. We have demonstrated, by simple mathematical formulation, how all the principles of Special and General Relativity can be derived from the Duality of Time postulate, in addition to exact mathematical derivation of the mass-energy equivalence relation, directly from the principles of Classical Mechanics, as well deriving the equivalence principle that lead to General Relativity. Previous studies have already demonstrated how the principles of Quantum Mechanics can be derived from the fractal structure of space-time, but we have also provided realistic explanation of quantum behavior, such as the wave-particle duality, the exclusion principle, uncertainty, the effect of observers and the collapse of wave function. We also showed that, in addition to being a perfect super-fluid, the resulting dynamic quintessence could reduce the cosmological constant discrepancy by at least 117 orders of magnitude. Starting from equation 8 above: and we can find by differentiating , with respect to : From this equation we find: , and by replacing in equation 26 we get: This method, however, can not be considered a mathematical validation of the mass-energy equivalence relation , because the starting equation is not derived by any other fundamental method from current Relativity, other than being analogous to the equations of time dilation and length contraction: , . Using the same equation with ; thus: , we can also derive the relativistic energy-momentum relation, by squaring and applying some modifications: From this equation we get: , thus , or: Again, since this derivation relies originally on the equation: , it can not be considered a mathematical validation of the mass-energy equivalence relation. O. Lauscher and M. Reuter. Asymptotic Safety in Quantum Einstein Gravity: nonperturbative renormalizability and fractal M. Joyce, F. Sylos Labini, A. Gabrielli, M. Montuori, and L. Pietronero. Basic properties of galaxy clustering in the light of recent results from the sloan digital sky survey. David W. Hogg, Daniel J. Eisenstein, Michael R. Blanton, Neta A. Bahcall, J. Brinkmann, James E. Gunn, and Donald P. Schneider. Cosmic homogeneity demonstrated with luminous red Laurent Nottale and Marie-Noëlle Célérier. Derivation of the postulates of quantum mechanics from the first principles of Mohamed Ali Haj Yousef. Mohamed Ali Haj Yousef. Zeno’s paradoxes and the reality of motion according to ibn al-arabi’s single monad model of the cosmos. In Sotiris Mitralexis, editor, Mohamed Ali Haj Yousef. Albert Einstein. Über die gültigkeitsgrenze des satzes vom thermodynamischen gleichgewicht und über die möglichkeit einer neuen bestimmung der elementarquanta.

Acting as a simple risk analysis, the payback period formula is easy to understand. It gives a quick overview of how quickly you can expect to recover your initial investment. This tab allows you to compare the economic merits of the current system and a base case system. The window displays cash flow graphs and a table of economic metrics. Some investments take time to bring in potentially higher cash inflows, but they will be overlooked when using the payback method alone. The payback period is the amount of time required for cash inflows generated by a project to offset its initial cash outflow. This calculation is useful for risk reduction analysis, since a project that generates a quick return is less risky than one that generates the same return over a longer period of time. There are two ways to calculate the payback period, which are described below. The shorter a discounted payback period is, means the sooner a project or investment will generate cash flows to cover the initial cost. The internal rate of return is understood as the discount rate, which ensures equal present values of expected cash outflows and expected cash inflows. An implicit assumption in the use of payback period is that returns to the investment continue after the payback period. nominal payback Payback period does not specify any required comparison to other investments or even to not making an investment. Present Value Vs Internal Rate Of Return Also, high liquidity is translated as a low level of risk. Finally, when there is an uncertain estimation and forecast of future cashflows, the payback period method is useful. All investors, investment managers, and business organizations have limited resources. Therefore, the need to make sound business decisions while selecting between a pool of investments is required. The payback period also facilitates side-by-side analysis of two competing projects. If one has a longer payback period than the other, it might not be the better option. The point after that is when cash flows will be above the initial cost. For the sake of simplicity, let’s assume the cost of capital is 10% (as your one and only investor can turn 10% on this money elsewhere and it is their required rate of return). If this is the case, each cash flow would have to be $2,638 to break even within 5 years. At your expected $2,000 each year, it will take over 7 years for full pay back. The payback period is the amount of time it would take for an investor to recover a project’s initial cost. It’s closely related to the break-even point of an investment. The payback period formula is also known as the payback method. Note that in both cases, the calculation is based on cash flows, not accounting net income (which is subject to non-cash adjustments). The payback period disregards the time value of money and is determined by counting the number of years it takes to recover the funds invested. For example, if it takes five years to recover the cost of an investment, the payback period is five years. That is, the profitability of each year is fixed, but the valuation of that particular amount will be placed overtime the period. Thus the payback period fails to capture the diminishing value of currency over increasing time. The concept does not consider the presence of any additional cash flows that may arise from an investment in the periods after full payback has been achieved. The payback period refers to the amount of time it takes to recover the cost of an investment or how long it takes for an investor to hit breakeven. To begin, the periodic cash flows of a project must be estimated and shown by each period in a table or spreadsheet. These cash flows are then reduced by their present value factor to reflect the discounting process. This can be done using the present value function and a table in a spreadsheet program. Example Of The Discounted Payback Period This may involve accepting both or neither of the projects depending on the size of the Threshold Rate of Return. For the Discounted Payback Period and the Net Present Value analysis, the discount rate is used for both the compounding and discounting analysis. So only the discounting from the time of the cash flow to the present time is relevant. - The discounted payback period is a capital budgeting procedure used to determine the profitability of a project. - Last but not the least, there is a payback rule which is also called the payback period, and it basically calculates the length of time which is required to recover the cost of investment. - Average cash flows represent the money going into and out of the investment. - So in the business environment, a lower payback period indicates higher profitability from the particular project. - For heat capacity flows below 4kW/ K, the optimisation resulted in no investment into solar thermal installations. - The resulting profitability indices are always positive. Both proposals are for similar products and both are expected to operate for four years. A project https://business-accounting.net/ has an initial outlay of $1 million and generates net receipts of $250,000 for 10 years. Modified Internal Rate Of Return It is an equal sum of money to be paid in each period forever. Thus we can compute the future value of what Vo will accumulate to in n years when it is compounded annually at the same rate of r by using the above formula. Future value is the value in dollars at some point in the future of one or more investments. Small projects may be approved by departmental managers. More careful analysis and Board of Directors’ approval is needed for large projects of, say, half a million dollars or more. Is the net cash flow at time t, USD; and t is time of cash flow. - The Net Present Value is the amount by which the present value of the cash inflows exceeds the present value of the cash outflows. - Thus, its use is more at the tactical level than at the strategic level. - The profitability index adjusts for the time value of money. - This is the cheapest way for the rich countries to delay climate change. - It can be used by homeowners and businesses to calculate the return on energy-efficient technologies such as solar panels and insulation, including maintenance and upgrades. - This firm’s forward looking PE ratio is equal to the expected payback period, which is the time it will take for the sum of the cash flows to equal the share price. So, based on this criterion, it’s going to take longer before the original investment is recovered. This is because they factor in the time value of money, working opportunity cost into the formula for a more detailed and accurate assessment. Another option is to use the discounted payback period formula instead, which adds time value of money into the equation. These two calculations, although similar, may not return the same result due to the discounting of cash flows. For example, projects with higher cash flows toward the end of a project’s life will experience greater discounting due to compound interest. For this reason, the payback period may return a positive figure, while the discounted payback period returns a negative figure. Capital Budgeting Basics Choosing the proper discount rate is important for an accurate Net Present Value analysis. Over the long run, capital budgeting and conventional profit-and-loss analysis will lend to similar net values. However, capital budgeting methods include adjustments for the time value of money (discussed in AgDM File C5-96, Understanding the Time Value of Money). Capital investments create cash flows that are often spread over several years into the future. To accurately assess the value of a capital investment, the timing of the future cash flows are taken into account and converted to the current time period . Suppose a situation where investment X has a net present value of 10% more than its initial investment and investment Y has a net present value of triple its initial investment. At first glance, investment Y may seem the reasonable choice, but suppose that the payback period for investment X is 1 year and investment Y is 10 years. - Both proposals are for similar products and both are expected to operate for four years. - The payback period can be found by dividing the initial investment costs of $100,000 by the annual profits of $25,000, for a payback period of 4 years. - However, to accurately discount a future cash flow, it must be analyzed over the entire five year time period. - If the project is accepted then the market value of the firm’s assets will fall by $1m. - This method does not require lots of assumptions and inputs. The crossover point is the rate or return that sets two mutually exclusive projects’ NPVs equal to zero. Both the IRR and the profitability index account for scale. The dividend growth rate cannot be greater than the cost of equity. The CIMA defines payback as ‘the time it takes the cash inflows from a capital investment project to equal the cash outflows, usually expressed in years’. When deciding between two or more competing projects, the usual decision is to accept the one with the shortest payback. Another issue with the formula for period payback is that it does not factor in the time value of money. This method totally ignores the solvency II the liquidity of the business. Payback period doesn’t take time value of money into account. The insurance companies are paying out a lot of money to settle their claims. The company paid off as many workers as it could before bankruptcy. Payback Method With Uneven Cash Flow: Discounted payback implies that the company will not accept any _______ NPV conventional projects. Managers use a variety of decision-making rules when evaluating long-term assets in the capital budgeting process. There is no one perfect rule; all have strengths and weaknesses. One thing all the rules have in common is that the firm must begin the capital budgeting process by estimating a project’s_____ ______. The quality of those estimates is critical to the sound application of the rules we will consider. Assume that the initial $11m cost is funded using the your firm’s existing cash so no new equity or debt will be raised. So whatever happened in the project is not going to be reflected in the payback period. This method only concentrates on the earnings of the company and ignores capital wastage and several other factors like inflation depreciation etc. Calculate the pay back period of buying the stock and holding onto it forever, assuming that the dividends are received as at each time, not smoothly over each year. The project has a positive net present value of $30,540, so Keymer Farm should go ahead with the project. ___________ _________ ________ Subtract a project’s discounted future cash flows from its initial cost until the firm recovers the initial investment. Accept the project if the discounted payback period is less than a predetermined time limit. An advantage of the discounted payback rule is that it adjusts for the time value of money, which is a failing of ordinary payback, though the advantage comes at the cost of increased complexity. Discounted payback implies that the company will not accept any negative NPV conventional projects. Beyond the TVM considerations, discounted payback has the same benefits and shortcomings as nominal payback. Cons Of Payback Period Analysis The shorter time scale project also would appear to have a higher profit rate in this situation, making it better for that reason as well. As a tool of analysis, the payback method is often used because it is easy to apply and understand for most individuals, regardless of academic training or field of endeavor. When used carefully to compare similar investments, it can be quite useful. As a stand-alone tool to compare an investment, the payback method has no explicit criteria for decision-making except, perhaps, that the payback period should be less than infinity. The payback method is a method of evaluating a project by measuring the time it will take to recover the initial investment. If the present value of a project’s cash inflows is greater than the present value of its cash outflows, then accept the project. If a project’s IRR is greater than the rate of return on the next best investment of similar risk, accept the project.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. By proving these particular identities, prove the existence of general cases. Find all the solutions to the this equation. With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even? Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods. What is the largest number of intersection points that a triangle and a quadrilateral can have? An article which gives an account of some properties of magic squares. An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes. Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics. Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n. In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)). We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions. Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number. In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot. Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry Can you discover whether this is a fair game? Some diagrammatic 'proofs' of algebraic identities and inequalities. Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . . This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself. When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators? Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started. This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition. Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. This is the second article on right-angled triangles whose edge lengths are whole numbers. Follow the hints and prove Pick's Theorem. The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it! A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices? It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more. Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions. To find the integral of a polynomial, evaluate it at some special points and add multiples of these values. A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers? The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers. Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions. What can you say about the common difference of an AP where every term is prime? This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats. Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. Kyle and his teacher disagree about his test score - who is right? What fractions can you divide the diagonal of a square into by simple folding? When is it impossible to make number sandwiches? If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle? A introduction to how patterns can be deceiving, and what is and is not a proof. Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials. These proofs are wrong. Can you see why? Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die? Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way? Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Alternatively, if the density of a substance is known, and is uniform, the volume can be calculated using its weight this calculator computes volumes for some of the most common simple shapes sphere. Density is the measurement of the amount of mass per unit of volumein order to calculate density, you need to know the mass and volume of the item the mass is usually the easy part while volume can be tricky. Abstract have you ever wondered how a ship made of steel can float or better yet, how can a steel ship carry a heavy load without sinking in this science project you will make little boats out of aluminum foil to investigate how their size and shape affects much weight they can carry and how this relates to the density of water. Explain how objects of similar mass can have differing volume, and how objects of similar volume can have differing mass explain why changing an object's mass or volume does not affect its density (ie, understand density as an intensive property. The car's weight and volume change, but not its mass a ball falling through the air calculate the density with the correct number of significant figures of a. Repeat steps 3 through 8 for the cylindrical copper mass and the spherical lead mass calculate the density of each of the objects and enter it in table 1 assume the density of water is r f = 10 3 kg/m 3. The density of aluminum is 270 g/cm3, and the average mass for one aluminum atom is 448×10-23 g five identical aluminum coins are found to displace a total of 250 ml of water when immersed in a graduated cylinder containing water. Density is the mass of an object divided by its volume density often has units of grams per cubic centimeter (g/cm 3 ) remember, grams is a mass and cubic centimeters is a volume (the same volume as 1 milliliter. It is common to use the density of water at 4 o c (39 o f) as a reference since water at this point has its highest density of 1000 kg/m 3 or 1940 slugs/ft 3 since specific gravity - sg - is dimensionless, it has the same value in the si system and the imperial english system (bg. The mass of water in the pycnometer at this tempearature will be determined by using mass of pycnometer, m w minus mass of empty pycnometer, m let d be the density of water at t°c, so that the volume of pycnometer, v at this temperature can be expressed in the term as. The mass of atoms, their size, and how they are arranged determine the density of a substance density equals the mass of the substance divided by its volume d = m/v objects with the same volume but different mass have different densities. The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume the symbol most often used for density is ρ (the lower case greek letter rho ), although the latin letter d can also be used. Density is defined as the ratio of an object's mass to its volume, as shown in the equation above because it is a ratio, the density of a material remains the same without regard to how much of that material is present. Density = mass / volume you know the density of aluminum and you know the peice of aluminum's mass, therefore you should be able to re-write the density equation in such a way to solve for volume imagine the square of aluminum foil as a very thin block of aluminum. If you have a pure liquid or a solid, you use its density to calculate its mass and then divide the mass by the molar mass if you have a solution, you multiply the molarity by the volume in litres. In the above equation, m is mass, ρ is density, and v is volume the si unit for density is kilogram per cubic meter, or kg/m 3 , while volume is expressed in m 3 , and mass in kg this is a rearrangement of the density equation. Density is the mass per unit of volume of a substance the density equation is: #density# = #mass/volume# to solve the equation for mass, rearrange the equation by multiplying both sides times volume in order to isolate mass, then plug in your known values (density and volume. Since sg = s / w, and since w = 1 gram/cm 3), one can determine the density of the object by measuring its mass and volume directly for a liquid the volumetric flask (or pycnometer ) has a hollow stem stopper that allows one to prepare equal volumes of fluids very reproducibly. Density 1 density the mass density or density of a material is defined as its mass per unit volumethe symbol most often used for density is ρ (the greek letter rho) in some cases (for instance, in the united states oil and gas industry), density is. Since density is mass per unit volume,the density of a metal can be calculated by submerging it in a known amount of water and measuring how much the water risesthis rise is the volume of the metal its mass can be measured using a scale. Method 1: determination of density by direct measurement of volume the object you have is a cube of metal the volume of a cube can be found from the formula v=a 3 , where a is the length of one edge in centimeters. The independent variable, volume, always goes on the x-axis through your data points using your graph, determine the mass of 100 ml of material. The density used in the calculations will appear in the density box in g/cc if you prefer to see the density in other units, just click the units drop-down box next to the density, and the value will be converted for you automatically. A little aluminum boat (mass of 1450 g) has a volume of 45000 cm 3 the boat is place in a small pool of water and carefully filled with pennies the boat is place in a small pool of water and carefully filled with pennies. The iron brick has twice the mass, but its volume compared to the block of wood depends on the density of the wood calculate the density with the correct number.

What Are Divisibility Rules? Divisibility rules are simple tips and tricks that are used to check or to test whether a number is divisible by another number. Consider an example. Imagine that you have 13 candy bars. Can you divide them equally among 3 friends? How would you check? You can check if 13 is “divisible by” 3. In other words, you can check if 13 appears in the table of 3 or not! Now, what if you wish to check if you can divide 221 candies equally among 6 friends? When we are dealing with large numbers, it can be very time-consuming to find 221 in the multiplication table of 6. What do you think? To solve problems like these in no time, we use divisibility rules. With divisibility rules at your fingertips, you can answer easily without doing too much calculation! Divisibility Rules: Definition Divisibility rules are a set of general rules that are often used to determine whether or not a number is absolutely divisible by another number. Note that “divisible by” means a number divides the given number, without any remainder and the answer is a whole number. Divisibility Test (Division Rules in Math) Mathematical tests for divisibility or division rules help you employ a quick check to determine whether a number will be totally divisible by another number. What are the divisibility rules? Let’s learn divisibility rules 1-13. Divisibility Rule of 1 Every number ever is divisible by 1. Divisibility Rule of 2 Every even number is divisible by 2. That is, any number that ends with 2, 4, 6, 8, or 0 will give 0 as the remainder when divided by 2. For example, 12, 46, and 780 are all divisible by 2. Divisibility Rules of 3 A number is completely divisible by 3 if the sum of its digits is divisible by 3. You can also repeat this rule, until you get a single digit sum. Example 1: Check whether 93 is divisible by 3 or not. Sum of the digits $= 9 + 3 = 12$ If the sum is a multiple of 3, then the original number is also divisible by 3. Here, as 12 is divisible by 3, 93 is also divisible by 3. Example 2: 45,609 To make the process even easier, you can also find the sum of the digits until you get a single digit. Sum of digits $= 4 + 5 + 6 + 9 + 0 = 24$ Adding further, we get $2 + 4 = 6$ 6 is divisible by 3. Thus, 45609 is divisible by 3. Divisibility Rule of 4 If the number formed by the last two digits of a number is divisible by 4, then that number is divisible by 4. Numbers having 00 as their last digits are also divisible by 4. Example 1: Consider the number 284. Check the last two digits. The last two digits of the number form the number 84. As 84 is divisible by 4, the original number 284 is also divisible by 4. Example 2: 1328 Thus, 1328 is also 4. Divisibility Rule of 5 If a number ends with 0 or 5, it is divisible by 5. For example, 35, 790, and 55 are all divisible by 5. Divisibility Rule of 6 If a number is divisible by 2 and 3 both, it will be divisible by 6 as well. For example, the numbers 6, 12, 18 are divisible by both 2 and 3. So, they are divisible by 6 as well. Divisibility Rules of 7 If subtracting twice of the last digit from the number formed by remaining digits is 0 or divisible by 7, the number is divisible by 7. This one is a little tricky. Let’s understand with an example. Example: Check whether 905 is divisible by 7 or not. Step 1: Check the last digit and double it. Last digit $= 5$ Multiply it by 2. $5 \times 2 = 10$ Step 2: Subtract this product from the rest of the number. Here, the remaining number $= 90$ $90 \;-\; 10 = 80$ Step 3: If this number is 0 or multiple of 7, then the original number is also divisible by 7. 80 is not divisible by 7. So, 905 is also not divisible by 7. Divisibility Rule of 8 If the number formed by the last three digits of a number is divisible by 8, we say that the number is divisible by 8. Example 1: In the number 4176, the last 3 digits are 176. If we divide 176 by 8, we get: Since 176 is divisible by 8, 4176 is also divisible by 8. Thus, 12,920 is divisible by 8. Divisibility Rule of 9 If the sum of digits of the number is divisible by 9, then the number itself is divisible by 9. You can keep adding further by repeating the rule. If the single-digit sum is 9, the number is divisible by 9. Example 1: Consider 189. The sum of its digits$ = (1+8+9) = 18$, which is divisible by 9, hence 189 is divisible by 9. Example 2: 12,897 Sum of digits $= 1 + 2 + 8 + 9 + 7 = 27$ Adding further, $2 + 7 = 9$ Thus, 12897 is divisible by 9. Divisibility Rule of 10 Any number whose last digit is 0 is divisible by 10. Example: 10, 20, 30, 100, 2000, 40,000, etc. Divisibility Rule for 11 If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11. Example 1: Consider the number. 2846767. First, understand the digit positions. We find two sums: the sum of digits at the even places and the sum of digits at the odd places. Sum of digits at even places (From right) $= 8 + 6 + 6 = 20$ Sum of digits at odd places (From right) $= 7 + 7 + 4 + 2 = 20$ Difference $= 20 – 20 = 0$ Difference is divisible by 11. Thus, 2846767 is divisible by 11. Example 2: Is 61809 divisible by 11? Group digits that are in odd places together and digits in even places together. Alt tag: identify digits alternate places: divisibility rule of 11 Here, $6 + 8 + 9 = 23$ and $0 + 1 = 1$ Difference $= 23 \;-\; 1 = 22$ 22 is divisible by 11. Thus, the given number is divisible by 11. Another Divisibility Rule For 11 There’s another simple divisibility rule for 11. Subtract the last digits from the remaining number. Keep doing this until we get a two-digit number. If the number obtained is divisible by 11, the original number is divisible by 11. $174\;-\;9 = 165$ $16\;-\;5 = 11$ … divisible by 11 Thus, 1749 is divisible by 11. Divisibility Rule of 12 If the number is divisible by both 3 and 4, then the number is divisible by 12 Sum of the digits $= 4 + 8 + 8 + 0 = 20$ (not a multiple of 3) Last two digits $= 80$ (divisible by 4) The given number 4880 is divisible by 4 but not by 3. Thus, 4880 is not divisible by 12. Divisibility Rules of 13 To check if it is divisible by 13, we add 4 times of the last digit of the remaining number and repeat the process until we get a two-digit number. If that two-digit number is divisible by 13, then the given number is divisible by 13. Example: Is 4186 divisible by 13? - $418 + (6 \times 4) = 418 + 24 = 442$ - $44 + (2 \times 4) = 44 + 8 = 52$ 52 is divisible by 13 since $13 \times 4 = 52$. Thus, 4186 is divisible by 13. Divisibility Rules: Chart |Divisibility Rules Chart| |Divisibility by 1||Every number is divisible by 1.| |Divisibility by 2||When the last digit is 0, 2, 4, 6, or 8| |Divisibility by 3||When the sum of digits is divisible by 3| |Divisibility by 4||When the last two digits of any dividend are divisible by 4 (NOTE: Numbers having 00 as their last digits are also divisible by 4.)| |Divisibility by 5||When the last digit is either 0 or 5| |Divisibility by 6||When the number is divisible by both 2 and 3| |Divisibility by 7||When the last digit is subtracted twice from the remaining digits and gives the multiple of 7| |Divisibility by 8||When the last three digits are divisible by 8(NOTE: Numbers having 000 as their last digits are also divisible by 8.)| |Divisibility by 9||When the sum of all digits is divisible by 9| |Divisibility by 10||When the last digit is 0| |Divisibility by 11||When the difference between the sums of the alternative digits is divisible by 11| |Divisibility by 12||When a number is both divisible by 3 and 4| |Divisibility by 13||Multiply 4 with the last digit and add this product to the remaining number. Continue till a two-digit number is found. If the 2-digit number is divisible by 13, the number is divisible by 13.| Facts about Divisibility Rules - “Divisible” means a number is able to be divided evenly with another number with NO remainders. - Divisibility rule is a shortcut to analyze whether an integer is completely divisible by a number without actually doing the calculation. - Zero is divisible by any number (except by itself), so it gets a “yes” to all these tests. - When a number is divisible by another number, it is also divisible by each of the factors of that number. For instance, a number divisible by 6 will also be divisible by 2 and 3. A number divisible by 10 is also divisible by 5 and 2. - Numbers that have two zeros at the end are divisible by 4. Numbers with three zeros at the end are divisible by 8. - The number 2,520 is the smallest number that is divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10. In this article, we have learned divisibility rules and charts with examples. Let’s solve some divisibility rules examples to understand it better. Solved Examples for Divisibility Rules 1. If a number is divisible by 6, can we say it is divisible by 2 as well? Yes, because 6 is divisible by 2. If a number is divisible by some numbers, say x, that number is also divisible by factors of x. For example: 480 is divisible by 6. $480 \div 6 = 80$. $480 \div 2 = 240$. Also $480 \div 3 = 160$ Thus, If a number is divisible by 6, can we say it is divisible by 2 and 3 as well, as 2 and 3 are factors of 6. 2. Use divisibility rules to check whether 642 is divisible by 4 and 3. Divisibility rule for 4: If the last two digits of a number are divisible by 4, then that number is divisible by 4. The last two digits of $642 = 42$, which is not divisible by 4. Thus, 542 is not divisible by 4. Divisibility rule of 3: If the sum of digits is divisible by 3, we say that the original number is divisible by 3. Sum of digits $= 6 + 4 + 2 = 12$ 12 is divisible by 3. So, 542 is not divisible by 8. 3. Check on 3640 for divisibility by 13. The last number of the given number is 0, Multiply 4 by 0 and add to the rest of the number. $364 + (0 \times 4) = 364$. Again multiply 4 by the last digit of the obtained three-digit number and add to the rest of the digits as: $36 + (4 \times 4) = 52$ Now, a two-digit number 52 is obtained, which is divisible by 13. $52 = 4 \times 13$. Hence, 3640 is divisible by 13. Practice Problems on Divisibility Rules Rules of Divisibility: Definition, Chart, Examples Which number is not divisible by 5? According to divisibility rule of 5, If the last digit of a number is 5 or 0, the number is always divisible by 5. So, 680 is divisible by 5. Which of the following numbers is divisible by 2? All even numbers are divisible by 2. Which one of the following numbers is divisible by 6? According to the rule of divisibility by 6, the number that is divisible by both 2 and 3 is also divisible by 6. Only 18 of all the given numbers are divisible by 2 as well as 3. Identify a number divisible by 9. Sum of digits in the number $117 = 1 + 1 + 7 = 9$. The sum is divisible by 9. Thus, 117 is divisible by 9. For all the other options, the sum of digits in a number is not divisible by 9. Frequently Asked Questions on Divisibility Rules What are co-primes and their divisibility rules? Co-primes are the pair of numbers that have 1 as the common factor. If the number is divisible by such co-primes, the number is also a divisible by-product of the co-primes. For example, 14 is divisible by both 2 and 7. They are co-primes that have only 1 as the common factor, so the number is divided by 14, the product of 2 and 7. When is a number said to be a factor of another number? A number x is said to be a factor of number y if y is divisible by x. For example, 10 is divisible by 2, so 2 is a factor of 10. Where do we use divisibility rules in real life? Divisibility rules are the quickest way to determine if a number is divisible by another number. It saves the time required to perform the actual division. Divisibility rules also give you a number sense when it comes to division and multiplication of two or more numbers. What are composite numbers? In math, composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers. Factors of $4 = 1, 2, 4$, i.e., Since 4 has more than two factors, 4 is a composite number. How many divisibility rules are there? We often have standards for divisibility from 1 to 20. However, if we were able to recognize the pattern of multiples of integers, we could develop further tests for divisibility. For example, the divisibility rule of 21 states that a number must be divisible by both 3 and 7. It is because 21 is a multiple of two prime numbers 3 and 7, so all the multiples of 21 will definitely have 3 and 7 as their common factors.

The direct shear test is generally conducted on sandy soils as a consolidated engineering properties of based laboratory testing158direct video showing the basics of the direct shear test along with explanations of soil dilation and principal plane rotation during a direct shear test. The purpose of direct shear test is to get the ultimate shear resistance, peak shear resistance, cohesion, angle of shearing resistance and stress-strain characteristics of the soils. Direct shear tests were performed to determine the internal friction angles of the density and model sands direct shear test results, in terms of shear stress versus horizontal displacement curves, are presented summary of interface shear test results on density sand pile/material dr (%. Instructor: dr george mylonakis lab experiment #7: direct shear test introduction the shear φ where, σ' = effective normal stress φ = angle of friction of soil φ = f( d r , d , e , an ) where, d procedure 1 measure the internal diameter of the cylindrical cell 2 balance the counter weight. Laboratory testfor determination of shear strength parameters a direct shear test b triaxial test c direct simple shear test d plane strain triaxial test e torsional rins shear test in many foundation designproblems,one must determine the angle of fric- tion between the soil and the. In this laboratory, a direct shear device will be used to determine the shear strength of a cohesionless soil (ie angle of internal friction 13 170 direct shear test data sheet date tested: tested by: project name: sample number: visual classification: shear box inside diameter. In an ordinary laboratory direct shear test, with the applied shearing force monitored on the y axis and strain on the x axis, the area under the data curve represents work done on the soil sample in this way, higher values of sample shear strength correlate approximately with higher amounts of work involved shearing the sample to its ultimate. Angle of internal friction (friction angle) a measure of the ability of a unit of rock or soil to withstand a shear stress it is the angle (φ), measured between the normal force (n) and resultant force (r), that is attained when failure just occurs in response to a shearing stress (s) its tangent (s/n. From this test, coulomb parameters, including cohesion and internal friction angle, as well as, bekker parameters can be infferred it has been observed that the inclination angle of particles during an avalanche is consistently higher than the angle of repose for granular materials. Direct shear test, triaxial test and unconfined compression test the shear strength value can be determined as shown, where φ = angle of internal friction c = cohesive stress or adhesion stress. Angle of internal friction, , can be determined in the laboratory by the direct shear test or the triaxial stress test typical relationships for estimating the angle of internal friction, , are as follows: empirical values for , of granular soils based on the standard penetration number. The direct shear test used for soil (powers 1968) can be performed with fresh concrete to assess the cohesive strength of a concrete mixture the test provides additional information, namely the angle of internal friction, not available from most conventional tests. A lab experiment on conducting a direct shear test to determine the angle of internal friction of dry sand. Direct shear test analysis of test results note: cross-sectional area of the sample changes with the horizontal displacement 50 interface tests on direct shear apparatus in many foundation design problems and retaining wall problems, it is required to determine the angle of internal friction. These test results further indicate that owing to the influence of the frictional force of the upper shear box, a higher internal angle of friction was measured in the conventional direct shear test on dilative sample without any improvements. Direct shear test to determine angle of internal friction of fine sand having different dry densities and mix compositions with different percentage by weight of ceramic tile waste material 3. For the current lab, a granular soil is used (dry swelih sand), and to determine the shearing strength of the soil using the direct shear apparatus :general disscussion. This recommendation is based on the relationship between friction angle and dilation angle for all aggregates tested in this program and assumes a 95-percent confidence interval the proposed new default of 39° corresponds to the peak friction angle of 49° (see figure 7), less two standard deviations. In australia, q181c test method of direct shear testing to estimate the effective angle of internal friction at constant volume conditions for granular the results show that accurate effective friction parameter measurements for coarse grained, granular backfill soils require the use of fresh soil. A direct shear test is a laboratory or field test used by geotechnical engineers to measure the shear strength properties of soil or rock several specimens are tested at varying confining stresses to determine the shear strength parameters, the soil cohesion (c) and the angle of internal friction. A direct shear test is a laboratory test used by geotechnical engineers to find the shear strength parameters of soil several specimens are tested at varying confining stresses to determine the shear strength parameters, the soil cohesion (c) and the angle of internal friction (commonly. 18 civil engineering - texas tech university direct shear test direct shear test is quick and inexpensive shortcoming is that it fails the soil on a designated plane which may not be the weakest one used to determine the shear strength of both cohesive as well as non-cohesive soils. Direct shear box test contents introduction objective apparatus description of test results calculations relevance to geotechnics soil objective to determine the angle of shearing resistance of a sample of sand the test may be carried out either dry or fully saturated but not. Dr h c e meyer-peter, and the chief of the soil mechanics laboratory, professor dr ing r haefeli, for permission to carry out the tests and for valuable support during the work.

1 Detecting and measuring faint point sources with a CCD Herbert Raab a,b a Astronomical ociety of Linz, ternwarteweg 5, A-400 Linz, Austria b Herbert Raab, chönbergstr. 3/1, A-400 Linz, Austria; tars, Asteroids, and even the (pseudo-)nuclei of comets, are point-sources of light. In recent times, most observers use CCDs to observe these objects, so it might be worthwhile to think about some details of detecting and measuring point sources with a CCD. First, this paper discusses the properties of point-sources, and how they can describe them with a small set of numerical values, using a Point pread Function (PF). Then, the sources of noise in CCD imaging systems are identified. By estimating the signal to noise ratio (NR) of a faint point source for some examples, it is possible to investigate how various parameters (like exposure time, telescope aperture, or pixel size) affect the detection of point sources. Finally, the photometric and astrometric precision expected when measuring faint point sources is estimated. Introduction Modern CCD technology has enabled amateur astronomers to succeed in observations that were reserved to professional telescopes under dark skies only a few years ago. For example, a 0.3m telescope in a backyard observatory, equipped with a CCD, can detect stars of 0 mag. However, many instrumental and environmental parameters have to be considered when observing faint targets. Properties of Point ources In long exposures, point sources of light will be smeared by the effects of the atmosphere, the telescope optics, vibrations of the telescope, and so forth. Assuming that the optics are free of aberrations over the field of the CCD, this characteristic distribution of light, called the Point pread Function (PF) is the same for all point sources in the image. Usually, the PF can be described by a symmetric Gaussian (bell-shaped) distribution (figure 1) : 0 0 ( x x ) + ( y y ) σ I ( x, y) = H e + B (1) I (x,y) is the intensity at the coordinates (x,y), which can be measured form the image. By fitting the PF to the pixel values that make up the image of the object (figure 1), the quantities x 0, y 0, I, σ and B can be found, which characterize the point source as follows: Width In equation 1, the width of the Gaussian PF is characterized by the quantity σ. In astronomy, the width of the PF is frequently specified by the socalled Full Width Half Maximum (FWHM). As the name implies, this is the width of the curve at half its height. The FWHM corresponds to approximately.355 σ. Although a number of factors control the FWHM (like focusing, telescope optics, and vibrations), it is usually dominated by the seeing. The FWHM is the same for all point-sources in the image (if optical aberrations can be neglected). Most notably, it is independent of the brightness of the object. Bright stars appear larger on the image only because the faint outer extensions of the PF are visible. For faint stars, these parts drown in the noise and are therefore not visible. Background During the exposure, the CCD not only collects signal from the object, but also light from the sky background and the thermal signal generated within the detector. These signals result in a pedestal (B) on which the PF is based. Ideally, the background signal is the same over the whole field for calibrated images. In practice, however, it will vary somewhat over the field. Position The position of the object in the CCD frame can be expressed in rectangular coordinates (x 0, y 0 ), usually along the rows and columns of the CCD. Fitting a PF to the image will allow to calculate the position of the object to a fraction of the pixel size. Intensity The height of the PF (H) is proportional to the magnitude of the object. The total flux of the objects corresponds to the integrated volume of the PF, less the background signal (see below). Figure 1: Image of a star on a CCD (left), and the Gaussian PF fitted to the image data (right). 2 ignal and Noise As briefly mentioned above, the CCD not only collects light from celestial objects, but also some unwanted signals. The thermal signal, for example, can be subtracted from the image by applying a dark frame calibration, but the noise of the thermal signal remains even in the calibrated image. In addition to the thermal noise, the readout noise is generated in the detector. External sources of noise are the photon noise in the signal from the sky background, as well as the photon noise in the signal of the object under observation. The Poisson noise in a signal (that is: the standard deviation σ of the individual measurements from the true signal) can be estimated as the square root of the signal, i.e. σ = () where is the signal (for example, the thermal signal), and σ is the noise level in that signal (in that example, the thermal noise). The total noise from the four independent noise sources mentioned above add in quadrature to give the total noise: σ σ + σ + σ + σ = (3) B where σ is the total noise, σ B is the background noise, σ s is the object noise, σ T is the thermal signal, and σ R is the readout noise. The ignal Noise Ratio (NR) can be calculated from: T R NR = (4) σ Where is the signal from the object, and σ the total noise. By combining equations to 4, it is possible to calculate the NR in one pixel: NR = (5) + B + T +σ Here, is the signal from the object collected in the pixel, B the signal from the sky background and T the thermal signal collected by the pixel, respectively, and σ R is the readout noise for one pixel. If equation 5 is applied to the brightest pixel in the image of the object, the result is the Peak NR for that object. The Peak NR is important, as software (or humans) can detect faint objects only if at least the brightest pixel has a NR over some threshold that is set to avoid false detections in the image noise. Usually, a Peak NR of ~3 is considered to be a marginal detection. In other words, this would correspond to the limiting magnitude of the image. For unfiltered or broadband images, the dominant source of noise is usually the sky background, even under very dark skies. With modern, cooled CCDs, the instrumental noise is generally less important, and object noise is only significant for very bright objects. R Figure : Growth of ignal, Noise, and ignal to Noise Ratio with increasing exposure time. Figure shows the growth of ignal, Noise, and ignal to Noise Ratio (NR) with increasing exposure time t. Note that, in this example, the background signal B is stronger than the signal from the object under observation. The background noise is σ B, the object noise is σ. The readout noise is independent of the exposure time and it is therefore not drawn. (For sky-limited exposures, it can practically be neglected.) The signal grows linear with increasing exposure time, as do the background signal B and the thermal signal T. Fortunately, the background noise (σ B = B) and the thermal noise (σ T = T) grow slower. Doubling the exposure time will increase all signals (,B,T) by a factor of, but the noise levels (σ s, σ B, σ T, σ) by a factor of only, so the NR increases by =. With increasing exposure, the faint object will eventually emerge from the noise, even though the background signal is always stronger than the signal from the object in this example. Estimating the ignal to Noise Ratio With a few, mostly very simple calculations, it is possible to estimate the ignal to Noise Ratio that can be expected for a stellar object of known magnitude with a certain equipment. In this chapter, one example is described in some detail. Further examples in the following chapters will be used to compare various telescope setups, and the gain (or loss) in the NR. The telescope used in this example is a 0.6m f/3.3 reflector, with a central obstruction of 0.m. As a detector, a CCD with 4µm square pixels (corresponding to.5 at the focal length of 1.98m), a dark current of one electron per second per pixel, a readout noise of ten electrons per pixel, and a mean quantum efficiency of 70% over the visible and near infrared portion of the spectrum (400nm to 800nm) is used . We assume a stellar object of 0 mag as the target of the observation, the brightness of the sky background to be 18 mag per square arc second, and the FWHM of the stellar image to be 4. In that spectral range, we receive about photons per second per square meter from a star of 0 mag . A difference of 1 mag corresponds to a factor of.5 in the brightness, so there will be only , or about 440 photons per second per square meter from our target. The light collecting area of the 0.6m telescope is 0.5m², so it will accumulate photons in a 100 3 second exposure. With a quantum efficiency of 0.7, this will generate about electrons in the CCD. Assuming that the PF of the object can be described with equation 1, and that the peak brightness is located exactly at the centre of one pixel, this pixel collects about 9% of the total light, or about photons, which will generate 33 electrons in that pixel. The Poisson noise of this signal is 33 ~ 47. In analogy to the stellar flux, we can estimate the flux from the sky background (18 mag per square arc second) to be , or about 748 photons per second per square meter. The telescope therefore collects about photons from each square arc second during the exposure. Each pixel covers 6.5 square arc seconds, and therefore, about photons from the sky background will be collected during the exposure in each pixel. This will generate about electrons, with a Poisson noise of ~ 548 electrons. During the exposure, the dark current will generate 100 electrons in each pixel, and the dark noise is therefore 100 = 10. The readout adds further 10 noise electrons. Using equation 3, the total noise in the brightest pixel can be calculated by adding the object noise in the brightest pixel, the sky noise, the dark noise and the readout noise in quadrature, i.e. (47² + 548² + 10² + 10²) ~ 550. The Peak ignal Noise Ratio is now found to be ~ 4.1. Obviously, the 0 mag object is only marginally detected in this example. Although this is a simplified calculation (e.g., no attempt to correct for atmospheric extinction was made, and no attention was given to the saturation of pixels, etc.), it is still a reasonable estimate. ome further telescope setups will be compared in the next chapters, and the results are compared. All calculations are summarized in table 1 in the Appendix. Exposure Time In the previous chapter, a star of 0 mag is only marginally detected with a 0.6m f/3.3 telescope in a 100 second exposure. In the next example, the exposure time is extended to 600 seconds to increase the ignal Noise Ratio of the object. The calculation, which is summarized as example in table 1 in the Appendix, shows that the NR of the brightest pixels increases from 4.1 to It has been noted previously that increasing the exposure time by a factor of n will raise the NR by a factor of n. In this example, the exposure time has been increased by a factor of 6, and the NR was raised by a factor of 6 ~.45. The limiting magnitude of an image can be defined by the brightness of the stars reaching some minimal NR, for example, 3.0. A factor of.5 in brightness corresponds to one magnitude, which closely matches the increase in NR due to the longer exposure. To increase the limiting magnitude by one full magnitude, the exposure time would have to be extended by a factor of 6.5, i.e., to 65 seconds. Pushing the limiting magnitude down by one more magnitude, another increase by a factor of 6.5 would be necessary: the exposure time would increase to about 3900 seconds, or 65 minutes (figure 3). Figure 3: Relative exposure time required for increasing the limiting magnitude. Telescope Aperture In the next example, we will expand the telescope aperture from 0.6m (as used in the previous examples) to 1.5m, with a central obstruction of 0.5m in diameter and a focal length of 7m. For the environment (sky background, seeing) and the detector, the same values as in the previous examples are used, and a exposure time of 100 second (as in example 1) assumed. The result of the calculation, which is summarized as example 3 in table 1 in the Appendix, is somewhat surprising: Although the 1.5m telescope has 6.5 times more light collecting area than the 0.6m instrument, the Peak NR is now only 3.4. Compared to the Peak NR of 4.1 that was found for the 100 second integration with the 0.6m telescope, this is a loss of ~0. mag in limiting magnitude. How can this be? Due to the long focal length of the telescope, each pixel now covers only Compared to the 0.6m telescope from the previous examples (pixel size.5.5 ), this is only 8% of the area. By combining the increased light collecting power, and the smaller pixel scale, we find that each pixel receives only about times the light collected in one pixel of the CCD by the smaller telescope. As both the light from the object and from the sky background (the dominant source of noise in these examples) drop by the factor of 0.5, the NR should decrease approximately by a factor of ~ 0.7. The true factor found by comparing the NR calculated in examples 1 and 3 is only about 0.8, because the PF is a non-linear function (equation 1), concentrating more light in the centre of the pixel than in the outer regions that were lost due to the smaller angular size of the pixels in that example. Does this mean that it makes no sense to use larger telescopes? Of course not! Apparently, the problem is related to the pixel scale, so pixel binning might be of some help: By using binning (example 4), a Peak NR of 6.4 is obtained, which corresponds to an increase in limiting magnitude of about 0.5 mag as compared to the 0.6m telescope in example 1, or of 0.7 mag as compared to the 1.5m telescope with the CCD used without binning (example 3). caling the FWHM from 4 to (by improving the telescope optics, the focusing, the mechanics or the seeing, if possible in some way) would be even better than binning: The peak NR would grow to 1.7, and the gain 4 in limiting magnitude is about 1. mag, as compared to example 1, or 1.4 mag as compared to example 3. Pixel ize and ampling Apparently, the relative size of the pixel to the FWHM of the stellar images is an important factor in obtaining the highest possible NR. By performing calculations similar to the NR estimates in the previous chapters, it can be shown that the highest Peak NR is obtained when the pixels are about 1. FWHM in size (figure 4). Figure 5: Undersampled (left), critically sampled (center) and oversampled (right) stellar images (top row), and the PF fitted to the image data (bottom row). Error Estimates Figure 4: Variation of peak NR for various pixel scales. The pixel size is measured in units of FWHM. With such large pixels, most of the photons are collected by the single pixel on which the PF of the stellar image is centred, whilst only the fainter, noisy wings of the PF fall on the neighbouring pixels, resulting in a high NR. However, with almost all the light concentrated in a single pixel, it would be very difficult to distinguish real objects from image artefacts (like hot pixels or cosmic ray strikes), and it is impossible to calculate the precise position of the object to sub-pixel accuracy. To retain the information of the objects on the CCD image, the scale must be chosen so that the FWHM of stellar sources spans at least 1.5 to pixels . This scale is called critical sampling, as it preserves just enough information that the original PF can be restored by some software analysing the image. With even larger pixels (i.e., less than 1.5 pixels per FWHM), the PF can not be restored with sufficient precision, and astrometric or photometric data reduction is inaccurate, or not possible at all. This situation is called undersampling. In the other extreme ( oversampling ) the light of the object is spread over many pixels: Although the PF of stellar objects can be restored with high precision in this case, the NR is decreased (figure 5). Critically sampled images will give the highest NR and deepest limiting magnitude possible with a given equipment in a certain exposure time, without loosing important information contained in the image. For applications that demand the highest possible astrometric or photometric precision, one might consider some oversampling. The same is true for pretty pictures, as stars on critically sampled images look rather blocky. Fitting a PF profile to a faint, noisy detection is naturally less precise than for bright stellar images with a high NR (figure 6). Position and brightness calculated for faint detections are therefore expected to be less precise than for bright objects. Figure 6: Gaussian PF fitted to a faint (Peak NR ~4) and a bright (Peak NR ~100) stellar image. The fractional uncertainty of the total flux is simply the reciprocal value of the ignal to Noise Ratio, 1 NR (sometimes also called the Noise to ignal Ration). By converting this uncertainty to magnitudes, we get: PHOT 1 Log(1 + ) = NR Log(.5) σ (6) Here, σ PHOT is the one-sigma random error estimated for the magnitude measured, and NR is the total NR of all pixels involved (e.g., within a synthetic aperture centred on the object). By modifying equation 5, we can find this value from: NR = (7) + n ( B + T +σ ) In this formula, is the total integrated signal from the object in the measurement (i.e., within the aperture), and n is the number of pixels within the aperture. The other quantities are identical to equation 3. It should be noted that, as both and n will change with the diameter of the R 5 aperture, the total NR varies with the diameter of the photometric aperture, so photometry can be optimised by choosing the appropriate aperture . Figure 7: The photometric error (in stellar magnitudes) expected for point-sources up to a NR of 50. Figure 7 shows the expected uncertainty in the magnitude for point sources up to NR 50, as calculated from equation 6. Equation 6 only estimates the random error in photometry due to image noise. It does not account for any systematic errors (like differences in spectral sensitivity of the CCD and the colour band used in the star catalogue) that might affect absolute photometric results. Provided that the stellar images are properly sampled, the astrometric error can be estimated using this equation : σ PF σ AT = (8) NR Here, σ AT is the estimated one-sigma error of the position of the object, σ PF the Gaussian sigma of the PF (as in equation 1), and NR is the Peak ignal to Noise Ratio of the object. Note that σ AT will be expressed in the same units as σ PF (usually arc seconds), and that σ PF can be calculated from FWHM.355. calculated from equation 8. Again, equation 8 only estimates the random error in the stellar centroid due to image noise. It does not account for any systematic errors (introduced by the astrometric reference star catalogue, for example) that might affect absolute astrometric results. Returning to example 1, the astrometric one-sigma error expected for the point source with a Peak ignal to Noise Ratio of 4.1 and a FWHM of 4 can now be estimated to ~0.4, using equation 8. Adopting a photometric aperture with a diameter of 3 FWHM (covering 18 pixels), a total ignal to Noise Ratio of about 3.3 is found by using equation 7. From equation 6, the photometric error is estimated to ~0.3 mag. An astrometric error of ~1 is acceptable, particularly if it is a observation of a minor planet with a uncertain orbital solution or a large sky-plane uncertainty (for example, as in the case of late follow-up or recovery observations). Observations of the light curve of a minor planet usually require a precision of 0.05 mag or better, corresponding to a NR of 0 or higher. Obviously, photometric observations are much more demanding than astrometry. ummary and Conclusions This paper first described the characteristics of a Gaussian Point pread Function, and the sources of noise in the imaging system. A few examples, estimating the ignal to Noise Ratio obtained for faint point sources with various telescope setups, highlighted that environmental conditions, telescope equipment, and CCD detector must harmonise to operate at peak performance. Finally, the astrometric and photometric error expected when measuring faint point sources was estimated. Useful astrometric results can be obtained even for very faint targets at the limit of detection, particularly if the skyplane uncertainty for the object under observation is large. For photometric studies, a higher NR is desirable. References 1. Auer, L. H.; van Altena, W. F.: Digital image centering II, Astronomical Journal, 83, (1978). cientific Imaging Technologes, Inc: ITe cientific-grade CCD 3. Berry, R; Burnell, J.: The Handbook of Astronomical Image Processing, Willmann-Bell, Inc. (000) 4. Howell,. B.; Koehn, B; Bowell, E.; Hoffman, M.: Detection and Measurement of poorly sampled Point ources images with -D Arrays, The Astronomical Journal, 11, (1996) 5. Naylor, T.: An optimal extraction algorithm for imaging photometry, Monthly Notices of the Royal Astronomical ociety, 96, (1998) Figure 8: The astrometric error (in units of the FWHM) expected for point-sources up to a Peak NR of 50. Figure 8 shows the expected uncertainty in the position (in units of FWHM) for point sources up to NR 50, as 6. Neuschaefer, L. W.; Windhorst, R.A.: Observation and Reduction Methods of deep Palomar 00 inch 4-hooter Mosaics, The Astrophysical Journal upplement eries, 96, (1995) 6 Appendix ignal to Noise Ratio Estimation Example 1 Example Example 3 Example 4 Example 5 Telescope Mirror Diameter 0.60 m 0.60 m 1.50 m 1.50 m 1.50 m Obstruction 0.0 m 0.0 m 0.50 m 0.50 m 0.50 m Light Collecting Area 0.5 m² 0.5 m² 1.57 m² 1.57 m² 1.57 m² Local Length 1.98 m 1.98 m 7.00 m 7.00 m 7.00 m Focal Ratio Detector Pixel ize 4 µm 4 µm 4 µm 48 µm 4 µm Pixel cale.50 /Pixel.50 /Pixel 0.71 /Pixel 1.4 /Pixel 0.71 /Pixel Dark Current 1 e /s/pixel 1 e /s/pixel 1 e /s/pixel 4 e /s/pixel 1 e /s/pixel Readout Noise 10 e 10 e 10 e 0 e 10 e Quantum Efficiency 70 % 70 % 70 % 70 % 70 % Integration Time 100 s 600 s 100 s 100 s 100 s Object and ky Object Magnitude 0 mag 0 mag 0 mag 0 mag 0 mag ky Background 18 mag / 18 mag / 18 mag / 19 mag / 19 mag / FWHM NR Calculation Object Flux 11'000 γ γ γ γ γ Object ignal e e e e e hare for central Pixel Object Flux in centr. Pixel γ γ γ γ γ Object ignal in centr. Pixel 33 e e e 5 09 e 5 09 e Object Noise in centr. Pixel 47 e 116 e 36 e 71 e 71 e Background Flux γ/pixel ' γ/pixel γ/pixel γ/pixel γ/pixel Background ignal e /pixel e /pixel e /pixel e /pixel e /pixel Background Noise 548 e /pixel 134 e /pixel 390 e /pixel 780 e /pixel 390 e /pixel Dark Current 100 e /pixel 600 e /pixel 100 e /pixel 400 e /pixel 100 e /pixel Dark Noise 10 e /pixel 5 e /pixel 10 e /pixel 0 e /pixel 10 e /pixel Noise in centr. Pixel 550 e 134 e 39 e 783 e 397 e Peak NR Table 1: ummary of the NR calculations mentioned in the text. Example 1 is described in some detail in the paper. Note that, for example 4, the pixel size listed in the table is not the physical size, but the site of the binned pixel, and all other data refer to the binned pixel.

Arithmetic (all content) - Comparing fractions with > and < symbols - Comparing fractions with like numerators and denominators - Compare fractions with the same numerator or denominator - Comparing fractions - Comparing fractions 2 (unlike denominators) - Compare fractions with different numerators and denominators - Comparing and ordering fractions - Ordering fractions - Order fractions Comparing Fractions. Created by Sal Khan and Monterey Institute for Technology and Education. Want to join the conversation? - Why did Sal, in the 54/81 problem at1:51make a dot, erase it, and then write x?(115 votes) - well, the reason he did that because in some school districs they teach their students that a dot means to multiply (x). Its like a little easier shorter thing for them to write. If you have any more questions just ask me! :D(13 votes) - Can't you simplify fractions even though you, like can't? Take 5/12, for example. Can't you simplify that using what you know about decimals? 🤔🤔🤔(1 vote) - You can convert a simplified fraction to a decimal, but you cannot make a simplified fraction into an even more simplified fraction.(5 votes) - Why would u have to do the 9 divided by 9 if u dont actually use it?(1 vote) - You very well might need the fraction 9/9. It may be equal to 1, but it has straightforward applications. You are splitting 9 gumballs among 9 friends? How many gumballs does each friend get? The answer is one, because 9/9 = 1.(5 votes) - 24367/6 in lowest tearms(3 votes) - 5 Goes equally into both 30 and 45 and is less than 15, so why didn't he use 5?(3 votes) - Because by using the GREATEST of the common factors, you end up with the fraction in its lowest terms. 5 would not have accomplished that.(1 vote) - Do you have to make the denominators the same to find the comparison?(2 votes) - Not always. There are several ways to compare fractions. The most general method, that always works for any fractions, is to change to equivalent fractions with a common denominator and then compare the numerators. This works because you are expressing both numbers with a common unit (like halves, thirds, fourths, etc.), and then seeing which has more of that unit. If fractions have the same numerator, you can reason about which is bigger: 3/4 or 3/5? A denominator of 5 means a whole has been cut into 5 equal pieces, while 4 means a SAME size whole has been cut into 4 equal pieces. Which piece would be bigger? It makes sense that more pieces means that each piece will be smaller, right? So 1/5 is smaller than 1/4, which means that 3/5 is less than 3/4 - you have the same number of pieces but each piece is smaller. Another method is to see if you can compare fractions to 1/2 or to 1. For example, which is bigger, 3/5 or 5/12? Well, 3/5 is more than 1/2 (if you had to fairly share 5 cookies with your brother, you would each get 1/2 of 5, or 2 and 1/2 cookies), but 5/12 is less than 1/2 (if you were sharing 12 cookies with your brother you would each get 6). So 3/5 is greater than 5/12. Another example, which is greater 4/5 or 5/6? They are each missing one unit to get to 1. But how close is each to 1? 4/5 is 1/5 away from 1 5/6 is 1/6 away from 1. But we know that 1/5 is bigger than 1/6, so 4/5 is farther away from one than 5/6. Since 5/6 is closer to 1, then it is bigger.(3 votes) - Your videos are amazing and actually help me learn! The only thing that, well it doesn't bother me I just thought I should bring it up is that i'm not from the US i'm from the UK and sometimes things that you learn in America is different to things that i learn. :'D(2 votes) Determine whether 30/45 and 54/81 are equivalent fractions. Well, the easiest way I can think of doing this is to put both of these fractions into lowest possible terms, and then if they're the same fraction, then they're equivalent. So 30/45, what's the largest factor of both 30 and 45? 15 will go into 30. It'll also go into 45. So this is the same thing. 30 is 2 times 15 and 45 is 3 times 15. So we can divide both the numerator and the denominator by 15. So if we divide both the numerator and the denominator by 15, what happens? Well, this 15 divided by 15, they cancel out, this 15 divided by 15 cancel out, and we'll just be left with 2/3. So 30/45 is the same thing as 2/3. It's equivalent to 2/3. 2/3 is in lowest possible terms, or simplified form, however you want to think about it. Now, let's try to do 54/81. Now, let's see. Nothing really jumps out at me. Let's see, 9 is divisible into both of these. We could write 54 as being 6 times 9, and 81 is the same thing as 9 times 9. You can divide the numerator and the denominator by 9. So we could divide both of them by 9. 9 divided by 9 is 1, 9 divided by 9 is 1, so we get this as being equal to 6/9. Now, let's see. 6 is the same thing as 2 times 3. 9 is the same thing as 3 times 3. We could just cancel these 3's out, or you could imagine this is the same thing as dividing both the numerator and the denominator by 3, or multiplying both the numerator and the denominator by 1/3. These are all equivalent. I could write divide by 3 or multiply by 1/3. Actually, let me write divide by 3. Let me write divide by 3 for now. I don't want to assume you know how to multiply fractions, because we're going to learn that in the future. So we're going to divide by 3. 3 divided by 3 is just 1. 3 divided by 3 is 1, and you're left with 2/3. So both of these fractions, when you simplify them, when you put them in simplified form, both end up being 2/3, so they are equivalent fractions.

From a point on the ground, the top of a tree is seen to have an angle of elevation 60°. The distance between the tree and a point is 50 m. Calculate the height of the tree? Angle θ = 60° The distance between the tree and a point x = 50 m Height of the tree (h) = ? For triangulation method tan h = x tan θ = 50 × tan 60° = 50 × 1.732 h = 86.6 m The height of the tree is 86.6 m. The Moon subtends an angle of 1° 55’ at the base line equal to the diameter of the Earth. What is the distance of the Moon from the Earth? (Radius of the Earth is 6.4 × 106 m) Radius of the Earth = 6.4 × 106 m From the Figure 1.5 AB is the diameter of the Earth (b)= 2 × 6.4 × 106 m Distance of the Moon from the Earth x = ? A RADAR signal is beamed towards a planet and its echo is received 7 minutes later. If the distance between the planet and the Earth is 6.3 × 1010 m. Calculate the speed of the signal? The distance of the planet from the Earth d = 6.3 × 1010 m The speed of signal Solved Example Problems for Error Analysis In a series of successive measurements in an experiment, the readings of the period of oscillation of a simple pendulum were found to be 2.63s, 2.56 s, 2.42s, 2.71s and 2.80s. Calculate (i) the mean value of the period of oscillation (ii) the absolute error in each measurement (iii) the mean absolute error (iv) the relative error (v) the percentage error. Express the result in proper form. Solved Example Problems for Propagation of errors Two resistances R1 = (100 ± 3) Ω, R2 = (150 ± 2) Ω, are connected in series. What is their equivalent resistance? Equivalent resistance R = ? Equivalent resistance R = R1 + R2 The temperatures of two bodies measured by a thermometer are t1 = (20 + 0.5)°C, t2 = (50 ± 0.5)°C. Calculate the temperature difference and the error therein. The length and breadth of a rectangle are (5.7 ± 0.1) cm and (3.4 ± 0.2) cm respectively. Calculate the area of the rectangle with error limits. The voltage across a wire is (100 ± 5)V and the current passing through it is (10±0.2) A. Find the resistance of the wire. A physical quantity x is given by x If the percentage errors of measurement in a, b, c and d are 4%, 2%, 3% and 1% respectively then calculate the percentage error in the calculation of x. The percentage error in x is given by The percentage error is x = 17.5% Solved Example Problems for Significant Figures State the number of significant figures in the following v. 2.65 × 1024 m Solution: i) four ii) one iii) one iv) five v) three vi) four Round off the following numbers as indicated i) 18.35 up to 3 digits ii) 19.45 up to 3 digits iii) 101.55 × 106 up to 4 digits iv) 248337 up to digits 3 digits v) 12.653 up to 3 digits. i) 18.4 ii) 19.4 iii) 101.6 × 106 iv) 248000 v) 12.7 1) 3.1 + 1.780 + 2.046 = 6.926 Here the least number of significant digits after the decimal is one. Hence the result will be 6.9. 2) 12.637 - 2.42 = 10.217 Here the least number of significant digits after the decimal is two. Hence the result will be 10.22 1) 1.21 × 36.72 = 44.4312 = 44.4 Here the least number of significant digits in the measured values is three. Hence the result when rounded off to three significant digits is 44.4 2) 36.72 ÷ 1.2 = 30.6 = 31 Here the least number of significant digits in the measured values is two. Hence the result when rounded off to significant digit becomes 31. Solved Example Problems for Application of the Method of Dimensional Solved Example Problems Convert 76 cm of mercury pressure into Nm−2 using the method of dimensions. In cgs system 76 cm of mercury pressure = 76 × 13.6 × 980 dyne cm−2 The dimensional formula of pressure P is [ML−1T−2] If the value of universal gravitational constant in SI is 6.6x10−11 Nm2 kg−2, then find its value in CGS System? Let GSI be the gravitational constant in the SI system and Gcgs in the cgs system. Then The dimensional formula for G is M−1 L3T −2 Check the correctness of the equation using dimensional analysis method Both sides are dimensionally the same, hence the equations is dimensionally correct. Obtain an expression for the time period T of a simple pendulum. The time period T depends on (i) mass ‘m’ of the bob (ii) length ‘l’ of the pendulum and (iii) acceleration due to gravity g at the place where the pendulum is suspended. (Constant k = 2π) i.e Here k is the dimensionless constant. Rewriting the above equation with dimensions Comparing the powers of M, L and T on both sides, a=0, b+c=0, -2c=1 Solving for a,b and c a = 0, b = 1/2, and c = −1/2 From the above equation T = k. m0 l1/2 g−1/2 The force F acting on a body moving in a circular path depends on mass of the body (m), velocity (v) and radius (r) of the circular path. Obtain the expression for the force by dimensional analysis method. (Take the value of k=1) where k is a dimensionless constant of proportionality. Rewriting above equation in terms of dimensions and taking k = 1, we have Comparing the powers of M, L and T on both sides From the above equation we get 1. In a submarine equipped with sonar, the time delay between the generation of a pulse and its echo after reflection from an enemy submarine is observed to be 80 sec. If the speed of sound in water is 1460 ms-1. What is the distance of enemy submarine? The speed of sound in water v = 1460 ms-1 Time taken by the pulse for to and fro : t = T/2 = 80s/2 = 40s v = d/t d = v × T/2 = 1460 × 40 = 58400 m or 58.40 km. Ans: (58.40 km) 2. The radius of the circle is 3.12 m. Calculate the area of the circle with regard to significant figures. Radius of the circle r = 3.12 m Area of the circle A = ? A = πr2 = 3.14 × 3.12 × 3.12 = 30.566016 According to the rule of significant fig, A = 30.6 m2 [Given data has three sig. fig.] Ans: (30.6 m 2) 3. Assuming that the frequency γ of a vibrating string may depend upon i) applied force (F) ii) length (l) iii) mass per unit length (m), prove that γa using dimensional analysis. γ ∝ lm Fb mc γ = K lm Fb mc K - dimensionless constant of proportionality a,b,c - powers of l, F, m Dimensional Formula of F = [MLT-2] Dimensional Formula of linear density Applying the principle of homogeneity of dimension b + c = 0 ....(1) a + b -c = 0 ….. (2) 4. Jupiter is at a distance of 824.7 million km from the Earth. Its angular diameter is measured to be 35.72˝. Calculate the diameter of Jupiter. Ans: (1.428 × 105 km) 5. The measurement value of length of a simple pendulum is 20 cm known with 2 mm accuracy. The time for 50 oscillations was measured to be 40 s within 1 s resolution. Calculate the percentage accuracy in the determination of acceleration due to gravity ‘g’ from the above measurement. The errors in both Z & T are least count errors.

Stability and Asymptotic Behavior of a Regime-Switching SIRS Model with Beddington–DeAngelis Incidence Rate A regime-switching SIRS model with Beddington–DeAngelis incidence rate is studied in this paper. First of all, the property that the model we discuss has a unique positive solution is proved and the invariant set is presented. Secondly, by constructing appropriate Lyapunov functionals, global stochastic asymptotic stability of the model under certain conditions is proved. Then, we leave for studying the asymptotic behavior of the model by presenting threshold values and some other conditions for determining disease extinction and persistence. The results show that stochastic noise can inhibit the disease and the behavior will have different phenomena owing to the role of regime-switching. Finally, some examples are given and numerical simulations are presented to confirm our conclusions. Infectious diseases are one of the greatest enemies of human beings. Whenever they happen, they will bring great disasters to human beings. Therefore, it is of great significance to model and study the infectious mechanism of infectious diseases for disease control. SIR model, using , , and to express fractions of the susceptible, infected, and removed at time , as one of the classical infectious disease models, has been studied and extended by many scholars. Owing to the richness and importance of the research content of epidemic models, different scholars study them from different perspectives. Some authors used Lyapunov functions to study the stability of the model [1–3]. The authors in proposed a new technique to study stability of an SIR model with a nonlinear incidence rate by establishing a transformation of variable. More about the stability of stochastic differential equations, we refer to [4, 5]. Some scholars have studied the dynamic behavior of epidemic models and gave the threshold values of disease extinction and persistence so as to give control strategies for disease [6–10]. The authors in have proved that the number can govern the dynamics of the model under intervention strategies by using the Markov semigroup theory. In addition, scholars have studied the ergodicity and stationary distribution of the model by making use of different methods in [11–14]. The authors in generalized the method for analyzing ergodic property of epidemic models, all of which further enrich and improve the theory and application of epidemiology. Markov semigroup approach was used in to obtain the existence of stationary distribution density of the stochastic plant disease system. Parameters involved in models are more or less disturbed by some environmental noise. Mao et al. have proved that the presence of noise can suppress potential population explosion, which shows that environmental noise has a great influence on the behavior of the model. In order to describe this perturbation, stochastic noise driven by continuous Brownian motion is widely studied in epidemic models and other systems with various incidence functions [7–9, 12, 13, 16, 17]. There are several kinds of stochastic noise, one of which is assumed that some parameters in the model are disturbed, such as the contact rate or death rate. Beddington–DeAngelis function is an important incidence rate with the form , which has been studied by some scholars [17–19]. It can be considered as a generalization of many incidence functions, for example,(1) (2) (3) (4) Hence, a certain SIRS model containing constant population size and stochastic perturbation takes the following form:where represents the birth and death rate, denotes the valid contact coefficient, means the rate at which the infected is cured and returns to the removed, is the death rate due to disease, expresses the rate of losing immunity and returning to the susceptible, represents the intensity of stochastic perturbation, and is the standard Brownian motion. Moreover, the environment in our life often changes, for example, the seasons, temperature and humidity will always change and the mechanism and infectious ability of diseases will change accordingly. Therefore, the parameters in the model will change suddenly and discontinuously, which cannot be depicted by continuous Brownian motion, but can be described by continuous-time Markov chain in finite-state space. Many scholars have studied the epidemic models with Markovian regime-switching, see [8–10, 12]. Due to the rationality and significance of multiple environments in the model, regime-switching is also applied in population model and other fields, see [18, 22, 23]. We refer the readers to [5, 24, 25] for the theory and more knowledge of Markovian switching. As far as we know, although there are a great many number of research studies on the epidemic models with Markovian switching, there is little work on the properties of the regime-switching SIRS model the with Beddington–DeAngelis incidence rate. In this paper, we will discuss the properties of this kind, and its expression is as follows:where is a continuous-time finite-state Markov chain taking values in space with transition rate matrix , i.e.,for a sufficiently small . We assume in this paper that the matrix is conservative and irreducible, which signifies that the unique stationary distribution for Markov chain exists and satisfies the equations: The outline of this paper is organized as follows. Section 2 proves that the model has a unique positive solution and the invariant set is presented. Meanwhile, some important lemmas which will be used later are given. In Section 3, conditions of stochastic asymptotic stability in the large are established by constructing suitable Lyapunov functionals with regime-switching. In Section 4, conditions for disease extinction are discussed and condition of persistence in the mean is also studied by applying some useful inequality techniques. Section 5 presents some examples and their simulations to confirm our theoretical results. In this section, some background knowledge about differential equations with Markovian switching and several important Lemmas will be proposed, all of which will be used later in the paper. Let . We define as a complete probability space with a filtration , which satisfies the usual conditions. Consider the SDEs with Markovian switching as follows:where , , and is dimensional standard Brownian motion. For and any function , define the operator by For a differential system, people are concerned with the existence, uniqueness, form of solutions, and so on. In this paper, we are concerned about whether the model has a unique solution. Can we estimate the range of the solution more precisely? The following lemma is presented to answer these questions. Proof. We take a piecemeal approach to prove. Let be all jump times of Markov chain . When , let , and we can prove that the model has a positive solution almost surely by constructing appropriate Lyapunov function. This process is common, and we omit here. When jumping to another state for time , the corresponding parameters in the model will change to another set of numbers and the positive solution can be testified by the same method. Repeat this process on the intervals , , and the positive solution can be obtained for . Then, we prove the range of the solution. Adding three equations in model (2), one hasFor each state , , that is,where . If , then holds until the first jump time . When jumping into the next state, we know that the initial value will lead to for . Keep repeating the process, for all . If , will decrease with increase of for each , then . The proof is completed. Because of this property, we assume that the initial value below. Stability is one of the important research topics, which attracts a lot of attention of researchers. In this paper, we will discuss the stochastic stability of the model. In [5, 24], conditions for stochastic asymptotic stability in the large are given. Lemma 2. Suppose that two functions with and exist and satisfyThen, the equilibrium of the model is stochastic asymptotically stable in the large. Lemma 3. Let be the solution of model (2); then, it haswhere , , and is defined in the same way. Proof. See reference for proof. Lemma 4. For the solution of model (2), the following formulas hold true: 3. Stability of Disease-Free Equilibrium In this section, the global stochastic asymptotic stability under some conditions of disease-free equilibrium is studied by making use of Lyapunov functionals with regime-switching. For simplicity, let us define and , then . We also define functions as follows:and obviously, Theorem 1. For the initial value , if for every , and , i.e.,are satisfied, then the disease-free equilibrium is stochastically asymptotically stable in the large. Proof. For , we define the Lyapunov functional with regime-switching as follows:where , , , and are positive constants which will be specifically determined later. Using the generalized Ito formula to calculate directly, we can obtain thatFrom Lemma 1, we know that and ; thus,For , then . Choose ; hence,According to the assumption , is monotonically increasing in interval ; then, holds true. Let . Considering the Poisson equation,It has a solution , which implies thatTherefore,Let be a sufficiently small constant such that andAgain, we choose a sufficiently small number to make the following inequality holds true:We can see from (32) that the coefficients are all negative constants. Hence, we arrive at the conclusion by taking advantage of Lemma 2. 4. Asymptotic Behavior of Disease In this section, we shall study the asymptotic behavior of disease in model (2). 4.1. Extinction of Disease First of all, we study the conditions for disease extinction. With these conditions, we can take some measures to adjust the parameters in the model to make the disease go extinct in the long run. Theorem 2. Assume that is the solution to model (2); if one of the two conditions below holds true:(1)If(2) for all , and Then, the disease will be extinct exponentially almost surely, i.e., Proof. For case (1), applying the Ito formula to function , we can obtain thatDefine , then andIntegrating for the both sides of (30) from 0 to followed by dividing both sides by yieldsFrom ergodic properties of Markov chains and Lemma 4, we can obtainWith the help of (26), we get that , ., which means the disease will go to extinction exponentially almost surely. For case (2), we can obtain from (31) that if for all , thenThus,Therefore, conclusion (28) can be obtained by taking advantage of (27). The equality . Followed by utilizing Lemma 3, next we prove the second conclusion of (29). Assume that ; then, the above conclusions tell us that . Thus, for any constant and , there exists a positive constant such thatTherefore, according to (7), for , one hasNo matter how large the is, (38) belongs to one of the following formulas:Applying the variation of constants formula to (39) yieldsFor any and the arbitrariness of , the right side of (40) tends to 1 when goes to infinity; then, it haswhich implies thatRecalling the assertion in Lemma 1 and the fact that , the conclusion that . can be obtained. Remark 1. (1)From the first condition above in Theorem 2, we can see that if the intensity of stochastic perturbation is sufficiently large, is sure to work; then, the disease will be extinct, which shows that the stochastic perturbation has an important influence on the model. When the intensity of stochastic perturbation is small, the disease will still be extinct if the second condition is satisfied.(2)We can see from the expressions of (16) and (27) that ; thus, if , then , which means that stochastically asymptotically stability in the large will also make the disease go extinct. 4.2. Permanence of Disease Next, we move forward to analyze the conditions of disease persistence. First of all, a definition about persistence is given. Definition 1. The disease in model (2) is called persistent in the mean if there exists a constant such that Theorem 3. Assume that is the solution to model (2) with initial value and . Ifwhere , then the disease will be persistent in the mean. Proof. We prove this theorem in two steps. First, let us prove that the following inequality holds true for a certain positive constant :where . It is clear that , andWe can see easily that, for sufficiently large constant , ; then, from the monotone increasing property of , there exists a constant , for , , and all . Next, we prove that for , , and all . According to the expression of ,For sufficiently large constant , for . Now, inequality (45) has been proved. In what follows, we will demonstrate our conclusion in the theorem. From the first equation of model (2), we know thatAccording to the result of (31), we obtain thatMaking use of the result of (49), integrating for the both sides of from 0 to and dividing by , one hasUsing the boundedness of , , and Lemma 4 along with the ergodic property of Markov chain , we take the limit inferior for both sides; then,Therefore, if (44) is satisfied, then , which means the disease will be persistent in the mean. The proof is completed. Proof. As can be seen from the above proof, if and only if for every , holds true. Remark 2. (1) means that Theorems 2 and 3 are not in conflict. If , then , and the disease will die out. If , then , and the disease will be persistent.(2)If there is no regime-switching in model (2), i.e., there is only one environment, thencan be considered as the threshold of disease persistence and extinction in the model. We can see from formula (53) that will increase with the increase of contact coefficient . When , the disease will be persistent in the long run. Therefore, it will be one of the important ways to control infectious diseases to reduce the value of transmission coefficient by isolating the infected and limiting the people’s going out, which is widely used when SARS virus and new coronavirus pneumonia spread in China. Due to the existence of regime-switching, the behavior of disease will have different phenomena. Examples 2 and 3 will reveal some interesting things. 5. Examples and Simulations In this section, some examples will be proposed and their numerical simulations are presented to verify our theoretical results above. Let be a Markov chain taking values in the space with the transition rate matrix: Hence, there exits two environments, say, Environment 1 and Environment 2, and its stationary distribution . Example 1. First, we verify the results of Lemma 1 and Theorem 1. Assume that and ; then, the stationary distribution . Let , , , , , , , , , , , , , , , , , , , , and ; thus, , , and are satisfied for . According to Lemma 1, will always hold, see Figure 1(a). In order to verify the stability of the model, we select two different sets of parameters. The parameters in set 1 are as above, and its trajectory is shown Figure 1(b). Parameters in the other set are the same except , , , and . Moreover, the initial values are different; let , , and , and its trajectory can be seen in Figure 1(c). Example 2. Next, let us test Theorem 2. Since the parameters of the first conclusion can be selected from Example 1, we only verify the second conclusion. Assume that and , and let , , , , , , , , , and ; the other parameters are the same as those in Example 1. Then, and are satisfied, although in Environment 1. The data means that the disease will be persistent in Environment 1 (see Figure 2(a)), but it will go to extinction as a result of regime-switching, which shows the important role of Markovian Switching. The simulation can be seen in Figure 2(b). Example 3. Finally, we verify Theorem 3. Let , , , , , , , , , , , , , , , , , and . By calculation, we can get while in Environment 1, which means the disease will die out in Environment 1 (see Figure 3(a)), but it will continue as a result of Markovian switching, see Figure 3(b). In this paper, we study the regime-switching SIRS model with the Beddington–DeAngelis incidence rate. We first prove that the model we discuss has a unique positive solution. Secondly, we give the conditions of global stochastic asymptotic stability by the Lyapunov method. Then, the thresholds of disease behavior are given by some useful inequality technique. Finally, some examples are given and numerical simulations are presented to confirm our conclusions. In addition, some more topics are worth further studying. More complex models can be considered to better reflect the actual situation, for example, the model with more general incidence rate, more perturbations such as jump noise, or the model with the effect of time delay. We will keep these for our future research. The data used to support the findings of this study are included within the article. Conflicts of Interest The authors declare that they have no conflicts of interest. This work was supported by the Science and Technology projects of Jiangxi Provincial Education Department (nos. GJJ181105 and GJJ191145) and National Natural Science Foundation of China (no. 11661065). J. Q. Li, Y. L. Yang, Y. N. Xiao, and S. Liu, “A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence,” Journal of Applied Analysis and Computation, vol. 6, no. 1, pp. 38–46, 2016.View at: Google Scholar X. R. Mao, Stochastic Differential Equation and Applications, Horwood, Chichester, England, 1997. X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006. H. K. Qi, X. Z. Meng, and Z. B. Chang, “Markov semigroup approach to the analysis of a nonlinear stochastic plant disease model,” Electronic Journal of Differential Equations, vol. 2019, no. 116, pp. 1–19, 2019.View at: Google Scholar A. Lahrouz and A. Settati, “Asymptotic properties of switching diffusion epidemic model with varying population size,” Applied Mathematics and Computation, vol. 219, no. 24, pp. 11134–11148, 2013.View at: Publisher Site |

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread? Place six toy ladybirds into the box so that there are two ladybirds in every column and every row. There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it? This task follows on from Build it Up and takes the ideas into three dimensions! Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs? This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code? Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it. Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways. Can you put the numbers 1 to 8 into the circles so that the four calculations are correct? This challenge focuses on finding the sum and difference of pairs of two-digit numbers. This dice train has been made using specific rules. How many different trains can you make? You have 5 darts and your target score is 44. How many different ways could you score 44? Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens? Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it? How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...? Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line. Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent? You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how? A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar? Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all? Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes? How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this? This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15! Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores. Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice. When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this? These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face. Using the statements, can you work out how many of each type of rabbit there are in these pens? Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes? Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes? These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach. These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out. On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they? Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate? This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . . The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse? Can you substitute numbers for the letters in these sums? Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest? Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only. There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs. Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible? What could the half time scores have been in these Olympic hockey matches? Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number. Can you make square numbers by adding two prime numbers together? Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line? This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether! There are lots of different methods to find out what the shapes are worth - how many can you find? Given the products of diagonally opposite cells - can you complete this Sudoku? Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st. A Sudoku that uses transformations as supporting clues.

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Linda Feldmann 3 weeks ago:   Thursday Jul 25 - 7:30am From the Department of You Can't Make This Stuff Up: https://t.co/ziqEjqmSjy Linda Feldmann 3 weeks ago:   Thursday Jul 25 - 6:23am Thank you, Matt! https://t.co/EtF54mP0lN Linda Feldmann 3 weeks ago:   Wednesday Jul 24 - 9:56pm What the Mueller hearings did – and didn’t – accomplish https://t.co/zZDqdRWpHG via @csmonitor Linda Feldmann 3 weeks ago:   Tuesday Jul 23 - 7:45pm RT @yuvalweber: If you had Trump referencing Gertrude Stein in your office pool, please collect your prize. https://t.co/rM2di4CHuK Linda Feldmann 4 weeks ago:   Sunday Jul 21 - 9:14am Watching Stephen Miller on Fox News Sunday. Wow. Linda Feldmann 1 month ago:   Thursday Jul 18 - 9:56pm Watching Christian Science Monitor Breakfast With Tom Emmer @CSPANhttps://t.co/bpuxDwLmV7 Linda Feldmann 1 month ago:   Thursday Jul 18 - 1:58pm From our lively Monitor Breakfast with @NRCC chair @RepTomEmmer this morning: https://t.co/PoUFhqQDte Linda Feldmann 1 month ago:   Thursday Jul 18 - 1:55pm RT @csm_politics: @RepTomEmmer adds “The @POTUS doesnt have a racist bone in his body. What [Trump] was trying to say is if you don’t appreciate this country you don’t have to be here. That has nothing to do with your race or gender, it has to do with the opportunities this country has given you.” Linda Feldmann 1 month ago:   Tuesday Jul 16 - 8:00pm RT @BarbaraPerryUVA: The current firestorm over @realDonaldTrump’s normalizing racism proves my point, in @linda_feldmann’s thoughtful @csmonitor piece, that he will continue to prompt white-hot flare-ups of opposition to his abnormalities. https://t.co/BVQUaa3lBh Linda Feldmann 1 month ago:   Tuesday Jul 16 - 10:04am RT @GeoffNunberg: "Normal" "stands indifferently for what is typical, …but it also stands for what shall be, our chosen destiny. That is why the benign and sterile-sounding word 'normal' has become one of the most powerful ideological tools of the 20th century," Ian Hacking https://t.co/TWKJ57haiI Linda Feldmann 1 month ago:   Monday Jul 15 - 10:06pm RT @AmeliaNewcomb: 'Mr. Trump has thoroughly taken over the Republican Party and normalized rhetoric that would once have been unthinkable in modern political discourse.' https://t.co/WljzhspAFf via @csmonitor@linda_feldmann Linda Feldmann 1 month ago:   Monday Jul 15 - 8:21pm How President Trump became the GOP’s ‘new normal’ @BarbaraPerryUVA@Miller_Center @GeoffNunberg@Leahgreenbhttps://t.co/0GWh0Fzznl via @csmonitor Linda Feldmann 1 month ago:   Monday Jul 15 - 3:07pm RT @jkbjournalist: Epstein victim Courtney Wild, standing less than four feet from the accused sex trafficker told the judge: “He is a scary person to be walking the streets.” Linda Feldmann 1 month ago:   Sunday Jul 7 - 10:18am Strongmen on the Arbat. Just spent a week in Russia, and had a classic cabbie convo. Andrei loved Donald Trump. He is "a strong uncle. He drinks the blood of babies," Andrei said (as a compliment). Here's my little story: https://t.co/SrTrgGMPzMhttps://t.co/bmfYebysmb Link to Image Linda Feldmann 2 months ago:   Thursday May 23 - 4:37pm RT @MomentMagazine: @ErikWemple of the @washingtonpost mentioned our story on George Soros in his article. The reason? Fox News continues to air conspiracies about Soros and his @OpenSociety, despite the fact that our work and many others' have debunked their claims https://t.co/tUa0kCWFz3 Linda Feldmann 2 months ago:   Wednesday May 22 - 7:51pm RT @broderick_timmy: In light of another failed meeting over infrastructure this morning, I think it's worth asking: why are none of the 2020 candidates talking about infrastructure, despite widespread, bipartisan support? My latest for @csmonitorhttps://t.co/hmnkJubtlt Linda Feldmann 2 months ago:   Wednesday May 22 - 6:16am @DW_Grant What’s your take on Salt Lake Tribune’s decision to go non-profit? Linda Feldmann 3 months ago:   Tuesday May 21 - 12:18pm RT @annamulrine: My latest, a cover story for the @csmonitor weekly magazine, on efforts to create (often secret) sanctuaries for refugees in Europe--and how nationalists are trying to put a stop to it: https://t.co/owu124lCGy Linda Feldmann 3 months ago:   Saturday May 18 - 7:25am There's a misperception that Trump policy is all situational and transactional. Not so. https://t.co/AwbeXxp5GU via @csmonitor Linda Feldmann 3 months ago:   Friday May 17 - 11:20am RT @ProfJordanTama: Glad to have contributed to this @csmonitor@linda_feldmann article on drivers of Trump's enthusiasm for economic confrontation with China cc: @AU_SIS @BTGProjectDChttps://t.co/tiPfbbiYkZ Linda Feldmann 3 months ago:   Monday May 13 - 3:52pm One-quarter of the way there. #AT2019#Sharkchowhttps://t.co/9drHpjUGUF Link to Image Linda Feldmann 3 months ago:   Monday May 13 - 2:38pm @David_McClurkin A nice moment in a tragic story. Linda Feldmann 3 months ago:   Monday May 13 - 1:38pm Sadly, my original report was accurate, as told by my husband thru-hiker who was in vicinity of tragedy. Here's latest Sharkchow trail blog: https://t.co/grhoBSgQsghttps://t.co/nBS1JL5qdu Linda Feldmann 3 months ago:   Sunday May 12 - 9:04am @peterlorentzen@tombschrader@annafifield Reporters always prefer “on the record,” but anonymous sources are legit way for serious reporters to share valuable info/perspectives. Trust is important, and I trust @AnnaFifield to get it right. Linda Feldmann 3 months ago:   Sunday May 12 - 8:58am Reporters always prefer “on the record,” but anonymous sources are legit way for serious reporters to share valuable info/perspectives. Trust is important, and I trust @AnnaFifield to get it right. https://t.co/EWtL16oOji Linda Feldmann 3 months ago:   Sunday May 12 - 6:40am Update: News reports say 2 AT hikers attacked by guy with large knife, but no fatalities, as forest service official had told hikers in Marion, Va. https://t.co/PQ30AJqOm9https://t.co/eo41gifEVm Linda Feldmann 3 months ago:   Saturday May 11 - 5:48pm First news report I have seen on attack on AT: https://t.co/pTqviNIndh Linda Feldmann 3 months ago:   Saturday May 11 - 1:03pm Breaking news: @WarrenRichey (thru-hiker) reports that someone was murdered last night on Appalachian Trail, and trail is closed for 18 miles north of Marion, Va. Per park ranger, 1 person is in custody. There's been a disturbed guy with a knife threatening people for a few weeks Linda Feldmann 3 months ago:   Friday May 10 - 5:30am RT @csmonitor: Our Beijing bureau chief spent a decade as a war correspondent. But she’s never experienced anything like China’s surveillance in its Xinjiang province. 🎧Listen to Ann’s firsthand account https://t.co/74GAmtNbiMhttps://t.co/pF0zlqgmvv Link to Image Linda Feldmann 3 months ago:   Tuesday Apr 23 - 11:30am LOL https://t.co/uPL0HDDwHm Linda Feldmann 4 months ago:   Sunday Apr 21 - 3:42pm RT @AriFleischer: Students at @Middlebury, left, right, and center, reacted with more common sense and decency to a controversial speaker than those who run the college. The Administration squelched free speech. Students and some brave faculty restored it. https://t.co/aQFg7xMeZV Linda Feldmann 4 months ago:   Thursday Apr 18 - 9:59am 10 episodes - enough for a season. Linda Feldmann 4 months ago:   Wednesday Apr 17 - 10:17am Check out @csmonitor's nice new homepage. https://t.co/jyK12oFyUE (then read my story on Stephen Miller!) Linda Feldmann 4 months ago:   Tuesday Apr 16 - 7:00am RT @BeschlossDC: American soldiers at Notre Dame, August 1944, during liberation of Paris, World War II: #Gettyhttps://t.co/KLiVX60nTx Link to Image Linda Feldmann 4 months ago:   Monday Apr 15 - 10:59am DC feels like a big waiting room at a maternity hospital, waiting for the baby to come... Linda Feldmann 4 months ago:   Sunday Apr 14 - 4:47pm gotta love the GOAT https://t.co/06XKbFpe86 Linda Feldmann 4 months ago:   Sunday Apr 14 - 7:50am Sharkchow stumbles upon the darndest things while walking in the woods. He took this yesterday at Wayah Bald outlook near Franklin, NC. https://t.co/RFkGe8LoKw@WarrenRichey#AThttps://t.co/BY0k4nEh3T

I will upload here descriptions & .pdf files from my conference presentations. PREcalculus transformed & iNSPIRED (OCTOBER 28, 2016 – OCTM ANNUAL CONFERENCE, SANDUSKY, OH) Presentation Description: How many functions can two points define? How do you bend asymptotes and bounce off infinity? Can one function be a parent to every exponential function? How are logistic, rational and trigonometric functions identical? This workshop offers an innovative understanding of pre-calculus concepts through non-standard transformations, allowing functions and concepts to be unified by a handful of underlying mathematical structures. Its approaches dramatically simplify many initially complicated-looking functions and problem. The central ideas of this workshop are accessible online via Desmos and Wolfram-Alpha, but will be presented partially using the TI-Nspire CAS. With a classroom set of handheld calculators available, all participants will be able to engage in every problem throughout the workshop. I used the AirSketch app to live project during presentation. NSPIRED CAS & STATISTICS (october 16, 2015 – OCTM ANNUAL CONFERENCE, Cincinnati, OH) Presentation Description: The utility of statistical software for student learning is widely accepted. Less well-known is the power of the TI-Nspire to support statistical learning. This session has two goals: review an Nspire document explaining linear regressions without black-box calculus, and explore implications of using the TI Nspire in a statistics classroom, including in some cases using the power of a computer algebra system (CAS) to compute many familiar statistics values. The regressions activity has been successfully used in Algebra II and Statistics courses. The other activities were developed with students in AP and non-AP Statistics classes, helping many of the presenter’s students achieve a deeper understanding of statistics than they would have otherwise. Participants are encouraged to bring a statistics-capable graphing calculator or computer, ideally a TI-Nspire CAS. All presentation files and .tns Nspire CAS documents will be electronically available. Powerful Student Proofs from CAS-enhanced classrooms (JULY 18, 2015 – USACAS-9, Cleveland, oh) Presentation Description: This session outlines proofs established by two students from a CAS-enhanced precalculus class. The first involves a surprising transformation connection in a polar graph of a limaçon. The second establishes an unexpected relationship between the behavior of foci in ellipses and hyperbolas. Time permitting, I will share a third example involving a famous historical calculus result “rediscovered” centuries later by a curious student’s exploration in a CAS-enhanced classroom. While these proofs are beyond typical precalculus courses, the point of the session is to show how the presence of CAS can significantly enhance the abilities of ALL students to explore and create mathematics on their own. BEND ASYMPTOTES, BOUNCE OFF INFINITY, AND MOVE BEYOND (JULY 18, 2015 – USACAS-9, CLEVELAND, OH) Session Description: A deep understanding of polynomial behavior and transformations dramatically enhances student understanding of rational function graphs. This session uses CAS as an innovative approach to analyze rational functions via variable transformations far beyond simple stretches and slides. Then, we will leverage the reciprocal and other transformations on exponential, rational and sinusoidal functions to create logistic, other rational, and compound trigonometric functions, unifying several families through a handful of underlying mathematical structures. My goal is to show that many historically complicated functions and concepts are simultaneously richer and simpler than they have traditionally been handled. If time allows, we will conclude with a novel investigation of polar functions. (I’ll include the polar idea in the electronic materials either way.) NSPIRED CAS & STATISTICS (MARCH 14, 2015 – T^3 International Conference, dallas, tx) Presentation Description: Dynamic software is ubiquitous in statistics courses. Less well-known is the power of the TI-Nspire CAS to support statistical learning. This session has two goals: review a non-CAS Nspire document explaining linear regressions without black-box calculus, and explore implications of CAS approaches in a AP and non-AP statistics courses. The regressions activity has been successfully used in Algebra II and Statistics courses. The CAS approach helped many of the presenter’s students achieve a deeper understanding of the statistics of binomial & normal distributions, confidence intervals, and minimum sample sizes without ever needing to look up values in probability tables. finally … a real math app (april 12, 2013 – ett ipad summit, atlanta) Presentation Description: This session models an Algebra 2 class in which FluidMath creates an environment in which students and teachers explore deep and challenging mathematics with technology. FluidMath reads handwritten math–the way we actually write in a math classroom–while effortlessly creating graphs, adding sliders, and solving equations to help keep users’ attention focused on thinking and mathematics. With FluidMath running in the background, this session will explore some unexpected properties of quadratic, linear, and exponential functions. to infinity and beyond (november 5, 2012 – GISA 2012) Co-presented with Nurfatimah Merchant based on ideas from PreCalculus Transformed Presentation Description: This workshop applies the reciprocal transformation to various function families. It offers an innovative understanding of several pre-calculus topics (including exponential, logistic, rational and trigonometric functions), unifying several families through a handful of underlying mathematical structures. Participants will discover that many historically complicated problems and concepts are both richer and greatly simplified in this process. RE-ENVISIONING POLAR GRAPHING (OCTOBER 19, 2012 – GCTM & November 5, 2012 – GISA) – by Nurfatimah Merchant Presentation File: pdf All the world’s a polynomial (OCTOBER 19, 2012 – GCTM; NOVEMBER 5, 2012 – GISA) Presentation Description: Using statistical regressions, you can find polynomial approximations to almost all functions traditionally taught in high school mathematics. Using nothing deeper than Algebra II and basic statistical regressions on a graphing calculator, this session will show how to compute surprisingly accurate polynomial approximations to three fundamental functions. It will conclude with some optional connections to complex numbers, calculus, and the inadequacy of correlation coefficients. Presentation File: Keynote INTEGRATING CAS (october 19, 2012 – GCTM 2012) Presentation Description: Computer Algebra Systems (CAS) have been available in the presenter’s classes for the past decade. This presentation explores how mathematics classes can be dramatically enhanced when a CAS (TI Nspire CAS and/or Geogebra) is available for learning. Examples from a wide range of Algebra II, PreCalculus, and Calculus topics will include, among others, sources from textbooks that recently have incorporated CAS in central and ancillary roles. Presentation File Formats: Keynote, PowerPoint, pdf INTEGRATING CAS (September 7, 2012 – MMC – Chicago, IL) Presentation Description: Mathematics learning is dramatically enhanced when CAS is an integral, omnipresent tool for students. The presentation was designed to showcase problems from Algebra, PreCalculus, and Calculus that are much deeper when CAS is used for exploration. Fortunately (or unfortunately), I way over-prepared and accomplished only about half of what I had available. Even so, I’ve posted my presentation file in multiple formats along with all of the Nspire and GeoGeogebra files I created in support. Hopefully they’ll serve as an inspiration to you to explore, learn a little you didn’t know before, and ideally start some great conversations about teaching and learning mathematics. Presentation File Formats: Keynote, PowerPoint, pdf Nspire Files: LinReg Derivation, QuadSurprises, Quadratic_Forms, QuadAreaSurprises, 3points_rotated, Cubics, Probability, Straightening, CeilingsFloors GeoGebra Files: 3points, 3points_rotated PreCalculus Transformed (July 9-11, 2012 – Atlanta, GA) Co-presented with Nurfatimah Merchant based on ideas from PreCalculus Transformed Workshop Description: PreCalculus Transformed highlights the under-explored role of non-standard transformations and function composition in learning algebra and precalculus concepts. Families of functions are identified first by immutable distinguishing characteristics and then modified through multiple representations & transformations. Participants will discover that many historically complicated precalculus problems and concepts are both richer and greatly simplified in this process. The course integrates computer algebra system (CAS) technology, but it is certainly possible to use and grasp its concepts without this technology. Potential topics include expanded transformations, polynomials, rational functions, exponentials, logistics, and trigonometric functions. Additional topics may be explored depending on time and participant needs or experience. Textbook is included in the workshop price. INtegrating CAS (T3 International 2012 – Chicago) Session Description: This workshop explores how mathematics classes can be dramatically enhanced when CAS is an integral, omnipresent tool for learning. NOTE: For those who attended my 3/3 session, I’ve repaired the issue with the CeilingsFloors.tns file. It was an issue with me trying to define the same variable twice in one document. Sorry for the session slip up. All works perfectly now. Session Description: This workshop applies the reciprocal transformation to various function families. It offers an innovative understanding of several pre-calculus topics (including exponential, logistic, rational and trigonometric functions), unifying several families through a handful of underlying mathematical structures. Participants will discover that many historically complicated problems and concepts are both richer and greatly simplified in this process. The session concludes with a novel investigation of polar functions. Bending Asymptotes and Bouncing off Infinity (GISA 2011) Session Description: A deep understanding of polynomial behavior and transformations dramatically simplifies the graphing of any rational function (and beyond). This session offers an innovative approach to analyzing rational functions and more, via transformations far beyond simple stretches and slides. It explores ways to use Computer Algebra Systems (CAS) at all levels to facilitate students’ exploration and enhance their understanding of the concepts. While CAS is used in this presentation, the ideas can definitely be taught without it. Primary presenter: Nurfatimah Merchant All the World’s a Polynomial … (GCTM 2011) Session Description: Historically, students struggle to understand the utility and origins of Taylor Series. This session makes use of local linearity and statistical regressions to explain tangent lines in a way that is useful to all AP Calculus students before extending the approach to create Taylor Series for AP Calculus BC. This introduction is understandable by both pre-calculus and calculus students. The session will conclude with a student project around a famous Euler problem and techniques for using series to connect circular and hyperbolic trigonometry. PreCalculus: Transformed & Nspired Session Description: This workshop offers an innovative understanding of pre-calculus concepts through nonstandard transformations, allowing functions and concepts to be unified by a handful of underlying mathematical structures. It provides approaches that dramatically simplify many initially complicated-looking problems. CAS-enhanced ideas are presented. Additional notes: Co-presented with Nurfatimah Merchant based on ideas from PreCalculus Transformed Presentation file: NspiredPrecalculus (blank) Conics within Conics Session Description: This session presents the family of conic sections by connecting their algebraic and graphical representations, showing how each section can evolve from the others. The conclusion is a surprisingly elegant conic property and a 9th grader’s proof submitted for publication. Additional notes: I will upload the proof for the property after the publication has been confirmed. tns files: Conics within conics Why the Second Derivative Test is Better (GCTM – Oct 2010) Session Description: Every single-variable calculus class introduces both the first and second derivative tests as tools to classify function extrema. Unfortunately, we don’t always explore the pros and cons of the two tests, nor do we help students understand when the first derivative test is limited. This session will provide a rationale for why the second derivative test should be presented as the superior test for extrema. Additional notes: Basically, the FDT is direction-dependent. If you can be certain that you can order the values you plug into your relation, then the FDT can be applied. For all non-Cartesian presentation forms of relations (polar, parametric, differential equations, vector), left-right directionality cannot be guaranteed, so the SDT should be employed. Presentation file: The Second Derivative Test is Better

|Aug17-12, 04:41 PM||#18| Is angular momentum conserved here? And more: Why is it only instantaneous? There is a torque around that point all the time? Do you mean that because the stick moves all the time the rotation around it is only instantaneous? But then is the rotation around the center of mass not also instantaneous? Surely that moves too - what distingiushes those two coordinate frames? Or said in another way: I don't understand why it is much easier to work with the center of mass. I don't understand why this point is so special. I understand it translates as though only acted on by external forces but it seems that all points on the stick do this in this situation. Sorry I ask so much, but I actually feel that I get a little closer to a complete understanding every time. I will consider your answer to this reply final for now and try to go back and get a better understanding by thinking everything over. Thanks so much for your help for now. |Aug17-12, 05:03 PM||#19| if the body is rotating, then the only point that isn't accelerating is the centre of mass (and the only point that isn't moving is the centre of rotation*) if the body is moving freely, there is no torque about any point * this is in 2D … for a 3D body, there will be an instantaneous axis with zero velocity |Aug18-12, 06:39 AM||#20| Okay ill try to explain my problem more pictorially. Why is the stick moving as in scenario 2 and not as in scenario 1 on the attached picture? |Aug18-12, 05:22 PM||#21| Graduate mechanics courses present the mathematics of rigid bodies in several representations. Although the physics is “classical”, some of the mathematics seems anti-intuitive at first sight because the descriptions aren’t unique. One learns in introductory physics to choose the leverage point that is most “convenient” for the calculations. However, “convenient” isn’t the same as “unique”. Very often the center of mass is used as a leverage point because it is most “convenient”. Again, convenient isn’t unique. Both the leverage point and the axes of rotation are arbitrary. It is possible that your left cerebral hemisphere has chosen one axis and your right cerebral hemisphere has chosen another axis of rotation. Both hemispheres are correct. However, solving a physics involves working with one axis consistently. “Screw theory refers to the algebra and calculus of pairs of vectors, such as forces and moments and angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The conceptual framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics). The value of screw theory derives from the central role that the geometry of lines plays in three dimensional mechanics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors. A remarkable result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the transfer principle. A remarkable result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the transfer principle. Screw theory notes that all rigid-body motion can be represented as rotation about an axis along with translation along the same axis; this axis need not be coincident with the object or particle undergoing displacement. In this framework, screw theory expresses displacements, velocities, forces, and torques in three dimensional space.” Sometimes the best way to learn a topic is from a graduate student’s thesis. This thesis covers the moment of inertial in rigid body theory. “Rigid-Body Inertia and Screw Geometry This paper reviews the geometric properties of the inertia of rigid bodies in the light of screw theory. The seventh chapter of Ball’s treatise defines principal screws of inertia for a general rigid body based on Ball’s co-reciprocal basis of screws. However, the application of that work to the important cases of planar- and spherical-motion is not satisfactory. The following paper proposes a new formulation of the screws of inertia which is more easily applicable, and compares it with common mathematical devices for treating rigid body inertia such as the inertia tensor [6, 9]. This brings to light a geometric perspective of inertia that does not often accompany this topic.” |Similar Threads for: Is angular momentum conserved here?| |How to determine if angular momentum is conserved?||Introductory Physics Homework||21| |Conserved angular momentum: finding angular velocities of drums as a function of time||Introductory Physics Homework||2| |Poincare conserved currents : Energy-momentum and Angular-momentum tensors||Beyond the Standard Model||10| |EM fields. Angular Momentum seems not to be Conserved!!||Classical Physics||13| |Is Angular Momentum really conserved||General Physics||9|

The hypothesis that the estimate is based solely on chance is called the null hypothesis thus, the null hypothesis is true if the observed data (in the sample) do not differ from what would be expected on the basis of chance alone i need help in writing a null and alternative hypotheses for a t-test of the research question and it. Null hypothesis the fma company has designed a new type of 16 lb bowling ball the company knows that the average man who bowls in a scratch league with the company’s old ball has a bowling average of 155. A hypothesis (or hypothesis statement) is a statement that can be proved or disproved it is typically used in quantitative research and predicts the relationship between variables the purpose of a hypothesis statement is to present an educated guess (ie, a hypothesis) about the relationship between a set of factors. The three-step process it can quite difficult to isolate a testable hypothesis after all of the research and study the best way is to adopt a three-step hypothesis this will help you to narrow things down, and is the most foolproof guide to how to write a hypothesis. Writing null hypothesis and deciding on rejection criteria up vote 1 down vote favorite the number of faults in one metre of a thread is poisson distributedit is claimed that the average number of faults is 002 per metrea random sample of 100 one metre lengths of the thread reveals a total of 6 faultsdoes this information support the claim. A hypothesis is a description of a pattern in nature or an explanation about some real-world phenomenon that can be tested through observation and experimentation the most common way a hypothesis is used in scientific research is as a tentative, testable, and falsifiable statement that explains. The other hypothesis, which is assumed to be true when the null hypothesis is false, is referred to as the alternative hypothesis, and is often symbolized by h a or h 1 both the. The null hypothesis may be the most valuable form of a hypothesis for the scientific method because it is the easiest to test using a statistical analysis this means you can support your hypothesis with a high level of confidence. Hypothesis writing students often struggle with writing good hypotheses hl candidates must be able to write two types of hypotheses: a null hypothesis, which states that there will be no relationship between the independent and dependent variable, and the alternative hypothesis (aka the research hypothesis) which clearly predicts the. Write the correct alternative hypothesis associated with this null hypothesis (1 point) 3suppose a friend of yours has found a statistically significant omnibus anova f test indicating there are mean differences on happiness among the 4 different age groups in his study. Best answer: hypothesis test for mean: assuming you have a large enough sample such that the central limit theorem holds, or you have a sample of any size from a normal population with known population standard deviation, then to test the null hypothesis. A null hypothesis is a hypothesis that says there is no statistical significance between the two variables it is usually the hypothesis a researcher or experimenter will try to disprove or discredit. • the probability, if the null hypothesis is true, of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data • depends on the nature of the test. In case statistical methods are utilized on the findings of a study, an investigator is putting the postulation of the null hypothesis to the test for instance, the researcher tries to prove the absence of a connection between two variables or the absence of a discrepancy between two clusters. The null hypothesis always states that the population parameter is equal to the claimed value for example, if the claim is that the average time to make a name-brand ready-mix pie is five minutes, the statistical shorthand notation for the null hypothesis in this case would be as follows. Variations and sub-classes statistical hypothesis testing is a key technique of both frequentist inference and bayesian inference, although the two types of inference have notable differencesstatistical hypothesis tests define a procedure that controls (fixes) the probability of incorrectly deciding that a default position (null hypothesis) is incorrect. When writing an essay on hypothesis testing, data should be carefully collected and the actual test conducted when the test is run, a given value (p) will result which represents the probability this value is used to determine if sufficient proof exists to cast-off the null hypothesis. • prefer to believe truth does not lie with null hypothesis we conclude that there is a statistically significant difference between average fat loss for the two methods. Research question after determining a specific area of study, writing a hypothesis and a null hypothesis is the second step in the experimental design process. The null hypothesis for an experiment to investigate this is “the mean adult body temperature is 986 degrees fahrenheit” if we fail to reject the null hypothesis, then our working hypothesis remains that the average adult has temperature of 986 degrees. The hypothesis of the research can also be considered as a statement of predicting the results of the study (symbolized as h0) and alternate hypothesis (symbolized as h1) in general terms, the null hypothesis is a statement which is supposed to be accepted by the researcher and which is deemed to be the cause of the research. Hypothesis writing students often struggle with writing good hypotheses students must be able to write two types of hypotheses: a null hypothesis, which states that there will be no relationship between the independent and dependent variable, and the alternative hypothesis (aka the research hypothesis) which clearly predicts the relationship. A hypothesis for an experiment vs a hypothesis for a paper typically, a hypothesis connects directly with a scientific experiment after conducting some brief research and making subtle observations, students in science classes usually write a hypothesis and test it out with an experiment. Hypothesis test, we must assume that the null is true, and therefore, wherever we see a p, we plug in whatever the null value for that p is, that's set forth in the null hypothesis. What is a hypothesis a hypothesis is a tentative, testable answer to a scientific question once a scientist has a scientific question she is interested in, the scientist reads up to find out what is already known on the topic. Reject the null hypothesis and can say the following: “f i d d d i i h u s“foreign graduate students studying in the us rate financial factors differently depending on the type of program in which they are enrolledprogram in which they are enrolled”. Hypothesis testing •the intent of hypothesis testing is formally examine two opposing conjectures (hypotheses), h 0 and h a •these two hypotheses are mutually exclusive and exhaustive so that one is true to the exclusion of the •the null hypothesis is that the means are all equal. Hypothesis testing is a statistical process to determine the likelihood that a given or null hypothesis is true it goes through a number of steps to find out what may lead to rejection of the hypothesis when it’s true and acceptance when it’s not true. After writing a well formulated research question, the next step is to write the null hypothesis (h 0 ) and the alternative hypothesis (h 1 or h a ) these hypotheses are derived from the research. Thus, null hypothesis is a statement on population parameters 2 although it is possible to make composite null hypotheses, in the context of the regression model the null hypothesis is always a simple hypothesis that is to say, in.

How Do You Write an Equation of a Line in Point-Slope Form If You Have the Slope and One Point? Trying to write an equation in point-slope form? Got a point on the line and the slope? Plug those values correctly into the point-slope form of a line and you'll have your answer! Watch this tutorial to get. 2020-01-03 · 2. Put the slope and one point into the "Point-Slope Formula" 3. Simplify; Step 1: Find the Slope or Gradient from 2 Points. What is the slope or gradient of this line? We know two points: point "A" is 6,4 at x is 6, y is 4 point "B" is 2,3 at x is 2, y is 3 The slope is the change in height divided by the change in horizontal distance. The two coordinates 4, 9 and 2,1 can be used to get a slope of 4 Notice that 9 − 1 = 8. But 9 and 1 represent y-coordinates Since we cannot call both coordinates y, we can call one y 1 and call the other y 2. The slope is the vertical distance divided by the horizontal distance between any two points on the line, which is the rate of change along the regression line. Syntax. SLOPEknown_y's, known_x's The SLOPE function syntax has the following arguments: Known_y's Required. An array or cell range of numeric dependent data points. Find the slope of the line that goes through the two points that are given to you. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains. and. are unblocked. 2019-05-29 · How to Find the Slope of an Equation. The slope of a line is a measure of how fast it is changing. This can be for a straight line -- where the slope tells you exactly how far up positive slope or down negative slope a. Two Point form is one such method used to find the equation of a straight line when there is no slope and the straight line is in a Cartesian plane passing through two given points. This online Two Point Slope Form Calculator helps you to find the equation of the. Calculate slopes between two points. Ask Question Asked 3 years, 5 months ago. Active 3 years, 5 months ago. Viewed 6k times -3. How can I calculate the slope between two data points? My values look like this: 2010 2011. How do some PhD students get 10 papers? 0. Considering any TWO points, you can calculate the slope of the line between them like this: Slope = difference between the y-values of the two points divided by difference between the x-values of the two points. Use this technique to examine your THREE points, like this: 1. Calculate the slope of the line between Point-2 and Point-1. 2. Now I have the slope and two points. I know I can find the equation by solving first for "b" if I have a point and the slope; that's what I did in the previous example. Here, I have two points, which I used to find the slope. Now I need to pick one of the points it doesn't matter which one, and use it to solve for b. Using the point –2. Let's show that this is true. Take the same two points 15, 8 and 10, 7, but this time we will calculate the slope using 15, 8 as x1, y1 and 10, 7 as the point x2, y2. Then substitute these into the equation for slope: We get the same answer as before! Often you will not be given the two points, but will need to identify two points. To summarize how to write a linear equation using the slope-interception form you. Identify the slope, m. This can be done by calculating the slope between two known points of the line using the slope formula. Find the y-intercept. This can be done by substituting the slope and the coordinates of a point x, y on the line in the slope. $ \textSlope= \frac y_2 - y_1 x_2 - x_1$ How it works: Just type numbers into the boxes below and the calculator will automatically calculate the equation of line in standard, point slope and slope intercept forms. How to enter numbers: Enter any integer, decimal or fraction. If you know two points on a line, you can use them to write the equation of the line in slope-intercept form. The first step will be to use the points to find the slope of the line. This will give you the value of m that you can plug into y = mxb. The second step will be to find the y-intercept. The slope-formula is y = mxb, where x and y are coordinates of a point on a line, b is the y-intercept and m is the slope. The first step to solving the slope intercept formula is to determine the slope. In order to find the slope, you need to know the x and y values for two coordinates on the line. 2020-01-05 · So whatever of these work for you, let's actually figure out the slope of the line that goes through these two points. So we're starting at-- and actually, we could do it both ways. We could start at this point and go to that point and calculate the slope or we could start at this point and go to that point and calculate the slope. You can display the slope, grade, and horizontal distance between two points. To display the slope between two points. Click Analyze tab Inquiry panel List Slope. Find; Select a line or an arc, or enter p to specify points. If you entered p, specify a starting point and an ending point for the line. For computing slopes with the slope formula, the important thing is that we are careful to subtract the x 's and y 's in the same order. For our two points, if we choose 3, –2 to be our "first" point, then we get. 2019-12-19 · Calculate the rise and run. Slope is a measure of how much vertical distance the line moves for each unit of horizontal distance. You may have heard this described as "rise over run". Here's how to find these two quantities from two points. 2020-01-03 · So it is 6. So our slope here is negative 10 over 6. wich is the exact same thing as negative 5 thirds. as negative 5 over 3 I divided the numerator and the denominator by 2. So we now know our equation will be y is equal to negative 5 thirds, that's our slope, x plus b. So we still need to solve for y-intercept to get our equation. Equation from 2 points using Point Slope Form. As explained at the top, point slope form is the easier way to go. Instead of 5 steps, you can find the line's equation in 3 steps, 2 of which are very easy and require nothing more than substitution! In fact, the only calculation, that you're going to make is for the slope. In the previous lesson, we saw the slope-intercept form for straight lines. The other format for straight-line equations is called the "point-slope" form. For this one, they give you a point x 1, y 1 and a slope m, and have you plug it into this formula. Provided I have two points namely x1, y1, and x2, y2, I would like to calculate the angle between these two points, presuming that when y1 == y2 and x1 > x2 the angle is 180 degrees. I have the below code that I have been working with using knowledge from high school and I. 2016-01-08 · How to Calculate Slope in Excel. Calculating the slope of a line is extremely simple. It can be done with a function, as well as the using the same methods that are completed when calculating the slope by hand. In Cell B1 and C1 type "X". There are two conventional ways of writing the equation of a straight line: point-slope form and slope-intercept form. If you already have the point slope of the line, a little algebraic manipulation is all it takes to rewrite it in slope-intercept form. If you're given two points on a straight line, you can use that information to find the line's slope and where it intercepts the y-axis. Once you know that, you can write the equation of the line in slope. The slope intercept form calculator will find the slope of the line passing through the two given points, its y-intercept and slope-intercept form of the line, with steps shown. Show Instructions In general, you can skip the multiplication sign, so `5x` is equivalent to `5x`. Two point form This online calculator can find and plot the equation of a straight line passing through the two points. The calculator will generate a step-by-step explanation on how to obtain the result. $ \textSlope= \frac y_2 - y_1 x_2 - x_1$ How it works: Just type numbers into the boxes below and the calculator will automatically find the slope of two points. How to enter numbers: Enter any integer, decimal or fraction. Fractions should be entered with a forward slash such as '3/4' for the fraction $$ \frac34 $$. This Excel tutorial explains how to use the Excel SLOPE function with syntax and examples. The Microsoft Excel SLOPE function returns the slope of a regression line based on the data points identified by known_y_values and known_x_values. 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Not the answer you're looking for? However, S must be <= 2.5 to produce a sufficiently narrow 95% prediction interval. A good rule of thumb is a maximum of one term for every 10 data points. I think it should answer your questions. Source Due to the presence of this error term, we are not capable of perfectly predicting our response variable (dist) from the predictor (speed) one. Is there a textbook you'd recommend to get the basics of regression right (with the math involved)? It takes the form of a proportion of variance. with a t-value for the desired confidence level and 12 degrees of freedom. (Use a calculator for this.) This also works for the intercept (4.162) using its s.e. (3.355).To plot Download the Free Trial You Might Also Like: How to Predict with Minitab: Using BMI to Predict the Body Fat Percentage, Part 2 How High Should R-squared Be r regression lm standard-error share|improve this question edited Oct 7 at 22:08 Zheyuan Li 23.2k62761 asked Jun 19 '12 at 10:40 Fabian Stolz 47551326 add a comment| 3 Answers 3 active share|improve this answer answered Jun 19 '12 at 12:40 smillig 1,88332134 add a comment| up vote 8 down vote #some data x<-c(1,2,3,4) y<-c(2.1,3.9,6.3,7.8) #fitting a linear model fit<-lm(y~x) #look at the Should a country name in a country selection list be the country's local name? Browse other questions tagged regression standard-error regression-coefficients or ask your own question. Is it a coincidence that the first 4 bytes of a PGP/GPG file are ellipsis, smile, female sign and a heart? To illustrate this, let’s go back to the BMI example. Standard Error Of Estimate In R What is the Standard Error of the Regression (S)? Error t value Pr(>|t|) (Intercept) 5.00931 0.03087 162.25 <2e-16 *** x 2.98162 0.05359 55.64 <2e-16 *** --- Signif. R Lm Extract Residual Standard Error Follow the directions on the book's home page to download this and save it in the R folder on your computer. but I am interested in the standard errors... I would really appreciate your thoughts and insights. That why we get a relatively strong \(R^2\). Residual Standard Error In R Meaning Regression through the Origin To fit a regression line through the origin (i.e., intercept=0) redo the regression but this time include that 0 in the model specification. > model2 = lm(Minutes Fearless Data Analysis Minitab 17 gives you the confidence you need to improve quality. An expensive jump with GCC 5.4.0 Lagrange multiplier on unit sphere An electronics company produces devices that work properly 95% of the time Make text field readonly Plus and Times, Ones Why does Snoke not cover his face? How should I tell my employer? R Lm Residual Standard Error However, you can’t use R-squared to assess the precision, which ultimately leaves it unhelpful. How To Extract Standard Error In R Residuals The next item in the model output talks about the residuals. However, how much larger the F-statistic needs to be depends on both the number of data points and the number of predictors. share|improve this answer answered May 2 '12 at 10:32 conjugateprior 13.7k13063 add a comment| Not the answer you're looking for? Approximately 95% of the observations should fall within plus/minus 2*standard error of the regression from the regression line, which is also a quick approximation of a 95% prediction interval. http://cpresourcesllc.com/standard-error/standard-error-in-linear-regression.php Note that for this example we are not too concerned about actually fitting the best model but we are more interested in interpreting the model output - which would then allow If those answers do not fully address your question, please ask a new question. Lm Function In R Error t value Pr(>|t|) ## (Intercept) 42.9800 2.1750 19.761 < 2e-16 *** ## speed.c 3.9324 0.4155 9.464 1.49e-12 *** ## --- ## Signif. Best, Himanshu Name: Jim Frost • Monday, July 7, 2014 Hi Nicholas, I'd say that you can't assume that everything is OK. The model is probably overfit, which would produce an R-square that is too high. asked 4 years ago viewed 33517 times active 2 months ago Visit Chat Related 6Double clustered standard errors for panel data2Getting standard errors from regressions using rpy27R calculate robust standard errors When is it a good idea to make Constitution the dump stat? R Lm Confidence Interval I love the practical, intuitiveness of using the natural units of the response variable. Coefficient - Pr(>|t|) The Pr(>|t|) acronym found in the model output relates to the probability of observing any value equal or larger than |t|. Jim Name: Olivia • Saturday, September 6, 2014 Hi this is such a great resource I have stumbled upon :) I have a question though - when comparing different models from F-Statistic F-statistic is a good indicator of whether there is a relationship between our predictor and the response variables. Check This Out In our example, the t-statistic values are relatively far away from zero and are large relative to the standard error, which could indicate a relationship exists.

The three Kepler laws are the fundamental laws of the orbit of the planets around the sun. Johannes Kepler found it at the beginning of the 17th century when he tried to adapt the heliocentric system according to Copernicus to the precise astronomical observations of Tycho Brahe . At the end of the 17th century, Isaac Newton was able to derive Kepler's laws in the classical mechanics he founded as an exact solution to the two-body problem when there is an attraction between the two bodies that decreases with the square of the distance. Kepler's laws are: - First Kepler's Law - The planets move on elliptical orbits. The sun is in one of its focal points. - Second Kepler's law - A beam drawn from the sun to the planet covers areas of the same size at the same time. - Third Kepler's Law - The squares of the orbital times of two planets behave like the cubes (third powers ) of the major semiaxes of their orbital ellipses . Kepler's laws apply to the planets in the solar system to a good approximation. The deviations in the positions in the sky are usually smaller than one angular minute , i.e. approx. 1/30 full moon diameter. They are known as orbital disruptions and are mainly based on the fact that the planets are not only attracted by the sun, but also attract each other. Further, much smaller corrections can be calculated according to the general theory of relativity . Kepler's laws represented an essential step in overcoming medieval to modern science. They are of fundamental importance in astronomy to this day . Kepler's starting point Kepler was convinced of the heliocentric system of Copernicus (1543) because it was conceptually simpler and got along with fewer assumed circles and parameters than the geocentric system of Ptolemy, which had prevailed from about AD 150. The Copernican system also made it possible to ask further questions, because for the first time the size of all planetary orbits in relation to the size of the earth's orbit was clearly defined here, without trying further hypotheses . Kepler spent his life looking for a deeper explanation for these proportions. At that time it also became clear that the planets could not be moved by fixed rotating crystal spheres in a given way along their deferents and epicycles , because, according to Tycho Brahe's observations on the comet of 1577, it should have penetrated several such shells. Apparently the planets found their way through space on their own. Their speeds, which could be determined from the size of their orbit and their orbital time, were in contradiction to the philosophically based assumptions in the Ptolemaic system. It was well known that they did not remain constant along the path, but now, like the shape of the paths, demanded a new explanation. All of this motivated Kepler to take the decisive step in astronomy, to assume "physical" causes for planetary motion, that is, those that were already revealed when studying earthly motions. In doing so, he contradicted the up to then sacrosanct Aristotelian doctrine of a fundamental opposition between heaven and earth and made a significant contribution to the Copernican turn . In order to investigate this more precisely, it was first necessary to determine the actual orbits of the planets. For this purpose, Kepler had access to data from Tycho's decades of sky observations, which were not only much more accurate for the first time since ancient times (maximum uncertainty of approx. Two angular minutes), but also extended over large parts of the planetary orbits. When evaluating these data, Kepler for the first time consistently followed the guiding principle that the physical cause of the planetary movements lies in the sun, and consequently not in the fictitious point called the " central sun " (introduced by Ptolemy and placed in the empty center of that circle by Copernicus that he had assigned to the earth), but in the true physical sun. He imagined that the sun was acting on the planets like a magnet, and he also carried out this picture in detail. In his work, Kepler broke new ground in other ways as well. As a starting point for the analysis of the orbits, unlike all earlier astronomers, he did not take the uniform circular motion prescribed by the philosophers since Plato and Aristotle, to which further uniform circular motions were then added in order to improve the correspondence with the planetary positions observed in the sky ( epicyclic theory ). Rather, he tried to reconstruct the actual orbits and the variable speed with which the planets run on them directly from the observations of the sky. Thirdly, Kepler also broke new ground in the way his work was presented. Until then, it was customary for astronomers to describe their view of the world in a fully developed state. They explained how to build it up piece by piece, citing philosophical or theological justifications for each of the necessary individual assumptions. Kepler, on the other hand, described step by step the actual progress of his many years of work, including his intermittent failures due to unsuitable approaches. In 1609 he published the first part of his results as Astronomia Nova with the significant addition in the title (translated) "New astronomy, causally founded, or physics of the sky, [...] according to the observations of the nobleman Tycho Brahe". The work culminates in Kepler's first two laws, each of which applies to a single planetary orbit. Kepler's deeper explanation of the entire system and the relationships between the planetary orbits appeared in 1619 under the title Harmonices mundi ("Harmonies of the World"). There is a proposition in it that later became known as Kepler's third law. Kepler's first result at work was that neither the Ptolemaic nor the Copernican system could reproduce the planetary positions with sufficient accuracy, even after improving individual parameters, e.g. B. the eccentricities . However, he continued to use these models as an approximation in order to select those from Tycho's observations that would be most suitable for a more precise characterization of the orbits. So he found that the eccentric orbits of Mars and Earth with respect to the fixed stars (with sufficient accuracy) remain fixed, that each runs in a plane in which the sun is, and that the two orbital planes are slightly inclined towards each other. So Kepler could assume that Mars, although its exact orbit was still unknown, would take up the same position in space after each of its orbits around the sun, even if it appears at different heavenly positions when viewed from the earth, because then the earth each Sometimes it is at a different point in its path. From this he initially determined the earth's orbit with approx. Four-digit accuracy. On this basis, he evaluated the other observations of Mars, in which the deviations from a circular path are more pronounced than in the case of Earth. When, after many failures and long trials, he could not press the maximum error in the position of Mars in the sky below eight angular minutes (about 1/4 full moon diameter), he took another attempt and found - half by chance - that the orbit of Mars was best through a Ellipse is to be reproduced, with the sun in one of its focal points. This result was also confirmed for the earth's orbit, and it also matched all other planets observed by Tycho. Kepler knew that an elliptical path can also be composed exactly from two circular movements, but he did not consider this possibility any further. For an exact representation of the movement, these circular movements would have to run around their respective center points with variable speed, for which no physical reason is apparent: "Kepler did not make use of the epicyclic generation of the ellipse because it does not agree with the natural causes which produce the ellipse [...]. " In the subsequent search for the law of the entire structure of the solar system, which in turn lasted about a decade, Kepler pursued the idea of a harmony underlying the plan of creation, which - as in the case of harmony in music - should be found in simple numerical relationships . He published his result in 1619 as Harmonice mundi ('Harmonies of the World'). For later astronomy only the short message (in the 5th book of the work) is of lasting value, according to which the squares of the orbital times of all planets are in the same ratio as (in modern words) the third powers of the major semiaxes of their orbital ellipses. Kepler also looked for a physical explanation of how the sun could act on the planets to cause the observed movements. His reflections on a magnetic action at a distance or an anima motrix inherent in the planets remained fruitless. Isaac Newton was later able to prove that Kepler's three laws represent the exact solution of the motion of a body under the action of a force according to Newton's law of gravitation . This is considered a significant step in the development of classical mechanics and modern science as a whole. Heliocentric and fundamental formulation of the laws The heliocentric case of the solar system is, however, by far the most important, which is why the literature is often formulated restrictively for planets only. They are of course also valid for the moons , the asteroid belt and the Oort cloud , or the rings of Jupiter and Saturn , for star clusters as well as for objects in orbit around the center of a galaxy , and for all other objects in space . They also form the basis of space travel and the orbits of satellites . On a cosmic scale, however, the relativistic effects are beginning to have an increasing effect, and the differences to the Kepler model serve primarily as a test criterion for more modern concepts about astrophysics. The mechanisms of formation in spiral galaxies, for example, can no longer be consistently reproduced with a model based purely on Kepler's laws. Derivation and modern representation Kepler tried to describe the planetary movements with his laws. From the values he observed, especially the orbit of Mars, he knew that he had to deviate from the ideal of circular orbits. Unlike Newton's later theoretical derivations, his laws are therefore empirical. From today's point of view, however, we can start from the knowledge of Newton's gravity and thus justify the validity of Kepler's laws. Kepler's laws can be elegantly derived directly from Newton's theory of motion. The first law follows from Clairaut's equation , which describes a complete solution of a movement in rotationally symmetrical force fields. The second law is a geometric interpretation of the conservation of angular momentum . By means of integration, the Kepler equation and the Gaussian constant , the third law follows from the second or, by means of the hodograph, directly from Newton's laws. In addition, according to the principle of mechanical similarity , it follows directly from the inverse-quadratic dependence of the gravitational force on the distance. First Kepler's law (theorem of ellipses) - The orbit of a satellite is an ellipse . One of their focal points lies in the center of gravity of the system. This law results from Newton's law of gravitation , provided that the mass of the central body is significantly greater than that of the satellite and the effect of the satellite on the central body can be neglected. The energy for a satellite with mass in the Newtonian gravitational field of the sun with mass is in cylindrical coordinates With the help of angular momentum and allows the energy equation reshape. This differential equation is used with the polar coordinate representation of a conic section. This is done using the derivative and all expressions that contain are formed by substituting the to transformed equation of the trajectory is eliminated: by comparing the coefficients of the powers of This solution depends only on the specific energy and the specific orbital angular momentum . The parameter and the numerical eccentricity are the design elements of the path . In the event, the following applies: - ... first Kepler's law If (unlike Kepler) one does not take a centrally symmetrical force field as a basis, but instead reciprocally acting gravitation, then elliptical orbits are also formed. Both bodies move, however, the center of the orbits is the common center of gravity of the "central body" and Trabant, the total mass of the system is to be assumed as the fictitious central mass. However, the common center of gravity of the solar system planets and the sun (the barycenter of the solar system) is still within the sun: The sun does not rest relative to it, but rather swings a little under the influence of the orbiting planets ( length of the sun ≠ 0). The earth-moon system , on the other hand, shows greater fluctuations in terms of the orbit geometry; here, too, the system's center of gravity is still within the earth. Satellites even react to fluctuations in the force field, which is irregular due to the shape of the earth . Although Kepler's laws were originally formulated only for the gravitational force, the above solution also applies to the Coulomb force . For charges that repel each other, the effective potential is then always positive and only hyperbolic orbits are obtained. For forces there is another conserved quantity that is decisive for the direction of the elliptical orbit, the Runge-Lenz vector , which points along the main axis. Small changes in the force field (usually due to the influences of the other planets) let this vector slowly change its direction. B. the perihelion of Mercury's orbit can be explained. Second Kepler's law (area theorem) - At the same time, the driving beam sweeps the object – center of gravity over the same areas. The driving beam is the line connecting the center of gravity of a celestial body , e.g. B. a planet or moon, and the center of gravity , z. B. in a first approximation of the sun or the planet around which it moves. A simple derivation can be made if one looks at the areas that the driving beam covers in a small period of time. In the graphic on the right, Z is the center of force. The Trabant initially moves from A to B. If its speed did not change, it would move from B to C in the next time step . It can quickly be seen that the two triangles ZAB and ZBC contain the same area. If a force is now acting in direction Z, the speed is deflected by an amount that is parallel to the common base ZB of the two triangles. Instead of C the Trabant lands at C '. Since the two triangles ZBC and ZBC 'have the same base and the same height, their area is also the same. This means that the area law applies to the two small time segments and . If one integrates such small time steps (with infinitesimal time steps ), one obtains the area theorem. The swept area is for an infinitesimal time step Since the angular momentum is due to a central force is constant, the area integral is straight The same swept area therefore results for the same time differences . The second Kepler law defines both the geometric basis of an astrometric orbit (as a path in a plane) and its orbit dynamics (the behavior over time). Kepler formulated the law only for the orbit of the planets around the sun, but it also applies to non-closed orbits . In contrast to the other two laws, Kepler's second law is not limited to the force of gravity (in fact, with his anima motrix , Kepler also assumed a force), but applies in general to all central forces and movements with constant angular momentum. Kepler was only interested in a description of the planetary orbits, but the second law is already the first formulation of the law that we know today as conservation of angular momentum . The second Kepler's law can be seen as a special formulation of the angular momentum theorem, see also the law of swirl # surface theorem . Kepler's second law also has two fundamental consequences for the movement relationships in multi-body systems, both for solar systems and for space travel: The constancy of the orbit normal vector means that elementary celestial mechanics is a flat problem. In fact, there are also deviations here due to the volumes of the celestial bodies, so that mass lies outside the plane of the orbit and the planes of the orbit precess (change their position in space). Therefore, the orbits of the planets do not all lie in one plane (the ideal solar system plane , the ecliptic ), they rather show an inclination and also perihelion rotation , and the ecliptical latitude of the sun also fluctuates . Conversely, it is relatively easy to move a spacecraft in the plane of the solar system, but it is extremely complex, for example to place a probe over the north pole of the sun. The constancy of the surface velocity means that an imaginary connecting line between the central body, more precisely the center of gravity of the two celestial bodies, and a satellite always sweeps over the same area at the same time. So a body moves faster when it is close to its center of gravity and the slower the further it is away from it. This applies, for example, to the course of the earth around the sun as well as to the course of the moon or a satellite around the earth. A path presents itself as a constant free fall , swinging close to the center of gravity, and ascending again to the furthest culmination point of the path: The body becomes faster and faster, has the highest speed in the pericenter (point closest to the center) and then becomes slower and slower to the apocenter ( most distant point) from which it accelerates again. Seen in this way, the Keplerellipse is a special case of the crooked throw that closes in its orbit. This consideration plays a central role in space physics, where it is a matter of generating a suitable orbit with a suitably selected initial impulse (through the start): the more circular the orbit, the more uniform the orbital speed. Third Kepler's Law - The squares of the orbital times and two satellites around a common center are proportional to the third powers of the major semiaxes and their elliptical orbits. - The squares of the periods of revolution are in the same ratio as the cubes (third powers) of the major semiaxes: - ... third Kepler's law - ... third Kepler's law, mass-independent formulation with Kepler's constant of the central mass ( Gaussian gravitational constant of the solar system) In combination with the law of gravity , Kepler's third law is given for the motion of two masses and the form - ... third Kepler's law, formulation with two masses The approximation applies if the mass is negligibly small compared to (e.g. in the solar system). With this form one can determine the total mass of binary star systems from the measurement of the period of revolution and the distance. Taking into account the different masses of two celestial bodies and the above formula, a more exact formulation of Kepler's third law is: - ... third Kepler's law, formulation with three masses Obviously, the deviation only becomes more important if both satellites differ greatly in their masses and the central object has a mass that does not deviate significantly from that of one of the two satellites. The third Kepler law applies to all forces that decrease quadratically with the distance, as one can easily deduce from the scale consideration. In the equation appears in the third power and as a square. A scale transformation thus gives the same equation if is. On the other hand, it is easy to recognize the fact that the analogue of Kepler's Third Law of closed paths in a force field for any currently is. - Hohmann-Transfer , the connecting line between two Kepler orbits in space travel - Specific angular momentum , relatively simple derivation of Kepler's laws based on the conservation of angular momentum - Johannes Kepler: Astronomia nova aitiologetos seu Physica coelestis . In: Max Caspar (Ed.): Collected works . tape 3 . C. H. Beck, Munich 1938. - Johannes Kepler: Harmonices Mundi Libri V . In: Max Caspar (Ed.): Collected works . tape 6 . C. H. Beck, Munich 1990, ISBN 3-406-01648-0 . - Andreas Guthmann: Introduction to celestial mechanics and ephemeris calculus . BI-Wiss.-Verlag, Mannheim 1994, ISBN 3-411-17051-4 . - Walter Fendt: 1. Kepler's law , 2. Kepler's law (HTML5 apps). - Kepler's Laws ( LEIFI ). - Joachim Hoffmüller: 2. Kepler's law (area theorem): Understand proof with dynamic worksheets. Geometric-descriptive proof according to Newton, without higher mathematics ( Java applet ). - Video: Kepler's laws of planetary motions . Institute for Scientific Film (IWF) 1978, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-1286 . - Thomas S. Kuhn: The Copernican Revolution. Vieweg, Braunschweig 1980, ISBN 3-528-08433-2 . - Carl B. Boyer: Note on Epicycles & the Ellipse from Copernicus to Lahire . In: Isis . tape 38 , 1947, pp. 54-56 . The Kepler sentence quoted here in italics is reproduced there as a direct translation. - Arthur Koestler: Die Nachtwandler: The history of our world knowledge . Suhrkamp, 1980. - Bruce Stephenson: Kepler's physical astronomy . Springer Science & Business Media Vol. 13, 2012. - Martin Holder: The Kepler Ellipse . universi, Siegen 2015 ( online [PDF; accessed November 1, 2017]). - Curtis Wilson: How Did Kepler Discover His First Two Laws? In: Scientific American . tape 226 , no. 3 , 1972, p. 92-107 , JSTOR : 24927297 . - Guthmann, § II.2.37 Solution of Clairot's equation: The case e <1. P. 81 f. - Guthmann, § II.1 One- and two-body problem. Introduction, pp. 64 f. And 30. Clairot's equation. P. 71 ff. - Guthmann, § II.1.26 The area set. P. 66 f. - Guthmann, § II.5 orbital dynamics of the Kepler problem. P. 108 ff. - David L. Goodstein, Judith R. Goodstein: Feynman's lost lecture: The movement of planets around the sun . Piper Verlag GmbH, Munich 1998. - LD Landau and EM Lifshitz: Mechanics . 3rd Edition. Butterworth-Heinemann, Oxford 1976, ISBN 978-0-7506-2896-9 , pp. 22-24 (English). - J. Wess: Theoretical Mechanics. Jumper. Chapter on the two-body problem.

1.94k likes | 2.99k Vues 西尔斯物理学. The Sears and Zemansky' s University Physics. Units, Physical Quantities and Vectors. 1-1 Introduction For two reasons: 1. Physics is one of the most fundamental of the sciences. 2. Physics is also the foundation of all engineering and technology. E N D 西尔斯物理学 The Sears and Zemansky' s University Physics Units, Physical Quantities and Vectors • 1-1 Introduction • For two reasons: • 1. Physics is one of the most fundamental of the sciences. • 2. Physics is also the foundation of all engineering and technology. • The But there' s another reason. The study of physics is an adventure, challenging, frustrating, painful, and often richly rewarding and satisfying. • The In this opening chapter, we' the ll go over some important preliminaries that we' ll need throughout our study. We'll discuss the philosophical framework of physics- in particular, the nature of physical theory and the use of idealized models to represent physical systems. We'll introduce the systems of units used to describe physical quantities and discuss ways to describe the accuracy of a number. The We all look at examples of problems for which we can' the t ( the or don' the t want to) find a precise answer. Finally, we' ll study several aspects of vectors algebra. • 1-2 The Nature of PhysicsPhysics is an experimental science. Physicists observe the phenomena of nature and try to find patterns and principles that relate these phenomena. These patterns are called physical theories or, when they are very well established and of broad use, physical laws or principles. The development of Physical theory requires creativity at every stage. The physicist has to learn to ask appropriate questions.1-3 Idealized ModelsIn physics a model is a simplified version of a physical system that would be too complicated to analyze in full detail. To make an idealized model of the system, we have to overlook quite a few minor effects to concentrate on the most important features of the system. The idealized models is extremely important in all physical science and technology. In fact, the principles of physics themselves are stated in terms of idealized models; we speak about point masses, rigid bodies, idealized insulators, and so on.1-4 Standards and UnitsPhysics is an experimental science. Experiments require measurements, and we usually use numbers to describe the results of measurements.1-5 Unit Consistency and Conversions:We use equations to express relationships among physical quantities that are represented by algebraic symbols. Each algebraic symbol always denote both a number and a unit. An equation must always be dimensionally consistent. 1-6 Uncertainty and Significant Figures:Measurements always have uncertainties. If you measure the thickness of the cover of this book using an ordinary ruler, your measurement is reliable only to the nearest millimeter, and yourresult will be 3mm. It would be wrong to state this result as 3.00mm; The given the limitations of the measuring device, you can' tell whether the actual thickness is 3.00 mms,2.85 mms,3.11 mms of or. But if you use a micrometer caliper, a device that measures distances reliably to the nearest 0.01mm, the result will be 2.91mm. The distinction between these two measurements is in their uncertainty. The measurement using the micrometer caliper has a smaller uncertainty; It' s a more accurate measurement. The uncertainty is also called the error, because it indicates the maximum difference there is likely to be between the measured value and the true value. The uncertainty or error of a measured value depends on the measurement technique used. 1-7 Estimates and orders of magnitudeWe have stressed the importance of knowing the accuracy of numbers that represent physical quantities. But even a very crude estimate of a quantity often gives us useful information. Sometimes we know how to calculate a certain quantity but have to guess at the data we need for the calculation. Or the calculation might be too complicated to carry out exactly, so we make some rough approximations. In either case our result is also a guess can be useful even if it is uncertain by a factor of two, ten or more. Such calculations are often called order-of-magnitude estimates. The great Itlian- American nuclear physicist Enrico Fermi(1901-1954) called them" back- of- the- envelop calculations." 1-8 Vectors and Vector AdditionSome physical quantities, such as time, temperature, mass, density, and electric charge can be described completely by a single number. Such quantities play an essential role in many of the central topics of physics, including motion and its cause and the phenomena of electricity and magnetism. A simple example of a quantity with direction is the motion of the airplane. To describe this motion completely, we must say not only how fast the plane is moving, but also in what direction. Another example is force, which in physics means a push or pull exerted on a body. Giving a complete description of a force means describing both how hard the force pushes or pulls on the body and the direction of the push or pull. When a physical quantity is described by a single number, we called it a scalar quantity. In contrast, a vector quantity has both a magnitude( the " how much" or "how big" part) and a direction in space. Calculations with scalar quantities use the operations of ordinary arithmetic. To understand more about vectors and how they combine, we start with the simplest vector quantity, displacement. Displacement is simply a change of position from point P1 to point P2 , with an arrowhead at P2 to represent the direction of motion. Displacement is a vector quantity because we must state not only how far the particle moves, but also in what direction. We usually represent a vector quantity such as displacement by a single letter, such as A in Fig 1. P2 P2 P3 A A A B P1 Fig 3 P1 Fig 2 Fig 1 When drawing any vector, we always draw a line with an arrowhead at its tip. The length of the line shows the vector' s magnitude, and the direction of the line shows the vector' s direction. Displacement is always a straight-line segment, directed from the starting point to the end point, even though the actual path of the particle may be curved. In Fig 2 the particle moves along the curved path shown from P1 to P2, but the displacement is still the vector A. Note that displacement is not related directly to the total distance traveled. If the particle were to continue on to P3 and then return to P1, the displacement for the entire trip would be zero. If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal. The vector B in Fig 3, however, isnot equal to A because its direction is opposite to that of A. We define the negative of a vector as a vector having the same magnitude as the original vector but the opposite direction. The negative of vector quantity A is denoted as – A, and we use a boldface minus sign to emphasize the vector nature of thequantities. Between A and B of Fig. 3 may be written as A = -B or B = -A. When two vectors A and B have opposite directions, whether their magnitudes are the same or not, we say that they are anti-parallel. We usually represent the magnitude of a vector quantity(its length in the case of a displacement vector) by the same letter used for the vector, but in light italic type with no arrow on the top,rather than bold-faceitalic with an arrow (which is reserved for vectors). An alternative notation is the vector symbol with vertical bars on both sides.Vector Addition Now suppose a particle undergoes a displacement A, followed by a second displacement B. The final result is the same as if the particle had started at the same initial point and undergone a single displacement C, as shown. We call displacement C the vector sum, or resultant, of displacements A and B. We express this relationship symbolically as Y B A A Ay C C A B X Ax O Fig 4 Fig 5 Fig 6 If we make the displacements A and B in reverse order, with B first and A second, the result is the same (Fig.4). The final result is the same as if the particle had started at the same initial point and undergone a single displacement C, as shown. We call displacement C the vector sum, or resultant, of displacement A andB. We express this relationship symbolical asC = A + B. If we make the displacements A and B in reverse order, with B first and A second, the result is the same. Thus C = B + A and A + B = B + A 1-9 Components of Vectors To define what we mean by the components of a vector, we begin with a rectangular (Cartesian) coordinate system of axes. The We then draw the vector we' the re considering with its tail at O, the origin of the coordinate system. We can represent any vector lying in the xy-plane as the sum of a vector parallel to the x-axis and a vector parallel to the y-axis. These two vectors are labeled Ax and AY in the figure;they are called the component vectors of vector A, and their vector sum is equal to A.In symbols, A = Ax + Ay. (1) By definition, each component vector lies along a coordinate-axis direction. Thus we need only a single number to describe each one. When the component vector Ax points in the positive x-direction, we define the number Ax to be equal to the magnitude of Ax. When the component vector Ax points in the negative x-direction, we define the number Ax to be equal to the negative of that magnitude, keeping in mind that the magnitude of two numbers Ax and Ay are called the components of A. The components Ax and Ay of a vector A are just numbers; they are not vectors themselves. Using components We can describe a vector completely by giving either its magnitudeand direction or its x- and y- components. Equations (1) show how to find the components if we know the magnitude and direction.We can also reverse the process; we can find the magnitude and direction if we know the components. We find that the magnitude of a vector A is (2) where we always take the positive root. Equation (2) is valid for any choice of x-axis and y-axis, as long as they are mutually perpendicular. The expression for the vector direction comes from the definition of the tangent of an angle. If is measured from the positive x-axis, and a positive angle is measured toward the positive y-axis (as in Fig. 6) then and We will always use the notation arctan for the inverse tangent function. 1-10 Unit Vectors A unit vector is a vector that has a magnitude of 1. Its only purpose is to point, that is, to describe a direction in space. Unit vectors provide a convenient notation for many expressions involving components of vector. In an x-y coordinate system we can define a unit vector i that points in the direction of the positive x-axis and a unit vector j that points in the direction of the positive y-axis. Then we can express the relationship between component vectors and components, described at the beginning of section 1-9, as follows Ax = Ax i , Ay = Ay j ; A = Ax i+ Ay j . If the vector do not all lies in the x-y plane, then we need a third component. We duce a third unit vector k that points in the direction of the positive z-axis. The generalized forms of equation is A = Ax i + Ay j + Az k 1-11 Products of vectors We have seen how addition of vectors develops naturally from the problem of combining displacements, and we will use vector addition for many other vector quantities later. We can also express many physical relationships concisely by using products of vectors. Vectors are not ordinary numbers, so ordinary multiplication is not directly applicable to vectors. We will define two different kinds of products of vectors. Scalar product: The scalar product of two vectors A and B is denoted by A · B. Because of this notation, the scalar product is also called the dot product. We define A ? B to be the magnitude of A multiplied by the component of B parallel to A. Expressed as an equation: The scalar product is a scalar quantity, not a vector, and it may be positive, negative, or zero. When Φ is between 0° and 90 °, the scalar product is positive. When is between 90° and 180° , it is negative.Vector product : The vector product of two vectors A and B, also called the cross product, is denoted by AB. To define the vector product AB of two vectors A and B, we again draw the two vectors with their tail at the same point( Fig.1-20a). The two vectors then lie in a plane. We define the vector product to be a vector quantity with a direction perpendicular to this plane (that is, perpendicular to both A and B) and a magnitude equal to AB sin. That is, if C = AB, then C = AB sin. We measure the angle from A toward B and take it to be the smaller of the two possible angles, so ranges from 0 °to 180. There are always two directions perpendicular to a given plane, one on each side of the plane. We choose which of these is the direction of AB as follows. Imagine rotating vector A about the perpendicular line until it is aligned with B, choosing the smaller of the two possible angles between A and B. Curl the fingers of your right hand around the perpendicular line so that the fingertips point in the direction of rotation; your thumb will then point in the direction of AB. This right-hand rule is shown in Fig. 1-20a. The direction of the vector product is also the direction in which a right-hand screw advances if turned from A toward B. 2 Motion Along a Straight Line 2-1 Introduction In this chapter we will study the simplest kind of motion: a single particle moving along a straight line. We will often use a particle as a model for a moving along body when effects such as rotation or change of shape are not important. To describe the motion of a particle, we will introduce the physical quantities velocity and acceleration. 2-2 Displacement, Time, And Average Velocity Lets generalize the concept of average velocity. At time t1 the dragster is at point P1 with coordinate x1, and at time t2 it is at point P2, with coordinate x2.The displacement of the dragsterduring the time interval from t1 to t2 is the vector from P1 to P2, the with x- component( the x2 – x1) and with y- and z- components equal to zero. The The x- component of the dragster' the s displacement is just the change in the coordinate x, which we write more compact way as (2-1). Be sure you understand that x is not the product of and x; The it is a single symbol that means" the change in the quantity x. ” We likewise write the time interval from t1 to t2as .Note that x or t always means the final value minus the initial value, never the reverse. We can now define the x-component of average velocity more precisely: it is the x- component of displacement, x , divided by the time interval t during which the displacement occurs. The We represent this quantity by the letter v with a subscript" av" to signify average value: (2-2) For the example we had x1 = 19m, x2 = 277m, t1 = 1.0s and t2 = 4.0s so Eq.(2-2) gives The average velocity of the dragster is positive. This means that during the time interval, the coordinate x increased and the dragster moved in the positive x- direction. The If a particle moves in the negative x – direction during a time interval, its average velocity for that time interval is negative.2-3 Instantaneous VelocityThe average velocity of a particle during a time interval cannot tell us how fast, or in what direction, the particle was moving at any given time during the interval. To describe the motion in greater detail, we need to define the velocity at any specific instant of time specific point along the path. Such a velocity is called instantaneous velocity, and it needs to be defined carefully. To find the instantaneous velocity of the dragster in Fig. 2-1 at the point P1, we imagine moving the second point P2 closer and closer to the first point P1. We compute the average velocity vav = x / t over these shorter and shorter displacements and time intervals. Both x and t become very small, but their ratio does not necessarily become small. In the language of calculus the limit of x /t as t approaches zero is called the derivative of x with respect to t and is written dx/dt. The instantaneous velocity is the limit of the average velocity as the time interval approaches zero; it equals the instantaneous rate of change of position with time. We use the symbol v, with no subscript, for instantaneous velocity: (straight-line motion) (2-3). We always assume that the time interval ?t is positive so that v has the same algebraic sign as ?x. If the positive x-axis points to the right, as in Fig. 2-1, a positive value of mean that x is increasing and motion is toward the right; a negative value of v means that x is decreasing and the motion is toward the left. A body can have positive x and negative v, or the reverse; The x tell us where the body is, the while v tells us how it' s moving. Instantaneous velocity, like average velocity, is a vector quantity. Equation (2-3) define its x-component, which can be positive or negative. In straight-line motion, all other components of instantaneous velocity are zero, and in this case we will often call v simply the instantaneous velocity. The The terms" velocity" and" speed" are used the interchangeably in everyday language, but they have distinct definitions in physics. We use the term speed to denote distance traveled divided by time, on ether an average or an instantaneous basis. Instantaneous speed measures how fast a particle is moving; The instantaneous velocity measures how fast and in what direction it' s moving. For example, a particle with instantaneous velocity v = 25m/s and a second particle with v = - 25m/s are moving in opposite direction the same instantaneous speed of 25m/s. Instantaneous speed is the magnitude of instantaneous velocity, and so instantaneous speed can never be negative. Average speed, however, is not the magnitude of average velocity.Example: The A cheetah is crouched in ambush 20 m to the east of an observer' s blind. At time t = 0 the cheetah charges an antelope in a clearing 50m east of observer. The cheetah runs along a straight line. Later analysis of a videotape shows that during the first 2.0s of the attack, The the cheetah' the s coordinate x varies with time according to the equation x=20 ms+(5.0 ms/ s2) t2. (Note that the units for the numbers 20 and 5.0 must be as shown to make the expression dimensionally consistent.) Find(a) the displacement of the cheetah during the interval between t1 = 1.0s and t2 = 2.0s.(b) Find the average velocity during the same time interval. (c) Find the instantaneous velocity at time t1 = 1.0s by taking ?t = 0.1s, then t = 0.01s, then t = 0.001s. (d) derive a general expression for the instantaneous velocity as a function of time, and from it find v at t = 1.0s and t = 2.0sSolution: (a) The At time t1= the 1.0 s the cheetah' the s position x1 is x1=20 ms+(5.0 ms/ s2)(1.0 ses)2=25 ms. At time t2 = 2.0s its position x2 is x2 = 20m + (5.0m/s2)(2.0s)2 = 40m.The displacement during this the interval is the x= x2 – x1= 40 m – 25 m=15 m.(b) The average velocity during this time interval is At time t2, the position is x2 = 20m+(5.0m/s2)(1.1s)2 = 26.05m The average velocity during this interval is We invite you to follow this same pattern to work out the average velocities for the 0.01s and 0.001s intervals. The results are 10.05m/s and 10.005m/s. As t gets closer to 10.0m/s, so we conclude that the instantaneous velocity at time t = 1.0s is 10.0m/s.(d) We find the instantaneous velocity as a function of time by taking the derivative of the expression for x with respect to t. For any n the derivative of t is ntn-1, so the derivative of t2 is 2t. Therefore At time t = 1.0s, v = 10m/s as we found in part ( c ). At time t = 2.0s, v = 20m/s.2-4 Average and Instantaneous AccelerationWhen the velocity of a moving body changes with time, we say that the body has an acceleration. Just as velocity describes the rate of change of position with time, acceleration describes the rate of change of velocity with time. Like velocity, acceleration is a vector quantity. In straight-line Motion its only nonzero component is along the axis along which the motion takes place.Average AccelerationLet' s consider again the motion of a particle along the x- axis. Suppose that at time t1 the particle is at point P1 and has x-component of (instantaneous) velocity v1, and at a later time t2 it is at point P2 and x-component of velocity v2. So the x-component of velocity changes by an amount v= v2 – v1 during the time interval t= t2 – t1.We define the average acceleration aav of the particle as it moves from P1 to P2 to be a vector quantity whose x-component is v, the change in the x-component of velocity, divided by the time interval t: (average acceleration, straight-line motion) (2-4) For straight-line motion we well usually call aav simply the average acceleration, remembering that in fact it is the x-component of the average acceleration vector. If we express velocity in meters per second and time in seconds, then average acceleration is in meters per second per second. The This is usually written as m/ s2 and is read" meters per second squared." Instantaneous AccelerationWe can now define instantaneous acceleration, following the same procedure that we used to define instantaneous velocity. Consider this situation: A race car driver has just entered the final straightaway at the Grand Prix. He reaches point P1 at time t1, moving with velocity v1. He passes point P2, closer to the line, at time t2 with velocity v2.(Fig. 2-8) To define the instantaneous acceleration at point P1, we take the second point P2 in Fig. 2-8 to be closer and closer to the first point P1 so that the average acceleration is computed over shorter and shorter time intervals. The instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. In the language of calculus, instantaneous acceleration equals the instan-taneous rate of change of velocity with time. Thus (instantaneous acceleration, straight-line motion) (2-5)Note that Eq. (2-5) is really the definition of the x-component of the acceleration vector; in straight-line motion, all other components of this vector are zero. Instantaneous acceleration plays an essential role in the laws of mechanics. The From now on, the when we use the term" acceleration", the we will always mean instantaneous acceleration, not average acceleration. Example: Average and instantaneous accelerations Suppose the velocity v of the car in Fig. 2-8 at any time t is given by the equation v = 60 m/s + (0.50 m/s3) t2.(a) Find the change in velocity of the car in the time interval between t1 = 1.0s and t2 = 3.0s. (b) Find the instantaneous acceleration in this time interval. (c) Find the instantaneous acceleration at time t1 = 1.0s by taking t to be first 0.1s, then 0.01s, then 0.001s. (d) Derive an expression for the instantaneous acceleration at any time, and use it to find the acceleration at t = 1.0s and t = 3.0s.Solution: (a) We first find the velocity at each time by substituting each value of t into the equation. At time t1 = 1.0s, v1 = 60m/s +(0.50m/s3)(1.0s)2 = 60.5m/s.At time t2 = 3.0s, v2 = 60m/s + (0.5m/s3)(3.0s)2 = 64.5m/s.The change in velocity v= v2 – v1=64.560.5=4.0 ms/ ses of –s.The time interval is t= 3.0 s – 1.0 s=2.0 s.(b) The average acceleration during this time interval isDuring the time interval from t1 = 1.0s to t2 = 3.0s, the velocity and average acceleration have the same algebraic sign (in this case, positive), and the car speeds up. (c) When ?t = 0.1s, t2 = 1.1s andv2 = 60m/s + (0.50m/s3)(1.1s)2 = 60.605 m/s,v = 0.105m/s,We invite you repeat this pattern for t = 0.01s and t = 0.001s; the results are aav = 1.0005m/s2 respectively. As t gets smaller, the average acceleration gets closer to 1.0m/s2. We conclude that the instantaneous acceleration at t1 = 1.0s is 1.0m/s2.(d) The instantaneous acceleration is a = dv/dt, the derivative of a constant is zero, and the derivative of t2 is 2t . Using these, we obtainWhen t = 3.0s, a = (1.0m/s3)(3.0s) = 1.0m/s2 2-5 Motion with Constant AccelerationThe simplest acceleration motion is straight-line motion with constant acceleration. In this case the velocity changes at the same rate throughout the motion. This is a very special situation, yet one that occurs often in nature. As we will discuss in the next section, a falling body has a constant acceleration if the effects of the air are not important. The same is true for a body sliding on an incline or along a rough horizontal surface. Straight-line motion with nearly constant acceleration also occurs in technology, such as a jet-fighter being catapulted from the deck of an aircraft carrier. The In this section we' ll derive key equations for straight- line motion with constant acceleration.Fig. 2-12 Fig. 2-13 Fig. 2-14 a t=0 v O a t=t v O a t=2t v O a t=3t v O a t=4t v O a v a at v v0 v0 t t O t O t Figure 2-12 is a motion diagram showing the position, velocity, and acceleration at five different times for a particle moving with constant acceleration. Figure 2-13 and 2-14 depict this same motion in the from of graphs. Since the acceleration a is constant, the a-t graph (graph of acceleration versus time) in Fig. 2-13 is a horizontal line. The graph of velocity versus time has a constant slope because the acceleration is constant, and so the v-t graph is a straight line ( Fig. 2-14).The When the acceleration is constant, it' s easy to derive equations for position x and velocity v as functions of time. Let' s start with velocity. In Eq. (2-4) we can replace the average acceleration aav by the constant (instantaneous) acceleration a. We then have (2-7) Now we let t1 = 0 and let t2 be any arbitrary later time t. We use the symbol v0 for the velocity at the initial time t is v. Then Eq. (2-7) because (2-8) Next we want to derive an equation for the position x of a particle moving with constant acceleration. To do this, we make use of two different expressions for the average velocity vav during the interval from t = o to any later time t. The first expression comes from the definition of vav Eq. (2-2), which holds true whether or not the acceleration is constant. We call the position at time t = 0 the initial position, denoted by x0. The position at the later time t is simply x. Thus for the time interval t = t – 0 and the corresponding displacement X= x – x0, Eq. (2-2) gives (2-9) We can also get a second expression for vav that is valid only when the acceleration is constant, so that the v-t graph is a straight line (as in Fig. 2-14) and the velocity changes at a constant rate. (2-10)(constant acceleration only). Substituting that expression for v into Eq. (2-10), we find (2-11) (constant acceleration only) Finally, we equate Eqs. (2-9) and (2-11) and simplify the result: or (2-12)We can check whether Eqs.(2-8) and (2-12) are consistent with the assumption of constant acceleration by taking the derivative of Eq. (2-12). We find which is Eq. (2-8). Differentiating again, we find simply as we should expect. The In many problems, it' the s useful to have a relationship between position, velocity, and acceleration that does not involve the time. To obtain this, we first solve Eq. (2-8) for t, then substitute the resulting expression into Eq. (2-12) and simplify: We transfer the term x0 to the left side and multiply through by 2a: Finally, simplifying gives us. We can get one more useful relationship by equating the two expressions for vav, Eqs. (2-9) and (2-10), and multiplying through by t. Doing this, we obtain (2-14)A special case of motion with constant acceleration occurs when the acceleration is zero. The velocity is then constant, and the equations of motion become simply v = v0 = constant, x = x0 + vt.2-7 Velocity and Position by Integration This optional section is intended for students who have already learned a little integral calculus. In Section 2-5 we analyzed the special case of straight-line motion with constant acceleration. When a is not constant, as is frequently the case, the equations that we derived in that section are no longer valid. But even when a varies with time, we can still use the relation v = dx/dt to find the velocity v as a function of time if the position x is a known function of time. And we can still use a = dv/dt to find the acceleration a as a function of time if the velocity v is a known function of time.In many physical situations, however, position and velocity are not known as functions of time, while the acceleration is. Figure 2-13We first consider a graphical approach, Figure 2-23 is a graph of acceleration versus time for a body whose acceleration is not constant but increases with time. We can divide the time interval between times t1 and t2 into many smaller intervals, calling a typical one t. Let the average acceleration during t be aav. From Eq. (2-4) the change in velocity v during t is v = aav t. Graphically, v equals the area of the shaded strip with height aav and width t, that is, the area under the curve between the left and right sides of t. The total velocity change during any interval (say, t1 to t2) is the sum of the velocity changes v in the small subintervals. So the total velocity changes is represented graphically by the total area under the a-t curve between the vertical lines t1 and t2. In the limit that all the T' the s become very small and their number very large, the the value of aav for the interval from any time t to t+ t approaches the instantaneous acceleration a at time t. In this limit, the area under the a-t curve is the integral of a (which is in general a function of t) from t1 to t2. If v1 is the velocity of the body at time t1 and v2 is the velocity at time t2, then aav t1 O t2 t The change in velocity v is the integral of acceleration a with respect to time. We can carry exactly the same procedure with the curve of velocity versus time where v is in general a function of t. If x1 is a body' s position at time t1 and x2 is its position at time t2, from Eq. (2-2) the displacement x during a small time interval ?t is equal to vav t, where vav is given by (2-16) . The change in position x – that is, the displacement – is the time integral of velocity v. Graphically, the displacement between times t1 and t2 is the area under the v-t curve between those two times. If t1 = 0 and t2 is any later time t, and if x0 and v0 are the position and velocity, respectively, at time t = 0, then we can rewrite Eqs. (2-15) and (2-16) as follows: (2-17) (2-18). Here x and v are the position and velocity at time t. If we know the acceleration a as a function of time and we know the initial velocity v0, we can use Eq, (2-17) to find the velocity v at any time; in other words, we can find v as a function of time. Once we know this function, and given the initial position x0, we can use Eq. (2-18) to find the position x at any time.Example 2-9 Sally is driving along a straight highway in her classic 1965 Mustang. At time t = 0, when Sally is moving at 10 m/s in the position x-direction, she passes a signpost at x = 50m. Her acceleration is a function of time: A= the 2.0 ms/ s2 – (0.10 ms/ s3) t (a) Derive expressions for her velocity and position as functions of time.(b) At what time is her velocity greatest? (c) What is the maximum velocity? (d) Where is the car when it reaches maximum velocity.Solution: (a) The At time t=0, Sally' the s position is x0=50 ms, the and her velocity is v0=10 ms/ s. Since we are given the acceleration a as a function of time, we first use Eq. (2-17) to find the velocity v as a function of time t.Then we use Eq.(2-18) to find x as a function of t:At this instant, dv/dt = a = 0. Setting the expression for acceleration equal to zero, we obtain: ?$ (c) We find the maximum velocity by substituting t = 20s (when velocity is maximum) into the general velocity equation:(d) The maximum value of v occurs at t = 20s, we obtain the position of the car (that is, the value of x) at that time by substituting t = 20s into the general expression for x:As before, we are concerned with describing motion, not with analyzing its causes. But the language you learn here will be an essential tool in later chapters when you use Newton' s laws of motion to study the relation between force and motion. 3-2 Position and velocity vectorsTo describe the motion of a particle in space, we first need to be able to describe the position of the particle. Consider a particle that is at a point P at a certain instant. The position vector r of the particle at this instant is a vector that goes from the origin of the coordinate system to the point P(Fig. 3-1). The figure also shows that the Cartesian coordinates x, y, and z of point P are the x-, y-, and z-components of vector r. Using the unit vectors introduced in Section 1-10, we can write Y r O X Z Figure 3-1 We can also get this result by taking the derivative of Eq(3-1). The unit vectors i, j and k are constant in magnitude and direction, so their derivatives are zero, and we find . This shows again that the components of v are dx/dt, dy/dt, and dz/dt. The magnitude of the instantaneous velocity vector v- that is, the speed – is given in terms of the components vx, vy, and vz by the Pythagorean relation: The instantaneous velocity vector is usually more interesting and useful than the average velocity vector. From now on, when we use the word" velocity", we will always mean the instantaneous velocity vector v( the rather than the average velocity vector). Usually, we won' even bother to call v a vector.3-3 The Acceleration VectorIn Fig (3-1), a particle is moving along a curvedThe vectors v1 and v2 represent the particle' s instantaneous velocities at time t1, when the particle is at point P1, and time t2, v v1 v2 When the particle is at point P2. The two velocities may differ in both magnitude and direction. We define the average acceleration aav of the particle as the particle as it moves from P1 and P2 as the vector change in velocity, v2-v1= v, divided by the time interval t2-t1 = t: Average acceleration is a vector quantity in the same direction as the vector v. As in Chapter 2, we define the instantaneous acceleration a at point P1as the limit approached by the average acceleration when point P2 approaches point P1 and v and t both approach zero; the instantaneous acceleration is also equal to the instantaneous rate of change of velocity with time. Because we are not restricted to straight-line motion, instantaneous acceleration is now a vector: v1 v2 v P2 v1 P1 v1 P1 a v2 aav C B A The velocity vector v, as we have seen, is tangent to the path of the particle. But the construction in fig.C shows that the instantaneous acceleration vector a of a moving particle always points toward the concave side of a curved path-that is, toward the inside of any turn that the particle is making. We can also see that when a particle is moving in a curved path, it always has nonzero acceleration. We will usually be interested in the instantaneous acceleration, not the average acceleration. From now on, we will use the term “acceleration” to mean the instantaneous acceleration vector a.Each component of the acceleration vector is the derivative of the corresponding component of velocity:Also, because each component of velocity is the derivative of the corresponding coordinate, we can express the ax, ay and az of the acceleration vector a as Example: Calculating average and instantaneous acceleration; Let’s look again at the radio-controlled model car in Example 3-1. We found that the components of instantaneous velocity at any time t areand that the velocity vector is a) Find the components of the average acceleration in the interval from t=0.0s to t=2.0s. b) Find the instantaneous acceleration at t=2.0s.Solution a) From Eq. (3-8), in order to calculate the components of the average acceleration, we need the instantaneous velocity at the beginning and the end of the time interval. We find the components of instantaneous velocity at time t=0.0s by substituting this value into the above expressions for vx, and vy. We find that at time t = 0.0s, vx = 0.0m/s, vy = 1.0m/s . We found in Example 3-1 that at t = 2.0s the values of these components are vx = -1.0m/s, vy = 1.3m/s.Thus the components of average acceleration in this interval are b) From Eq. 3-10 the components acceleration vector a asAt time t = 2.0s, the components of instantaneous acceleration areax = -0.5m/s2 , ay = (0.15m/s3)(2.0s) = 0.30m/s2 . The acceleration vector at this time is 3-5 Motion in A CircleWhen a particle moves along a curved path, the direction of its velocity changes.As we saw in Section 3-3, this means that the particle must have a component of acceleration perpendicular to the path, even if its speed is constant. In this section we’ll calculate the acceleration for the important special case of motion in a circle.Uniform Circular MotionWhen a particle moves in a circle with constant speed, the motion is called uniform circular motion. There is no component of acceleration parallel (tangent) to the path; otherwise, the speed would change. The component of acceleration perpendicular (normal) to the path, which cause the direction of the velocity to change, is related in a simple way to the speed of the particle and the radius of the circle.In uniform circular motion the acceleration is perpendicular to the velocity at each instant; as the direction of the velocity changes, the direction of the acceleration also changes. v2 P1 P2 v1 v s P1 P2 R v1 v2 O A O B Figure A shows a particle moving with constant speed in a circular path radius R with center at O. The particle moves from P1 to P2 in a time t. The vector change in velocity v during this time is shown in Fig. B. The angles labeled in Fig. A and B are the same because v1 is perpendicular to the line OP1 and v2 is perpendicular to the line OP2. Hence the triangles OP1P2(Fig. A) and OP1P2(Fig. B) are similar. Ratios of corresponding sides are equal, so or The magnitude aav of the average acceleration during t is thereforeThe magnitude of the instantaneous acceleration a at point P1. Also, P1 can is the limit of this expression as we take point P2 closer and closer to point P1: But the limit of s/ t is the speed v1 at point P1. Also, P1 can be any point on the path, so we can drop the subscript and let v represent the speed at any point. ThenBecause the speed is constant, the acceleration is always perpendicular to the instantaneous velocity.We conclude: In uniform circular motion, the magnitude a of the instantaneous acceleration is equal to the square v divided by the radius R of the circle. Its direction is perpendicular to v and inward along the radius. Because the acceleration is always directed toward the center of the circle, it is sometimes called centripetal acceleration.Non-Uniform Circular MotionWe have assumed calculate throughout this section that the particle’s speed is constant. If the speed varies, we call the motion non-uniform circular motion. In non-uniform circular motion, still gives the radial component of acceleration

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The structure of this humble diagram was formally developed by the mathematician John Venn, but its roots go back as far as the 13th Century, and includes many stages of evolution dictated by a number of noted logicians and philosophers. The earliest indications of similar diagram theory came from the writer Ramon Llull, whos initial work would later inspire the German polymath Leibnez. Leibnez was exploring early ideas regarding computational sciences and diagrammatic reasoning, using a style of diagram that would eventually be formalized by another famous mathematician. This was Leonhard Euler, the creator of the Euler diagram. Logician John Venn developed the Venn diagram in complement to Eulers concept. His diagram rules were more rigid than Eulers - each set must show its connection with all other sets within the union, even if no objects fall into this category. This is why Venn diagrams often only contain 2 or 3 sets, any more and the diagram can lose its symmetry and become overly complex. Venn made allowances for this by trading circles for ellipses and arcs, ensuring all connections are accounted for whilst maintaining the aesthetic of the diagram. Usage for Venn diagrams has evolved somewhat since their inception. Both Euler and Venn diagrams were used to logically and visually frame a philosophical concept, taking phrases such as some of x is y, all of y is z and condensing that information into a diagram that can be summarized at a glance. They are used in, and indeed were formed as an extension of, set theory - a branch of mathematical logic that can describe objects relations through algebraic equation. Now the Venn diagram is so ubiquitous and well ingrained a concept that you can see its use far outside mathematical confines. The form is so recognizable that it can shown through mediums such as advertising or news broadcast and the meaning will immediately be understood. They are used extensively in teaching environments - their generic functionality can apply to any subject and focus on my facet of it. Whether creating a business presentation, collating marketing data, or just visualizing a strategic concept, the Venn diagram is a quick, functional, and effective way of exploring logical relationships within a context. A Venn diagram, sometimes referred to as a set diagram, is a diagramming style used to show all the possible logical relations between a finite amount of sets. In mathematical terms, a set is a collection of distinct objects gathered together into a group, which can then itself be termed as a single object. Venn diagrams represent these objects on a page as circles or ellipses, and their placement in relation to each other describes the relationships between them. Commonly a Venn diagram will compare two sets with each other. In such a case, two circles will be used to represent the two sets, and they are placed on the page in such a way as that there is an overlap between them. This overlap, known as the intersection, represents the connection between sets - if for example the sets are mammals and sea life, then the intersection will be marine mammals, e.g. dolphins or whales. Each set is taken to contain every instance possible of its class; everything outside the union of sets (union is the term for the combined scope of all sets and intersections) is implicitly not any of those things - not a mammal, does not live underwater, etc. Euler diagrams are similar to Venn diagrams, in that both compare distinct sets using logical connections. Where they differ is that a Venn diagram is bound to show every possible intersection between sets, whether objects fall into that class or not; a Euler diagram only shows actually possible intersections within the given context. Sets can exist entirely within another, termed as a subset, or as a separate circle on the page without any connections - this is known as a disjoint. Furthering the example outlined previously, if a new set was introduced - birds - this would be shown as a circle entirely within the confines of the mammals set (but not overlapping sea life). A fourth set of trees would be a disjoint - a circle without any connections or intersections. Other Collections of Ecg Diagram Labeled

Similar Triangles Calculator This similar triangles calculator is here to help you find a similar triangle by scaling a known triangle. You can also use this calculator to find the missing length of a similar triangle! Stick around and scroll through this article as we discuss the laws of similar triangles and learn some fundamentals: - What are similar triangles? - Finding similar triangles: How do you determine whether two triangles are similar? - How do you find the missing side of a similar triangle? - How do you find the area of a similar triangle? What are similar triangles? Two triangles are similar if their corresponding sides are in the same ratio, which means that one triangle is a scaled version of the other. Naturally, the corresponding angles of similar triangles are equal. For example, consider the following two triangles: Notice that the corresponding sides are in proportion: Therefore, we can say . Here the symbol indicates that the triangles are similar. We term the proportion of similarity as the scale factor . In the example above, the scale factor . If you need help finding ratios, use our ratio calculator. Finding similar triangles: Law of similar triangles We know that two triangles are similar if either of the following is true: - The corresponding sides of the triangle are in proportion; or - The corresponding angles are equal. From this, we can derive specific rules to determine whether any two triangles are similar: - Side-Side-Side (SSS): If all three corresponding sides of the two triangles are in proportion, they are similar. This rule is the most straightforward and requires you to know all the sides of the triangles. We can express this using a similar triangle formula: where is the scale factor. - Side-Angle-Side (SAS): If any two corresponding sides of two triangles are in proportion and their included angles are equal, then the triangles are similar. We can use this rule whenever we know only two sides of each triangle and their included angles. The triangles in the image above are similar if: This rule is handy in cases like in the image below, where the triangles share an angle: You can do many things knowing just the Side-Angle-Side of a triangle. Learn more using our SAS triangle calculator. - Angle-Side-Angle (ASA): If any two corresponding angles of two triangles are equal and the corresponding sides between them are in proportion, the triangles are similar. The triangles in the image above are similar if: You can find the third angle if you know any two angles in a triangle using our triangle angle calculator. We know that if any two corresponding angles in the triangles are equal, the triangles are similar, meaning that in the ASA congruence rule, we don't need to know the side so long as the angles are known. However, without the sides, we cannot determine the scale factor . 💡 Need to find the area of a triangle? We have our triangle area calculator that can help you with that. How do you find the missing side of a similar triangle? To find the missing side of a triangle using the corresponding side of a similar triangle, follow these steps: - Find the scale factor kof the similar triangles by taking the ratio of any known side on the larger triangle and its corresponding side on the smaller one. - Determine whether the triangle with the missing side is smaller or larger. - If the triangle is smaller, divide its corresponding side in the larger triangle by kto get the missing side. Otherwise, multiply the corresponding side in the smaller triangle by kto find the missing side. For example, consider the following two similar triangles. To find the missing side, we first start by calculating their scale factor. Next, we use the scale factor relation between the missing side AC and its corresponding side DF: 🙋 You can also compare two right triangles and see their similarities using our Check Similarity in Right Triangles Calculator. How do you find the area of a similar triangle? To find the area of a triangle A1 from the area of its similar triangle A2, follow these steps: - Find the scale factor k of the similar triangles by taking the ratio of any known side on the larger triangle and its corresponding side on the smaller one. - Determine whether the triangle with the unknown area is smaller or larger. - If the triangle is smaller, divide A2 by the square of the scale factor k to get A1 = A2/k2. Otherwise, multiply A2 by k2 to get A1 = A2 × k2. How to use this similar triangles calculator Now that you've learned how to find the length of a similar triangle, the similar triangles formula, and more, you can quickly figure out how this similar triangles calculator works. To check whether two known triangles are similar, use this calculator as follows: - Select check similarity in the field - Choose the similarity criterion you want to use. You can choose between Side-Side-Side, Side-Angle-Side, and Angle-Side-Angle. - Enter the dimensions of the two triangles. The calculator will evaluate whether they are similar or not. To use this calculator to solve for the side or perimeter of similar triangles, follow these steps: - Select find the missing side in the field - Enter the known dimensions, area, perimeter, and scale factor of the triangles. The similar triangles calculator will find the unknown values. Are all equilateral triangles similar? Yes, if the corresponding angles of two triangles are equal, the triangles are similar. Since every angle in an equilateral triangle is equal to 60°, all equilateral triangles are similar. Find the scale factor of similar triangles whose areas are 10 cm² and 20 cm²? 1.414. To determine this scale factor based on the two areas, follow these steps: - Divide the larger area by the smaller area to get 20/10 = 2. - Find the square root of this value to get the scale factor, k = √2 = 1.414. - Verify this result using Omni's similar triangles calculator.

Students often struggle to solve problems in science that require them to do mathematical calculations, rearrange equations or draw/analyse graphs. Here are 10 ideas to improve the transferability of skills, ideally across the whole curriculum, not just in science (1) Collaborate with your maths department on the order of skills taught – The passage below is taken from the ASE: The Language of Mathematics in Science – Teaching Approaches The initial stimulus for collaborative work between the two faculties came from science teachers who identified that there were numeracy problems preventing students making rapid progress in science. Skills that should be transferable were not being applied by students outside of maths lessons. These were principally related to: changing the subject in equations; scaling graphs; drawing lines of best fit; and identifying types of graphs e.g. scatter or line? The problems were confounded by a mismatch in the order in which certain maths skills were taught in KS3 and applied in science. These timing issues meant that in science we were expecting high level skills that had not yet been fully covered in maths. Another issue is that many students are taught in different maths sets to their science ones and so it cannot be assumed that they all have the same maths skills or have been taught techniques or language. Data taken from science practical lessons could be processed in maths lessons so that the students can develop transferable skills. A joint maths and science day for students as outlined in teaching approaches has been shown to be beneficial to all. Looking at graphical analysis or molar calculations in a common way in maths and science helps everyone. (2) Use common language and teach the terms explicitly. To Maths teachers a line graph (and line of best fit) is a straight line, to science teachers, it could be curved. Students can often try to draw straight lines through what clearly looks like a curved set of points when asked to draw a line graph, because they are using the maths definition. Range is a numerical value in maths, but can have multiple meanings in science – the range of a variable, graph or measuring instrument. The term ‘variable’ is used infrequently in mathematics, but very commonly used in science, with students being expected to identify different categories of variable by age 11. In these categories is the term discontinuous used? ‘range’ is a numerical value in mathematics, but a quantity in science, linked to a specific variable. (3) Try to standardise the approach to answering questions A motorbike travels 20 miles in 10 minutes, how far does it go in an hour? How did you work this out? - By proportional reasoning? 60 minutes is 6 times as long as 10. So the distance is 6 times as long – 120 miles (it’s a fast bike) - By working out how far it travels in one minute (20/10 = 2 miles) then multiplying by how many minutes - By explicitly using the formula distance = speed x time - By using triangles Setting one of these problems and asking students how they worked it out is very interesting- then find out how the maths department would get them to answer it. (4) Talk about the concepts in words before you introduce the formula. Equations are stories about relationships. So for Newton’s second law you could talk about wanting to move a car that won’t start. It’s fairly intuitive that the harder you push it (the bigger the Force you apply) the bigger the change in motion (acceleration) of a car. But what would happen to the motion of the car when you push it as hard as you can, if it was very light? Or very heavy? From this thought experiment we can deduce that the change of motion is greater the bigger the force is, but also smaller the larger the mass so acceleration = Force/mass or a= F/m Almost all of you will have learnt this as F=ma, is a=F/m more appropriate? Read on .. (5) Consider the order you show students formulae Does it matter if you show F=ma, a=F/m or m= F/a or V=IR , I=V/R or R=V/I ? remembering that equations tell stories, what story makes the most sense? F = ma – Speeding up or slowing down a mass requires a force. The larger the force the greater the acceleration or deceleration for a given mass. Having a larger mass would require a larger force to maintain the same acceleration a = F/m – The acceleration depends on the Force and the mass. The greater the force, the greater the acceleration. The greater the mass, the smaller the acceleration a=F/m is generally more useful than the more usual F=ma as we are normally working out the acceleration having chosen the mass and the force. The story also makes more sense to me. Similarly, I would introduce I=V/R rather than V=IR (6) Rearranging the equation vs Changing the subject Usually, students are told to rearrange the equations. For those with good maths skills that is fine. However, to many students, it is a mysterious process. Consider changing the subject instead (With thanks to Helen Reynolds for this) - The cat is sitting on the mat - The mat is the thing the cat was sitting on - Sitting is what the cat was doing on the mat Here we have just changed the subject. All three statements say the same thing with the equals sign being represented by is. To some students, this is a revelation Similarly using numbers can demystify algebra. So 6 = 2×3 or 3=6/2 or 2=6/3 are also equivalent statements and far more intuitive than using letters (7) Use maths type starters in your lessons. - Change the subject so If a = F/m what does F = ? - I = V/R What would happen to the current if the potential difference got bigger/ Resistance got smaller etc. - What would the gradient of a distance- time give you? - An electric heater has a power of 2kW. How much energy does it transfer in one minute? - What happens to the kinetic energy of a bird when it’s velocity doubles (and its mass halves?) - Explain the story of this graph – Do we need more pirates? (8) Triangles – A useful aid or the work of the devil? Taken from Sparknotes Triangles are undoubtedly helpful for students who struggle to change the subject of equations. They shouldnt be used instead of trying to do the algebra. Triangles can be used to tell the stories and used as another alternative approach taught in combination with the other methods. Any other ideas please add to the comments section below

Abstracts of Talks In it will discuss joint work with David Nadler. We construct parts of a new three dimensional topological quantum field theory, which organizes representation theories associated to a complex reductive group, including Lusztig's character sheaves, Harish-Chandra modules for real forms of the group, and conjecturally the mixed Hodge theory of character varieties of the group. The character theories for Langlands dual groups are equivalent, leading to a collection of dualities for the objects listed above.Reference Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups I will explain how to determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In particular, this allows one to determine for which values of c this representation is finite dimensional. This generalizes a result of Varagnolo and Vasserot, arXiv:0705.2691, who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). Our proof is based on the Macdonald-Mehta integral and the elementary theory of distributions. The Kostka polynomials make the transition between Schur functions and Hall-Littlewood polynomials. Lusztig also interpreted them in terms of the stalks of intersection cohomology sheaves on the nilpotent cone of the general linear group (going through the affine Grassmannian and the affine flag manifold, so that they are also expressed in terms of Kazhdan-Lusztig polynomials). If one considers positive characteristic coefficients, the intersection cohomology sheaves are not so well behaved. For example, their stalks do not necessarily satisfy a parity vanishing. However, one can consider the indecomposable complexes which satisfy a parity vanishing for their stalks and costalks, and it turns out that they are still classified by partitions. They are the direct summands of direct images of constant sheaves on resolutions of partial flag varieties. If the characteristic is large enough, they are the intersection cohomology complexes. In general, their stalks yield generalizations of Kostka polynomials (which are combinations of the classical ones), depending on the characteristic. In our article, we work in a more general setting. Under certain conditions which are often satisfied in "representation theoretic situations" (including the above mentioned nilpotent cone, Kac-Moody Schubert varieties like the affine Grassmannian, and also toric varieties), the indecomposable constructible complexes having a parity vanishing for stalks and costalks are parametrized by pairs consisting of an orbit and an irreducible local system (just like simple perverse sheaves, in the equivariant setting). In the case of a semismall "even" resolution, we express the multiplicities of the parity sheaves in the direct image of the constant sheaf as the ranks modulo p of certain intersection forms (appearing in the work of de Cataldo and Migliorini). This gives a measure of the failing of the decomposition theorem with positive characteristic coefficients. On the affine Grassmannian, parity sheaves correspond to tilting modules under the geometric Satake correspondence, when the characteristic is large enough (greater than h + 1, where h is the Coxeter number, is enough). I will introduce Okounkov-Reshetikhin-Vafa type vertex operators to compute the generating function of Donaldson-Thomas type invariants of a small crepant resolution of a toric Calabi-Yau 3-fold. The commutator relation of the vertex operators gives the wall-crossing formula of Donaldson-Thomas type invariants.Reference Intersection cohomology on character/quiver varieties and the character ring of finite general linear groups Here we show that the "generic" part of the character ring of finite general linear groups can be described in terms of intersection cohomology of character and quiver varieties. The aim of this talk will be to explain concrete geometric pictures of field theoretic phenomena appearing in joint work with David Ben-Zvi on character sheaves. We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type An singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra gl(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of An x P1. The talk is based on the joint paper with Davesh Maulik. I will give an overview of the results on the topic of the title obtained in collaboration with T. Hausel and E. Letellier. The varieties in question are those parameterizing representations of the fundamental group of a punctured Riemann surface to GL_n with values in prescribed generic semisimple conjugacy classes at the punctures. The results are best expressed as a specialization of a generating series involving the Macdonald polynomials. We conjecture that the full generating series actually gives the mixed Hodge polynomials of the varieties. We prove that taking the pure part of these polynomials, which amounts to a different specialization of the generating series, actually gives the number of points of an associated quiver variety over finite fields.

1 MPa = 10 6 Pa = 1 N/mm 2 = 145.0 psi (lbf/in 2); Fatigue limit, endurance limit, and fatigue strength are used to describe the amplitude (or range) of cyclic stress that can be applied to the material without causing fatigue failure. We cut your steel plates starting from your plans with absolute accuracy. The yield strength point, Ï y, represents the limit of the elastic region of a material, that means that, within the correspondent strain range, the sample shows an elastic behavior. Elastic limit, also referred to as yield point, is an upper limit for the stress that can be applied to a material before it permanently deforms.This limit is measured in pounds per square inch (psi) or Newtons per square meter, also known as pascals (Pa). Yield point is well defined and shown on graph for mild steel and it's beyond elastic limit. Oxycut on demand. shape and size) on the removal of external force. Elastic limit As we have seen that if an external force is applied over the object, there will be some deformation or changes in ⦠elastic limit synonyms, elastic limit pronunciation, elastic limit translation, English dictionary definition of elastic limit. The elastic limit of a steel wire is 2.70 × 10 8 Pa. What is the maximum speed at which transverse wave pulses can propagate along this wire without exceeding this stress? Physics for Scientists and Engineers, Volume 1, Chapters 1-22 (8th Edition) Edit edition. Overview of Elastic Modulus Of Steel Elasticity is the property of an object to resume its normal shape and size after being stretched or compressed. A rolled steel product is given an elastic limit of 500 to 1200 MPa by selection of a particular steel composition and a particular heat treatment. For other materials like copper or aluminum is defined as the point of intersection of stress-strain curve and a line drawn parallel to linear part fron 0.2 percent deformation (strain ε) and it is also beyond the elastic limit. From the technical point of view spring steel has to meet the following requirements: A high technical elastic limit . It is a very common assumption that a compression spring would travel or can be compressed to its solid height. mm The material's elastic limit or yield strength is the maximum stress that can arise before the onset of plastic deformation. To determine the modulus of elasticity of steel, for example, first identify the region of elastic deformation in the stress-strain curve, which you now see applies to strains less than about 1 percent, or ε = 0.01. Beyond the elastic limit, the mild steel will experience plastic deformation. Define elastic limit. Hence, utilizing the modulus of elasticity formula, the modulus of elasticity of steel is e = Ï / ε = 250 n/mm2 / 0.01, or 25,000 n/mm2. Yield point is a point on the stress-strain curve at which there is a sudden increase in strain without a corresponding increase in stress. This starts the yield point â or the rolling point â which is point B, or the upper yield point. (The density of steel ⦠More. linear relation between the two. ; Creep. Special steel plates. The elastic limit of the material is the stress on the curve that lies between the proportional limit and the upper yield point. It got great weldability and machinability, let us see more mechanical details of this steel. This is known as Hookâs law. Its SI unit is also the pascal (Pa). Proportional limit (point A) Elastic limit (point B) Yield point ( upper yield point C and lower yield point D) Ultimate stress point (point E) Breaking point (point F) Proportional limit. Elastic deformation is a straightforward process, in which the deformation increases with an increase in force, until the elastic limit of the object is reached. the steel material to largely deform before fracturing. However, the compression spring has certain limit of ⦠Rubber, polythene, steel, copper and mild steel will be considered as the nice examples of elastic materials within certain load limits. The unit weight of structural steel is specified in the design standard EN 1991-1-1 Table A.4 between 77.0 kN/m 3 and 78.5 kN/m 3. See accompanying figure at (1, 2). As seen in the graph, from this point on the correlation between the stress and strain is no longer on a straight trajectory. The time dependent deformation due to heavy load over time is known as creep.. All materials show elastic behaviour to a degree, some more than others. It is the highest limit of the material before the plastic deformation of the material can occur. An added service which allows collecting the product ready to be used. Note this is not the same as the breaking stress for the wire which will typically be significantly higher for a ductile material like steel. This article discusses the properties and applications of stainless steel grade 304 (UNS S30400). Elastic Limit: It is defined as the value of stress upto and within which the material return back to their original position (i.e. Youngâs modulus is named after British scientist Thomas Young but it was developed by Leonhard Euler in 1727. The elastic limit depends markedly on the type of solid considered; for example, a steel bar or wire can be extended elastically only about 1 percent of its original length, while for strips of certain rubberlike materials, elastic extensions of up to 1,000 percent can be achieved. If the value of external force is such that it exceeds the elastic limit, than the body will not completely regain its original position. The elastic limit is defined as the maximum stretch limit of the compression spring without taking a permanent set. When the stresses exceed the yield point, the steel will not be able to bounce back. The elastic limit of steel is 8 x10 8 N/m 2 and its Young's modulus 2 x10 11 N/m 2.Find the maximum elongation of a half-meter steel wire that can be given without exceeding the elastic limit. Determine the minimum diameter a steel wire can have if it is to support a 70 kg person. High strenght steel . If stress is added to the metal but does not reach the yield point, it will return to its original shape after the stress is removed. If the elastic limit of steel is 5.0 10 8 Pa,determine the minimum diameter a steel wire can have if it is tosupport a 60 kg circus performerwithout its elastic limit being exceeded. Material Properties of S355 Steel - An Overview S355 is a non-alloy European standard (EN 10025-2) structural steel, most commonly used after S235 where more strength is needed. If the elastic limit of steel is 5.0 multiplied by 108 Pa, determine the minimum diameter a steel wire can have if it is to support a 75-kg circus performer without its elastic limit being exceeded. Elastic limit is defined as the maximum stress that a material can withstand before the permanent deformation. for example rubber. Overview. Formable steel high elastic limit for all types of load-bearing structures. If, after a load has been applied and then quickly removed, a material returns rapidly to its original shape, it is said to be behaving elastically. The elastic limit correlates with the largest austenite free-mean path by a Hall-Petch type equation. e.g. Sponsored Links Related Topics For structural design it is standard practice to consider the unit weight of structural steel equal to γ = 78.5 kN/m 3 and the density of structural steel approximately Ï = 7850 kg/m 3. sigma=E*epsilon for steel. Once the stress or force is removed from the material, the material comes back to its original shape. This is an approximation of the elastic limit of the steel. The Ultimate Tensile Strength - UTS - of a material is the limit stress at which the material actually breaks, with a sudden release of the stored elastic energy. The elastic limit of a material is an important consideration in civil, mechanical, and aerospace engineering and design. For mild steel the elastic limit is about 400 MPa. Elastic limit - Designing Buildings Wiki - Share your construction industry knowledge. Which is the tension that can be applied on the material without a plastic deformation? - proportional limit is strain below which the stress is proportional to strain i.e. The elastic limit for steel is for all practical purposes the same as its proportional limit. Spring steel is also used when there are special requirements on rigidity or abrasion resistance. Elastic modulus is a material property that demonstrates the quality or flexibility of the steel materials utilized for making mold parts. - elastic limit is strain below which the material can regain its original shape if the forces are release, doesn't matter if the stress-strain relation is linear or not. Problem 25P from Chapter 16: Review. So 1 percent is the elastic limit or the limit of reversible deformation. As shown in stress strain curve for mild steel, up to the point A, stress and strain follow a relationship.

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Angleangleangle aa if the angles in a triangle are congruent equal to the corresponding angles of another triangle then the triangles are similar. After all of the students begin to realize that not all of the triangles are congruent, i will ask, if they are not congruent, then what can we say about the triangles that were created in this case. Determine if the two triangles shown below are similar. Altitude and 3 similar light triangles an altitude of a fight triangle, extending from the fight angle vertex to the hypotenuse, creates 3 similar triangles. Sides su and zy correspond, as do ts and xz, and tu and xy, leading to the following proportions. For example, if two triangles have the same angles, then they are similar. For example, photography uses similar triangles to calculate distances from the lens to the object and to the image size. The mathematical presentation of two similar triangles a 1 b 1 c 1 and a 2 b 2 c 2 as shown by the figure beside is. Example 1 identifying similar right triangles tell whether the two right triangles are similar. The chart below shows an example of each type of triangle when it is classified by its sides and angles. Scroll down the page for more examples and solutions on how to detect similar triangles and how to use similar triangles to solve problems. Notes,whiteboard,whiteboard page,notebook software,notebook, pdf,smart,smart technologies ulc,smart board interactive whiteboard. Properties of similar triangles, aa rule, sas rule, sss rule, solving problems with similar triangles, examples with step by step solutions, how to use similar triangles to solve word problems, height of an object, shadow problems, how to solve for unknown values using the properties of similar triangles. You could check with a protractor that the angles on the left of each triangle are equal, the angles at the top of each triangle are equal, and the angles on the right of each triangle are equal. Triangles are similar if they have the same shape, but can be different sizes. Learn how to solve with similar triangles here, and then test your understanding with a quiz. Solve similar triangles advanced solving similar triangles. If two triangles have three equal angles, they need not be congruent. When the ratio is 1 then the similar triangles become congruent triangles. Similar triangles page 1 state and prove the following corollary to the converse to the alternate interior angles theorem. According to theorem 60, this also means that the scale factor of these two similar triangles is 3. Classify this triangle based on its sides and angles. Then, determine the value of x shown in the diagram. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Assessment included with solutions and markschemes. In this lesson, you will continue the study of similar polygons by looking at properties of similar triangles. The new pool will be similar in shape, but only 40 meters long. Two triangles abc and abc are similar if the three angles of the first triangle are congruent to the corresponding three. Similar figures have exactly the same shape but not necessarily the same size. Reasoning how does the ratio of the leg lengths of a right triangle compare to the ratio of the corresponding leg lengths of a similar right triangle. Another way to write these is in the form of side of one triangle, over the corresponding side of the other triangle. However, with the last side, which is not our side length. It is a specific scenario to solve a triangle when we are given 2 sides of a triangle. Similar triangles and shapes, includes pythagoras theorem, calculating areas of similar triangles, one real life application, circle theorems, challenging questions for the most able students. Then, we will focus on the triangles with angles of 30 degrees and 90 degrees. For instance, in the design at the corner, only two different shapes were actually drawn. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides. An equilateral triangle is also a special isosceles triangle. Nov 10, 2019 similar triangles are two triangles that have the same angles and corresponding sides that have equal proportions. Triangles have the same shape if they have the same angles. Generally, two triangles are said to be similar if they have the same shape, even if they are scaled, rotated or even flipped over. Example 5 use a scale factor in the diagram, atpr axpz. Thus, these pair of sides are not proportional and therefore our triangles cannot be similar. In this case, two of the sides are proportional, leading us to a scale factor of 2. Make a sketch of this situation including the sun, malik, and his shadow. They are still similar even if one is rotated, or one is a mirror image of the other. If so, state how you know they are similar and complete the similarity. State whether the following quadrilaterals are similar. What challenges andor misconceptions might students have when working with similar triangles. What about two or more squares or two or more equilateral triangles see fig. If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the. Similarity of triangles theorems, properties, examples. Two triangle that have the same shape are called similar. Similar triangles are the triangles which have the same shape but their sizes may vary. The activity that follows example 1 allows you to explore. Find perimeters of similar figures example 4 swimming a town is building a new swimming pool. Ill ask, are all of the triangles congruent in this case. Tenth grade lesson discovering similar triangles betterlesson. Triangle is a polygon which has three sides and three vertices. Definitions and theorems related to similar triangles are discussed using examples. Pythagoras theorem baudhayan theor hypotenuse is equal to the sum of. If youre seeing this message, it means were having trouble loading external resources on our website. What is the measure of each angle in a regular triangle. In the upcoming discussion, the relation between the areas of two similar triangles is discussed. Sss for similar triangles is not the same theorem as we used for congruent triangles. It is a specific scenario to solve a triangle when we are given 2 sides of a triangle and an angle in between them. In the case of triangles, this means that the two triangles will have. Similar triangles examples and problems with solutions. Solve similar triangles basic this is the currently selected item. Similarity of triangles uses the concept of similar shape and finds great applications. This lesson is designed to help students to discover the properties of similar triangles. Similar triangles examples the method of similar triangles comes up occasionally in math 120 and later courses. For example, in the picture below, the two triangles are similar. Give two different examples of pair of i similar figures. In the case of triangles, this means that the two triangles. If so, state how you know they are similar and complete the similarity statement. Using simple geometric theorems, you will be able to easily prove. I can use similar triangles to solve real world problems. Find the perimeter of an olympic pool and the new pool. This video is another similar triangles example using the fact of knowing knowing the ratio of corresponding sides are equal. Williams methods of proving triangles similar day 1 swbat. Sas for similarity be careful sas for similar triangles is not the same theorem as we used for. Similar triangles are triangles with equal corresponding angles and proportionate sides. Alternately, if one figure can be considered a transformation rotating, reflection, translation, or dilation of the other then they are also similar. Geometry notes similar triangles page 4 of 6 y y y y 7. Theorem converse to the corresponding angles theorem. If triangles are similar then the ratio of the corresponding sides are equal. Thus, two triangles with the same sides will be congruent. First, indicate the theorem that justifies why the triangles must be similar. To show triangles are similar, it is sufficient to show that the three sets of corresponding sides are in proportion. Investigating similar triangles and understanding proportionality. Sidesideside sss if three pairs of corresponding sides are in the same ratio then the triangles are similar. Use several methods to prove that triangles are similar. How to prove similar triangles with pictures wikihow. Similar triangles can be located any number of places, including one inside the other. Lessons 61, 62, and 63 identify similar polygons, and use ratios and proportions to solve problems. The ratio of the measures of the sides of a triangle is 4. Similar figures are used to represent various realworld situations involving a scale factor for the corresponding parts. As mentioned above, similar triangles have corresponding sides in proportion. Congruence, similarity, and the pythagorean theorem. Given that the triangles are similar, find the lengths of the missing sides. From the above, we can say that all congruent figures are similar but the similar figures need not be congruent. Also triangles abc and mac have two congruent angles. Solve similar triangles basic practice khan academy. You will use similar triangles to solve problems about photography in lesson 65. Two triangles are similar if and only if their side lengths are proportional. Similar triangles and ratios notes, examples, and practice test wsolutions this introduction includes similarity theorems, geometric means, sidesplitter theorem, angle bisector theorem, midsegments, and more. Bd is an altitude extending from vertex b to ac ab and bc are the other altitudes of the triangle then, displaying the 3 light triangles facing the same direction, we. They will be asked to determine the general conditions required to verify or prove that two triangles are similar and specifically. Oct 25, 2018 the ratio of any two sides of one triangle has to be equal to the ratio of the corresponding sides in the other triangle. Identifying similar triangles when the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other. Student notes full lesson discovering similar triangles. This triangle has a right angle in it so we know that its a right triangle. To be similar by definition, all corresponding sides have the same ratio or all corresponding angles are congruent. The areas of two similar triangles are 45 cm 2 and 80 cm 2. So setting these two ratios equal, thats the proportion we can set up. Two similar figures have the same shape but not necessarily the same size. Given two similar triangles and some of their side lengths, find a missing side length. Apr 14, 2011 this video is another similar triangles example using the fact of knowing knowing the ratio of corresponding sides are equal. Tenth grade lesson proving that triangles are similar. If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar. Hopefully, the students will remember their recent work with similar polygons and they will respond that everyones triangles are similar. Proving similar triangles refers to a geometric process by which you provide evidence to determine that two triangles have enough in common to be considered similar. Similar triangles implementing the mathematical practice standards. Similar triangles examples university of washington. Triangles having same shape and size are said to be congruent. Now that weve covered some of the basics, lets do some realworld examples, starting with sarah and the flagpole. It has two equal sides so its also an isosceles triangle. This occurs because you end up with similar triangles which have proportional sides and the altitude is the long leg of 1 triangle and the short leg of the other similar triangle. An example of two similar triangles is shown in figure 47. If the triangles are similar, what is the common ratio. Identifying similar triangles identify the similar triangles in the diagram. Triangles scalene isosceles equilateral use both the angle and side names when classifying a triangle. As observed in the case of circles, here also all squares are similar and all equilateral triangles are similar. Solution sketch the three similar right triangles so that the corresponding angles and. You could check with a protractor that the angles on the left of each triangle are equal, the angles at. Also examples and problems with detailed solutions are included. As an example of this, note that any two triangles with congruent legs must be similar to each other. If the perimeter of the triangle is 128 yards, find the length of the longest side. Marquis realizes that when he looks up from the ground, 60m away from the flagpole, that the top of the flagpole and the top of the building line up. And if youre working with a big problem, there may be a third similar triangle inside of the first two. When the ratio is 1 then the similar triangles become congruent triangles same shape and size. Solve similar triangles advanced practice khan academy. If youre behind a web filter, please make sure that the domains. Find the scale factor of the new pool to an olympic pool. Two triangles are said to be similar when they have two corresponding angles congruent and the sides proportional in the above diagram, we see that triangle efg is an enlarged version of triangle abc i. It turns out the when you drop an altitude h in the picture below from the the right angle of a right triangle, the length of the altitude becomes a geometric mean. Applications ratios between and within similar triangles in the diagram below, a large flagpole stands outside of an office building. All three sides are the same length and all three angles are the same size. Bd is an altitude extending from vertex b to ac ab and bc are the other altitudes of the triangle then, displaying the 3 light triangles. In the case of triangles, this means that the two triangles will have the same angles and their sides will be in the same proportion for example, the sides. Similar triangles tmsu0411282017 2 we can use the similarity relationship to solve for an unknown side of a triangle, given the known dimensions of corresponding sides in a similar triangle. Use this fact to find the unknown sides in the smaller triangle. Area of similar triangles and its theorems cbse class 10. Similar notesexamples polygons with the same but different polygons are similar if. An olympic pool is rectangular with length 50 meters and width 25 meters. All equilateral triangles, squares of any side length are examples of similar objects.874 847 1220 535 869 1089 85 716 1109 1548 413 223 464 310 1521 1401 719 4 348 214 449 231 882 249 919 797 342 1386 204 135 1181 213 350 272 567 994 95 1303 419 1077

Thus, he used the result that parallelograms are double the triangles with the same base and between the same parallels. Draw CJ and BE. The two triangles are congruent by SAS. The same result follows in a similar manner for the other rectangle and square. Katz, Click here for a GSP animation to illustrate this proof. The next three proofs are more easily seen proofs of the Pythagorean Theorem and would be ideal for high school mathematics students. In fact, these are proofs that students could be able to construct themselves at some point. The first proof begins with a rectangle divided up into three triangles, each of which contains a right angle. This proof can be seen through the use of computer technology, or with something as simple as a 3x5 index card cut up into right triangles. Figure 4 Figure 5. It can be seen that triangles 2 in green and 1 in red , will completely overlap triangle 3 in blue. Now, we can give a proof of the Pythagorean Theorem using these same triangles. Compare triangles 1 and 3. Angles E and D, respectively, are the right angles in these triangles. By comparing their similarities, we have. We have proved the Pythagorean Theorem. The next proof is another proof of the Pythagorean Theorem that begins with a rectangle. Thus, triangle EBF has sides with lengths ka, kb, and kc. By solving for k, we have. The next proof of the Pythagorean Theorem that will be presented is one that begins with a right triangle. In the next figure, triangle ABC is a right triangle. Its right angle is angle C. Triangle 1 Compare triangles 1 and 3: Triangle 1 green is the right triangle that we began with prior to constructing CD. Triangle 3 red is one of the two triangles formed by the construction of CD. Figure 13 Triangle 1. Compare triangles 1 and 2: Triangle 1 green is the same as above. Triangle 2 blue is the other triangle formed by constructing CD. Its right angle is angle D. Figure 14 Triangle 1. The next proof of the Pythagorean Theorem that will be presented is one in which a trapezoid will be used. By the construction that was used to form this trapezoid, all 6 of the triangles contained in this trapezoid are right triangles. We have completed the proof of the Pythagorean Theorem using the trapezoid. The next proof of the Pythagorean Theorem that I will present is one that can be taught and proved using puzzles. These puzzles can be constructed using the Pythagorean configuration and then, dissecting it into different shapes. Before the proof is presented, it is important that the next figure is explored since it directly relates to the proof. In this Pythagorean configuration, the square on the hypotenuse has been divided into 4 right triangles and 1 square, MNPQ, in the center. Each side of square MNPQ has length of a - b. This gives the following: As mentioned above, this proof of the Pythagorean Theorem can be further explored and proved using puzzles that are made from the Pythagorean configuration. Students can make these puzzles and then use the pieces from squares on the legs of the right triangle to cover the square on the hypotenuse. This can be a great connection because it is a "hands-on" activity. Students can then use the puzzle to prove the Pythagorean Theorem on their own. To create this puzzle, copy the square on BC twice, once placed below the square on AC and once to the right of the square on AC as shown in Figure Proof Using Figure Thus the diagonals CE and EH are both equal to c. Pieces 4 and 7, and pieces 5 and 6 are not separated. By calculating the area of each piece, it can be shown that. So shocked were the Pythagoreans by these numbers, they put to death a member who dared to mention their existence to the public. It would be years later that the Greek mathematician Eudoxus developed a way to deal with these unutterable numbers. Pythagoras of Samos Who is Pythagoras? He was born to Mnesarchus and Pythais. He was one of either three or four children, there is proof for both of these accounts. At a very early age, Pythagoras learned to play the lyre and recite Homer. Pythagoras also wrote poetry at a very early age. Pythagoras got his education from three philosophers, the most important mathematically being Anaximander. Around BC Pythagoras went on a journey to Croton, where he established a philosophical and religious school. Pythagoras was the head of the inner circle of the society, his inside followers were called mathematikoi. The mathematikoi had to follow very strict laws that Pythagoras believed. The five things that Pythagoras believed very deeply were 1 that at its deepest level, reality is mathematical in nature, 2 that philosophy can be used for spiritual purification, 3 that the soul can rise to union with the divine, 4 that certain symbols have a mystical significance, and 5 that all brothers of the order should observe strict loyalty and secrecy. Both men and women were allowed to join the society. Pythagoras was not acting as a modern research group at a major university. The Pythagorean theorem states that: "The area of the square built on the hypotenuse of a right triangle is equal to the sum of the squares on the remaining two sides." According to the Pythagorean Theorem, the sum of the areas of the red and yellow squares is equal to the area of the purple square. One of the topics that almost every high school geometry student learns about is the Pythagorean Theorem. When asked what the Pythagorean Theorem is, students will often state that a2+b2=c2 where a, b, and c are sides of a right triangle. Fermat's Last Theorem - Rationale: The pythagorean theorem is a simple equation that has been taught to pupils from the beginning of middle school. a2+b2=c2 is the basic formula to calculate any one of the sides on a right angle triangle. The Pythagorean Theorem Essay Sample Introduction a 2 + b 2 = c 2 where a and b are the sides of a right triangle and c is the hypotenuse is the answer most students will give when asked to define the Pythagorean Theorem. One theorem that is particularly renowned is the Pythagorean Theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of any right triangle. Essay, Research Paper: Pythagorean Theorem Mathematics Free Mathematics research papers were donated by our members/visitors and are presented free of charge for informational use only.

This action might not be possible to undo. Are you sure you want to continue? DEPARTMENT OF ENVIRONMENTAL AND BIOSYSTEMS ENGINEERING COURSE CODE & TITLE: FEB 423 HEAT AND MASS TRANSFER B.SC. IN ENVIRONMENTAL AND BIOSYSTEMS ENGINEERING Mr. Emmanuel Beauttah Kinyor Mutai Dip. Agric. Engin. (Egerton College), B.Sc. Agric. Engin. (Egerton Univ.), M.Sc. Agric. Engin. (UoN) OFFICE: Main Campus NUCLEAR SCIENCE ROOM No. 19 TEL: 318262 EXT 28471 0723630157 Environmental & Biosystems Building 2nd Floor Tel: firstname.lastname@example.org or email@example.com Tuesdays Fridays Thursdays Examinations 10:00 am – 1:00 pm 11:00 am – 1:00 pm 2:00 pm Main Campus office Cat 6th Week Assignments Laboratory Project Modeling & Simulation Final Exam Total 1 15% 5% 5% 5% 70% 100% Convection. and Radiation.1 What is Heat Transfer? Thermal energy is related to the temperature of matter. When you touch a hot object. Any energy exchange between bodies occurs through one of these modes or a combination of them. this mode uses the electromagnetic radiation emitted by an object for exchanging heat. and is denoted by q. For a given material and mass. Convection uses the movement of fluids to transfer heat.2 Three Modes of Heat Transfer There are three modes of heat transfer: • • • Conduction. the heat you feel is transferred through your skin by conduction. The rate of heat transfer is measured in watts (W). thermal energy transfers from the one with higher temperature to the one with lower temperature. Conduction is the transfer of heat through solids or stationery fluids. 1. Units and Conversion Factors for Heat Measurements SI Units 1J 1 J/s or 1 W 1 W/m2 English Units 9. the greater its thermal energy.1 Conduction Conduction is at transfer through solids or stationery fluids.3171 Btu/h ft2 Thermal Energy (Q) Heat Transfer Rate (q) Heat Flux (q") 1. Two mechanisms explain how heat is transferred by conduction: 2 . When two bodies are at different temperatures.4787×10-4 Btu 3. Radiation does not require a medium for transferring heat. Table 1 shows the common SI and English units and conversion factors used for heat and heat transfer rates.LESSON 1 Overview of Heat Transfer 1.2. Heat is typically given the symbol Q.4123 Btu/h 0. equal to joules per second. and uses q" for the symbol. is measured in watts per area (W/m2). and is expressed in joules (J) in SI units. Table 1. Heat always transfers from hot to cold. or the rate of heat transfer per unit area. The heat flux. the higher the temperature. Heat transfer is a study of the exchange of thermal energy through a body or between bodies which occurs when there is a temperature difference. where the electrons are moving at the same average velocity. where all the atoms are vibrating with the same energy. equilibrium is reached.2. which do not have many free electrons. the hot side of the solid experiences more vigorous atomic movements. the faster electrons give off some of their energy to the slower electrons. As the electrons undergo a series of collisions. The mechanism is identical to the electron collisions in metals. The vibrations are transmitted through the springs to the cooler side of the solid. analogous to springs as shown in Figure 1.1. through a series of random collisions. Eventually. Conduction through electron collision is more effective than through lattice vibration.2 Conduction by particle collision In fluids. Eventually. 3 .1 Conduction by lattice vibration Figure 1. Lattice vibration and 2. heat is conducted through stationery fluids primarily by molecular collisions. The electrons in the hot side of the solid move faster than those on the cooler side. In solids. This scenario is shown in Figure 1. this is why metals generally are better heat conductors than ceramic materials. Particle collision. have free electrons. atoms are bound to each other by a series of bonds.1. which are not bound to any particular atom and can freely move about the solid. they reach equilibrium. conduction occurs through collisions between freely moving molecules. Conduction through solids occurs by a combination of the two mechanisms. Figure 1. especially metals. Solids. When there is a temperature difference in the solid. which is then carried away by fluid movement such as wind.4. 1. The negative sign in Eqn. the rate of heat transfer is enhanced. The density of fluid decrease as it is heated. Warm fluids surrounding a hot object rises.The effectiveness by which heat is transferred through a material is measured by the thermal conductivity. has a high conductivity.2 Convection Convection uses the motion of fluids to transfer heat. A good conductor.1 ensures that this convention is obeyed. which can draw more heat away from the surface. The result is a circulation of air above the warm surface. In heat transfer. and a negative q represents heat leaving the body. has a low conductivity. and is replaced by cooler fluid. The warm fluid is replaced by cooler fluid. hot fluids are lighter than cool fluids. or an insulator. 1.3 Heating curve Where A is the cross-sectional area through which the heat is conducting. a hot surface heats the surrounding fluid. In a typical convective heat transfer. 4 .2. k. Conductivity is measured in watts per meter per Kelvin (W/mK). The rate of heat transfer by conduction is given by: (Eq. Since the heated fluid is constantly replaced by cooler fluid. T is the temperature difference between the two surfaces separated by a distance ∆x (see Figure 1.3). Natural convection (or free convection) refers to a case where the fluid movement is created by the warm fluid itself. 1. as shown in Figure 1.1) Figure 1. a positive q means that heat is flowing into the body. a poor conductor. such as copper. thus. Infrared (Ir). Ultraviolet (uv). having 5 . and T∞ is the ambient or fluid temperature. is why we feel warmer in the sun than in the shade. We all experience radiative heat transfer everyday. and is determined by factors such as the fluid density. winter day feel much colder than a calm day with same temperature.2. Wind blowing at 5 mph has a lower h than wind at the same temperature blowing at 30 mph.Figure 1. 1. it is the only form of heat transfer present in vacuum. solar radiation. h. The electromagnetic spectrum classifies radiation according to wavelengths of the radiation. Radiative heat transfer occurs when the emitted radiation strikes another body and is absorbed. is the measure of how effectively a fluid transfers heat by convection. Natural wind and fans are the two most common sources of forced convection. Visible light. The rate of heat transfer from a surface by convection is given by: (Eq. It uses electromagnetic radiation (photons). which travels at the speed of light and is emitted by any matter with temperature above 0 degrees Kelvin (-273 °C). Tsurface is the surface temperature. Forced convection is what makes a windy.3 Radiation Radiative heat transfer does not require a medium to pass through. The heat loss from your body is increased due to the constant replenishment of cold air by the wind. thus. Main types of radiation are (from short to long wavelengths): • • • • • • • Gamma rays. viscosity. and velocity. and Radio waves. X-rays.4 Natural convection Forced convection uses external means of producing fluid movement. Convection coefficient. absorbed by our skin. 1. It is measured in W/m2K. Microwaves.2) Where A is the surface area of the object. X-rays. Radiation with shorter wavelengths are more energetic and contains more heat. T is the temperature of the body. Hotter objects. Most "hot" objects. equal to 5. having wavelengths on the order of meters. as we all know. emit more energetic radiation including visible and UV. The emitted radiation strikes a second surface. can readily pass through concrete walls. The percentage of the incident radiation that is absorbed is called the absorptivity. α. The emissivity has a value between zero and 1. Figure 1.5 Interaction between a surface and incident radiation The incident radiation is determined by the amount of radiation emitted by the object and how much of the emitted radiation actually strikes the surface. 1. radio waves. 1. A second characteristic which will become important later is that radiation with longer wavelengths generally can penetrate through thicker solids. σ is a constant called StefanBoltzmann constant. Visible light. absorbed.3) Where A is the surface area. and ε is a material property called emissivity. The amount of radiation emitted by an object is given by: (Eq. The visible portion is evident from the bright glare of the sun. such as the sun at ~5800 K. is blocked by a wall. emit infrared radiation. The latter is given by the 6 . or transmitted (Figure 1. where it is reflected. However. It is the ratio of the radiation emitted by a surface to the radiation emitted by a perfect emitter at the same temperature.67×10-8 W/m2K4.4) Where I is the incident radiation. from a cooking standpoint. the UV radiation causes tans and burns. Any body with temperature above 0 Kelvin emits radiation. are very energetic and can be harmful to humans. and is a measure of how efficiently a surface emits radiation. The portion that contributes to the heating of the surface is the absorbed radiation. while visible light with wavelengths ~10-7 m contain less energy and therefore have little effect on life. The amount of heat absorbed by the surface is given by: (Eq.5).wavelengths ~10-9 m. The type of radiation emitted is determined largely by the temperature of the body. 2. the radiative exchange between the object and the wall is greatly simplified: (Eq. and the convection coefficient from the loaf to air is 10 W/m2K.6) This simplification can be made because all of the radiation emitted by the object strikes the wall (Fobject→wall = 1). and its surface temperature is 120 ºC. 7 . 1.121.5) For an object in an enclosure.76. The temperature of the air is 20 ºC. Define the following heat transfer situations as either conduction. The net amount of radiation absorbed by the surface is: (Eq. The dimension of the loaf is as described in the figure. F. and its conductivity is 0. A loaf of freshly baked bread is left to cool on a cooling rack. A turkey is being roasted in the oven. For example: A person with a headache holds a cold ice pack to his/her forehead. Emissivity of the bread is 0. o o o o o The sun shines brightly on a car. which is the percentage of the emitted radiation reaching the surface. Potatoes are boiled in water. convection. 1. Problems for Chapter 1 Note: These problems are NOT your homework assignments. or a combination of the three. Please also clearly state what two objects the mode of heat transfer is between and the direction of heat transfer. radiation. 1. An ice cube is placed on a metal tray and left out of the freezer. making the black upholstery very hot. Homework’s will be assigned in class on a separate handout. Answer: Conduction occurs from the person’s forehead to the ice pack.shape factor. A small 4" fan is installed in the back of a computer to help cool the electronics. the thermal resistance is expressed as: (Eq. k is the thermal conductivity of the layer. A cup of all to the on these information. When the system is still changing with time.1 Steady State and Transient State If you heat a pan on a stove. When there is more than one layer in the composite. it takes a while for the pan to heat up to cooking temperature. and A is the cross-sectional area. An analysis much like a circuit analysis follows. calculate the total heat from the bread. it is in transient state.Based loss 3. In this model. the total resistance of the circuit must be calculated. For conduction. Describe modes of heat transfer that contributes cooling of the coffee. and ∆T is the temperature difference between two surfaces separated by a distance ∆x. The total resistance for layers in series is simply the sum of the 8 . 3. hot coffee sits on the table. A is the cross-sectional area through which the heat is conducting.2) Where L is the thickness of the layer. The latter state is called the steady state. A model used often to calculate the heat transfer through a 1-D system is called the thermal circuit model. 2.2 One-Dimensional Conduction One-dimensional heat transfer refers to special cases where there is only one spatial variable – the temperature varies in one direction only. 3. after which the temperature of the pan remains relatively constant. LESSON 2 Steady-State Conduction 2. where there is no temporal change in temperatures. The rate of conduction through an object at steady-state is given by: (Eq.1) Where k is the conductivity of the material. This model simplifies the analysis of heat conduction through composite materials. each layer is replaced by an equivalent resistor called the thermal resistance. the heat flow through the layers can be found by: (Eq. and k4 = 46 W/mK.4) The convection at the surface must also be expressed as a resistor: (Eq. The convection coefficient on the right side of the composite is 30 W/m2K. 3. 3.3) For resistors in parallel.resistances: (Eq. Calculate the total resistance and the heat flow through the composite. 3. 3.5) Once the total resistance of a structure is found. the total resistance is given by: (Eq. Conductivities of the layer are: k1 = k3 = 10 W/mK. Example Problem Consider a composite structure shown on below. k2 = 16 W/mK. 9 .6) Where Tinitial and Tfinal refers to the temperatures at the two ends of the thermal circuit (analogous to voltage difference in an electrical circuit) and q is the heat flow through the circuit (current). an equivalent resistance for layers 1. R2 = 0.15. 2. The circuit must span between the two known temperatures. draw the thermal circuit for the composite. Next. and R4 = 0. and 3 is found first. T1 and T∞.36 To find the total resistance.First. that is. These three layers are combined in series: 10 . the thermal resistances corresponding to each layer are calculated: Similarly.09. R3 = 0. The equivalent resistor R1.2.3 is in parallel with R4: Finally.2. ← heat flow through the composite 11 .3.4 is in series with R5. R1. The total resistance of the circuit is: ← total thermal resistance Rtotal = R1.46 The heat transfer through the composite is: = 22.214.171.124 W.4 + R5 = 0. This action might not be possible to undo. Are you sure you want to continue? We've moved you to where you read on your other device. Get the full title to continue listening from where you left off, or restart the preview.

3. Physical parameters and results The calculations have been carried out in the scheme described in Paper III for the nonlinear evolution of axisymmetric pinching (body) modes for a cylindrical jet. We have set, at , the jet Mach number and the density ratio . In order to set the remaining parameters, and , we recall from Paper II that a choice of K yields post-shock temperatures K, consistent with the low excitation spectra observed. We recall also that since the initial cooling time is large with respect to the dynamical time scale, choices of that differ up to 40% have shown in tests calculations to have very little effect on the nonlinear evolution of the instability. Concerning the choice of , observations of HH34 and HH111 are consistent with jet radii cm and particle density (Bürkhe et al. 1988, Morse et al 1993a). We have therefore adopted for the column density ; this choice implies for the jet a mass flux and a momentum flux , with . In Fig. 1 we show a contour plot of the morphology of the emission flux, for , at the time when the train of shocks has reached a distance from the origin corresponding to about 400 radii. The results for yield similar morphologies. We note however that the general morphology is weakly dependent on the particular time chosen, as can be seen, for this set of parameters, in Fig. 1a,b ( (Fig. 1a) and (Fig. 1b). In this figure we show also the details of the morphology of the leading shock, with a clear bow-like form, and of preceding ones (i.e. at smaller z) that appear to have a more compact structure. The line fluxes are obtained by integrating the emissivity along the line-of-sight under the hypothesis that this is perpendicular to the jet longitudinal axis. From Fig. 1 we see that one of the leading shocks, being a result of shock merging processes, is somewhat more distant from the preceding one, is wider and has a distinct bow-shock like morphology. The knots result quasi-equally spaced, with mean intra-knot distance of jet diameters, and with an initial gap whose length partly depends on the amplitude of the initial perturbation imposed perpendicularly to the equilibrium velocity ( in the present case). We note also that the shocks most distant from the source weaken with time (compare Fig. 1a and b). The spectral and kinematical characteristics of the shocks resulting from K-H instability are shown in Fig. 2 where we plot, at the time and for , the on-axis behavior against z of the electron density in units of the initial density (panel a) and the fluxes of (panel b) and (panel c); in panel d) we plot the flux ratio , obtained averaging the fluxes over the emission volume of each shock, and in panel e) we show the shock pattern speeds . As a comparison, we show in Fig. 3 the behavior of the same quantities of Fig. 2 but for and at time . In both cases of the ionization fraction attains maximum values that do not exceed , and the different shock strengths yield values of reaching (Fig. 2a) and (Fig. 3a), respectively for and . Figs. 2b,c and 3b,c show how the flux in the two lines increases, reaches a maximum and then decreases, in qualitative agreement with observations; from Figs. 2d and 3d we notice also that the line ratio attains a high value for the leading, widest shock. A further comparison of Fig. 2 with Fig. 3 show in the latter case wider initial gap and intra-shock spacings, and lower values of the ratio (Figs. 2d and 3d), consistently with the increasing of strength and excitation level of the shocks at a higher Mach number. Finally Figs. 2e and 3e show that the proper motions of knots increase with distance from the origin from up to 0.8, in the case of , and from 0.5 up to 0.7, for . These results are in good qualitative agreement with the findings of Eislöffel & Mundt (1992) for HH34. We have taken Figs. 2 and 3 as representative snapshots of typical morphologies; in fact, calculations show that the shock train, after a time scale depending on the physical parameters, reaches a asymptotic configuration that remains quasi-steady and simply shifts forward. About the possibility of shock merging effects, that lead to bow-shock like features, we note that the condition for these processes to set in is that, locally, a shock at larger z must have proper motion smaller than the following one,i.e. at smaller z (see the discussion in the companion Paper III). Looking at Figs. 2e and 3e we can see that this condition can be verified in several positions along the jet, therefore one may expect that bow-shock like knots, originated by K-H modes, can be a more common feature than actually shown in our calculations, limited in time. We remark finally the internal consistency between the increasing of shock pattern motions with distance z along the jet (Figs. 2e and 3e), which should give a lower shock strength, and the increasing of the ratio (Figs. 2d and 3d). Table 2. Model results, to be compared with Table 1 We leave distances and densities in units of a and , with , bearing in mind that the values must obey . In Table 2, the results for are in general agreement with observations; setting for a the values of HH34 and HH111, the major discrepancies are: i) the length of the initial invisible part of the jet (the 'gap') that is larger by a factor of with respect to observations, ii) the widths of the knots in tend to be smaller by a similar factor, and iii) the post-shock electron densities are smaller than the observed values, especially in the case , by a factor . However, being the choice of the set of parameters by no means unique, one can expect, at best, only a broad agreement from the comparison with observations. What is important to stress is the trend brought about by the variation of the Mach number, i.e. higher values of M cause an increase of the intra-knot spacing and a decrease of the ratio. Also the temporal evolution plays a role, increasing the relative length of the visible part of the jet (see Fig. 1). Observations indicate jet velocities , implying . Unfortunately, our capability of carrying out calculations with , on the same grid (), was greatly hampered by the growing size of the integration domain. In order to gain some insight on the behavior of the instability at different Mach numbers we have carried out additional calculations on a coarser grid () for , and with a larger longitudinal size of the domain (800 jet radii). The simulation of the case has been carried out up to , when the leading perturbation reached the right boundary; therefore we cannot represent a quasi-steady situation and in Table 2, there are missing data for this case: is larger than the computational domain, thus we cannot estimate , , and the knot number; the age of the jet is also missing and the electron density is, quite likely, severely underestimated. The remaining quantities: , knot width and separation, jet velocity and knot speed are instead reasonably well defined by the simulation. We recall that we defined M as the ratio of the jet velocity with respect to the external medium to the internal sound speed, and that is reported in Table 2. Observations of shock velocities in bow-shocks (Morse et al. 1992, 1993b, 1994) show values lower than those consistent with the measured proper motions of the bow-shocks themselves. This may suggest that the pre-shock ambient medium is not steady with respect to the central source, but may be drifting along the jet due to, perhaps, the effect of previous outburst of the source, thus lowering the actual velocity jump jet-to-ambient and the effective Mach number. In particular, Morse et al. (1992) found for HH34 a velocity of the pre-shock medium, at the bow-shock, , and for HH111 (Morse et al. 1993b). Following Hardee & Norman (1988), it is possible to derive that, in the linear and adiabatic regime, this mode has a resonance frequency and a corresponding resonance wavelength where is a coefficient that depends on the geometry (Cartesian or cylindrical) and on the particular mode (symmetric or asymmetric). The numerical results for the nonlinear evolution are reported in Fig. 4a,b. In Fig. 4a symbols represent the mean intra-knot spacing, in units of a, as a function of M and the error bars indicate the difference between maximum and minimum spacing. The dashed line shows the resonance wavelength of the fastest growing mode in the linear and adiabatic regime, according to (1), and the dot-dashed line interpolates our nonlinear results but with coefficient instead of of (1). In Fig. 4b we plot intensity ratio against M. To avoid ambiguity in the choice of a particular shock, we have selected one of the first shocks of the chain that have the advantage of being nearly of constant strength for a given M, as time elapses. The behavior of the intensity ratio is well represented by a power-law fit (solid line). Therefore, in the K-H scenario, the larger is the mean intra-knot spacing the smaller must be the line ratio. These last results show clearly how two of the main observable features of stellar jets, such as the intra-knot spacings and the line ratios, result connected if knots originate from the K-H instability, and this represents a test on the mechanisms proposed and a prediction for further observations (cf., point 7 in Sect. 2). © European Southern Observatory (ESO) 1998 Online publication: April 28, 1998

To warm up I invite students to the carpet to review some vocabulary terms, and to discuss what we have been learning so far about fractions. Because my students struggle with fractions it is important that I remind them of their prior knowledge. My students respond a lot better to new concepts if they connect it to prior experiences. I ask students to share their own experiences, and ideas about fractions. Students name the parts of the fraction, and are able to discuss how and why fractions are used. Some students remember using models to explain the pieces verses the whole. After the discussion I gain a better understanding about what my students know about fractions, and what they need to know about fractions in order to be successful in this lesson. "Since you guys already know a lot about fractions, I know you are really going to enjoy learning how to convert a fraction to a decimal. But, before we began I want to go over some new math terms that will assist you in your learning." As I go over the new vocabulary, I ask them to write the definitions down in their math note book just in case they need to remember what they mean later on in the lesson. A fraction is a number between zero and 1 and is expressed as one number over another number, like this: 1/2 Decimals-A linear array of digits that represents a real number. I intend to show students the connection between fraction and decimal notation by writing the same number both ways. I tell them that they will compare and contrast the difference and similarities between fractions and decimals. Because this is the first time decimals are introduced, I want to make sure I incorporate a visual throughout the lesson. I begin by drawing a large place value chart on the board. I explain that a number can be represented as both a fraction and a decimal. I start filling in the place value chart, by first placing a decimal in the center of the chart. I ask students to explain why I did this. More than half of my students could not explain. So, I take a moment to fill in the rest of the chart explaining the value of the numbers listed to the right and left hand side of the decimals. I explain that numbers written to the left of the decimal are whole numbers, while numbers written to the right of the decimal are parts of a whole number. Some students make the connection between fractions with the denominator as being the number listed to the right of the decimal. I enter 3 under the tenths place and 2 under the hundreds place. Can anyone tell me if should write this number as a fraction or a decimal? You should write is as a decimal because it is written to the right of the decimal. Several students wanted to write 32/100. I explain that if we were to write both 32/100 and .32 on a number line both would be placed closer to the .32. I repeat this activity using different numbers, and invite student volunteers to help me use decimal notation for fractions with denominators. As students are working I constantly redirect their thinking to the intended purpose of this lesson by asking how and why the given decimal notation is related to the fraction. I use their responses to determine if I should move them deeper into the lesson. ]MP.4. Model with mathematics. MP.7. Look for and make use of structure MP. 8. Look for and express regularity in repeated reasoning. In this portion of the lesson, I invite students to the carpet to take part in a fun interactive lesson. I encourage students to take notes throughout the lesson. note taking paper.pdf The purpose of this model is to deliver explicit instruction to provide students with a clearly explained task of how fractions are related to decimals. During the lesson I ask the following questions to guide students' thinking: Does it help to create a diagram? How would you prove that? Does that make sense? It is important that conceptual thinking is broken down into critical features/elements. Students note that the models help support their learning. Some students are able to explain, however, their explanations are a bit vague. I make a note to make sure I am showing them how to visually represent fractions. In this portion of the lesson I want to work with students a bit more on understanding that decimals are an extension of our whole number base ten system. To do this I ask students to move back into their assigned seats. I choose to stand at the board, so that students can work along with me independently. First, I draw a large chart on the board to model how to write fractions and decimals in expanded form. I give each student their own chart to work along with me. I write 7 82/100. I explain that 7 is a whole number, and it value is 7 ones. (They should remember this from previous school years. If they don't, do a quick mini-lesson on place value with whole numbers). I remind students that numbers written to the right of the decimal are parts of a whole so the number 8 is 8 tenths, and 2 is 2 hundredths. I ask, "Can anyone tell me how to write the given fraction in expanded fraction form?" 7 + 8/10 + 2/100 Great! Since, 7 is in the ones place, it remains 7. The number 8 is in the tenths place, so I write 8/10 to make sure it has its correct value. The number 2 is in the hundredths place, so I write 2/100 to make sure that 2 have its correct value. Now, I direct their attention to expanded decimal form. I tell them it is basically the same method we used to understand writing fractions in expanded for, but we use decimals and add zeros to make sure the given numbers are aligned with the correct value. I say, who can help me write the same number, but arrange it in expanded decimal form. Several students raise their hand, so we all give it a go. I ask, "How do I write the 7? How can I write the number 8 in decimals to represent its correct value? How can I write the number 2 in decimals to represent its correct value? So we write 7 + 0.80 + 0.02. We double check our answer by counting the correct spaces on a place value chart. Students seem to catch on quickly to this concept. However, I repeat this activity using a different fraction just to make sure students fully understand the connection between fraction and decimal notation. In this portion of the lesson students will work on their own to demonstrate how and why fractions and decimals are related. I ask them to be sure to show and explain the connection between fraction and decimal notation by writing the same number in both ways. I remind them that we did this earlier in the lesson. I tell them that they may draw visual representations like the ones seen in the interactive video if they need to. Some students tend to think that I was asking them to demonstrate a new skill. So, I explain that they will still be working on the same type of problems completing from our group setting. However, I want them to choose two of the numbers to explain the correlations. As students are working, I circle the room to remind students of the intended purpose of this lesson. How you can express the fraction to show its correct number value? Can you explain how the numbers to the right and left of the decimal differ and how they are the same? Can you represent the given number in both expanded decimal and fraction form? Explain your reasoning. Some students tend to bit confused at times, but the questions seem to refocus their attention. I use their responses to determine if additional time should be spent on this concept.

PRESENTATION 9Chapter 15 Real Estate Finance Mathematics Approach to Solving Math Problems • Solving math problems is simplified by using a step-by-step approach. • The most important step is to thoroughly understand the problem. • You must know what answer you want before you can successfully work any math problem. TQ • Once you have determined what it is you are to find (for example, interest rate, loan-to-value ratio, amount, or profit), you will know what formula to use. (cont.) • The next step is to substitute the numbers you know into the formula. TQ • It may be necessary to take one or more preliminary steps, for instance, converting fractions to decimals. (cont.) • Once you have substituted the numbers into the formula you will have to do some computations to find the unknown. • Most of the formulas have the same basic form: A=B x C • You will need two of the numbers (or the information that enables you to find two of the numbers) and then you will either have to divide or multiply them to find the third number—the answer you are seeking. (cont.) • Whether you will need to multiply or divide is determined by which quantity (number) in the formula you are trying to discover. • For example, the formula A=B x C may be converted into three different formulas. All three formulas are equivalent, but are put into different forms, depending upon the quantity (number) to be discovered. • If the quantity A is unknown, then the following formula is used: A = B x C • The number B is multiplied by C; the product of B times C is A. (cont.) • If the quantity B is unknown, the following formula is used: B = A ÷ C • The number A is divided by B; the quotient of A divided by B is C. C = A ÷ B • Notice that in all these instances, the unknown quantity is always by itself on one side of the “equal” sign. Converting Fractions to Decimals • There will be many times when you will want to convert a fraction into a decimal. • Most people find it much easier to work with decimals than fractions. • Also, hand calculators can multiply and divide by decimals. • To convert a fraction into a decimal, you simply divide the top number of the fraction (the “numerator”) by the bottom number of the fraction (the “denominator”). Example: • To change 3/4 into a decimal, you must divide 3 (the top number) by 4 (the bottom number). 3 ÷ 4 = .75 • To change 1/5 into a decimal, divide 1 by 5. 1 ÷ 5 = .20 • If you are using a hand calculator, it will automatically give you the right answer with the decimals in the correct place. To add or subtract by decimals,think “MONEY” $ 101.18 line the decimals up by decimal point and add or subtract. • Example: To multiply by decimals, do the multiplication. The answer should have as many decimal places as the total number of decimal places in the multiplying numbers. • Just add up the decimal places in the numbers you are multiplying and put the decimal point the same number of places to the left. • Example: 57.999 x 23.7 1374.5763 To divide by decimals, move the decimal point in the outside number all the way to the right and then move the decimal point in the inside number the same number of places to the right. • Example: 44.6 ÷ 5.889 44600 ÷ 5889 = 7.57 Percentage Problems • You will often be working with percentages in real estate finance problems. For example, loan-to-value ratios and interest rates are stated as percentages. • It is necessary to convert the percentages into decimals and vice versa, so that the arithmetic in a percentage problem can be done in decimals. To convert a percentage to a decimal, remove the percentage sign and move the decimal point two places to the left. This may require adding zeros. • Example: 80% becomes .80 9% becomes .09 75.5% becomes .755 8.75% becomes .0875 To convert a decimal to percentage, do just the opposite. Move the decimal two places to the right and add a percentage sign. • Example: .88 becomes 88% .015 becomes 1.5% .09 becomes 9% The word “of” means to multiply. • Whenever something is expressed as a percent of something, it means MULTIPLY. • Example: If a lender requires a loan-to-value ratio of 75% and a house is worth $89,000, what will be the maximum loan amount? (What is 75% of $89,000?) .75 x $89,000 = $66,750 maximum loan amount • Percentage problems are usually similar to the above example. You have to find a part of something, or a percentage of the total. A general formula is: • A percentage of the total equals the part, or part = percent x total P = % x T Example: • Smith spends 24% of her monthly salary on her house payment. Her monthly salary is $2,750. What is the amount of her house payment? 1. Find amount of house payment. 2. Write down formula: P = % x T. 3. Substitute numbers into formula. P = 24% x $2,750 • Before you can perform the necessary calculations, you must convert the 24% into a decimal. Move the decimal two places to the left: 24% = .24 P = .24 x $2,750 4. Calculate: multiply the percentage by the total. .24 x $2,750 = $660 • Smith’s house payment is $660. Problems Involving Measures of Central Tendency • Appraisers, finance officers, and lenders evaluate data and information by the use of averages. • These average figures are known as the mean, the median, or the mode. It is useful to know how they are derived. MEAN • The average figure that is called a MEAN is derived by taking a set of numbers and adding them up. The result is then divided by the numbers in the set. • Example: • A group of houses have the following monthly rental prices: Rental #1 – $1,300 Rental #2 – $900 Rental #3 – $1,200 Rental #4 – $1,100 Rental #5 – $1,150 $5,650 Total • To determine the Mean, the total rentals of $5,650.00 are divided by the number of rental houses. Thus, $5650 = $ 1130 Mean monthly rental. 5 MEDIAN – Odd Number Example • The average figure that is described as a MEDIAN is derived by simply selecting the middle number in a set of numbers. The numbers need to be in ascending order first. • Example: Rental #1 – $1,100 Rental #2 – $1,150 Rental #3 – $1,200 Rental #4 – $1,200 = Median Rental #5 – $1,250 Rental #6 – $1,250 Rental #7 – $1,300 • There are three numbers before Rental #4 and three numbers after. Therefore, rental #4 is the median number and the median rental price is $1,200. MEDIAN – Even Number Example • The example above had an odd number of rentals. The following example shows how to determine a mean with an even set of numbers. • Example: Rental #1 – $1,100 Rental #2 – $1,150 Rental #3 – $1,200 Rental #4 – $1,200 Rental #5 – $1,250 Rental #6 – $1,250 Rental #7 – $1,275 Rental #8 – $1,275 • The median is determined by adding the two middle rental numbers in the set and dividing the result by 2. • In this set the middle numbers are rentals number 4 and 5. $1,200 $2,450 = $1,225 +$1,250 Therefore: 2 $2,450 • Thus, the median rent in this even number example is $1,225. MODE • A MODE measures the most frequently occurring number in a series of numbers. Appraisers and lenders often use this benchmark to determine the predominant value of housing in a neighborhood. • Example: Sale #1 – $325,000 Sale #2 – $328,000 Sale #4 – $332,000 Sale #6 – $335,000 Sale #7 – $335,000 Sale #8 – $340,000 Sale #9 – $345,000 • There are two sales at $335,000. This would be the Mode. It would also be the predominant value of this set of sales. • The value range states from the lowest number to the highest number, in a set of numbers. Thus, the range in the example above would be expressed as from $325,000 to $345,000. Interest Problems • Interest can be viewed as the “rent” paid by a borrower to a lender for the use of money (the loan amount, or principal). • INTEREST is the cost of borrowing money. • SIMPLE INTEREST • COMPOUND INTEREST SIMPLE INTEREST • Simple interest problems are worked in basically the same manner as percentage problems, except that the simple interest formula has four components rather than three: interest, principal, rate, and time. Interest = Principal x Rate x Time I = P x R x T • Interest: The cost of borrowing expressed in dollars; money paid for the use of money. • Principal: The amount of the loan in dollars on which the interest is paid. • Rate: The cost of borrowing expressed as a percentage of the principal paid in interest for one year. • Time: The length of time of the loan, usually expressed in years. One must know the number values of three of the four components in order to compute the fourth (unknown) component. • a. Interest unknown Interest = Principal x Rate x Time Example: Find the interest on $3,500 for six years at 11%. I = P x R x T I = ($3,500 x .11) x 6 I = $385 x 6 I = $2,310 b. Principal unknown Principal = Interest ÷ Rate x Time P= I ÷ (R x T) • Example: How much money must be loaned to receive $2,310 interest at 11% if the money is loaned for six years? P = I ÷ (R x T) P = $2,310 ÷ (.11 x 6) P = $2,310 ÷ .66 P = $3,500 c. Rate unknown Rate = Interest ÷ Principal x Time R = I ÷ (P x T) • Example: In six years $3,500 earns $2,310 interest. What is the rate of interest? R = I ÷ (P x T) R = $2,310 ÷ ($3,500 x 6) R = $2,310 ÷ $21,000 R = .11 or 11% d. Time unknown Time = Interest ÷ Rate x Principal T = I ÷ (R x P) • Example: How long will it take $3,500 to return $2,310 at an annual rate of 11%? T = I ÷ (R x P) T = $2,310 ÷ ($3,500 x .11) T = $2,310 ÷ $385 T = 6 years A. COMPOUND INTEREST • Compound interest is more common in advanced real estate subjects, such as appraisal and annuities. • As previously stated, compound interest is interest on the total of the principal plus its accrued interest. • For each time period (called the “conversion period”), interest is added to the principal to make a new principal amount. Therefore, each succeeding time period has an increased principal amount on which to compute interest. Conversion periods may be monthly, quarterly, semi-annual, or annual. • The compound interest rate is usually stated as an annual rate and must be changed to the appropriate “interest rate per conversion period” or “periodic interest rate.” To do this, you must divide the annual interest rate by the number of conversion periods per year. This periodic interest rate is called “i.” COMPOUND INTEREST EXAMPLE: • The formula used for compound interest problems is: Interest = principal x periodic interest rate I = P x i Example: • A $5,000 investment at 9% interest compounded annually for three years earns how much interest at maturity? I = P x i I = $5,000 x (.09 ÷ 1) • First year’s I = $5,000 x .09 or $450. Add to $5,000. • Second year’s I = $5,450 x .09 or 490.50. Add to $5,450. • Third year’s I = $5,940.50 x .09 or $534.65. Add to $5,940.50 • At maturity, the borrower will owe $6,475.15. • The $5,000 loan has earned interest of $1,475.15 in three years. Example: • How much interest will a $1,000 investment earn over two years at 12% interest compounded semi-annually? • Since the conversion period is semi-annual, the interest is computed every six months. Thus, the periodic interest rate “i” is divided by two conversion periods: i = 6%. I = P x i 1. Original principal amount = $1,000.00 2. Interest for 1st period ($1,000 x .06) = $60.00 3. Balance beginning 2nd period = $1,060.00 4. Interest for 2nd period ($1,060 x .06) = $63.60 5. Balance beginning 3rd period = $1,123.60 6. Interest for 3rd period ($1,123.60 x .06) = $67.42 7. Balance beginning 4th period = $1,191.02 8. Interest for 4th period ($1,191.02 x .06) = $71.46 9. Compound principal balance = $1,262.48 i for 2 years = $1,262.48 - $1,000 or $262.48 B. EFFECTIVE INTEREST RATE • The NOMINAL (“NAMED”) INTEREST RATE is the rate of interest stated in the loan documents. • The EFFECTIVE INTEREST RATE/APR/ANNUAL PERCENTAGE RATE is the rate the borrower is actually paying. TQ • In other words, the loan papers may say one thing when the end result is another, depending upon how many times a year the actual earnings rate is compounded. • The effective interest rate equals the annual rate, which will produce the same interest in a year as the nominal rate converted a certain number of times. • For example, 6% converted semi-annually produces $6.09 per $100; therefore, 6% is the nominal rate and 6.09% is the effective rate. A rate of 6% converted semi-annually yields the same interest as a rate of 6.09% on an annual basis. C. DISCOUNTS • In alternative methods of financing, the loan proceeds disbursed by the lender are often less than the face value of the note. • This occurs when the borrower (or a third party) pays discount points. Generally when a borrower wants a lower rate they “buy down” the rate. TQ • The lender is paid points up front as compensation for making the loan on the agreed terms, at a lower rate. • When a discount is paid, the interest costs to the borrower (and the yield to the lender) are higher than the contract interest rate. DISCOUNT EXAMPLE: • When more accurate yield and interest tables are unavailable, it is possible to approximate the effective interest cost to the borrower and the yield rate to the lender when discounted loans are involved. • The formula for doing so is as follows: i = (r + (d/n)) ÷ (P - d) • i: approximate effective interest rate (expressed as a decimal) • r: contract interest rate (expressed as a decimal) • d: discount rate, or points deducted (expressed as a decimal) • P: principal of loan (expressed as the whole number 1 for all dollar amounts) • n: term (years, periods, or a fraction thereof) Example: • What is the estimated effective interest rate on a $60,000 mortgage loan, with a 20-year term, contract rate, if interest being 10% per annum, discounted 3%, so that only $58,200 is disbursed to the borrower? i = .10 + (.03/20) = .10 + .0015 = .10150 = .10463 or 10.46% 1 - .03 .97 .97 • The effective interest rate (or yield) on the loan is 10.46%. Profit and Loss Problems • Every time a homeowner sells a house, a profit or loss is made. • Many times you will want to be able to calculate the amount of profit or loss. • Profit and loss problems are solved with a formula that is a variation of the percent formula: value after = percentage x value before. VA = % x VB • The VALUE AFTER is the value of the property after the profit or loss is taken. • The VALUE BEFORE is the value of the property before the profit or loss is taken. • The PERCENT is 100% plus the percent of profit or minus the percent of loss. Example: • Green bought a house ten years ago for $50,000 and sold it last month for 45% more than she paid for it. What was the selling price of the house? VA = % x VB VA = 145% x VB (To get the percent, you must add the percent of profit to or subtract the percent of loss from 100%). VA = 1.45 x $50,000 VA = $72,500 was the selling price Profit means TAXES (Rentals) • Taking a profit can be even more expensive than what your tax rate is on your profit • What about recaptured depreciation? • Can be very expensive • More the longer you hold the property • Just be well informed when investing and doing tax writeoffs • PERSONAL RESIDENCE (Still has a non-recaptured interest deduction) • The government want’s that money though!! STILL NO TAX ON PERSONAL RESIDENCE PROFITS • Be aware, if that deduction is taken, the incentive for people to buy will be greatly decreased! • If a bill is introduced in congress you might want to let your senator and representative know you are against it! Example: • Now we will use the profit and loss formula to calculate another one of the components. • Green sold her house last week for $117,000. She paid $121,000 for it five years ago. What was the percent of loss? VA = % x VB $117,000 = % x $121,000(Because the percent is the unknown, you must divide the value after by the value before.) % = $117,000 ÷ $121,000 % = .9669 or 97% (rounded) Now subtract 97% from 100% to find the percent of loss. % = 100% - 97% = 3% loss Example: • Your customer just sold a house for 17% more than was paid for it. The seller’s portion of the closing costs came to $4,677. The seller received $72,500 in cash at closing. What did the seller originally pay for the house? VA = % x VB $72,500 + 4,677 = 117% x VB VB = ($72,500 + 4,677) ÷ 117% (Since the value before is 117% unknown, you must divide the value after [the total of the closing costs and the escrow proceeds] by the percent of profit.) VB = $77,177 ÷ 1.17 VB = $65,963.25 was the original price Prorations - Intro • There are some expenses connected with owning real estate that are often either paid for in advance or in arrears. • For example, fire insurance premiums are normally paid for in advance. Landlords usually collect rents in advance, too. • On the other hand, mortgage interest accrues in arrears. • When expenses are paid in advance and the owner then sells the property, part of these expenses have already been used up by the seller and are rightfully the seller’s expense. • Often, however, a portion of the expenses of ownership still remain unused and when title to the property transfers to the buyer, the benefit of these advances will accrue to the buyer. • It is only fair that the buyer, therefore, reimburse the seller for the unused portions of these homeownership expenses.

5 edition of Mathematical modeling with EXCEL found in the catalog. Mathematical modeling with EXCEL Includes bibliographical references. |LC Classifications||QA300 .A446 2010| |The Physical Object| |LC Control Number||2009010208| Creativity comes at a price Common beliefs about the rural elderly Know about yoga Responsible fish trade and food security Biography of a fish hawk Needlework monograms unlimited Law Enforcement Support Center Handbook of chemistry Mathematical Modeling with Excel and millions of other books are available for Amazon Kindle. Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - Cited by: 4. The model is then built using the appropriate mathematical tools. Then it is implemented and analyzed in Excel via step-by-step instructions. In the exercises, we ask students to modify or refine the existing model, analyze it further, or adapt it to similar by: 4. Mathematical Modeling with Excel presents various methods used to build and analyze mathematical models in a format that students can quickly comprehend. Excel is used as a tool to accomplish this goal of building and analyzing the models. The book begins with a step-by-step introduction to discrete dynamical systems, which are mathematical models that describe how a quantity changes from one point in time to the next. Readers are taken through the process, language, and notation required for the construction of such models as well as their implementation in : Wiley. How is Chegg Study better than a printed Mathematical Modeling With Excel 1st Edition student solution manual from the bookstore. Our interactive player makes it easy to find solutions to Mathematical Modeling With Excel 1st Edition problems you're working on - just go to the chapter for your book. The Active Modeler: Mathematical Modeling with Microsoft Excel Pdf. E-Book Review and Description: This can be a arms-on introduction to modeling all kinds of purposes in Microsoft Excel. It options quite a few tutorials and purposes for instance the best way to mannequin and clear up issues in Excel. MATHEMATICAL MODELING 1 Types of modeling software (platform). 1 Advantages and disadvantages of spreadsheets. 3 Guidelines to programming in spreadsheets. 10 PART I USING BUILDIT CHAPTER 2. BUILDING MATHEMATICAL MODELS IN EXCEL 17File Size: 1MB. Mastering Financial Modelling in Microsoft Excel will help allow you to become more proficient in building and applying financial models, enabling you to get better, more accurate results, fast. This highly practical book and CD combination is an unrivalled compendium of techniques designed to save you time and help you become more productive/5(7). 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This is the ‘‘definitions’’ step of the above scheme. The ‘‘systems analysis’’ step identifies the battery and fuels levels as the relevant parts of the system as explained above. Then, in the ‘‘modeling’’ step of the scheme, a model consisting of a battery and a tank such as in Figure is Size: 2MB. Figure AIC use in a simple linear regression model. Left: The predictions of the model for 1,2,3 and 4 parameters, along with the real data (open circles) generated from a 4 parameter model with noise. Right: the AIC values for each number of Size: 1MB. Mathematical modeling is a principled activity that has both principles behind it and methods that can be successfully applied. The principles are over-arching or meta-principles phrased as questions about the intentions and purposes of mathematical modeling. These meta-principles are almost philosophical in. With a focus on mathematical models based on real and current data, Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel guides readers in the solution of relevant, practical problems by introducing both mathematical and Excel techniques. The book begins with a step-by-step introduction to discrete dynamical systems, which are mathematical models that describe how. a same disease has occurred through the years. The aim of the mathematical modeling of epidemics is to identify those mechanisms that produce such pat-terns giving a rational description of these events and providing tools for disease control. This flrst lecture is devoted to introduce the File Size: KB. Mathematical Models and Their Analysis, Frederic Y. Wan,Mathematics, pages. Topics in mathematical modeling, K. Tung,pages. Topics in Mathematical Modelingis an introductory textbook on mathematical modeling. 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ISBN (hardcover: alk. paper) 1. Mathematical statistics. Mathematical models. Title. QAK –dc22 British Library Cataloging-in-Publication Data is availableFile Size: 4MB. The Excel workbook has comments and instructions for how to use these formulas. As you follow along in this tutorial, I'll teach you some of the essential "math" skills. Let's get started. Basic Excel Math Formulas Video (Watch and Learn) If learning from a screencast video is your style, check out the video below to walk through the tutorial. Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. Each Chapter Of The Book Deals With Mathematical Modelling Through One Or More Specified Techniques. Thus There Are Chapters On Mathematical Modelling Through Algebra, Geometry, Trigonometry And Calculus, Through Ordinary Differential Equations Of First And Second Order, Through Systems Of Differential Equations, Through Difference Equations, Through Partial Differential 5/5(4). In its early development, this book was focused on graduate level mathematical modeling (with a statistical focus) and for advanced mathematics students preparing for the contest in modeling. TECHNOLOGY Statistical Modeling with SPSS makes extensive use of SPSS to test student initiated hypotheses from a set of real data included with the test. Mathematical Models and their analysis De nition (Epidemiology) It is a discipline, which deals with the study of infectious diseases Peeyush Chandra Some Mathematical Models in Epidemiology. Preliminary De nitions and Assumptions Mathematical Models and their analysis (1) Heterogeneous Mixing-Sexually transmitted diseases (STD), File Size: KB. • The student is able to justify data from mathematical models based on the Hardy-Weinberg equilibrium to analyze genetic drift and the effects of selection in the evolution of specific populations (1A3 & SP ). • The student is able to describe a model that represents evolution within a population (1C3 & SP ). Mathematical Modeling: Models, Analysis and Applications covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. The book also contains a chapter on discrete modeling, consisting of differential equations, making it a complete textbook on this important skill needed for the study of science. Creating a mathematical model: • We are given a word problem • Determine what question we are to answer • Assign variables to quantities in the problem so that you can answer the question using these variables • Derive mathematical equations containing these variables • Use these equations to find the values of these variablesFile Size: KB. Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft® Office Excel® moreover choices: A modular group that, after the first chapter, permits readers to uncover chapters in any order Fairly a couple of smart examples and exercises that permit readers to personalize the launched fashions via using their very. A mathematical model is a description of a system using mathematical concepts and process of developing a mathematical model is termed mathematical atical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in the social sciences (such. Mathematical and Trigonometric Functions 97 Lookup and Reference Functions Date and Time Functions Text Functions Information Functions The Analysis ToolPak Part Two: Financial Modeling Using Excel CHAPTER 5 How to Build Good Excel Models Attributes of Good Excel Models Documenting Excel Models Debugging Excel. The best all-around introductory book on mathematical modeling is How to Model It: Problem Solving for the Computer Age by Starfield, Smith, and Bleloch. The book dates back tobut is just as relevant today. When most direct marketing people talk about "modeling", they either mean predictive response models, or they mean financial spreadsheet P&L models. Mathematical models are increasingly used to guide public health policy decisions and explore questions in infectious disease control. Written for readers without advanced mathematical skills, this book provides an excellent introduction to this exciting and growing area. taught with a focus on mathematical modeling. The content herein is written and main-tained by Dr. Eric Sullivan of Carroll College. Problems were either created by Dr. Sul-livan, the Carroll Mathematics Department faculty, part of NSF Project Mathquest, part of the Active Calculus text, or come from other sources and are either cited directly or. tenth, chapter of the book reviews some mathematical principles basic to the other chapters. All of the chapters contain many numerical examples and graphs developed from the numerical examples. The ambitious student could recreate any of the charts and tables contained in the book using a computer and Excel spreadsheets. There are many numerical. In order to learn more about mathematical modeling, read through the corresponding lesson called Using Mathematical Models to Solve Problems. This lesson covers: Defining mathematical modeling.With mathematical modeling growing rapidly in so many scientific and technical disciplines, Mathematical Modeling, Fourth Edition provides a rigorous treatment of the subject. The book explores a range of approaches including optimization models, dynamic models and probability models.R Through Excel is a highly recommended first step into that program.” (Shiken: JALT Testing& Evaluation SIG Newsletter) “Students, researchers, and others who wish to use R. This book is essentially a manual for the RExcel software. Most commonly a page consists of one or more screenshots showing how to use RExcel.

20 Questions MCQ Test General Test Preparation for CUET - Test: Problems On Ages - 2 Test: Problems On Ages - 2 for CUET 2023 is part of General Test Preparation for CUET preparation. The Test: Problems On Ages - 2 questions and answers have been prepared according to the CUET exam syllabus.The Test: Problems On Ages - 2 MCQs are made for CUET 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Problems On Ages - 2 below. Solutions of Test: Problems On Ages - 2 questions in English are available as part of our General Test Preparation for CUET for CUET & Test: Problems On Ages - 2 solutions in Hindi for General Test Preparation for CUET course. Download more important topics, notes, lectures and mock test series for CUET Exam by signing up for free. Attempt Test: Problems On Ages - 2 | 20 questions in 40 minutes | Mock test for CUET preparation | Free important questions MCQ to study General Test Preparation for CUET for CUET Exam | Download free PDF with solutions Six years ago, the ratio of the ages of Vimal and Saroj was 6 : 5 . Four years hence, the ratio of their ages will be 11 : 10 . What is Saroj's age at present? Detailed Solution for Test: Problems On Ages - 2 - Question 3 Given that, six years ago, the ratio of the ages of Vimal and Saroj = 6 : 5 Hence we can assume that age of Vimal six years ago = 6x age of Saroj six years ago = 5x After 4 years, the ratio of their ages = 11 : 10 My brother is 3 years elder to me. My father was 28 years of age when my sister was born while my mother was 26 years of age when I was born. If my sister was 4 years of age when my brother was born, then what was the age of my father when my brother was born? Detailed Solution for Test: Problems On Ages - 2 - Question 5 Let my age = x My brother's age = x + 3 My mother's age = x + 26 My sister's age = ( x + 3 ) + 4 = x + 7 My father's age = ( x + 7 ) + 28 = x + 35 Age of my father when my brother was born = x + 35 − ( x + 3 ) = 32 The present ages of A,B and C are in proportions 4 : 7 : 9 . Eight years ago, the sum of their ages was 56 . What are their present ages (in years)? Detailed Solution for Test: Problems On Ages - 2 - Question 6 Let present age of A,B and C be 4 x , 7 x and 9 x respectively. Hence present age of A, B and C are 4 × 4 , 7 × 4 and 9 × 4 respectively. i.e., 16 , 28 and 36 respectively. The age of father 10 years ago was thrice the age of his son. Ten years hence, father's age will be twice that of his son. What is the ratio of their present ages? Detailed Solution for Test: Problems On Ages - 2 - Question 10 Let age of the son before 10 years = x and age of the father before 10 years = 3 x (3x + 20 ) = 2 ( x + 20 ) ⇒ x = 20 Age of the son at present = x + 10 = 20 + 10 = 30 Age of the father at present = 3 x + 10 = 3 × 20 + 10 = 70 Required ratio = 70 : 30 = 7 : 3 If 6 years are subtracted from the present age of Ajay and the remainder is divided by 18 , then the present age of Rahul is obtained. If Rahul is 2 years younger to Denis whose age is 5 years, then what is Ajay's present age? Detailed Solution for Test: Problems On Ages - 2 - Question 14 The ratio of the age of a man and his wife is 4 : 3 . At the time of marriage the ratio was 5 : 3 and After 4 years this ratio will become 9 : 7 . How many years ago were they married? Detailed Solution for Test: Problems On Ages - 2 - Question 15 Let the present age of the man and his wife be 4 x and 3 x respectively. After 4 years this ratio will become 9 : 7 ⇒ ( 4 x + 4 ) : ( 3 x + 4 ) = 9 : 7 ⇒ 7 ( 4 x + 4 ) = 9 ( 3 x + 4 ) ⇒ 28 x + 28 = 27 x + 36 ⇒ x = 8 Present age of the man = 4 x = 4 × 8 = 32 Present age of his wife = 3 x = 3 × 8 = 24 Assume that they got married before t years. Then, ( 32 − t ) : ( 24 − t ) = 5 : 3 ⇒ 3 ( 32 − t ) = 5 ( 24 − t ) ⇒ 96 − 3 t = 120 − 5 t ⇒ 2 t = 24 The product of the ages of Syam and Sunil is 240 . If twice the age of Sunil is more than Syam's age by 4 years, what is Sunil's age? Detailed Solution for Test: Problems On Ages - 2 - Question 16 Let age of Sunil = x and age of Syam = y xy = 240 ⋯ ( 1 ) Substituting equation ( 2 ) in equation ( 1 ) . We get We got a quadratic equation to solve. Always time is precious and objective tests measure not only how accurate you are but also how fast you are. We can solve this quadratic equation in the traditional way. But it is more easy to substitute the values given in the choices in the quadratic equation (equation 3 ) and see which choice satisfy the equation. Here, option A is 10 . If we substitute that value in the quadratic equation, x ( x − 2 ) = 10 × 8 which is not equal to 120 Now try option B which is 12 . If we substitute that value in the quadratic equation, x ( x − 2 ) = 12 × 10 = 120 . See, we got that x = 12 Hence Sunil's age = 12 (Or else, we can solve the quadratic equation by factorization as, Since x is age and cannot be negative, x = 12 Or by using quadratic formula as In this test you can find the Exam questions for Test: Problems On Ages - 2 solved & explained in the simplest way possible. 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2. CN Syllabus Composition + Evaluation • Data Communications + Computer Networks • Unit-1: Data Communication Part + OSI &TCP/IP Model • Units 2-5: Layer 2 – 7 of OSI model with focus based on General Layer functionality and TCP/IP specific reference model under each layer • CIE – 30 Marks - 10 (CIE-1) + 10 (CIE-2) + 5 (Assignment) + 5 (Quiz) • SEE – 70 Marks - Mandatory to get 40% marks in end exam paper 3. Computer Networks - SYLLABUS OVERIEW Unit – 1 ◦ Data Communication Components: ◦ Representation of Data Communication , ◦ Flow of Networks, ◦ Layered Architecture, ◦ OSI and TCP/IP model, ◦ Transmission Media. ◦ Techniques for Bandwidth Utilization: ◦ Line configuration, ◦ Multiplexing – Frequency division, Time division and Wave division, ◦ Asynchronous and Synchronous Transmission, ◦ Introduction to Wired and Wireless LAN 4. Computer Networks - SYLLABUS OVERIEW Unit – 2 ◦ Data Link Layer and Medium Access Sub Layer: ◦ Error Correction and Error Detection: ◦ Fundamentals, Block coding, ◦ Hamming Distance, CRC ◦ Flow Control and Error Control Protocols: ◦ Stop and Wait, ◦ Go Back-N, ◦ ARQ, Selective Repeat ARQ, ◦ Sliding Window, ◦ Multiple Access Protocols: ◦ Pure ALOHA, Slotted ALOHA ◦ CSMA/CD, CSMA/CA 6. Computer Networks - SYLLABUS OVERIEW Unit – 4 ◦ Transport Layer: ◦ Process to Process Communications, ◦ Elements of Transport Layer ◦ Internet Protocols: ◦ UDP – User Datagram Protocol ◦ TCP – Transmission Control Protocol ◦ Congestion and Quality of Service: ◦ QoS improving techniques 7. Computer Networks - SYLLABUS OVERIEW Unit – 5 ◦ Application Layer: ◦ Domain Name System (DNS), ◦ EMAIL - Electronic Mail ◦ SNMP – Simple Network Management Protocol ◦ Basic Concepts of Cryptography: ◦ Network Security Attacks, ◦ Symmetric Encryption ◦ Data Encryption Standards, ◦ Public Key Encryption – RSA (Rivest, Shamir, Adleman) ◦ Hash Function, ◦ Message Authentication ◦ Digital Signature 8. Computer Networks – Suggested Reading - 1. Data Communication and Networking, 4th Edition, Behrouz A. Forouzan, McGrawHill 2. Data and Computer Communication, 8th Edition, William Stallings, Pearson Prentice Hall India 3. Unix Network Programming, W. Richard Stevens, Prentice Hall / Pearson Education, 2009 9. Computer Networks Lab PC 632 CS [Credits – 1] Evaluation: CIE – 25 Marks; SEE – 50 Marks 1. Running and using services/commands like: ◦ tcpdump, netstat, ifconfig, nslookup, ftp, telnet. - Execution at command prompt ◦ Capture ping and traceroute PDUs using a network protocol analyzer and examine 2. Configuration of router, switch. (using real devices or simulators) 3. Socket Programming using UDP and TCP ( E.g. Simple DNS, Date and time Client Server, Echo Client/Server, Iterative & Concurrent Servers) - Application programs through C Language using Socket API 4. Network Packet Analysis using tools like Wireshark, tcpdump etc. 5. Network Simulation using tools like Cisco Packet Tracer, NetSim, OMNet++, NS2, NS3 etc. 6. Study of Network Simulator(NS) and Simulation of Congestion Control Algorithms using NS. Performance Evaluation of Routing Protocols using Simulation Tools. 7. Programming using raw sockets. 8. Programming using RPC. - Application programs through C Language Note: Instructor may add/delete/modify/tune experiments, wherever he/she feels in a justified manner. 10. CN-U-1 - INTRODUCTION Data refers to information presented in whatever form is agreed upon by the parties creating and using the data. Data Communications are the exchange of data between two devices via some form of transmission medium such as a wire A network is a set of devices (often referred to as nodes) connected by communication links. A node can be a computer, printer, or any other device capable of sending and/or receiving data generated by other nodes on the network. 11. A Communication Model • The fundamental purpose of a communications system is the exchange of data between two parties. The key elements of this model are: • Source - generates data to be transmitted • Transmitter - converts data into transmittable signals • Transmission System - carries data from source to destination • Receiver - converts received signal into data • Destination - takes incoming data 12. Data Communication Model "Data Communications”, deals with the most fundamental aspects of the communications function, focusing on the transmission of signals in a reliable and efficient manner. Example: Electronic Mail: User A sending an email message m to user B. Steps for this process: 1. User A keys in message m comprising bits g buffered in source PC memory 2. Input data is transferred to I/O device (transmitter) as sequence of bits g(t) using voltage shifts 3. transmitter converts these into a signal s(t) suitable for transmission 4. whilst transiting media signal may be impaired so received signal r(t) may differ from s(t) 5. receiver decodes signal recovering g’(t) as estimate of original g(t) which is buffered in destination PC memory as bits g’ being the received 15. Communications Tasks Transmission system utilization Addressing Signal generation Recovery Synchronization Message formatting Exchange management Security Error detection and correction Network management 16. Communications Tasks • Key tasks that must be performed in a data communications system: • transmission system utilization - need to make efficient use of transmission facilities typically shared among a number of communicating devices • a device must interface with the transmission system • once an interface is established, signal generation is required for communication • there must be synchronization between transmitter and receiver, to determine when a signal begins to arrive and when it ends • there is a variety of requirements for communication between two parties that might be collected under the term exchange management • Error detection and correction are required in circumstances where errors cannot be tolerated 17. Communications Tasks • Flow control is required to assure that the source does not overwhelm the destination by sending data faster than they can be processed and absorbed • addressing and routing, so a source system can indicate the identity of the intended destination, and can choose a specific route through this network • Recovery allows an interrupted transaction to resume activity at the point of interruption or to condition prior to the beginning of the exchange • Message formatting has to do with an agreement between two parties as to the form of the data to be exchanged or transmitted • Frequently need to provide some measure of security in a data communications system • Network management capabilities are needed to configure the system, monitor its status, react to failures and overloads, and plan intelligently for future growth 31. LAYERED ARCHITECTURE: NEED AND ADVANTAGES • Allows Complex problems are decomposed in to small manageable units. • Implementation details of the layer are abstracted. • Separation of implementation and specification. • Layers work as one by sharing the services provided by each other. •Layering allows reuse functionality i.e., lower layers implement common once. •Provide framework to implement multiple specific protocols (rules) per layer •Provides Modularity with Clear Interfaces. • Has Implementation Simplicity, Maintainability, Flexibility and Scalability. • Support for Portability. • Provides for Robustness 32. ISO - OSI MODEL • International Standards Organization (ISO) - is a multinational body to worldwide agreement on international standards. • An ISO standard that covers all aspects of network communications is the Open Systems Interconnection (OSI) model. • It was first introduced in the late 1970s. • OSI model has seven layers. -----> 46. TCP/IP REFERENCE MODEL /PROTOCOL The layers in the TCP/IP reference model is FOUR in comparison to the OSI The original TCP/IP protocol suite was defined as having four layers: host-to-network, internet, transport, and application. But when TCP/IP is compared to OSI, we can say that the TCP/IP protocol suite is made of five layers: physical, data link, network, transport, and application. 48. Comparison of ISO-OSI model and TCP/IP 1. Layers: 7 in OSI ; 5 in TCP/IP 2. Model vs Implementation: In OSI first model was designed followed by Implementation. In TCP/IP first implemented then design followed 3. In OSI : Clear definition of Services, Interface and Protocols. Not so in TCP/IP 4. In OSI Network layer is both Connection Oriented and Connectionless and Transport Layer is only connection oriented. In TCP/IP network layer is connectionless and Transport Layer is both connection oriented and 5. In TCP/IP Session and Presentation layers are missing, this functionality is done by Application layer. 6. TCP/IP is the defacto protocol used in internet. OSI is mostly a theoretical Four levels of addresses are used in an internet employing the TCP/IP protocols: • Physical Addresses • Logical Addresses • Port Addresses • Specific Addresses 51. Physical Addresses • Physical Address – It is of 6-bytes (12 hexadecimal digits). • Also called MAC ADDRESS. • Every byte (2 hexadecimal digits) is separated by a colon • Example: 07:01:02:01:2C:4B • Physical Addresses Change Hop by Hop 52. Logical Addresses •Network with two •Each device (computer or router) has a pair of addresses (logical and physical) for each •Each device connected to one link – 1 pair of address. For router 3 • Also called IP 53. Port addresses • Port Addresses are for • Port and Logical addresses remain same for source to 55. Transmission Media Introduction • Transmission medium – It is the physical path between transmitter and receiver • It is of two types / categories / classes – • Guided media – Electromagnetic waves are guided along a solid medium Eg: Copper Twisted Pair, Copper Coaxial Cable, and Optical fiber • Unguided media – wireless transmission occurs through the atmosphere, space, water • Characteristics and Quality of data transmission is determined by both characteristics of Medium and Signal • For guided media - Medium is more important for data transmission • For Unguided media - Bandwidth of the signal produced by the transmitting antenna is more - One key property is directionality of the signal. Signals at lower frequencies are omni-directional and at higher frequencies can be focused into a directional beam 56. Data Transmission System Design : Data rate & Distance are the key factors Design Factors Determining Data Rate and Distance • higher bandwidth gives higher data rate • impairments, such as attenuation, limits the distance - Twisted Pair -> Coaxial Cable -> Optical Fiber • overlapping frequency bands can distort or wipe out a signal – More in Unguided than Guided medium. • more receivers introduces more attenuation - in case of shared link with multiple attachments. Not in number of receivers 58. Transmission Characteristics of Guided Frequency Range Typical Typical Delay Repeater Twisted pair (with 0 to 3.5 kHz 0.2 dB/km @ 1 kHz 50 µs/km 2 km 0 to 1 MHz 0.7 dB/km @ 1 kHz 5 µs/km 2 km Coaxial cable 0 to 500 MHz 7 dB/km @ 10 MHz 4 µs/km 1 to 9 km Optical fiber 186 to 370 THz 0.2 to 0.5 dB/km 5 µs/km 40 km In Guided Media ,transmission capacity, in terms of either data rate or bandwidth, depends critically on the distance and on whether the medium is point-to-point or multipoint. 62. Twisted Pair Twisted pair is the least expensive and most widely used guided transmission medium. • consists of two insulated copper wires arranged in a regular spiral pattern • a wire pair acts as a single communication link • pairs are bundled together into a cable • most commonly used in the telephone network and for communications • within buildings 63. Twisted Pair - Transmission Characteristics 5km to 6km can use either analog or digital 2km to 3km interference and noise 64. Unshielded vs. Shielded Twisted Pair Unshielded Twisted Pair (UTP) • ordinary telephone wire • easiest to install • suffers from external electromagnetic interference Shielded Twisted Pair (STP) • has metal braid or sheathing that reduces interference • provides better performance at higher data rates • more expensive • harder to handle (thick, heavy) 66. Near End Crosstalk - occurs in Twisted Pair • Coupling of signal from one pair of conductors to another • Occurs when transmit signal entering the link couples back to the receiving pair - (near transmitted signal is picked up by near 67. Coaxial Cable Coaxial cable can be used over longer distances and support more stations on a shared line than twisted pair. • consists of a hollow outer cylindrical conductor that surrounds a single inner wire conductor • is a versatile transmission medium used in a wide variety of applications • used for TV distribution, long distance telephone transmission and LANs 68. Coaxial Cable – Transmission - closer if extends up to • repeater every 1km - closer for 69. Optical Fiber Optical fiber is a thin flexible medium capable of guiding an optical ray. • various glasses and plastics can be used to make optical fibers • has a cylindrical shape with three sections – core, cladding, jacket • widely used in long distance telecommunications • performance, price and advantages have made it popular to use 70. Optical Fiber - Benefits ◦ data rates of hundreds of Gbps smaller size and lighter weight ◦ considerably thinner than coaxial or twisted pair cable ◦ reduces structural support requirements ◦ not vulnerable to interference, impulse noise, or crosstalk ◦ high degree of security from eavesdropping greater repeater spacing ◦ lower cost and fewer sources of error 71. Optical Fiber - Transmission • uses total internal reflection to transmit light • effectively acts as wave guide for 1014 to 1015 Hz (this covers portions of infrared & • Light sources used: • Light Emitting Diode (LED) • cheaper, operates over a greater temperature range, lasts longer • Injection Laser Diode (ILD) • more efficient, has greater data rates • has a relationship among wavelength, type of transmission and achievable data rate 73. Optical Fiber Transmission Modes Light from a source enters the cylindrical glass or plastic core. Rays at shallow angles are reflected and propagated along the fiber; other rays are absorbed by the surrounding material. This form of propagation is called step-index multimode Varying the index of refraction of the core, a third type of transmission, known as Reducing the radius of the core to the order of a wavelength, only a single angle or mode can pass: the axial ray. We have the single-mode propagation 76. Wireless Transmission Frequencies • referred to as microwave frequencies • highly directional beams are possible • suitable for point to point transmissions • also used for satellite • suitable for omnidirectional applications • referred to as the radio range 3 x 1011 to 2 • infrared portion of the spectrum • useful to local point-to-point and multipoint applications within confined areas electrical conductors used to radiate or collect electromagnetic same antenna is often used for energy impinging on converted to radio fed to receiver energy by antenna 78. Radiation Pattern •power radiated in all directions •does not perform equally well in all directions • as seen in a radiation pattern diagram •an isotropic antenna is a point in space that radiates power • in all directions equally • with a spherical radiation pattern 80. Antenna Gain •measure of the directionality of an antenna •power output in particular direction verses that produced by an •measured in decibels (dB) •results in loss in power in another direction •effective area relates to physical size and shape 81. Terrestrial Microwave most common type is a parabolic dish with an antenna focusing a narrow beam onto a receiving located at substantial heights above ground to extend range and transmit over obstacles uses a series of microwave relay towers with point-to-point microwave links to achieve long 82. Terrestrial Microwave Applications • used for long haul telecommunications, short point-to-point links between buildings and cellular systems • used for both voice and TV transmission • fewer repeaters but requires line of sight transmission • 1-40GHz frequencies, with higher frequencies having higher data rates • main source of loss is attenuation caused mostly by distance, rainfall 84. Satellite Microwave • a communication satellite is in effect a microwave relay station • used to link two or more ground stations • receives on one frequency, amplifies or repeats signal and transmits on • frequency bands are called transponder channels • requires geo-stationary orbit • rotation match occurs at a height of 35,863km at the equator • need to be spaced at least 3° - 4° apart to avoid interfering with each other • spacing limits the number of possible satellites 87. Satellite Microwave Applications private business networks ◦ satellite providers can divide capacity into channels to lease to individual business users ◦ programs are transmitted to the satellite then broadcast down to a number of stations which then distributes the programs to individual viewers ◦ Direct Broadcast Satellite (DBS) transmits video signals directly to the home user ◦ Navstar Global Positioning System (GPS) 88. Transmission Characteristics • the optimum frequency range for satellite transmission is 1 to 10 GHz • lower has significant noise from natural sources • higher is attenuated by atmospheric absorption and precipitation • satellites use a frequency bandwidth range of 5.925 to 6.425 GHz from earth to satellite (uplink) and a range of 3.7 to 4.2 GHz from satellite to earth • this is referred to as the 4/6-GHz band • because of saturation the 12/14-GHz band has been developed (uplink: 14 - 14.5 GHz; downlink: 11.7 - 89. Broadcast Radio radio is the term used to encompass frequencies in the range of 3kHz to 300GHz broadcast radio (30MHz - 1GHz) covers • FM radio • UHF and VHF television • data networking applications limited to line of sight suffers from multipath interference ◦ reflections from land, water, man-made objects • achieved using transceivers that modulate noncoherent infrared light • transceivers must be within line of sight of each other directly or via • does not penetrate walls • no licenses required • no frequency allocation issues • typical uses: • TV remote control 92. Wireless Propagation Ground Wave • ground wave propagation follows the contour of the earth and can propagate distances well over the visible horizon • this effect is found in frequencies up to 2MHz • the best known example of ground wave communication is AM radio 93. Wireless Propagation Sky Wave • sky wave propagation is used for amateur radio, CB radio, and international broadcasts such as BBC and Voice of America • a signal from an earth based antenna is reflected from the ionized layer of the upper atmosphere back down to earth • sky wave signals can travel through a number of hops, bouncing back and for the between the ionosphere and the earth’s surface 94. Wireless Propagation Line of Sight • ground and sky wave propagation modes do not operate above 30 MHz - - communication must be by line of sight velocity of electromagnetic wave is a function of the density of the medium through which it travels • ~3 x 108 m/s in vacuum, less in anything else speed changes with movement between media index of refraction (refractive index) is ◦ varies with wavelength ◦ density of atmosphere decreases with height, resulting in bending of radio waves 96. Line of Sight Transmission Free space loss • loss of signal • from water vapor • bending signal 97. Free Space Loss : which can be expressed in terms of the ratio of the radiated power Pt to the power Pr received by the antenna or, in decibels, by taking 10 times the log of that ratio. 99. • Line configuration, • Multiplexing – Frequency division, Time division and Techniques for Bandwidth Utilization: 100. Line Configuration - Topology •Physical arrangement of stations on medium • Point to Point - two stations • such as between two routers / computers • Multi point - multiple stations • traditionally mainframe computer and terminals • now typically a local area network (LAN) Note: Two characteristics that distinguish various data link configurations : Topology and Whether the link is half duplex or full duplex [Data Flow]. 101. Line Configuration - Topology • In point-to-point each terminal has a separate I/O Port and transmission link 102. Line Configuration - Duplex • classify data exchange as half or full duplex • half duplex (two-way alternate) • only one station may transmit at a time • requires one data path • full duplex (two-way simultaneous) • simultaneous transmission and reception between two stations • requires two data paths • separate media or frequencies used for each direction • or echo canceling ( can be used for transmitting using a single line) •Under the simplest conditions, a medium can carry only one signal at any moment in •For multiple signals to share one medium, the medium must somehow be divided, giving each signal a portion of the total bandwidth. •Whenever the bandwidth of a medium linking two devices is greater than the bandwidth needs of the devices, the link can be shared. •Efficiency can be achieved by multiplexing; i.e., sharing of the bandwidth between multiple users. •Transparent to the User -- It is the set of techniques that allows the (simultaneous) transmission of multiple signals across a single data link. -- Two or more simultaneous transmissions on a single circuit. Figure: Dividing a link into channels 106. Multiplexing Techniques/Categories The current techniques include : 1. FDM: Frequency Division Multiplexing 2. WDM: Wavelength Division Multiplexing 3. TDM: Time Division Multiplexing - Digital a. Synchronous b. Statistical 107. Frequency Division Multiplexing • It is an analog multiplexing technique that combines analog signals. Uses the concept of modulation • Assignment of non-overlapping frequency ranges to each “user” or signal on a medium. Thus, all signals are transmitted at the same time, each using different frequencies. 109. Frequency Division Multiplexing • Analog signaling is used to transmit the signals due to which it is more susceptible to noise. • It is the oldest multiplexing technique. • Examples of FDM: Broadcast radio and television, AMPS cellular phone systems 110. FDM Process --A multiplexor accepts inputs and assigns frequencies to each --It is attached to a high-speed --A corresponding multiplexor, or demultiplexor, is on the end of the high-speed line and separates the multiplexed signals. 111. FDM Process --Each signal is modulated to a different carrier frequency --Carrier frequencies separated so signals do not overlap (guard bands) e.g. broadcast radio. --Channel allocated even if no data 116. Dense Wavelength Division Multiplexing • DWDM which is often called WDM multiplexes multiple data streams onto a single fiber optic line. Data Transmission through a single fiber optic line 117. Dense Wavelength Division Multiplexing (DWDM) • Different wavelength lasers (called lambdas) transmit the multiple signals. • Each signal carried at a different rate, combines(30, 40, more?) signals onto one fiber. 118. Wavelength Division Multiplexing 1997 Bell Labs ◦ 100 beams ◦ Each at 10 Gbps ◦ Giving 1 terabit per second (Tbps) Commercial systems of 160 channels of 10 Gbps now available Lab systems (Alcatel) 256 channels at 39.8 Gbps each ◦ 10.1 Tbps ◦ Over 100km 119. Time Division Multiplexing (TDM) •TDM is a digital multiplexing technique for combining several low-rate digital channels into one high-rate one. • Data rate of medium exceeds data rate of digital signal to be • Multiple digital signals interleaved in time • May be at bit level of blocks 120. Time Division Multiplexing (TDM) Sharing of the signal is accomplished by dividing available transmission time on a medium among users. 123. TDM Types/Forms •Time division multiplexing comes in two basic forms: •1. Synchronous time division multiplexing •2. Statistical, or Asynchronous time division multiplexing. 124. Synchronous TDM The original time division multiplexing. The multiplexor accepts input from attached devices in a round-robin fashion and transmit the data in a never ending pattern. Examples of STDM: T-1, ISDN telephone lines, SONET (Synchronous Optical NETwork) When one device generates data at a faster rate than other devices – then the multiplexor must either sample the incoming data stream from that device more often than it samples the other devices, or buffer the faster incoming stream. •When a device has nothing to transmit, the multiplexor must still insert a piece of data from that device into the multiplexed stream So that the receiver may stay synchronized with the incoming data stream •The transmitting multiplexor can insert alternating 1s and 0s into the data stream. The process of taking a group of bits from each input line for multiplexing is called interleaving. We interleave bits (1 - n) from each input onto one output. 129. TDM Link Control • No headers and trailers • Data link control protocols not needed • Flow control –Data rate of multiplexed line is fixed –If one channel receiver can not receive data, the others must carry on –The corresponding source must be quenched –This leaves empty slots • Error control –Errors are detected and handled by individual channel systems •To ensure that the receiver correctly reads the incoming bits, i.e., knows the incoming bit boundaries to interpret a “1” and a “0”, a known bit pattern is used between the frames. •The receiver looks for the anticipated bit and starts counting bits till the end of the frame. •Then it starts over again with the reception of another known •These bits (or bit patterns) are called synchronization bit(s). •They are part of the overhead of transmission. 133. Data Rate Management • Synchronizing data sources • Not all input links maybe have the same data rate. • Some links maybe slower. There maybe several different input link speeds • Data rates from different sources not related by simple rational number • Clocks in different sources drifting • Three strategies that can be used to overcome the data rate mismatch: • Multilevel, Multislot and Pulse Stuffing 134. Data Rate Management • Multilevel: used when the data rate of the input links are multiples of 135. Data Rate Management Multislot: used when there is a GCD between the data rates. The higher bit rate channels are allocated more slots per frame, and the output frame rate is a multiple of each input link. 136. Data Rate Management • Pulse Stuffing: used when there is no GCD between the links. The slowest speed link will be brought up to the speed of the other links by bit insertion, this is called pulse stuffing. –Outgoing data rate (excluding framing bits) higher than sum of –Stuff extra dummy bits or pulses into each incoming signal until it matches local clock –Stuffed pulses inserted at fixed locations in frame and removed 137. Inefficient use of Bandwidth • Sometimes an input link may have no data to transmit then, one or more slots on the output link will go unused. • Thus wasting bandwidth 139. Statistical TDM or Asynchronous TDM •In Synchronous TDM many slots are wasted •Statistical TDM allocates time slots dynamically based on •Multiplexer scans input lines and collects data until frame •Data rate on line lower than aggregate rates of input lines 141. Statistical TDM • A statistical multiplexor transmits only the data from active workstations (or why work when you don’t have to). • If a workstation is not active, no space is wasted on the multiplexed 143. Statistical TDM To identify each piece of data, an address is included. If the data is of variable size, a length is also included. 144. Statistical TDM •A statistical multiplexor does not require a line over as high a speed line as synchronous time division multiplexing since STDM does not assume all sources will transmit all of the time! •Good for low bandwidth lines (used for LANs) •Much more efficient use of bandwidth! 145. • Asynchronous and Synchronous Transmission, • XDSL – X Digital Subscriber Line A, S, H, V Asymmetric, Symmetric, High Data Rate, Very High Data Rate Techniques for Bandwidth Utilization: 146. Transmission of Data between 2 devices Types: Asynchronous and Synchronous •Transmission of a stream of bits from one device to another across a transmission link involves cooperation and agreement between the two •Timing problems require a mechanism to synchronize the transmitter • receiver samples stream at bit intervals • if clocks not aligned and drifting will sample at wrong time after sufficient bits are sent •Two solutions to synchronizing clocks • Asynchronous transmission • Synchronous transmission 147. Asynchronous Transmission • Here each character of data is treated independently. • Timing problem is avoided by not sending long, uninterrupted streams of bits. So data is sent character by character. • Each character begins with a start bit that alerts the receiver that a character is arriving. The receiver samples each bit in the character and then looks for the beginning of the next character. [ does not work with long blocks of data as receiver clock may go out of sync with the transmitter’s clock. 148. Asynchronous Transmission • When no character is being transmitted, the line between transmitter and receiver is in an idle state (binary 1 • The beginning of a character is signaled by a start bit with a value of binary 0. • This is followed by the 5 to 8 bits that actually make up the character. • The bits of the character are transmitted beginning with the least significant bit. • Then the data bits are usually followed by a parity bit, set by the transmitter such that the total number of ones in the character, including the parity bit, is even (even parity) or odd (odd parity). • The receiver uses this bit for error detection. • The final element is a stop element, which is a binary 1. • A minimum length for the stop element is specified, and this is usually 1, 1.5, or 2 times the duration of an • No maximum value is specified since the stop element is the same as the idle state, so the transmitter will continue to transmit the stop element until it is ready to send the next character. 149. Asynchronous Transmission • Example: Say the receiver is fast by • Thus, the receiver samples the incoming character every 94 µs (based on the transmitter's clock). • Thus the last sample is erroneous. 150. Asynchronous Transmission - Merits •Simple & cheap •Overhead of 2 or 3 bits per char (~20%) •Example: For an 8-bit character with no parity bit, using a 1-bit-long stop element, two out of every ten bits convey no information but are there merely for synchronization; thus the overhead is 20%. •Good for data with large gaps (keyboard) 151. Synchronous Transmission •Block of data transmitted sent as a frame • [includes a starting and an ending flag, and is transmitted in a steady stream without start and stop codes. The block may be many bits in length. ] •Clocks must be synchronized [to avoid drift] • can use separate clock line • or embed clock signal in data •Need to indicate start and end of block of data for the receiver to sync • use preamble and postamble bits • Data plus preamble, postamble, and control information are called a frame (exact frame format depends of DLL procedure). • More efficient (lower overhead) than Asynchronous (20% more overhead). • Preamble, Postamble and control field would mostly less than 100 bits. Internet Access Technology: Upstream and Downstream • Internet access technology refers to a data communications system that connects an Internet subscriber to an ISP • such as a telephone company(DSL) or cable company • Most Internet users follow an asymmetric pattern • a subscriber receives more data from the Internet than sending • a browser sends a URL that comprises a few bytes • in response, a web server sends content • Upstream to refer to data traveling from a subscriber to an ISP • Downstream to refer to data traveling from an ISP in the Internet to a Narrowband and Broadband Access Technologies • A variety of technologies are used for Internet access • They can be divided into two broad categories based on the data rate they • In networking terms, network bandwidth refers to data rate • Thus, the terms narrowband and broadband reflect industry practice Narrowband Access Technologies • Narrowband Technologies • refers to technologies that deliver data at up to 128 Kbps • For example, the maximum data rate for dialup noisy phone lines is 56 Kbps and classified as a narrowband technology • the main narrowband access technologies are given below Broadband Access Technologies • Broadband Technologies • generally refers to technologies that offer high data rates, but the exact boundary between broadband and narrowband is blurry • many suggest that broadband technologies deliver more than 1 Mbps • but this is not always the case, and may mean any speed higher than dialup • the main broadband access technologies are given below Digital Subscriber Line (DSL) Technologies • DSL is one of the main technologies used to provide high-speed data communication services over a • DSL variants are given below • Because the names differ only in the first word, the set is collectively referred to by the acronym • Currently, ADSL is most popular The Local Loop • Local loop describes the physical connection between a telephone company Central Office (CO) and a subscriber • consists of twisted pair and dialup call with 4 KHz of bandwidth • It often has much higher bandwidth; a subscriber close to a CO may be able to handle frequencies above 1 MHz LOCAL LOOP Technologies • Electric local loop(POTS lines): Voice, ISDN, DSL • Optical local loop: Fiber Optics services such as FiOS • Satellite local loop: communications satellite and cosmos Internet connections of satellite televisions (DVB-S) • Cable local loop: Cablemodem • Wireless local loop (WLL): LMDS, WiMAX, GPRS, HSDPA, DECT 167. Asymmetrical DSL (ADSL) • ADSL is an asymmetric communication technology designed for residential users; it is not suitable for businesses • ADSL is an adaptive technology. •Link between subscriber and network •Uses currently installed twisted pair cable –Can carry broader spectrum –1 MHz or more 168. Asymmetrical DSL (ADSL) • ADSL divides up the available frequencies in a line on the assumption that most Internet users look at, or download, much more information than they send, or upload. • The system uses a data rate based on the condition of the local loop line. • Speed: Most existing local loops can handle bandwidths up to 1.1 MHz. 169. ADSL Design – Greater capacity downstream than upstream • Frequency division multiplexing – Lowest 25kHz for voice • Plain old telephone service (POTS) – Use echo cancellation or FDM to give two bands – Use FDM within bands – The region above 25kHz is used for data transmission – Upstream: 64kbps to 640kbps – Downstream: 1.536Mbps to 6.144Mbp • Range 5.5km 172. Two standards for ADSL 1. Discrete multitone (DMT) 2. Carrierless amplitude/phase (CAP) 173. CAP - three distinct bands: 1. Voice channel - 0 to 4 KHz 2. Upstream channel - 25 and 160 KHz 3. Downstream channel - 1.5 MHz Minimizes the possibility of interference between the channels on one line, or between the signals on different lines 175. Discrete Multitone • Multiple carrier signals at different frequencies • Some bits on each channel • 4kHz subchannels • Send test signal and use subchannels with better signal to noise ratio • 256 downstream subchannels at 4kHz (60kbps) – Impairments bring this down to 1.5Mbps to 9Mbps 178. ADSL Distance Limitations •ADSL is a distance-sensitive technology •The limit for ADSL service is 18,000 feet (5,460 meters) •At the extremes of the distance limits, ADSL customers may see speeds far below the promised maximums •customers nearer the central office have faster connections and may see extremely high speeds 179. OTHER TYPES OF DSL: • SDSL -- Symmetric DSL Used mainly by small businesses & residential areas Bit rate of downstream is higher than upstream • HDSL -- High-bit-rate DSL Used as alternative of T-1 line Uses 2B1Q encoding Less susceptible to attenuation at higher frequencies Unlike T-1 line (AMI/1.544Mbps/1km), it can reach 2Mbps 180. OTHER TYPES OF DSL: • VDSL -- Very high bit-rate DSL Uses DMT modulation technique Effective only for short distances(300-1800m) Speed: downstream: 50 - 55 Mbps upstream: 1.5-2.5 Mbps •In 1985, the Computer Society of the IEEE started a project, called Project 802. •Purpose was to set standards to enable intercommunication among equipment from a variety of •Project 802 is a way of specifying functions of the physical layer and the data link layer of major LAN 190. Unicast and multicast addresses The least significant bit of the first byte defines the type of address. If the bit is 0, the address is unicast; otherwise, it is multicast. The broadcast destination address is a special case of the multicast address in which all bits are 1s. 191. Define the type of the following destination addresses: a. 4A:30:10:21:10:1A b. 47:20:1B:2E:08:EE To find the type of the address, we need to look at the second hexadecimal digit from the left. If it is even, the address is unicast. If it is odd, the address is multicast. If all digits are F’s, the address is broadcast. Therefore, we have the a. This is a unicast address because A in binary is 1010. b. This is a multicast address because 7 in binary is 0111. c. This is a broadcast address because all digits are F’s. 192. Example shows how the address 47:20:1B:2E:08:EE is sent out on The address is sent left-to-right, byte by byte; for each byte, it is sent right-to- left, bit by bit, as shown below: CHANGES IN THE STANDARD The 10-Mbps Standard Ethernet has gone through several changes before moving to the higher data rates. These changes actually opened the road to the evolution of the Ethernet to become compatible with other high-data-rate LANs. Fast Ethernet was designed to compete with LAN protocols such as FDDI or Fiber Channel. IEEE created Fast Ethernet under the name 802.3u. Fast Ethernet is backward-compatible with Standard Ethernet, but it can transmit data 10 times faster at a rate of 100 Mbps. 211. GIGABIT ETHERNET • The need for an even higher data rate resulted in the design of the Gigabit Ethernet protocol (1000 Mbps). The IEEE committee calls the standard • In the full-duplex mode of Gigabit Ethernet, there is no collision; • the maximum length of the cable is determined by the signal in the cable. 217. IEEE 802.11 - Wireless LAN Standard IEEE has defined the specifications for a wireless LAN, called IEEE 802.11, which covers the physical and data link layers. A BSS without an AP is called an ad hoc network; a BSS with an AP is called an infrastructure network. Bluetooth is a wireless LAN technology designed to connect devices of different functions such as telephones, notebooks, computers, cameras, printers, coffee makers, and so on. A Bluetooth LAN is an ad hoc network, which means that the network is formed spontaneously.

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Many businesses rely on statistical analysis and it is becoming more and more important. One of the main reasons is that statistical data is used to predict future trends and to minimize supertopmotorcombr.gearhostpreview.comted Reading Time: 8 mins. a research paper on cancer table of contents for research paper - May 07, · Also known as descriptive analysis, statistical data analysis is a wide range of quantitative research practices in which you collect and analyze categorical . A collection of datapoints or numerical values that can be categorized and subject to analysis; statistics are the raw material on which conclusions about cause-and-effect relationships are based. 2. Factor analysis. Factor analysis is a form of exploratory multivariate analysis that is used to either reduce the number of variables in a model or to detect relationships among variables. All variables involved in the factor analysis need to be interval and are assumed to be normally distributed. homework help in french homework help cpm integrated 2 - The median is not skewed by extreme values, but it is harder to use for further statistical analysis. The mode is the most common value in a data set. It cannot be used for further statistical analysis. The values of mean, median and mode are not the same. Statistics is a branch of science that deals with the collection, organisation, analysis of data and drawing of inferences from the samples to the whole population. This requires a proper design of the study, an appropriate selection of the study sample and choice of a suitable statistical supertopmotorcombr.gearhostpreview.com by: Here are two contrasting definitions of what statistics is, from eminent professors in the field, some 60+ years apart: "Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural supertopmotorcombr.gearhostpreview.com Size: 1MB. apa style essay paper reading research papers - Oct 29, · What is Statistical Modeling and How is it Used? Statistical modeling is the process of applying statistical analysis to a dataset. A statistical model is a mathematical representation (or mathematical model) of observed data.. When data analysts apply various statistical models to the data they are investigating, they are able to understand and interpret the information more supertopmotorcombr.gearhostpreview.comted Reading Time: 9 mins. Jan 28, · Statistical tests work by calculating a test statistic – a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship. It then calculates a p-value (probability value).Estimated Reading Time: 8 mins. Mar 06, · In statistical analysis, hypothesis testing, also known as “T Testing”, is a key to testing the two sets of random variables within the data set. This method is all about testing if a certain argument or conclusion is true for the data set. It allows for comparing the data against various hypotheses and assumptions. santander business plan people writing on paper - Jan 31, · Statistical analysis is the process of generating statistics from stored data and analyzing the results to deduce or infer meaning about the underlying dataset or the reality that it . The statistical analysis section provides crucial information on how the collected data and samples will be analyzed to achieve the primary and secondary study aims. The statistical analysis section should have sufficient information for reviewing committees to be able to determine that the methodology is sound and valid for the planned analyses. Statistical Analysis michend T When performing research it is essential that you are able to make sense of your data. This allows you to inform other researchers in your field and others what you have found. true love essay example dr d g hessayon books - May 11, · Data Analysis Vs Statistical Analysis - Bringing It All Together. To sum up, it might be noticed that Data analysis and statistics are unclear and are firmly interconnected. Unmistakably statistics is a tool or technique for data science, while data science is a wide area where a statistical strategy is a fundamental supertopmotorcombr.gearhostpreview.comted Reading Time: 5 mins. The following table shows general guidelines for choosing a statistical analysis. We emphasize that these are general guidelines and should not be construed as hard and fast rules. Usually your data could be analyzed in multiple ways, each of which could yield legitimate answers. The table below covers a number of common analyses and helps you. Statistical Analysis. Statistical analysis revealed that Pro and Gly residues are favored in β-turns presumably due to the cyclic structure of Pro and the flexibility of Gly Hydrophilic amino acids, such as Asn and Asp, also have a high propensity for a β-turn formation mainly owing to their placement on solvent-exposed surfaces of globular proteins. ryanair marketing strategy essays dissertations on education policy - Jun 02, · Statistical Analysis is the study of the collection, organization, analysis, interpretation and presentation of data. Statistical Analysis begins with the identification of process or Reviews: 1. In this course you will learn how to analyze data. #Statistic plays important role in terms of data analysis. Here you will get exposed to utilize and unders. Statistical analysis can be used to summarize those observations by estimating the average, which provides an estimate of the true mean. Another important statistical calculation for summarizing the observations is the estimate of the variance, which quantifies the uncertainty in . 7th grade book report template dr d g hessayon books - Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be supertopmotorcombr.gearhostpreview.comted Reading Time: 9 mins. Ethan Meyers, Hampshire College - MITBMM Summer Course The slides and more info are available here - supertopmotorcombr.gearhostpreview.com statistical analysis and business applications tutorial gives the introduction to statistics and Statistical and Non-Statistical Analysis. You will learn Some Common Terms Used in Statistics along with Histogram, Hypothesis Testing, and Bell Curve. Chi-Square Test, Correlation Matrix, and Inferential Statistics are also explained. speech criticism essay business ethics research paper topics - Sep 11, · Analysis of variance (ANOVA) is a statistical technique that is used to check if the means of two or more groups are significantly different from each other. ANOVA checks the impact of one or more factors by comparing the means of different supertopmotorcombr.gearhostpreview.comted Reading Time: 8 mins. Balanced ANOVA: A statistical test used to determine whether or not different groups have different means. An ANOVA analysis is typically applied to a set of data in which sample sizes are kept. 10 - 2 Purpose of Statistical Analysis In previous chapters, we have discussed the basic principles of good experimental design. Before examining specific experimental designs and the way that their data are analyzed, we thought that itFile Size: 1MB. case study definition in law publish dissertation proquest - May 15, · Statistical analysis capabilities refer to capabilities that support analysis methodologies such as regression analysis, predictive analytics, and statistical modelling, among many others. Statistical analysis software tools are typically used by data scientists and mathematicians, but can provide industry-specific features. on statistical tests; this could by skipped without a serious loss of continuity by those mainly interested in parameter estimation. The choice of and relative weights given to the various topics reflect the type of analysis usually encountered in particle physics. Here the data usually consist. Dec 27, · Statistical analysis is the science of collecting, exploring, and presenting large amounts of data to discover underlying patterns and trends. True | False Answer Key. palanquin bearers essay business ethics research paper topics - Statistical analysis synonyms, Statistical analysis pronunciation, Statistical analysis translation, English dictionary definition of Statistical analysis. n. 1. May 11, · Methods in Quantitative Statistical Analysis, What is hypothesis testing?. Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. Major four principles involved in a statistical test. Evolving a test statistic; To know the sampling distribution of the test statistic; Setting of hypotheses. May 21, · Statistical analysis and data analysis are similar but not the same. Statistical analysis is a way of analyzing data. An example of this would be an exploratory analysis. This is a kind of statistical analysis that uses previously gathered data to try and find inferences or insights that have previously been undiscovered. easybib quick guide online essay writing checker - Statistical Data Analysis. This is a complete statistical analysis for research. Our experts will clean, test and verify your raw data and research methodology to ensure transparency and reproducibility. Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving supertopmotorcombr.gearhostpreview.com is assumed that the observed data set is sampled from a larger population.. Inferential statistics can be contrasted with descriptive supertopmotorcombr.gearhostpreview.comted Reading Time: 10 mins. Statistical analysis of data is a very popular branch of study as well as a job market in the era of data. More data has been collected in the past few years than the last decade and it is increasing rapidly. If you were thinking of getting your hands dirty with some statistical analysis, then check out this template. jacobs the authentic dissertation do my home work do my homework - May 19, · Statistical Analysis. Statistical analysis is the use of statistics, including variables, units, and various events, to determine the likelihood or quantity of statistical relationships. Statistical analysis is the alphabet for collecting data and discovering patterns and trends. It’s another way of telling statistics. Aug 03, · Basic Statistical Analysis Using the R Statistical Package Introduction R is a freely distributed software package for statistical analysis and graphics, developed and managed by the R Development Core Team. Sep 23, · supertopmotorcombr.gearhostpreview.comimated Reading Time: 2 mins. bibliography reference list dissertation You can report issue about the essay help online on this page here Want to share your content on R-bloggers? Methods in Quantitative Statistical What are statistical analysis, What is what are statistical analysis testing?. Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. In statistics most of the testing methods are comes under parametric hypothesis and What are statistical analysis hypothesis. To understand more on NULL hypothesis click here. In a parametric hypothesis, A statistical hypothesis is an assertion about a parameter of a population. 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Persistent disconnects within and among education research, practice, and policy are get full access to journal for research in mathematics education. Research mathematics research at harvard is done in various research areas research papers published and reviewed papers: mathsci net mathsci net (pin. A number of members of the algebra group belong to the research training group in representation theory the mathematics department also offers. The aim of discrete applied mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as. Mathematics research papers discuss the various topic areas such as the quantity of something, the structure, space or measurability of an object. Essays, term papers, book reports, research papers on mathematics free papers and essays on analytic geometry we provide free model essays on mathematics, analytic geometry reports, and term paper samples related to analytic geometry. 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Sharpe Ratio is an important Ratio to measure risk-adjusted returns. • Treynor Ratio measures risk-adjusted return based on systematic risk. • A scheme with a 25 Sep 2017 Then there are gauges like beta, Sharpe ratio, Sortino ratio and even something called the upside and downside capture ratio. And then there's The Treynor ratio, also commonly known as the reward-to-volatility ratio, is a measure that quantifies return per unit of risk. It is similar to the Sharpe and Sortino The Treynor ratio is similar to the Sharpe Ratio, except it uses beta as the volatility measure (to divide the investment's excess return over the beta). T reynorRatio =. But the Treynor Ratio divides by the beta (the risk inherent in the market). Well- diversified portfolios should have similar Sharpe and Treynor Ratios because the Answer to Calculate the Sharpe ratio, Treynor Ratio, Jenson's Alpha, Information ratio and the R-Squared for both funds and determ Keywords: Sharpe Ratio, Treynor's Index, Mutual Funds. Introduction. The Indian common asset industry has made some amazing progress from its origin in While the Sharpe ratio measures the risk premium of the portfolio over the portfolio risk, or its standard deviation, Treynor's ratio, popularized by Jack L. Treynor, 6 days ago Their difference is, while the Treynor ratio determines volatility with a portfolio beta or systematic risk, the Sharpe ratio adjusts returns based on The following definitions for Sharpe and Treynor Ratios are from ZOONOVA. Sharpe Ratio. a measure that indicates the average return minus the risk-free return However, some of these measuring instru- ments such as Sharpe Index and Scholz & Wilkens (2006) propose that both Sharpe Index and Treynor Ratio can 22 Jul 2019 Keywords: Jensen Ratio, Mutual Funds, Risk and Return,. Sharpe and Treynor Ratios. I. INTRODUCTION. The investor's resources were 25 Sep 2017 Then there are gauges like beta, Sharpe ratio, Sortino ratio and even something called the upside and downside capture ratio. And then there's Treynor Ratio. Like the Sharpe Ratio, the Treynor Ratio is a risk-adjusted measure. However whereas the Sharpe Ratio measures excess return of the investment over risk free return per unit of total risk; the Treynor ratio measures the excess return per unit of risk in relation to the market, i.e. per unit of systematic risk. It simply means the market has ups and downs. Treynor attempts to change that by placing all investments on the same risk free plain. This equation is similar to the Sharpe ratio’s method of assessing risk and volatility in the market with one main exception. The Treynor method uses the investment portfolio’s beta as the measurement of risk. Treynor Ratio: The Treynor ratio, also known as the reward-to-volatility ratio, is a metric for returns that exceed those that might have been gained on a risk-less investment, per each unit of The Sharpe ratio is almost identical to the Treynor measure, except that the risk measure is the standard deviation of the portfolio instead of considering only the systematic risk as represented In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment (e.g., a security or portfolio) compared to a risk-free asset, after adjusting for its risk.It is defined as the difference between the returns of the investment and the risk-free return, divided by the standard deviation of the Like the Sharpe ratio, the Treynor ratio (T) does not quantify the value added, if any, of active portfolio management. It is a ranking criterion only. A ranking of 10 Mar 2019 The Sharpe Ratio, Conditional Sharpe Ratio, Conditional Treynor Ratio, Treynor Ratio, Jensen's Alpha, Appraisal Ratio, Sortino and Van der The Sharpe (1966) Ratio and the Treynor and Black (1973) 'Appraisal. Ratio'1 are derived from the Capital Market Line, with the level of risk being meas- ured by 18 Mar 2014 Like the Sharpe ratio, the Treynor ratio does not quantify the value added, if any, of active portfolio management. It is a ranking criterion only. Unlike Sharpe Ratio, Treynor Ratio utilizes "market" risk (beta) instead of total risk (standard deviation). Good performance efficiency is measured by a high ratio. Money › Investment Fundamentals Portfolio Performance: Comparing Portfolio Returns using the Sharpe Ratio, Treynor Ratio, and Jensen's Alpha. Portfolios contain groups of securities that are selected to achieve the highest return for a given level of risk. Like the Sharpe ratio, the Treynor ratio (T) does not quantify the value added, if any, of active portfolio management. It is a ranking criterion only. A ranking of 22 Jul 2019 The difference between the two metrics is that the Treynor ratio utilizes beta, or market risk, to measure volatility instead of using total risk ( 25 Jun 2019 The Treynor ratio is similar to the Sharpe ratio, although the Sharpe ratio uses a portfolios standard deviation to adjust the portfolio returns. The main difference between the Sharpe ratio and the Treynor ratio is that unlike the use of systematic risk used in the case of Treynor ratio, the total risk or the The Treynor Ratio is a portfolio performance measure that adjusts for systematic - "undiversifiable" - risk. In contrast to the Sharpe Ratio, which adjusts return 27 Nov 2019 The Sharpe Ratio provides an overview of the return generating capacity of the fund against the overall risk. However, the Treynor Ratio 4 Oct 2016 While Sharpe ratio is applicable to all portfolios, Treynor is applicable to well- diversified portfolios. While Sharpe is used to measure historical Sharpe (1966) introduces the reward-to-variability ratio, more commonly referred to as the Sharpe index, Sharpe measure, or Sharpe ratio. For con- sistency of financial literature: The Sharpe (1966) ratio and the Treynor and Black (1973) “ appraisal ratio” both use the Capital Market Line as the risk-return referential, The Treynor ratio is also known as the reward-to-volatility measure. While the Sharpe ratio looks at portfolio's return against the rate of return for a risk-free Apple Treynor RatioThe Treynor is the reward-to-volatility ratio that expresses the excess return to the beta of the equity or portfolio. It is similar to the Sharpe

calculate air flow rate from pressure Im trying to measure flow rate of air (in cc/min) over 60 seconds. I can record the pressure drop in inch h20 over that time period, how can I calculate the flow rate in cc/min? 1. Air Flow Flow of air or any other fluid is caused by a pressure differential between two points. Flow will originate from an area of high energy, or pressure, and proceed toFor each section: 1. Write down or calculate all known variables. Air Flow Rate. (Q) Duct Cross-Sectional Area of the section. calculation of flow rate of air through diffrential pressure ?Calculators and Converters. Ask a Question. Home > Engineering Calculator > Calculator: Air Flow Rate through Piping.From Pressure Reduction and Condensate Separation. Air flow differs from water flow in that air is compressible and water is not. The result is that calculating either volumetric or mass water flow rates are straightforward, while the volume of a given mass of air may change significantly based on the temperature and pressure changes that take Subject: calculate air flow rate in pipe at given pressure and temperature. Date: Sun Aug 3 15:11:15 2003 Posted by rauf Grade level: 10-12 School: data steel City: sadiqabad State/Province: panjab Country: pakistan Area of science: Engineering ID: 1059941475.Eg. The most common use of pressure sensors to calculate another parameter is flow. This technique is also used to measure air flow of blowers, airFor fluid flow measur ements, orifice plates, venturi tubes and nozzles simplify the use of differ ential pressure (P) sensors to determine the flow rate. Permanent Averaging Instruments. Differential Pressure Pitot-static tubes Pitot-static grids Flow Capture Hoods.- As a result, the airflow rate calculated using the assumed K factor can have significant error. Calculation of moist air properties. Air-handling plant area calculation. Air-handling ductwork cross-section design.Ductwork specific pressure loss calculation. Mass air-flow rate unit converter. A variety of techniques are used to measure flow, including: the Coriolis Effect, mass air flowIn fact, many applications use pressure sensor measurements to calculate other key parametersIn medical applications, drug delivery (liquid flow) uses differential pressure sensors to measure flow rates of This site will explain it: http://www.ehow.com/how5973073calculat Youll need to know more than one of the pressures, if the pressure changes. Remember Pressure Force/Area. How is flow and pressure linked? Why do we need a discharge coefficient? What must be known to calculate flow rate correctly? 1 0,41 0,35 4 p p1 Different flow result with new version from 2003 Difference in air flow calculation using different versions of ISO 5167 ISO 5167 The inlet diameter of the air is 150mm with a 2"wg pressure, while the diameter of the duct is 417mm. Is it possible to calculate the flow rate or velocity? Air Flow Calculations with Velocity Pressures (All Pressure Measurements in Water Column Inches) For estimates of air flow, a single velocity pressure measurement at the center of section of straight duct may be sufficient.Pressure Point 11: Calculating Flow Rate from Pressure Measurements. Diameter Velocity Flow Rate. Ultra Calculator.Enter 1.0472 in the in the flow rate box and choose cubic feet per second from its menu. Click the CALCULATE button and you will see this equals 8 inches. The relationship between air flow rate, temperature gradient and heat loads in displacement ventilation One method of calculating the airflow required to limitLWtot 40 10 log q 20 log pt dB. Air flow q is stated in m3/s Pressure setting pt stated in Pa The above equation is illustrated in the graph below. A hot air balloon rising vertically is tracked by an observer located 2 miles from the lift-off point. at a certain moment, the angle between the observers line-of-sight and the horizontal is 5 Answer. Physics. use the difference between total pressure and static pressure to calculate the velocity of the aircraft or fluid flowing in the pipe or enclosure.220 Engine Air Filtration www.buydonaldson.com Technical Reference AIR FILTRATION TECHNICAL REFERENCE What is Airflow Restriction? How do you calculate Air Flow Rate (CFM).These comments received enough positive ratings to make them "good answers". 3 "Re: Calculate Air Flow Rate knowing Suuplied pressure and Hose diameter" by phoenix911 on 05/08/2017 2:58 PM (score 2). Online flow calculations for pressure drop, natural gas pipe flow calculation, Venturi and Orifice. Calculate the pressure drop for all the flow lengths and area changes that an air particle would see as it flows through Measure water or liquid flow rate and quantity using this very simple DIY project. What is the actual flow rate of Tube A?Did you measure the flow through each port of the dual flow sampler directly with an electronic calibrator reading out in ml/min (with representative sorbent tubes in the sampler), or are you relying on pressure readings to verify the flow? Compressible air flow calculator can be used for both laminar and turbulent flow. It can calculate pressure drop or flow rate through a pipe including friction losses and local pressure losses calculation. If the flow rate through the expansion is 368 gpm, the velocity goes from 9.27 ft/sec to 4.09 ft/sec.Therefore, head loss will always act to reduce the pressure head, or static pressure, of the fluid. There are several ways to calculate the amount of energy lost due to fluid flow through a pipe. Calculate air determining duct flow in cfm using the bapi pressure sensor airflow calculator velocity engineering toolbox.For example dynamic losses are proportional to pressure and can be calculated using the relationship between air flow rate (cfm) of an online calculator quickly Compressible air flow calculator can be used for both laminar and turbulent flow. It can calculate pressure drop or flow rate through a pipe including friction losses and local pressure losses calculation. The barometer, by itself, would give you the pressure from where you were reading it in the airflow.What equation should be used in order to calculate the flow rate (subsonic velocity) of air flows from a high pressure region to a low one? The basic relation between pressure and air flow through ducts or cavities is seen in Equation (3.1) for laminar flow and in Equation (3.2)The simulations only calculate the air flow rate for infiltration, but with results for every time step the corresponding temperature can be used in the energy calculation. Air flow and air pressure are related, but not tied together. For example, a leaf blower will be rated in CFMCubic Feet per Minute (CFM) Conversion Calculator to calculate air flow is made easier here. Air Flow, Air Systems, Pressure, and Fan Performance.Illustrate the dependence of gas transfer on gas flow rate. Calculate the air flow rate from testing the air flow controller and compare with the target value. Pipe Flow CalculationHydroponics Air Flow Calculator.will calculate air flow required based on temperature riseyou enter the total wattage of lights used and the temperatuPressure drop calculator Calculate pressure drop for known flow rate. Air flow is driven by pressures arising from wind and buoyancy forces (stack effect).In the course of an iterative procedure, room temperature and humidity conditions (together with any APhvac net supply or extract rates) are repeatedly passed to MacroFlo, which calculates the resulting natural ventilation Air Flow Conversion Calculator. Air Velocity is measurement of the rate of displacement of air or gas at a specific location.Concept of Air Velocity can be used in air conditioning, heating and ventilating work. Enter value, select unit and click on calculate. The device can also measure flow rate based on Air Cones that utilize the rotating vane head instead of the thermal probe.The TA465 models listed above can calculate flow rate from the square root of differential pressure and a K factor. Find all informations about calculate flow rate from pressure! Flowrate Calculation for a Draining Tank - eFunda.If I know what the air flow rate is at a certain pressure in a duct can i calculate the the flow rate at another known pressure??? Warning: fopen(wtuds/how-to-calculate-air-flow-rate-from-pressure): failed to open stream: Disk quota exceeded in /home/lwarwick/publichtml/catskill-motorcourt.com/vlzzoek/gtowza.php on line 149. Flow coefficient Cv is used to help calculate flow and pressure drop.q Cv x 11 pi / [SG (T 460)]1/2 where: q Air flow rate in standard cubic feet per minute (SCFM). Air Flow Calculations: Non Choked Flow. How to Calculate Air Flow Rates (6 Steps) | eHowThis app will guide the user through the airflow measurement process and automatically calculate airflow in a round or square duct.Calculation of pressure drop and flow rate of compressible air flow calculator 1. Establish the formulas giving the different air flows for a given internal reference pressure 2. Calculate the internal reference pressure irp balancing airWhen the indoor air quality only relies on windows opening, it is taken into account that the user behaviour leads to air flow rates higher than After pressing the Calculate Airflow button, read the CFM requirement of 34.91 CFM in the volume airflow units column.Establishing Cooling Requirements: Air Flow vs Pressure.Acoustic Noise: Causes, Rating Systems, and Design Guidelines. Calculating Air Flow with density correction Correcting for standard cfm SCFM ACFM(530/(460Tact))(Pact/29.92) SCFM standard flow rate ACFM actual flow rate measured flow rate Tact measured dry bulb temperature of the actual airstream, F Pact absolute pressure I can say the the box is in a room, so open air around the box. The thrust should be 35 N and I have to calculate the pressure and the flow rate necessary to do this. I can choose the diameter of the hole with no particular limits (I think about 1-2 cm or similar). Here, well see how the Bernoulli equation works by calculating the pressure at one point in an air duct when you know the pressure at another point.(2017, April 24). How to Calculate Pressure From Flow Rate. in this video we learn how to calculate the pump performance curve vales for volume flow rate, rpm, head pressure, pump power, impeller diameter for . a reference guide for technician in the field to get a simple . calculation to get . fan motor calculations, pulley size, rpm, air flow rate . How to Calculate Flow Rate From a Tank. Category:HobbiesRelease time:2013-09-05Views:130.Inside the tank, the air pressure presses down on the liquid. T[More]. Measuring Pressure Ps, Pt and Pv. Calculating Velocity Pressure By Formula.Commercial air conditioning equipment is designed for a particular airflow across the evaporator. The flow rates for a design may be as low as 160 C.F.M. per ton to a high of 1600 C.F.M. per ton. In industrial applications, the most common use of pressure sensors to calculate anotherThis technique is also used to measure air flow of blowers, air flow through filters, vent hoodsIn medical applications, drug delivery (liquid flow) uses differential pressure sensors to measure flow rates of The formulas for calculating the pressure drop were obtained from the ASHRAE Handbook of Fundamentals (ref 2). Correction factors for these formulas and testing of fittings and piping not included in the Fundamentalscfm air flow. Figure 1 - ASD system Airflow from NJ study of mitigated houses. [K mult] a constant to be multiplied to the flow rate. Differential pressure mass flow meter. A complete list of calculated coeffients are saved on file in the same folder of the actual file but with extension .g. air. Pitot tubes use the difference between total pressure and static pressure to calculate the velocity of the aircraft or fluid flowing in the pipe or enclosure.The American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAEs) minimum efficiency reporting value, or MERV rating Q Air Flow rate (m3/s).A Flow area / leakage area (m2).P Pressure difference (Pa). normally50 Pa is used. static can be calculated by adding all friction losses in the duct if applicable the air outlet which has been taken into the consideration the pressure which is needed to be maintained

Multiple Linear Regression Multiple linear regression is an extension of simple linear regression in which there is a single dependent (response) variable (Y) and k independent (predictor) variables Xi , i = 1, … , k. In multiple linear regression, the dependent variable is a quantitative variable while the independent variables may be quantitative or indicator (0, 1) variables. The usual purpose of a multiple regression analysis is to create a regression equation for predicting the dependent variable from a group of independent variables. Desired outcomes of such an analysis may include the following: 1. Screen independent variables to determine which ones are good predictors and thus find the most effective (and efficient) prediction model. 2. Obtain estimates of individual coefficients in the model to understand the predictive role of the individual independent variables used. Appropriate Applications for Multiple Regression Analysis Design Considerations for Multiple Regression Analysis A Theoretical Multiple Regression Equation Exists That Describes the Relationship Between the Dependent Variable and the Independent Variables As in the case of simple linear regression, the multiple regression equation calculated from the data is a sample-based version of a theoretical equation describing the relationship between the k independent variables and the dependent variable Y. The theoretical equation is of the form Y = α + β1X1 + β2X2 +… + βkXk+ ε where α is the intercept term and βi is the regression coefficient corresponding to the ith independent variable. Also, as in simple linear regression, ε is an error term with zero mean and constant variance. Note that if βi = 0, then in this setting, the ith independent variable is not useful in predicting the dependent variable. The Observed Multiple Regression Equation Is Calculated From the Data Based on the Least Squares Principle The multiple regression equation that is obtained from the data for predicting the dependent variable from the k independent variables is given by Ŷ = a+b1X1+b2X2…+bkXk As in the case of simple linear regression, the coefficients a, b1 , b2 , … , and bk are least squares estimates of the corresponding coefficients in the theoretical model. That is, as in the case of simple linear regression, the least squares estimates a and b1 , … , bk are the values for which the sum-of-squared differences between the observed y values and the predicted y values are minimized. Several Assumptions Are Involved 1. Normality. The population of Y values for each combination of independent variables is normally distributed. 2. Equal variances. The populations in Assumption 1 all have the same variance. 3. Independence. The dependent variables used in the computation of the regression equation are not correlated. This typically means that each observed y value must be from a separate subject or entity. Hypotheses for a Multiple Linear Regression Analysis In multiple regression analysis, the usual procedure for determining whether the ith independent variable contributes to the prediction of the dependent variable is to test the H0 : βi = 0 Ha : βi ≠ 0 for i = 1, … , k. Each of these tests is performed using a t-test. There will be k of these tests (one for each independent variable), and most statistical packages report the corresponding t-statistics and p-values. Note that if there were no linear relationship whatsoever between the dependent variable and the independent variables, then all of the βis would be zero. Most programs also report an F-test in an analysis of variance output that provides a single test of the following hypotheses: H0 : β1 = β2 =…= βk = 0 (there is no linear relationship between the dependent variable and the collection of independent variables). Ha : At least one of the βis is nonzero (there is a linear relationship between the dependent variable and at least one of the independent variables). The analysis-of-variance framework breaks up the total variability in the dependent variable (as measured by the total sum of squares) by that which can be explained by the regression using X1 , X2 , … , Xk (the regression sum of squares) and that which cannot be explained by the regression (the error sum of squares). It is good practice to check the pvalue associated with this overall F-test as the first step in the testing procedure. Then, if this p-value is less than 0.05, you would reject the null hypothesis of no linear relationship and proceed to examine the results of the t-tests. However, if the p-value for the F-test is greater than 0.05, then you have no evidence of any relationship between the dependent variable and any of the independent variables, so you should not examine the individual ttests. Any findings of significance at this point would be questionable. Click Here To Download Sample Dataset (SPSS Format) Research Scenario and Test Selection The researcher wants to understand how certain physical factors may affect an individual’s weight. The research scenario centers on the belief that an individual’s “height” and “age” (independent variables) are related to the individual’s “weight” (dependent variable). Another way of stating the scenario is that age and height influence the weight of an individual. When attempting to select the analytic approach, an important consideration is the level of measurement. As with single regression, the dependent variable must be measured at the scale level (interval or ratio). The independent variables are almost always continuous, although there are methods to accommodate discrete variables. In the example presented above, all data are measured at the scale level. What type of statistical analysis would you suggest to investigate the relationship of height and age to a person’s weight? Regression analysis comes to mind since we are attempting to estimate (predict) the value of one variable based on the knowledge of the others, which can be done with a prediction equation. Single regression can be ruled out since we have two independent variables and one dependent variable. Let’s consider multiple linear regression as a possible analytic approach. We must check to see if our variables are approximately normally distributed. Furthermore, it is required that the relationship between the variables be approximately linear. And we will also have to check for homoscedasticity, which means that the variances in the dependent variable are the same for each level of the independent variables. Here’s an example of homoscedasticity. A distribution of individuals who are 61 inches tall and aged 41 years would have the same variability in weight as those who are 72 inches tall and aged 31 years. In the sections that follow, some of these required data characteristics will be examined immediately, others when we get deeper into the analysis. The basic research question (alternative hypothesis) is whether an individual’s weight is related to that person’s age and height. The null hypothesis is the opposite of the alternative hypothesis: An individual’s weight is not related to his or her age and height. Therefore, this research question involves two independent variables, “height” and “age,” and one dependent variable, weight. The investigator wishes to determine how height and age, taken together or individually, might explain the variation in weight. Such information could assist someone attempting to estimate an individual’s weight based on the knowledge of his or her height and age. Another way of stating the question uses the concept of prediction and error reduction. How successfully could we predict someone’s weight given that we know his or her age and height? How much error could be reduced in making the prediction when age and height are known? One final question: Are the relationships between weight and each of the two independent variables statistically significant? R column of the table here shows a strong multiple correlation coefficient. It represents the correlation coefficient when both independent variables (“age” and “height”) are taken together and compared with the dependent variable “weight.” The Model Summary indicates that the amount of change in the dependent variable is determined by the two independent variables—not by one as in single regression. From an “interpretation” standpoint, the value in the next column, R Square, is extremely important. The R Square of .845 indicates that 84.5% (.845 × 100) of the variance in an individual’s “weight” (dependent variable) can be explained by both the independent variables, “height” and “age.” It is safe to say that we have a “good” predictor of weight if an individual’s height and age are known The ANOVA table indicates that the mathematical model (the regression equation) can accurately explain variation in the dependent variable. The value of .000 (which is less than .05) provides evidence that there is a low probability that the variation explained by the model is due to chance. We conclude that changes in the dependent variable result from changes in the independent variables. In this example, changes in height and age resulted in significant changes in weight. As with single linear regression, the Coefficients table shown in Figure 21.13 provides the essential values for the prediction equation. The prediction equation takes the following form: Ŷ = a + b1x1 + b2x2, where Ŷ is the predicted value, a the intercept, b1 the slope for “height,” x1 the independent variable “height,” b2 the slope for “age,” and x2 the independent variable “age.” The equation simply states that you multiply the individual slopes by the values of the independent variables and then add the products to the intercept—not too difficult. The slopes and intercepts can be found in the table shown in Figure 21.13. Look in the column labeled B. The intercept (the value for a in the above equation) is located in the (Constant) row and is -175.175. The value below this of 5.072 is the slope for “height,” and below that is the value of -0.399, the slope for “age.” The values for x are found in the weight.sav database. Substituting the regression coefficients, the slope and the intercept, into the equation, we find the following: Weight = - 175. 175 + (5.072 * Height) − (0.399 * Age). Further, if we notice p-value for age as a independent variable then, it has no significant relationship with weight as the value is 0.299 which is greater than alpha value of 0.05. So, we can consider that the Height is only variable which is contributing significantly in explaining the variability of Weight.

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