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The principal objective of this book is to present a collection of challenging test problems arising in literature studies and a wide spectrum of applications. Due to the non-differentiability of the perturbed solutions in equilibrium constraints, a non-smooth optimization model is established. A generalized bundle subgradient projection is presented to effectively solve the network design problem. Global convergence analysis for the proposed method is also delivered. Network design problems NDP consist of identifying an optimal subgraph of a graph, subject to side constraints. In generalized NDP, the vertex set is partitioned into clusters and the feasibility conditions are expressed in terms of the clusters. Fuzzy generalized network approach for solving. Formulation and feasibility test of optimal road network design model with endogenously determined travel demand. Proceedings of the 5th World Conference on Transport. Abstract: Many combinatorial optimization problems can be considered as nding a subgraph of a given graph with di erent requirements of graph properties e. They can be generalized for network design in di erent application areas, such as power. Solving the Transit Network Design problem with Constraint. Modeling and solving a distribution network design problem. Wiley Online Library. Several results Generalized network design problems : modeling. Chapter 11 Net w ork Optimization Net ork mo dels ha v e three main adv an tages o v er linear programming: 1. They can b e solv ed v ery quic kly. Approximating the Generalized Capacitated Tree-Routing Problem | SpringerLink Problems whose linear program w ould ha v e ro ws and 30, columns can b e solv ed in a matter. Generalized Network Design Problems : Modeling. Network Optimization: Continuous and Discrete Models. The monograph describes in a unified manner a series of mathematical models, methods, propositions, and algorithms developed. The LLamasoft Digital Design and Decision Center provides end-to-end supply chain perspective to quickly pinpoint inefficiencies to offer solutions to create the network you want. Create digital models to evaluate and compare trade-offs of potential supply chain network changes. By modeling in a no-risk environment, you can make high-stakes. General Optimization Methods for Network Design of different notational conventions for modeling and formulating network design problem and introduces the. A variety of practical network optimization problems arising in the context of models and the analysis of five such problems, and the subsequent design and new algorithm for solving the generalized lp distance location-allocation problem. In what it follows we describe an application encountered in the design of regional network design. Network models Network representation is widely used in:. Alggorithms and software are beingg used to solve hugge network problems on a routine basis. Many network problems are special cases of linear. Design network inserting links in order. This review provides an overview of the queueing modeling issues and the related performance evaluation and optimization approaches framed in a joined manufacturing and product engineering. Such networks are represented as queueing networks. The performance of the queueing networks is evaluated using an advanced queueing network analyzer: the generalized expansion method. - Taste of Home Brunch Favorites: 201 Delicious Ideas to Start your Day. - Farmacotherapie Bij Kinderen: Kennislacunes in Beeld Gebracht. - Practical Pulmonary Pathology: A Diagnostic Approach! - Generalized Network Design Problems: Modeling and Optimization. - Search form. Sustainable supply chain network design: An optimization. Transportation models play an important role in logistics and. The express purpose of this monograph is to describe a series of mathematical models, methods, propositions, algorithms developed in the last years on generalized network design problems in a unified. After separately studying these two pooling problem instantiations, we have unified our work by developing APOGEE Algorithms for Pooling-problem Optimization in GEneral and Extended classes , a generic computational tool that globally optimizes standard, generalized, and extended pooling problems. Generalized network design problems : modeling and optimization Generalized Network Design Problems Modeling and Optimization In combinatorial optimization, many network design problems can be generalized in a natural way by considering a related problem on a clustered graph, where the original problem s feasibility constraints are expressed in terms of the clusters, i. Decomposition Methods and Network Design Problems. Network Optimization by Generalized Methodology - wseas. Introduction to Network Models Network models are applicable to an enormous variety of decision problems that can be modeled as networks optimization problems and solved efficiently and effectively. Some of these decision problems are really physical problems such as transportation or flow of commodities. Integer programming models and branch-and-cut approaches. The generalized methodology for the electronic networks optimization was elaborated by means of the optimal control theory approach. In this case the problem of the electronic system design. Mathematical optimization alternatively spelled optimisation or mathematical programming is the selection of a best element with regard to some criterion from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods Ordinary network flow models require flow conservation on all arcs: The amount of flow entering an arc equals the amount of flow leaving the arc. Generalized network flow models, on the other hand, is a generalization of standard network flow models in which each arc of the underlying graph has an associated positive gain or loss factor. Generalized Network Design Problems by Petrica The Modeling and Optimization: Theory and Applications MOPTA conference is an annual event aiming to bring together a diverse group of people from both discrete and continuous optimization, working on both theoretical and applied aspects. Optimal solutions generated turn out to be integer if the relevant constraint data are integer. A common. PAPER Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures There are various ways that network functionalities can be allocated to different layers and to different network elements, some being more desirable than others. The intellectual goal of the research surveyed by this article is to provide. Model and heuristic for a generalized access network. - Book design! - Product details. - Spermatogenesis Genetic Aspects! - Free pdf Generalized Network Design Problems Modeling And Optimization Free Download. - Additional Information? - UNIX System Administration Handbook. - Generalized Network Design Problems: Modeling and Optimization by Petrica C. Pop - ifedugadokir.gq; Model predictive control MPC is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models. Authors; In this paper the routing problem is formulated as a fuzzy multiobjective optimization model. The fuzzy approach allows for the inclusion and evaluation of several criteria simultaneously. The express purpose of this monograph is to describe a series of mathematical models, methods, propositions, algorithms developed in the last years on generalized network design problems in a unified manner. The book consists of seven chapters, where in addition to an introductory chapter, the following generalized network design problems are formulated and examined: the generalized minimum spanning tree problem, the generalized traveling salesman problem, the railway traveling salesman problem, the generalized vehicle routing problem, the generalized fixed-charge network design problem and the generalized minimum vertex-biconnected network problem. The book will be useful for researchers, practitioners, and graduate students in operations research, optimization, applied mathematics and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will find this book useful, too. Petrica C. Related Generalized Network Design Problems: Modeling and Optimization Copyright 2019 - All Right Reserved

Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses OA to mean it is obligatory that A (or it ought to be (the case) that A), and PA to mean it is permitted (or permissible) that A, which is defined as . Note that in natural language, the statement "You may go to the zoo OR the park" should be understood as instead of , as both options are permitted by the statement; See Hans Kamp's paradox of free choice for more details. When there are multiple agents involved in the domain of discourse, the deontic modal operator can be specified to each agent to express their individual obligations and permissions. For example, by using a subscript for agent , means that "It is an obligation for agent (to bring it about/make it happen) that ". Note that could be stated as an action by another agent; One example is "It is an obligation for Adam that Bob doesn't crash the car", which would be represented as , where B="Bob doesn't crash the car". In Georg Henrik von Wright's first system, obligatoriness and permissibility were treated as features of acts. Soon after this, it was found that a deontic logic of propositions could be given a simple and elegant Kripke-style semantics, and von Wright himself joined this movement. The deontic logic so specified came to be known as "standard deontic logic," often referred to as SDL, KD, or simply D. It can be axiomatized by adding the following axioms to a standard axiomatization of classical propositional logic: In English, these axioms say, respectively: FA, meaning it is forbidden that A, can be defined (equivalently) as or . where . It is generally assumed that is at least a KT operator, but most commonly it is taken to be an S5 operator. In practical situations, obligations are usually assigned in anticipation of future events, in which case alethic possiblities can be hard to judge; Therefore, obligation assignments may be performed under the assumption of different conditions on different branches of timelines in the future, and past obligation assignments may be updated due to unforeseen developments that happened along the timeline. The other main extension results by adding a "conditional obligation" operator O(A/B) read "It is obligatory that A given (or conditional on) B". Motivation for a conditional operator is given by considering the following ("Good Samaritan") case. It seems true that the starving and poor ought to be fed. But that the starving and poor are fed implies that there are starving and poor. By basic principles of SDL we can infer that there ought to be starving and poor! The argument is due to the basic K axiom of SDL together with the following principle valid in any normal modal logic: If we introduce an intensional conditional operator then we can say that the starving ought to be fed only on the condition that there are in fact starving: in symbols O(A/B). But then the following argument fails on the usual (e.g. Lewis 73) semantics for conditionals: from O(A/B) and that A implies B, infer OB. Indeed, one might define the unary operator O in terms of the binary conditional one O(A/B) as , where stands for an arbitrary tautology of the underlying logic (which, in the case of SDL, is classical). The accessiblity relation between possible world is interpreted as acceptibility relations: is an acceptable world (viz. ) if and only if all the obligations in are fulfilled in (viz. ). Alan R. Anderson (1959) shows how to define in terms of the alethic operator and a deontic constant (i.e. 0-ary modal operator) standing for some sanction (i.e. bad thing, prohibition, etc.): . Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction. In addition to the usual modal axioms (necessitation rule N and distribution axiom K) for the alethic operator , Anderson's deontic logic only requires one additional axiom for the deontic constant : , which means that there is alethically possible to fulfill all obligations and avoid the sanction. This version of the Anderson's deontic logic is equivalent to SDL. However, when modal axiom T is included for the alethic operator ( ), it can be proved in Anderson's deontic logic that , which is not included in SDL. Anderson's deontic logic inevitably couples the deontic operator with the alethic operator , which can be problematic in certain cases. An important problem of deontic logic is that of how to properly represent conditional obligations, e.g. If you smoke (s), then you ought to use an ashtray (a). It is not clear that either of the following representations is adequate: Under the first representation it is vacuously true that if you commit a forbidden act, then you ought to commit any other act, regardless of whether that second act was obligatory, permitted or forbidden (Von Wright 1956, cited in Aqvist 1994). Under the second representation, we are vulnerable to the gentle murder paradox, where the plausible statements (1) if you murder, you ought to murder gently, (2) you do commit murder, and (3) to murder gently you must murder imply the less plausible statement: you ought to murder. Others argue that must in the phrase to murder gently you must murder is a mistranslation from the ambiguous English word (meaning either implies or ought). Interpreting must as implies does not allow one to conclude you ought to murder but only a repetition of the given you murder. Misinterpreting must as ought results in a perverse axiom, not a perverse logic. With use of negations one can easily check if the ambiguous word was mistranslated by considering which of the following two English statements is equivalent with the statement to murder gently you must murder: is it equivalent to if you murder gently it is forbidden not to murder or if you murder gently it is impossible not to murder ? Some deontic logicians have responded to this problem by developing dyadic deontic logics, which contain binary deontic operators: (The notation is modeled on that used to represent conditional probability.) Dyadic deontic logic escapes some of the problems of standard (unary) deontic logic, but it is subject to some problems of its own.[example needed] Philosophers from the Indian Mimamsa school to those of Ancient Greece have remarked on the formal logical relations of deontic concepts and philosophers from the late Middle Ages compared deontic concepts with alethic ones. In his Elementa juris naturalis (written between 1669 and 1671), Gottfried Wilhelm Leibniz notes the logical relations between the licitum (permitted), the illicitum (prohibited), the debitum (obligatory), the, and the indifferens (facultative) are equivalent to those between the possibile, the impossibile, the necessarium, and the contingens respectively. Ernst Mally, a pupil of Alexius Meinong, was the first to propose a formal system of deontic logic in his Grundgesetze des Sollens (1926) and he founded it on the syntax of Whitehead's and Russell's propositional calculus. Mally's deontic vocabulary consisted of the logical constants U and ∩, unary connective !, and binary connectives f and ∞. Mally defined f, ∞, and ∩ as follows: Mally proposed five informal principles: He formalized these principles and took them as his axioms: From these axioms Mally deduced 35 theorems, many of which he rightly considered strange. Karl Menger showed that !A ↔ A is a theorem and thus that the introduction of the ! sign is irrelevant and that A ought to be the case if A is the case. After Menger, philosophers no longer considered Mally's system viable. Gert Lokhorst lists Mally's 35 theorems and gives a proof for Menger's theorem at the Stanford Encyclopedia of Philosophy under Mally's Deontic Logic. The first plausible system of deontic logic was proposed by G. H. von Wright in his paper Deontic Logic in the philosophical journal Mind in 1951. (Von Wright was also the first to use the term "deontic" in English to refer to this kind of logic although Mally published the German paper Deontik in 1926.) Since the publication of von Wright's seminal paper, many philosophers and computer scientists have investigated and developed systems of deontic logic. Nevertheless, to this day deontic logic remains one of the most controversial and least agreed-upon areas of logic. G. H. von Wright did not base his 1951 deontic logic on the syntax of the propositional calculus as Mally had done, but was instead influenced by alethic modal logics, which Mally had not benefited from. In 1964, von Wright published A New System of Deontic Logic, which was a return to the syntax of the propositional calculus and thus a significant return to Mally's system. (For more on von Wright's departure from and return to the syntax of the propositional calculus, see Deontic Logic: A Personal View and A New System of Deontic Logic, both by Georg Henrik von Wright.) G. H. von Wright's adoption of the modal logic of possibility and necessity for the purposes of normative reasoning was a return to Leibniz. Although von Wright's system represented a significant improvement over Mally's, it raised a number of problems of its own. For example, Ross's paradox applies to von Wright's deontic logic, allowing us to infer from "It is obligatory that the letter is mailed" to "It is obligatory that either the letter is mailed or the letter is burned", which seems to imply it is permissible that the letter is burned. The Good Samaritan paradox also applies to his system, allowing us to infer from "It is obligatory to nurse the man who has been robbed" that "It is obligatory that the man has been robbed". Another major source of puzzlement is Chisholm's paradox. There is no formalisation in von Wright's system of the following claims that allows them to be both jointly satisfiable and logically independent: Responses to this problem involve rejecting one of the three premises.

integrate sin 4x cos 6x dx. My WordPress Blog. Sample Page.4x dx- cos4x sin2x dx let us do one by one Math2.org Math Tables: Table of Integrals Power of x. x n dx x n1 (n1)-1 C (n -1) d/dx [-sin(x)] -cos(x). The thing is though, that all of these arent just the derivative of the trig functions, if you do it step by step you would get intermediated/dx [cos(x)] d/dx [x] . -sin(x) 1 . Type in any integral to get the solution, steps and graph int(sin2x )/(1cos2x)dx-ln(1cos2x)C At first glance, this one seems like a toughie Get the answer to Integral of sin(x)2 with the Cymath math problem solver - a free math equation solver and math solving app for calculus and algebra. Possible intermediate steps: integral sin(x) sin(2 x) dx Use the trigonometric identity sin(alpha) sin(x)-cos(3 x)) dx Integrate the sum term by term and factor out constants: 1/2 integral cos( x) dx-1/2 Повторите попытку позже. Опубликовано: 4 сент. 2013 г. This video is about integrate sin2 x dx. cos X sin X dx. so the original integral becomes.There are a LOT Of sin Xs. so lets say sinX u. it becomes. Lets use integration by parts: If we apply integration by parts to the rightmost expression again, we will get cos 2(x)dx cos2(x)dx, which is not very useful. The solution of the differential equation cos x cos y dx sin x sin y dy 0 is. Question Posted / rajasekar. Yes, thats correct! Basically, usinx du/dxcos x. Put this back into the equation, you get integrate (u3). then the final answer is .25(sin4 x). int x2sin2x dx. I am given to evaluate: Please, the result is 1/4I cant get thisshould it be just normal multiplying rule for integrals or something else? (n-1)INTEGRAL sin(n-2)(x)(1-sin2(x))dxAnother approach is to use a trigonometric identity to express sin10(x) as a sum of terms of the form. integral of x sin 2 (x) dx. attempt: Let u sin2 (x) v x u 2sinxcosx v x2 / 2.For this one: Using the identity, [tex]sin2(x)frac1-cos(2x)2[/tex], rewrite as This video is about integrate sin2 x dx. The full question is 2xsin(3x2). sin x ] sin 3x dx [ 2 sin 2x. Write back if you need more help, Penny . Simple harmonic motion: the swing of the pendulum The Guardian The slope of the curve is the rate of change ( dx/dt). When the wave is at its highest, the slope is changing quickest How to integrate sin6 x cos2 x dx. I do not know of an easy way to evaluate this integral. But heres one way that isnt too bad. Another method for integrating sec x dx, that is more tedious, but less dependent on a memorized trick, is to convert sec x dx into the integral of a rational function using the substitution y sin x int: x sin(2x) dx. This can be done using integration by parts-x/2 cos(2x) 1/2 int: cos(2x) dx. Again, by simple substitution this second integral is done and we get our final answer xsin(2x) dx (-1/2)xcos2x (1/4)sin2x You get this by using Integration by Parts. An integral in the form udv can be written as uv-vdu In.thus the given integral becomes: sinx cosx dx (1/4)sin(2x) dx (1/4) sin(2x) dx now you can reduce the order of the integrand using the half-angle identity: sinx (1/2) [1 - cos(2x) Compute sin4(x) cos2(x) dx. Solution. c2. We conclude that: sin4 x cos2 x dx. Write integrand as (sin x cos x)2 (frac12sin 2x)2 . Then use the following facts: sin2 2x 1-cos2 2x. cos2 2x frac12(cos 4x 1). Note: The original question asked for the A Reduction Formula Problem: Integrate I (sin x)n dx. Try integration by parts with. We get.du (n 1)(sin x)n2 cos x dx dv sin x dx. sin(x y) dx dy | sin(y) cos(y) dy | / /. 0 0 0. and this last integral is equal to int fracdxcos3x fracsin x2cdot cos2x frac12lnleft|tanleft(frac x2fracpi2right)right| . Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Type in any integral to get the solution, steps and graph sin x dx cos x C. 163. 164 Chapter 8 Techniques of Integration.2x cos(x2) dx. This is not a simple derivative, but a little thought reveals that it must have come from. It means we have to find the sin2x dx. As we know the trigonometry identity of sin( xy)sinxcosy cosxsiny. By putting xy we will get. Answer: d/dx(sinx) cosx, so use substitution.intu(-2) du which evaluates to -1/u C. Reverse the substitution to finish: int cosx/ sin2x dx -1/sinx -cscx. The integral int sin6 x dx has to be determined. (Some more advanced calculators can.) integral of (cos 2x cos x) dx How to integrate sin(2x) / cos3(x)? SOLUTION Simply substituting u cos x isnt helpful, since then du sin x dx. In order to integrate powers of cosine, we would need an extra sin x factor. To integrate sin2x cos2x, also written as cos2x sin2x dx, sin squared x cos squared x, sin2(x) cos2(x), and (sin x)2 (cos x)2, we start by using standard trig identities to to change the form. Z Z integral of sin 2x /8 - 1/16 cos 2x to integrate sin 2x cos 2 2x put cos 2x t then 2 sin 2x dx dt we get t2 dt/2 integrating we get t3/6 Free derivative calculator - differentiate functions with all the sin 3x cosx dx ( integrate sin x/cos3 x dx ) No. Though youre just a 10th grade student, you know how to solve integrals. 2. Relevant equations I know the integral of sin(x)dx -cos(x) C. 3. The attempt at a solution What I did was to say that the integral is -cos( 2x) C, which isnt the correct answer Rahul Pillai. how to integrate. f (log(sin x ))dx. Varun Nagarajan. what is integral of log( x)? cos(x)dx d(sin(x)). giving 2sin(x)d(sin(x)), or 2zdz by replacing sin( x) by z. >>also is there a general pattern for integral like this ? Int dx/(sinxcosx) Int (cosx-Sinx)dx/(Cos2x-sin2x) Int cosx dx/ (1-2Sin 2x) - Int sinx dx/ (2cos2x-1) (.1).lets now plug-in the value of sin x and cosx in (1). sin x cos x. Integral. sin x dx - cos x C. Eulers formula. For each of these, we simply use the Fundamental of Calculus, because we know their corresponding derivatives. cos( x) sin(x), cos(x) dx sin(x) c. The good news is that your indefinite integral is correct, but off by 1/24, which can be absorbed in the integration constant. Depending on how you integrated, it could give equivalent results of: (3sin( x) There are Cinfty test functions in L1(0, infty) that make the integral value of int0infty(sin x/x) phi(x) dx range from 0 to infty. What does narrowing these test For each of these, we simply use the Fundamental of Calculus, because we know their corresponding derivatives. cos( x) sin(x), cos(x) dx sin(x) c. Find: (intergrate) 2sin3x sin 2x dx? Type in any integral to get the solution, steps and graph Find: (intergrate) 2sin3 x sin 2x dx? Related QuestionsMore Answers Below. How can I integrate sin(x) /sin(4x)?What is integral sin (cos x).dx? integrate 1/(sinx cosx) dx. For the second integral, use u sin(x), du cos(x)dx. and now the substitution u cos(x), so that du -sin(x)dx, makes that a polynomial integtration: Of course, if m is odd, you can do the same thing, switching "sine" and "cosine". Use the half angle formula, sin2(x) 1/2(1 - cos(2x)) and substitute into the integral so it becomes 1/2 times the integral of (1 - cos( 2x)) dx. Example: sin3(x) sin2(x) sin(x). Hence the given integral may be written as followsWe now let u cos(x), hence du/dx -sin(x) or -du sin(x)dx and substitute in the given intergral to obtain.

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A general formula for duration Working with time duration, not time of day ask question i have to look at the formula to check whether i entered the data correctly [and didn't screw up the formula] your data fields where you are entering in your times should need to be changed from general or number format to the custom format of hh:mm. General format as hh:mm:ss00 formula or constant as long as you have date & time in each cell as entered by excel you simply subtract the earlier timestamp from the later timestamp to get a differences in days and/or hours and/or minutes but since months do not have the same number of days and years don't have the same number of. Quadratic functions(general form) quadratic functions are some of the most important algebraic functions and they need to be thoroughly understood in any modern high school algebra course the properties of their graphs such as vertex and x and y intercepts are explored interactively using an html5 applet. Projectile motion, general solution back projectile motion curved motion physics contents index home what follows is a general solution for the two dimensional motion of an object thrown in a gravitational field. A general formula for computing the coefficients of the correlation connecting global solar radiation to sunshine duration. Loan balance situation: a person initially borrows an amount a and in return agrees to make n repayments per year, each of an amount pwhile the person is repaying the loan, interest is accumulating at an annual percentage rate of r, and this interest is compounded n times a year (along with each payment)therefore, the person must continue paying these installments of amount p until the. So this is a geometric series with common ratio r = –2 (i can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2) the first term of the sequence is a = –6plugging into the summation formula, i get. The formula for the energy of motion is ke = 5 × m × v 2 where ke is kinetic energy in joules, m is mass in kilograms and v is velocity in meters per second force and work newton's three laws of motion form the basis for classical physics. Applying the damages formula to her claim, an insurance adjuster would begin with a figure of between $900 and $3,000 (15 to 5 x $600) this would then be added to mary’s lost income of $400 to get the figure from which negotiations would begin as compensation for mary’s injuries. This is the aptitude questions and answers section on time and distance important formulas with explanation for various interview, competitive examination and entrance test solved examples with detailed answer description, explanation are given and it would be easy to understand. The general format is the default number format that excel applies when you type a number for the most part, numbers that are formatted with the general format are displayed just the way that you type them however, if the cell is not wide enough to show the entire number, the general format rounds numbers that have decimals the general number format also uses scientific (exponential. A general formula for the generation time fran˘cois bienvenu 1, lloyd demetrius2,3, and st ephane legendre 1team of mathematical eco-evolution, ecole normale sup erieure, paris, france 2department of organismic and evolutionary biology, harvard university, usa 3max planck institute for molecular genetics, berlin, germany 10/06/2013. 21-110: finding a formula for a sequence of numbers it is often useful to find a formula for a sequence of numbers having such a formula allows us to predict other numbers in the sequence, see how quickly the sequence grows, explore the mathematical properties of the sequence, and sometimes find relationships between one sequence and another. This video explains how to use the distance formula on two points time is also taken to explain the exact answer it returns vs the decimal approximation which can be a bit more useful. A general formula for duration In this lesson we'll take a look at the number format called general the general format is excel's default format for all cells in a new worksheet, all cells have this format. Duration and convexity bond prices change inversely with interest rates, and, hence, there are several formulas for calculating the duration of specific bonds that are simpler than the above general formula the formula for the duration of a coupon bond is the following. The weighted average formula is a general mathematical formula, but the following information will focus on how it applies to finance use of weighted average formula the concept of weighted average is used in various financial formulas. Tip: if your cell is formatted as general and you enter the time function, excel will format your result as h:mm am/pm (column e) based on your regional settings if you wish to see the decimal result (column d) from the time function, you will have to change the format of the cell to general after entering the formula. Enter your desired formula into the 2nd row of the column you want to fill make sure to use $ for any referenced cells where the row stays the same for all equations select the cell containing the formula, and press ctrl + shift + down. General approach to measuring the duration of liabilities for property-liability insurers has been to calculate a weighted average of the time to payment for loss reserves (campbell, 1995 hodes and. How about the inverse problem, i have an elapsed time in format of hh:mm (hours and minutes), i would like to covert to a general number for costing purposes in other words 2:06 (2 hours six minutes) = 21 hours. (1) column 3a shows duration increasing with maturity, but less than proportionately (2) column 4a compared with 3a shows that a decline in yield to maturity (from 1175% to 675%) increases duration, especially for the longer maturities. A problem on how to determine the rate of a reaction given a) the balanced chemical equation, b) the change in concentration for any reactant or product, and c) the time interval during which the. Extract time from a date-time number in excel i have a worksheet that tracks start and stop times for different events throughout the day, all during the week sometimes i have to pull out the time of day, irrespective of the date, with the time function. Determining the length of an irregular arc segment is also called rectification of a curvehistorically, many methods were used for specific curves the advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. 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Quadratic Functions Worksheet With Answers. You can choose the magnitude of the “a” time period and the direction during which the parabola opens. With the help of the group we are ready to proceed to enhance our academic sources. The sides of an equilateral triangle are shortened by 12 items, 13 items and 14 units respectively and a right angle triangle is formed. Find the vertex of the given quadratic functions through the use of the technique of finishing the square. These Algebra 1 – Quadratic Functions Worksheets produces issues for fixing quadratic equations by factoring. The quadratic equations worksheet will help students apply the usual form of quadratic equations and discover methods to remedy the quadratic equation. These worksheets will help the students of sophistication 10 to practice more for board exams. These Algebra 1 – Quadratic Functions Worksheets produces problems for solving quadratic equations with the quadratic formula. - Bernadette throws the javelin for her school’s track and subject group. - You can select the magnitude of the “a” time period and the course by which the parabola opens. - Find the vertex of the given quadratic capabilities through the use of the strategy of finishing the square. - With the assistance of the neighborhood we can continue to enhance our academic assets. - The sides of an equilateral triangle are shortened by 12 items, 13 items and 14 models respectively and a proper angle triangle is shaped. Also you’ll find a way to change the assorted translated capabilities using the three other enter bins which are labeled a, b, and c. If you want to reposition the display you need to use the tool at the high of the screen that appears like 4 arrows to drag the display screen to a unique place. [newline]Also you have to use the pointer button at the prime of the display screen to tug the perform f to show how the opposite functions change. The file could be run by way of the free on-line application GeoGebra, or run domestically if GeoGebra has been put in on a pc. The sides of an equilateral triangle are shortened by 12 models, thirteen models and 14 units respectively and a proper angle triangle is formed. - 1 Example Query #1 : Graphing Parabolas - 2 A 7a Parts Of Quadratic Functions Scavenger Hunt - 3 Related posts of "Quadratic Functions Worksheet With Answers" Example Query #1 : Graphing Parabolas The point $(x,h)$ is identical as $(x,-2f)$ so the graph of $h$ is the same because the reflection of the graph of $2f$ concerning the $x$-axis. So the values of $f$ are first doubled, exaggerating the slope of the graph, after which the graph is mirrored concerning the $x$-axis. Engage your students with efficient distance learning resources. Deciphering Solutions Of Quadratic Features In the reasonable stage, the x-values are decimals or fractions. Factorize every quadratic perform and write the function in intercept kind. Practice this array of worksheets to realize expertise in factoring the function, finding zeros and converting quadratic function to intercept type. The graph below exhibits essential attributes of the graph of a parabola that you ought to use to research and interpret the graphs of quadratic capabilities. We’re going to investigate the graphs of quadratic capabilities. Google Sheets Digital Pixel Artwork Math Linear Equations: Identifying Key Options Substitute the values of x in the quadratic perform to determine the y values. To facilitate a simple practice, the coefficients and x-values are provided in integers. These Algebra 1 – Quadratic Functions Worksheets produces problems for finishing the sq.. A 7a Parts Of Quadratic Functions Scavenger Hunt Corbett Maths provides outstanding, unique exam type questions on any topic, in addition to videos, previous papers and 5-a-day. Find if the given values are the answer of the given equations. As a member, you will also get unlimited entry to over eighty four,000 classes in math, English, science, historical past, and more. Plus, get follow checks, quizzes, and personalized teaching that can assist you succeed. With a and c mounted, observe the impact of change of worth of ‘b’on the graph and reply the next questions. By setting every bracket equal to zero and fixing, we get the required solutions. By contrast, a parabola of the shape rotates concerning the vertical axis, not the horizontal axis. As the adverse check in entrance of theterm makes flips the parabola about the horizontal axis. If the parabola opens downward, the vertex is a maximum level, and if the parabola opens upward, the vertex is a minimal level. The x-intercept is the point, or factors, the place the parabola crosses the x-axis. There may be 0, 1, or 2 x-intercepts, depending on the parabola. Algebra 1 Unit 7: Quadratic Features As the title may counsel, this worksheet helps the student follow graphing a quadratic equation from each vertex and intercept form. Good for the Algebra 1 or Algebra 2 pupil, or as a refresher for the Geometry scholar. This set of quadratic operate worksheets incorporates workout routines on evaluating quadratic functions for the given x-values. The x-values are integers in the straightforward stage worksheets. Pick some earlier than and after the AOS and plug into your equation. This graph represents the peak of a diver vs. the time after the diver jumps from a springboard. Answer the next questions primarily based on the data. As you accomplish that, discover the parabolas being created by the experimenters. One may be represented with a quadratic perform, and one with a linear operate. Students must graph the heights of the objects over time and reply questions that may lead to important serious about this distinctive system of equations. Math worksheet on quadratic equations will assist the scholars to practice the usual type of quadratic equation. Practice the quadratic equation and discover methods to remedy the quadratic equation. This Algebra 1 – Quadratic Functions Worksheets will produce issues for working towards graphing quadratic perform from their equations. You can select the magnitude of the “a” time period and the path in which the parabola opens. Students can obtain the PDFs of quadratic equations worksheets here. We’re going to research and analyze graphs of quadratic capabilities and then interpret these graphs inside the context of the state of affairs. Analyze graphs of quadratic features and interpret those graphs throughout the context of the state of affairs. The students can both sketch the graphs by hand or use graphing calculators.

But before taking any final decision, I suggest to check the VSEPR structure and then decide as per the diagram. ), The Lewis structure of SF4 is the combination of 34 valence electron and 5 electron pairs around the Sulfur, in which there are four bonding pairs and one lone pair. or greater than it? In the geometry, three atoms are in the same plane with bond angles of 120°; the other two atoms are on opposite ends of the molecule. So, SF4 is polar. How scientists got that number was through experiments, but we don't need to know too much detail because that is not described in the textbook or lecture. The overall shape is described as see-saw. and types of electrons pairs; magnitude of repulsions between them to arrive at The appearance of SF4 is like a colorless gas. Hint: There is now lone pair on P. Question Determine the electron geometry (eg) and molecular geometry (mg) of SiF4. If the charge distribution is symmetric, it is non-polar. So, this was the explanation about SF4. increase in the volume occupied by electron pair(s). In NSF3, there is a triple bond between N and S. Hence the than in other cases. SO3 Molecular Geometry, Lewis Structure, and Polarity Explained, O3 Lewis Structure, Polarity, Hybridization, Shape and Much More, CS2 Lewis Structure, Hybridization, Polarity and Molecular Shape, NH3 Molecular Geometry, Hybridization, Bond Angle and Molecular Shape, PCL3 Molecular Electron Geometry, Lewis Structure, Bond Angles and Hybridization. question, Next question SF4 covers under ‘Trigonal Bipyramidal’ because of its electron arrangements. electrons pairs on atoms connected to central atom. Some elements in Group 15 of the periodic table form compounds of the type AX 5; examples include PCl 5 and AsF 5. It linked by lines i.e. 109 o 28' In POF 3 , there is a double bond between P and O, which also causes more repulsion than single bond, but less than the triple bond. Isn’t drago’s rule being violated in your answer? This electron arrangement is known as ‘Trigonal Bipyramidal.’. After all this process, the last hybrid orbital contains a lone pair. bonds are single bonds, which exert less repulsion on other bond pairs. Thanks a lot of helping out.. As we have discussed, SF4 has one lone pair and four sigma bonds of F. The central atom is S. So in simple terms, we can say that its bonding regions are four with the one lone pair. Preparation. In this structure, Sulfur is the least electronegative element and so transfers in the middle of the structure, and the diagram gives a three-dimensional structural information. But what about drago’s rule. You can also look at its molecular geometry. * First write the Lewis dot structures for the molecules and find the number The reason is that the lone pair prefers one of the equatorial positions. sif4 polar or nonpolar 3 November 2020 by The three-dimensional arrangement of the fragment or atoms which create a molecule by getting together is known as Molecular Geometry. So, this was the explanation about SF4. >. For bent molecular geometry when the electron-pair geometry is tetrahedral the bond angle is around 105 degrees. the bond angle is maximum i.e. the relative bond angles. This will reduce the bond angle more SF4 Molecular Geometry, Lewis Structure, and Polarity – Explained. Molecule polarity gives the acknowledgment regarding the molecule’s solubility, boiling point, etc. For other informative articles, kindly stay connected with geometry of molecules and if you have any other queries, leave a message in the comments section. valence electrons (associated with an atom. Here, there is only one lone pair around the central atom (Sulfur) which is an odd number. The bond angle is least affected in case of SiF 4, since all the Si-F bonds are single bonds, which exert less repulsion on other bond pairs. in triple bond occupies more space, it exerts more repulsion than that of double 4 I hope you got all the answers of what you were looking for! With 2P-orbitals, there are overlapped four of the hybrid orbitals. Using the example above, we would add that H 2 O has a bond angle of 109.5° and CO 2 would have a bond angle of 180°. A) eg=tetrahedral, mg=trigonal pyramidal B) eg=octahedral, mg=square planar C) eg=trigonal bipyramidal, mg=trigonal pyramidal ... Place the following in order of increasing X-Se-X bond angle, where X represents the outer atoms in each molecule. It is also hazardous as it is highly toxic and corrosive. The reason is that the lone pair prefers one of the equatorial positions. Just like this molecule – SF4. SF4 stands for Sulfur tetrafluoride. The equatorial F atoms are 120 from each other., so the axial/equatorial bond angle is … Here, SF4 bond angles are around 102 degrees in the equatorial plane and around 173 degrees between the axial and equatorial positions. Determine the electron geometry (eg) and molecular geometry (mg) of CO32⁻. A) eg=tetrahedral, mg=trigonal pyramidal B) eg=octahedral, mg=square planar C) eg=trigonal bipyramidal, mg=trigonal pyramidal ... Place the following in order of increasing X-Se-X bond angle, where X represents the outer atoms in each molecule. The bond angles of a molecule, together with the bond lengths (Section 8.8), define the shape and size of the mole-cule. Electron-pair Geometry: Molecular Geometry: Bond Angle: 2: 0: linear: linear: 180: 3: 0: … 1) Write the complete Lewis dot structures of above molecules indicating Therefore, tetrahedrals have a bond angle of 109.5 degrees. The bond angle is least affected in case of SiF4, since all the Si-F The axial F atoms are 180 degrees from each other. (VSEPR – Valence Shell Electron Pair Repulsion theory). The number of valence electrons is 34 and 5 electron pairs. paper, < Previous The other explanation goes like this: Two S-F bonds are opposite from each other, in complete 180 degrees. Amazing Explanation!!! Since the electron density Thanks for your article. You will get a reply from the expert as soon as possible. The advantage of this structure is that it shows the chemical connectivity and bonding of all the particles which are associated with atoms and the reactivity of a molecule. Hunting accurate information is among the biggest issues for the younger generation. Here, SF4 bond angles are around 102 degrees in the equatorial plane and around 173 degrees between the axial and equatorial positions. The molecular formula is number and varieties of particles available in the group of atoms. The shape is like a seesaw. If you want to know that the molecule is polar or nonpolar, first of all, you should draw the Lewis structure of the molecule. … Give the approximate bond angle for a molecule with a tetrahedral shape. But the other two S-F bonds are pointing down, and that is why their bond dipoles do not cancel. Chicken Waldorf Salad With Greek Yogurt, The Rock Als Hot Pepper Challenge, Trovita Orange Wikipedia, Two Gases Insoluble In Water, Legion Y740 Specs, Ibanez Grx70qa Tks, Kingfisher School Calendar 2020-2021, Shallow Mount Pioneer Sub, Red Rock Resorts Reopening, Wfdsa Member Companies List,

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Board Paper of Class 12-Science 2011 Chemistry (SET 3) - Solutions (i) All questions are compulsory (ii) Question numbers 1 to 8 are very short-answer questions and carry 1 mark each. (iii) Question numbers 9 to 18 are short-answer questions and carry 2 marks each. (iv) Question numbers 19 to 27 are also short-answer questions and carry 3 marks. (v) Question numbers 28 to 30 are long-answer questions and carry 5 marks each. (vi) Use Log Tables, if necessary. Use of calculators is not allowed. - Question 1 Define ‘activation energy’ of a reaction.VIEW SOLUTION - Question 2 What is meant by ‘reverse osmosis’?VIEW SOLUTION - Question 3 What type of ores can be concentrated by magnetic separation method?VIEW SOLUTION - Question 4 Write the IUPAC name of the following compound: CH2 = CHCH2BrVIEW SOLUTION - Question 5 What is meant by ‘lanthanoid contraction’?VIEW SOLUTION - Question 6 How would you convert ethanol to ethene?VIEW SOLUTION - Question 7 Draw the structure of 4-chloropentan-2-one.VIEW SOLUTION - Question 8 Give a chemical test to distinguish between ethylamine and aniline.VIEW SOLUTION - Question 9 Calculate the packing efficiency of a metal crystal for a simple cubic lattice.VIEW SOLUTION - Question 10 Explain how you can determine the atomic mass of an unknown metal if you know its mass density and the dimensions of unit cell of its crystal.VIEW SOLUTION - Question 11 Differentiate between molarity and molality values for a solution. What is the effect of change in temperature on molarity and molality values?VIEW SOLUTION - Question 12 The thermal decomposition of HCO2H is a first order reaction with a rate constant of 2.4 × 10−3 s−1 at a certain temperature. Calculate how long will it take for three-fourths of initial quantity of HCO2H to decompose. (log 0.25 = − 0.6021)VIEW SOLUTION - Question 13 What do you understand by the rate law and rate constant of a reaction? Identify the order of a reaction if the units of its rate constant are: (i) L−1 mol s−1 (ii) L mol−1 s−1VIEW SOLUTION - Question 14 Describe the principle controlling each of the following processes: (i) Preparation of cast iron form pig iron. (ii) Preparation of pure alumina (Al2O3) from bauxite ore.VIEW SOLUTION - Question 15 Explain giving reasons: (i) Transition metals and their compounds generally exhibit a paramagnetic behaviour. (ii) The chemistry of actinoids is not so smooth as that of lanthanoids.VIEW SOLUTION - Question 16 Complete the following chemical equations: State reasons for the following: (i) Cu (I) ion is not stable in an aqueous solution. (ii) Unlike Cr3+, Mn2+, Fe3+ and the subsequent other M2+ ions of the 3d series of elements, the 4d and the 5d series metals generally do not form stable cationic species.VIEW SOLUTION - Question 17 Write the main structural difference between DNA and RNA. Of the four bases, name those which are common to both DNA and RNA.VIEW SOLUTION - Question 18 Write such reactions and facts about glucose which cannot be explained by its open chain structure.VIEW SOLUTION - Question 19 A solution prepared by dissolving 8.95 mg of a gene fragment in 35.0 mL of water has an osmotic pressure of 0.335 torr at 25°C. Assuming that the gene fragment is a non-electrolyte, calculate its molar mass.VIEW SOLUTION - Question 20 Classify colloids where the dispersion medium is water. State their characteristics and write an example of each of these classes. Explain what is observed when (i) an electric current is passed through a sol (ii) a beam of light is passed through a sol (iii) an electrolyte (say NaCl) is added to ferric hydroxide solVIEW SOLUTION - Question 21 How would you account for the following: (i) NF3 is an exothermic compound but NCl3 is not. (ii) The acidic strength of compounds increases in the order: PH3 < H2S < HCl (iii) SF6 is kinetically inert. - Question 22 Write the state of hybridization, the shape and the magnetic behaviour of the following complex entities: (i) [Cr(NH3)4 Cl2] Cl (ii) [Co(en)3] Cl3 (iii) K2 [Ni(CN)4]VIEW SOLUTION - Question 23 State reasons for the following: (i) pKb value for aniline is more than that for methylamine. (ii) Ethylamine is soluble in water whereas aniline is not soluble in water. (iii) Primary amines have higher boiling points than tertiary amines.VIEW SOLUTION - Question 24 Rearrange the compounds of each of the following sets in order of reactivity towards SN2 displacement: (i) 2-Bromo-2-methylbutane, 1-Bromopentane, 2-Bromopentane (ii) 1-Bromo-3-methylbutane, 2-Bromo-2-methylbutane, 3-Bromo-2-methylbutane (iii) 1-Bromobutane, 1-Bromo-2, 2-dimethylpropane, 1-Bromo-2-methylbutaneVIEW SOLUTION - Question 25 How would you obtain the following: (i) Benzoquinone from phenol (ii) 2-methyl propan-2-ol from methyl-magnesium bromide (iii) Propane-2-ol from propeneVIEW SOLUTION - Question 26 Write the names and structures of the monomers of the following polymers: (iii) NeopreneVIEW SOLUTION - Question 27 What are the following substances? Give one example of each. (i) Food preservatives (ii) Synthetic detergents (iii) AntacidsVIEW SOLUTION - Question 28 (a) Draw the structures of the following molecules: (b) Complete the following chemical equations: (i) HgCl2 + PH3 → (ii) SO3 + H2SO4 → (iii) XeF4 + H2O → (a) What happens when (i) chlorine gas is passed through a hot concentrated solution of NaOH? (ii) sulphur dioxide gas is passed through an aqueous solution of a Fe (III) salt? (b) Answer the following: (i) What is the basicity of H3PO3 and why? (ii) Why does fluorine not play the role of a central atom in inter-halogen compounds? (iii) Why do noble gases have very low boiling points?VIEW SOLUTION - Question 29 (a) What type of a battery is lead storage battery? Write the anode and cathode reactions and the overall cell reaction occurring in the operation of a lead storage battery. (b) Calculate the potential for half-cell containing 0.10 M K2Cr2O7 (aq), 0.20 M Cr3+ (aq) and 1.0 × 10−4 M H+ (aq) The half-cell reaction is and the standard electrode potential is given as E0 = 1.33 V. (a) How many moles of mercury will be produced by electrolysing 1.0 M Hg (NO3)2 solution with a current of 2.00 A for 3 hours? [Hg(NO3)2 = 200.6 g mol−1] (b) A voltaic cell is set up at 25°C with the following half-cells Al3+ (0.001 M) and Ni2+ (0.50 M). Write an equation for the reaction that occurs when the cell generates an electric current and determine the cell potential. - Question 30 (a) Illustrate the following name reactions: (i) Cannizzaro’s reaction (ii) Clemmensen reduction (b) How would you obtain the following: (i) But-2-enal from ethanal (ii) Butanoic acid from butanol (iii) Benzoic acid from ethylbenzene (a) Given chemical tests to distinguish between the following: (i) Benzoic acid and ethyl benzoate (ii) Benzaldehyde and acetophenone (b) Complete each synthesis by giving missing reagents or products in the following:

Individual differences | Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology | A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement. For example, length is a physical quantity. The metre is a unit of length that represents a definite predetermined length. When we say 10 metres (or 10 m) or (1 dekameter), we actually mean 10 times the definite predetermined length called "metre". The definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common. Now there is a global standard, the International System of Units (SI), the modern form of the metric system. In trade, weights and measures is often a subject of governmental regulation, to ensure fairness and transparency. The Bureau international des poids et mesures (BIPM) is tasked with ensuring worldwide uniformity of measurements and their traceability to the International System of Units (SI). Metrology is the science for developing nationally and internationally accepted units of weights and measures. In physics units are standards for measurement of physical quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method. A standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights and measures developed long ago for commercial purposes. Scienceand medicine, often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely. The judicious selection of the units of measurement can aid researchers in problem solving (see, for example, dimensional analysis). - 1 Systems of units - 2 Base and derived units - 3 Calculations with units - 4 Real-world implications - 5 See also - 6 Notes - 7 External links Systems of units[edit | edit source] Traditional systems[edit | edit source] Historically many of the systems of measurement which had been in use were to some extent based on the dimensions of the human body according to the proportions described by Marcus Vitruvius Pollio. As a result, units of measure could vary not only from location to location, but from person to person. Metric systems[edit | edit source] A number of metric systems of units have evolved since the adoption of the original metric system in France in 1791. The current international standard metric system is the International System of Units. An important feature of modern systems is standardization. Each unit has a universally recognized size. Both the Imperial units and US customary units derive from earlier English units. Imperial units were mostly used in the British Commonwealth and the former British Empire. US customary units are still the main system of measurement used in the United States despite Congress having legally authorized metric measure on 28 July 1866. Some steps towards US metrication have been made, particularly the redefinition of basic US units to derive exactly from SI units, so that in the US the inch is now defined as 0.0254 m (exactly), and the avoirdupois pound is now defined as 453.59237 g (exactly) Natural systems[edit | edit source] While the above systems of units are based on arbitrary unit values, formalised as standards, some unit values occur naturally in science. Systems of units based on these are called natural units. Similar to natural units, atomic units (au) are a convenient system of units of measurement used in atomic physics. Legal control of weights and measures[edit | edit source] } To reduce the incidence of retail fraud, many national statutes have standard definitions of weights and measures that may be used (hence "statute measure"), and these are verified by legal officers. Base and derived units[edit | edit source] Different systems of units are based on different choices of a set of fundamental units. The most widely used system of units is the International System of Units, or SI. There are seven SI base units. All other SI units can be derived from these base units. For most quantities a unit is absolutely necessary to communicate values of that physical quantity. For example, conveying to someone a particular length without using some sort of unit is impossible, because a length cannot be described without a reference used to make sense of the value given. But not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Thus only a small set of units is required. These units are taken as the base units. Other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units. Which units are considered base units is a matter of choice. The base units of SI are actually not the smallest set possible. Smaller sets have been defined. For example, there are unit setsTemplate:Which? in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic field are actually different manifestations of the same phenomenon. Calculations with units[edit | edit source] Units as dimensions[edit | edit source] Any value of a physical quantity is expressed as a comparison to a unit of that quantity. For example, the value of a physical quantity Z is expressed as the product of a unit [Z] and a numerical factor: - For example, "2 candlesticks" Z = 2 [candlestick]. The multiplication sign is usually left out, just as it is left out between variables in scientific notation of formulas. The conventions used to express quantities is referred to as quantity calculus. In formulas the unit [Z] can be treated as if it were a specific magnitude of a kind of physical dimension: see dimensional analysis for more on this treatment. Units can only be added or subtracted if they are the same type; however units can always be multiplied or divided, as George Gamow used to explain: - "2 candlesticks" times "3 cabdrivers" = 6 [candlestick][cabdriver]. A distinction should be made between units and standards. A unit is fixed by its definition, and is independent of physical conditions such as temperature. By contrast, a standard is a physical realization of a unit, and realizes that unit only under certain physical conditions. For example, the metre is a unit, while a metal bar is a standard. One metre is the same length regardless of temperature, but a metal bar will be one metre long only at a certain temperature. Guidelines[edit | edit source] - Treat units algebraically. Only add like terms. When a unit is divided by itself, the division yields a unitless one. When two different units are multiplied, the result is a new unit, referred to by the combination of the units. For instance, in SI, the unit of speed is metres per second (m/s). See dimensional analysis. A unit can be multiplied by itself, creating a unit with an exponent (e.g. m2/s2). Put simply, units obey the laws of indices. (See Exponentiation.) - Some units have special names, however these should be treated like their equivalents. For example, one newton (N) is equivalent to one kg·m/s2. Thus a quantity may have several unit designations, for example: the unit for surface tension can be referred to as either N/m (newtons per metre) or kg/s2 (kilograms per second squared). Whether these designations are equivalent is disputed amongst metrologists. Expressing a physical value in terms of another unit[edit | edit source] Conversion of units involves comparison of different standard physical values, either of a single physical quantity or of a physical quantity and a combination of other physical quantities. just replace the original unit with its meaning in terms of the desired unit , e.g. if , then: Now and are both numerical values, so just calculate their product. Or, which is just mathematically the same thing, multiply Z by unity, the product is still Z: For example, you have an expression for a physical value Z involving the unit feet per second () and you want it in terms of the unit miles per hour (): - Find facts relating the original unit to the desired unit: - 1 mile = 5280 feet and 1 hour = 3600 seconds (3.6 kiloseconds) - Next use the above equations to construct a fraction that has a value of unity and that contains units such that, when it is multiplied with the original physical value, will cancel the original units: - Last,multiply the original expression of the physical value by the fraction, called a conversion factor, to obtain the same physical value expressed in terms of a different unit. Note: since valid conversion factors are dimensionless and have a numerical value of one, multiplying any physical quantity by such a conversion factor (which is 1) does not change that physical quantity. Or as an example using the metric system, you have a value of fuel economy in the unit litres per 100 kilometres and you want it in terms of the unit microlitres per metre: Real-world implications[edit | edit source] One example of the importance of agreed units is the failure of the NASA Mars Climate Orbiter, which was accidentally destroyed on a mission to Mars in September 1999 instead of entering orbit, due to miscommunications about the value of forces: different computer programs used different units of measurement (newton versus pound force). Considerable amounts of effort, time, and money were wasted. On April 15, 1999 Korean Air cargo flight 6316 from Shanghai to Seoul was lost due to the crew confusing tower instructions (in metres) and altimeter readings (in feet). Three crew and five people on the ground were killed. Thirty seven were injured. In 1983, a Boeing 767 (which came to be known as the Gimli Glider) ran out of fuel in mid-flight because of two mistakes in figuring the fuel supply of Air Canada's first aircraft to use metric measurements. This accident is apparently the result of confusion both due to the simultaneous use of metric & Imperial measures as well as mass & volume measures. See also[edit | edit source] Notes[edit | edit source] - US Metric Act of 1866. as amended by Public Law 110–69 dated August 9, 2007 - (2002). NIST Handbook 44 Appendix B. National Institute of Standards and Technology. - Emerson, W.H. (2008). On quantity calculus and units of measurement. Metrologia 45 (2): 134–138. - Unit Mixups. US Metric Association. - Mars Climate Orbiter Mishap Investigation Board Phase I Report. NASA. - NTSB. Korean Air Flight 6316. Press release. - Korean Air incident. Aviation Safety Net. - includeonly>Witkin, Richard. "Jet's Fuel Ran Out After Metric Conversion Errors", New York Times, July 30, 1983. Retrieved on 2007-08-21. “Air Canada said yesterday that its Boeing 767 jet ran out of fuel in mid-flight last week because of two mistakes in figuring the fuel supply of the airline's first aircraft to use metric measurements. After both engines lost their power, the pilots made what is now thought to be the first successful emergency dead stick landing of a commercial jetliner.” [edit | edit source] - A Dictionary of Units of Measurement - Center for Mathematics and Science Education, University of North Carolina - NIST Handbook 44, Specifications, Tolerances, and Other Technical Requirements for Weighing and Measuring Devices - NIST Handbook 44, Appendix C, General Tables of Units of Measurement - Official SI website - Quantity System Framework - Quantity System Library and Calculator for Units Conversions and Quantities predictions Legal[edit | edit source] Metric information and associations[edit | edit source] - Official SI website - UK Metric Association - US Metric Association - The Unified Code for Units of Measure (UCUM) Imperial measure information[edit | edit source] |This page uses Creative Commons Licensed content from Wikipedia (view authors).|

NCERT Class 10 Maths Chapter 6: Complete Resource for Triangles The benefits of using NCERT Solutions for Class 10 Maths Triangles PDF is profound. The PDF of Class 10 Maths Chapter 6 NCERT Solutions has been prepared by expert mathematicians at Vedantu after thorough research on the subject matter. All the solutions provided here are written in a simple and lucid manner. With the aid of these NCERT Solutions for Class 10 Chapter 6 of Maths, students can not only improve their knowledge but also aspire to score better in their examinations. What is even better is that you can now download these NCERT Solutions for Class 10 Chapter Triangles PDF for free. The PDF will allow you to refer to these solutions as per your need and convenience. Download NCERT Solution PDF today to have easy access to all subject solutions for free which also includes Class 10 Science NCERT Solutions. Exercises under NCERT Solutions for Class 10 Maths Chapter 6 Triangles NCERT Solutions for Class 10 Maths Chapter 6, "Triangles," is a chapter that deals with the properties and classification of triangles. The chapter contains six exercises, each covering a different aspect of the topic. Below is a brief explanation of each exercise: Exercise 6.1: In this exercise, you will be introduced to the basic concepts of triangles, including the definition, elements, types, and angles. You will also learn about congruent triangles and the criteria for their congruence. Exercise 6.2: This exercise focuses on the properties of triangles, such as the angle sum property, the exterior angle property, and the inequality theorem. You will also learn about the Pythagorean theorem and its applications. Exercise 6.3: In this exercise, you will learn about the similarity of triangles, including the criteria for similarity, the theorem of basic proportionality, and the application of similarity in practical situations. Exercise 6.4: This exercise covers the mid-point theorem, which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length. You will also learn about the converse of this theorem. Exercise 6.5: In this exercise, you will learn about the altitude and median of a triangle and their properties. You will also learn about the centroid and the orthocenter of a triangle. Exercise 6.6: This exercise covers the concept of the circumcenter and incenter of a triangle and their properties. You will also learn about the construction of circumcenter and incenter using various methods. NCERT Maths Class 10 Chapter 6 - Free PDF Download You can opt for Chapter 6 - Triangles NCERT Solutions for Class 10 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below. NCERT Solutions for Class 10 Maths Other Chapter Solutions PDF Download NCERT Solutions for Class 10 Maths Chapter 6 Triangles Details Given below are the details of the various sub-topics included in the Class 10 Chapter 6 Triangles NCERT Solutions: NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.1 Introduction The PDF of Class 10 Maths Triangles recalls students’ knowledge in this introduction part. Students were already introduced to the concept of Triangles in Class 9 wherein they studied properties such as congruence of Triangles. The introduction part of the chapter basically acts as a window for the students so that they are able to get an insight as to what would they be learning new under the topic of Triangles. NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.2 Similar Figures In this section of the Class 10 Maths Chapter 6, students are introduced to the concept of similar figures. Students are taught the basis of similarity in figures such as squares or equilateral triangles with the same lengths of the sides, circles with the same radii. As the students progress through this topic, they get to understand that similar figures can have the same shape but not necessarily the exact size. The questions from this topic mostly ask students to prove similarity between figures by applying the theorems. NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.3 Similarity of Triangles Once the students are made familiar with the concept of similarity, they are then introduced to the criteria under which two or more triangles are deemed similar. The NCERT Solutions for Class 10 Maths Chapter 6 PDF, in this section, explains the theorem of Basic Proportionality. A thorough understanding of this topic will allow students to form the base for solving complex problems in higher mathematics. NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.4 Criteria for Similarity of Triangles This section outlines and explains the criteria for the similarity of triangles. The basic criteria for two triangles to be called similar include: if their corresponding angles are equal and if the corresponding sides of the triangles are in the same ratio (or proportion). Students will be able to visualise the theorems as they are illustrated with the help of proper examples. NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.5 Areas of Similar Triangles Students can understand the formula and learn the process for finding the surface area of similar triangles in this section. Maths NCERT Class 10 Chapter 6 allows students to find the area of similar triangles with the utilisation of the different theorems. NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.6 Pythagoras Theorem The NCERT Solutions Class 10 Chapter 6 explores the use of the Pythagoras theorem in the case of similar triangles. Students have already learnt the theorem and its proof in Class 9. In this section, students will learn how to prove this theorem by employing the concept of similarity of triangles. NCERT Solutions for Class 10 Maths Chapter 6 Triangles: 6.7 Summary The summary comprises all the topics that you have studied in the chapter. Going through the summary will allow you to recollect all that you have learnt in the chapter including the important concepts, theorems, etc. Points to Remember 1. A triangle is a polygon with three angles and three sides. A triangle's interior angles add up to 180 degrees, whereas its exterior angles add up to 360 degrees. 2. A triangle can be classified into the following types based on its angle and sides. Scalene Triangle: All the three sides of this triangle have different measures. Isosceles Triangle: Any two sides of this triangle have equal length. Equilateral Triangle: All the three sides of this triangle are equal and each angle measures 60 degrees. Acute Angled Triangle: All the angles measure less than 90 degrees. Right Angle Triangle: Any one of the 3 angles is equal to 90 degrees. Obtuse-Angled Triangle: One of the angles is greater than 90 degrees. 3. Centroid of a Triangle The centroid of a triangle is the point where the medians of its three sides intersect. It will always be within the triangle. 4. Incenter of a Triangle The incenter of a triangle is defined as the point where the angle bisectors of the three angles intersect. It is the point in the triangle where the circle is inscribed. Drawing a perpendicular from the incenter to any of the triangle's sides gives the radius. 5. Circumcenter of the Triangle The circumcenter of a triangle is defined as the point where the perpendicular bisectors of its three sides intersect. It isn't necessarily located inside the triangle. For an obtuse triangle, it might be outside the triangle, but for a right-angled triangle, it could be at the midpoint of the hypotenuse. The orthocenter of a triangle is the point where the altitudes of the triangle intersect. It also falls outside the triangle in the case of an obtuse triangle and at the vertex of the triangle in the case of a right-angle triangle, just like the circumcenter. 7. Similarity of Triangles In triangles, we'll use the same condition that two triangles are similar if their respective angles are the same and their corresponding sides are proportionate. 8. Basic Proportionality Theorem According to Thales theorem, if a line is drawn parallel to any of the triangle's sides so that the other two sides intersect at a distinct point, the two sides are divided in the same ratio. 9. Converse of Basic Proportionality Theorem It is the inverse of the basic proportionality theorem, which states that if a straight line divides the two sides of a triangle in the same ratio, that straight line is parallel to the triangle's third side. Similarity Criteria of Triangles There are four criteria for determining if two triangles are similar or not. They are as follows: Side-Side- Side (SSS) Criterion Angle Angle Angle (AAA) Criterion Angle-Angle (AA) Criterion Side-Angle-Side (SAS) Criterion NCERT Solutions for Class 10 Maths Chapter 6 All Exercises Vedantu's NCERT Solutions for Class 10 Maths Chapter 6 - Triangles provide a comprehensive and accessible resource for students to grasp the intricacies of triangle geometry. With a diverse range of well-structured exercises and step-by-step explanations, these solutions promote a deeper understanding of key concepts. By incorporating real-life applications, students can appreciate the relevance of triangles in everyday scenarios. Vedantu's expertly crafted solutions foster self-confidence in solving complex problems, bolstering students' problem-solving abilities. The user-friendly platform encourages interactive learning, making the study process engaging and enjoyable. As a reliable aid, Vedantu's NCERT Solutions for Class 10 Maths Chapter 6 empower students to excel in their academics and develop a strong foundation in geometry. FAQs on NCERT Solutions for Class 10 Maths Chapter 6 - Triangles 1. How Many Exercises are There in NCERT Solutions for Class 10 Maths Chapter 6 Triangles? The Class 10 Maths NCERT Solutions for the Chapter Triangles contain exercises corresponding to each topic. The chapter contains a total of 6 exercises with a total of 65 questions. The questions include a mix of long and short type questions. Students should attempt to understand all the concepts and theorems given in the chapter and then solve the questions in the exercises. Solving these questions will definitely give the students a competitive edge in the exams. 2. How Many Marks are Allotted to the Class 10 Maths Chapter 6 Triangles in the Board Exam? The Class 10 Maths Chapter 6 Triangles is a part of a broader unit ‘Geometry’ in the Board exams. The unit of Geometry comprises a total of 15 marks in the Board exams. The Triangles chapter is an important chapter as per the examination point of view and as such is likely to carry around 5-6 marks in the Class 10 Board exams. 3. Which are the Important Topics to Remember Present in CBSE Class 10 Chapter 6 Triangles? In the CBSE Class 10 Maths Chapter 6, the topic discussed is Triangles. The topics that are important from this chapter are: Similarity theorems of triangles. Criteria for triangle similarity. Area calculation of similar triangles. Pythagoras theorem and the concept of similar triangles. Students should make sure that they are thorough with all these topics and should leave no stone unturned to practise as many questions as possible while preparing this topic for the exams. 4. Can the PDF of NCERT Solutions for Class 10 Maths Chapter 6 Triangles be Downloaded for Free? Yes, at Vedantu you can download the NCERT Solutions for Class 10 Maths Chapter 6 Triangles PDF for absolutely free of cost. The solutions of this chapter have been compiled by some of the best subject experts and provide a clear insight into the various concepts included in the chapter. To download the PDF of the Class 10 Maths Chapter 6, you will just be required to click on the link provided on this page. You can also choose to take a print out of the PDF and keep it handy for revision purposes. You can also download the Vedantu app n your phone from where you will be able to access the top-notch study material for your Class 10 exam preparation at one go. 5. Do I need to practice all the questions given in the NCERT Solutions Class 10 Maths Triangles? It is a good idea to practice every question given in the NCERT Solutions for the Class 10 Maths chapter on Triangles. This way you will understand all the topics and concepts clearly and solve all the problems easily. You will also gain confidence about the exam with increased speed and accuracy because the NCERT Solutions provided by Vedantu are curated by subject matter experts. These solutions are therefore guaranteed to help you to clear your concepts easily and effectively for your exam. 6. What are the important topics covered in Class 10 Maths NCERT Solutions Chapter 6? Chapter 6 of the Class 10 Maths NCERT book deals with Triangles. The most important topics that are covered in this chapter are: Definition of a triangle Similarity of two polygons with an equal number of edges Similarity of triangles Proving the Pythagorean Theorem The concepts of Class 10 Maths Chapter 6 may be a bit tricky to understand. Therefore it's a good idea to download and study the NCERT solutions for Class 10 Maths. These solutions are prepared by subject matter experts with decades of experience and will help you to understand all concepts thoroughly and easily. 7. How can I score the best in Class 10 Maths Chapter 6 Triangles? The following points will help you to score well and get to the best of your potential in Class 10 Maths Chapter 6 Triangles: Understand the concept of this chapter. Refer to extra materials like NCERT Solutions of Class 10 Maths Chapter 6 Triangles available at free of cost on the Vedantu app and on the Vedantu website. Solve model papers. Maintain a separate notebook for formulas and theorems. Practice graphs and diagrams. If you want to score better than all your peers, then your best shot is definitely to download the NCERT Solutions for Class 10 Maths by Vedantu. Vedantu’s NCERT solutions are prepared by the best Maths teachers in India and written in easy to understand language. 8. What are the most important theorems that come in Class 10 Chapter 6 Triangles? The most important theorems in class 10 Chapter 6 Triangles are: Angle Bisector Theorem Inscribed Angle Theorem To clearly understand the major theorems included in the Class 10 Chapter 6 Triangles, it's best to download the NCERT solutions for Class 10 Maths. These solutions will help you in learning advanced theorems like the ones present in this chapter. This way you can be sure that you will be able to score well in your Class 10 board exams as well.

DNC Deputy Chair Credibly Accused of Domestic Abuse Minnesota Democratic Representative and Deputy Chair of the DNC Keith Ellison has vowed to “keep fighting” in the wake of allegations that he physically and emotionally abused an ex-girlfriend. Ellison, who is stepping down from his congressional seat at the end of this term, on Tuesday won the Minnesota Democratic primary for Attorney General, just days after the allegations broke. The left, who traditionally state that victims’ statements must always be taken at face value, is split over the abuse allegations against Ellison with some doubting the veracity of the claims. The accusations against Ellison stem from a lengthy Saturday night Facebook post by the son of Ellison’s ex-girlfriend, environmental activist Karen Monahan. In the post, Monahan’s son described a video of Ellison abusing his mother, dragging her off a bed by her feet and calling her a “f***ing b**ch,” as well as a series of allegedly abusive text and twitter conversations. In a Sunday tweet Karen Monahan, whose long-term relationship with Ellison ended in 2016, confirmed her son’s statements saying, “What my son said is true. Every statement he made was true.” Despite Monahan’s statement, she has not released the recording, which Ellison says could not possibly exist as he has never abused an ex-girlfriend. In an interview with Minnesota public radio, Monahan said that she does not plan to release the recording as, “It’s humiliating, it’s traumatizing, for everyone’s family involved, and for me.” She has, however, released a batch of 100 text and Twitter messages between her and Ellison. However, according to reporter Briana Bierschbach, “there is no evidence in the messages reviewed by MPR News of the alleged physical abuse.” There are also no police reports or court documents relating to the alleged abuse. These new allegations against Ellison seem to support earlier allegations made by Left-wing activist Amy Alexander who claimed in 2006 that Ellison abused her. Writing at the time, Alexander said, "His anger kicked in. He berated me. He grabbed me and pushed me out of the way. I was terrified. I called the police.” Independent journalist Laura Loomer recently published evidence of a 2005 9-1-1 call regarding the abuse of Alexander. Despite the allegations against him, Ellison handily won the Democratic nomination for Minnesota Attorney General with 50% of the vote. His closest competitor in the crowded field, Debra Hilstrom, only garnered 19%. While the allegations gave many voters pause, most determined that there was a lack of credible evidence and that the timing of the allegations was too suspect. Some voters cited last year’s resignation of Minnesota Senator Al Franken as a motivating factor in their decision to continue to support Ellison. The Democratic candidate is also benefiting from institutional support in the face of the allegations. The DNC did not make a statement on the issue until late on Tuesday, hours before voting in the Minnesota primary ended. The DNC statement reads, “These allegations recently came to light and we are reviewing them. All allegations of domestic abuse are disturbing and should be taken seriously.” Former DNC communications director Luis Miranda was sterner on the matter telling NPR, “The party has no choice but to suspend him at a minimum until they figure out what’s going on.” DNC Chairman Tom Perez said Wednesday that while he takes the allegations “very seriously,” he does not believe that they threaten the party as it seeks to capitalize on opposition to President Trump. It is unlikely that the allegations against Ellison will pose a serious political threat to him in heavily Democratic Minnesota unless a criminal investigation uncovers further evidence of abuse. The lackluster response of party officials and the mainstream media demonstrates the double standard that exists regarding sexual misconduct on the part of the left. It is interesting that those who are so quick to convict others in the court of public opinion, without evidence, are so hesitant to accuse one of their own without further investigation. This just goes to show that so much of the social justice posturing that we see is not based in principle, but rather on a political tribalism that threatens to destabilize the Republic. 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Thank you for the opportunity to help you with your question! Calculation for GDP.... For year 2000...... For rum, cost of 5000 bottlles=15x5000=7500dollars Cost of 10000 bags of rice=10,000x200=2000,000dollars Cost of 800 bags of beans=250x8000=1400,000dollars Total GDP for year 2000=7500+2000,000+1400,000=3475,000dollars Now for year 2014...... Cost for 7500 bottles of rum=25x7500=187500dollar Cost of 25,000bags of rice=25,000x350=8750,000dollar Cost of 20,000 bags of beans=400x20,000=8000000dollar So GDP for 2014=187500+8750000+8000000=16937500dollarPlease let me know if you need any clarification. I'm always happy to answer your questions. Real GDP accounts for the price changes that may occur due to inflation.If prices change from one period to the next but actual output does not change,NGDP will also change even though output remains same.To adjust for price change,RGDP is calculated using prices from specific year , the base year. In this case base year is year 2000.so for year 2000 the NGDP and RGDP are same. For year 2014 ,RGDP and N GDP are different.We can calculate the cost of all items separately , using the cost price for each item for year 2000, though quantity will be for year 2014. And you can calculate in similar way and find RGDP for year 2014. I think this will help you. GDP deflator for year 2000 will be same as NGDP because our base year is year 2000 For year 2014, we can calculate GDP deflator.... Cost of 7500 bottles of rum=15x7500=112,500dollars Cost of 25,000 of bags of rice=25,000x200=5000,000dollars Cost of 20,000 bags of beans=20,000x250=5000,000dollars Total cost=GDP deflator for year2014=112,5000+5000,000+5000,000=10112,500dollars When we calculate GDP deflator, we take the cost price for the base year and quantity for the year for which we are calculating GDP deflator , here this is year 2014. CPI, consumer price index..... This is the estimate of the price level of consumer goods and services in an economy. A CPI takes certain basket of common goods and services and tracks the changes in the prices of that basket of goods over time. Calculation for CPI..... Step(1)....select a base year, say year 2000 Step(2)...select a basket of goods, say rum, beans bags and rice bags and add the prices of all goods in that year. Step(3)....select the year you want to calculate CPI, here year 2014 , and add all the prices of all goods in that basket for that year , 2014 Step(4)....calculate CPI as... CPI==price of goods in year 2014/price of goods in year2000x100 So inflation =487-100=387 percent I think now you can understand. Suppose that the closed economy of an island H is described by the following equations: GDP (Y) = 10000, government expenditures (G)= 600, (Taxes (T) = 2000, Consumption (C) =400 + 3/4 (Y-T), and investment (I) = 200 -1400 r 1. Private Sving 2. Public Saving 4. Equilibrium interest rate 5. What can you you conclude about the economy of island H? Hi, before attempting this I want to clear one thing , will I be paid for the correct answer? Also what is r in this question, it is not clear to me, waiting for your reply, thanks Ok, I can try... In closed economy, if Y is the national income(GDP), C is the consumption, I is the investment, and G is the government purchases, then.... National saving can be thought as the amount of remaining money that is not consumed or spent by government.In simple model of closed economy, anything which is not used is to be invested, so National saving should be split into private saving and public saving.A new term T is tax payed by consumers that goes directly to the government as shown here.... Y-C-G+T-T=I We add T and subtract T , so eq does not change Now (Y-T-C) is called private saving (T-G) is called the public savings , also the government revenue minus government expenditure.This is also known as Budget surplus. Government expenditure G=600 Now private saving=Y-T-C=10,000-2000-6400=10,000-8400=1600 National Saving=Private saving +Public saving=1600+1400=3000=I Now r is the equilibrium interest rate, we can calculate it as follows... Substituting the value of I, we get... r=-2800/1400=-2 Is same as -200percent The negative value of r is not a interesting figure, I am wondering whether eq I=200-1400r is written correct, pl check it again. About economy of the island H, I can say that when revenue of government is more than expenditure, which is public saving is 1400 , so there is Budget Surplus, this the situation when income exceeds the expenditure, it means government is being run efficiently, economy of the island is good, a Budget surplus might be used to pay off debts, for the development of the island. I think this will help you, regarding payments, whatever you feel ok, you can pay ,I will appreciate that. Its a spreadsheet assignment that needs to uplpoaded Its about 22 questions on what functions to use to fill out a worksheet/ spreadsheet are you familiar with spreadsheet assignments or should i ask another tutor Content will be erased after question is completed.

Cost/Benefit AnalysisReference: System Analysis and Design (Chapter 8) By Elias M. Awad Data Analysis The system requirements are: 1. Better customer service. 2. Faster information retrieval. 3. Quicker notice preparation. 4. Better billing accuracy. 5. Lower processing and operating cost. 6. Improve staff efficiency. 7. Consistent billing procedure to eliminate error. Data Analysis Several alternative must be evaluated. The approach can introduction of computer billing system, change in operation procedure, replacement of staff, improve billing system or combination of this approach. Cost and benefit categories In developing cost estimate for a system, we need to consider the following cost elements: 1. Hardware 2. Personals 3. Facility cost 4. Operating cost 5. Supply Procedure for cost/benefit determination Cost and benefit analysis procedure that gives a picture of various cost, benefits, and rules associated with a system Determination of cost and benefits uses the following steps: 1. Identifying the cost and benefit pertaining to a given project. 2. Categorize the various costs and benefits for analysis. 3. Select a method for evaluation. 4. Interpreted the result of analysis. 5. Take action. Procedure for Cost/Benefit Determination Cost and benefit identification: Certain cost and benefits easily identify than other. Example:- Direct cost. Categories of cost or benefits that is not easily identifiable is opportunity costs and benefit. Classification of Cost and benefits The next step is to categorize cost and benefits 1. Tangible or Intangible costs and benefits: Tangibility refers to the easy with which cost or benefit can be measure. Expenditure of cash for specific item or activity is known as tangible cost. Cost that are known to exist but whose financial value cannot be accurately measured are known as intangible cost. Classification of Cost and benefits1. Tangible or Intangible costs and benefits: Benefits are also classified as tangible on intangible. Management often ignore intangibles this may lead to Classification of Cost and benefits2. Direct or Indirect Cost and Benefits: Direct cost are those with which a dollar figure can be directly associated in the project. Direct benefits also can be specifically attributable to a given project. Indirect cost are the results of operations that are not directly associated with a given system or activity. Indirect benefits are realized as a bi product of another activity or system. Classification of Cost and benefits3. Fixed or Variable: Fixed cost are constant they do not change. Variable cost incurred on regular basis and they are proportional to work volume. Fixed benefits are constant they do not change. Variable benefit realized on regular basis. Saving versus Cost Advantage Saving are realized when there is some kind of cost advantage. Cost advantage reduce or eliminates expenditure. True saving reduce or eliminates various cost being incurred. There are also saving that do not directly reduce the Select Evaluation Method The common evaluation methods are: 1. Net benefit analysis. 2. Present value analysis. 3. Net present value. 4. Payback analysis. 5. Break even analysis. 6. Cash-flow 1. Net benefit analysis Net benefit= (Total benefit)- (Total cost) Advantage: Easy to calculate, easy to interpret and easy to present. Disadvantage: It does not account for time value of money. Time value of money is express as: 2. Present value Analysis Present value analysis calculate the cost and benefits of the system in terms of today's value of the Numerical based on Present value AnalysisQ. Suppose that $3,000 is to be invested in a project, and the expected annual benefit is $1,500 for four year life of the system. Determine the expected profit or loss. 3. Net Present Value Net Present value: The net present value is express as percentage of investment. This approach is relatively easy to calculate and accounts for the time value of money. 4. Payback Analysis It tells you how long it will take to earn back the money you'll spend on the project. The shorter the payback period, sooner a profit is realized and more attractive is the 4. Payback Analysis Payback formula: Elements of the formula:A. B. C. D. E. F. G. H. Capital investment(development cost). Investment credit difference(tax incentive). Cost investment(site preparation). Company federal income tax bracket difference. State and local tax. Life of capital. Time to install the system. Benefit 4. Payback Analysis Elements of the formula:1. 2. 3. 4. 5. Project benefits(H). Depreciation (A/F) State and local tax (A X E). Benefit from Federal Income Tax (FIT): 1 2 3 = 4 Benefits after FIT: 4 (4 X D) Payback Analysis Numerical ExampleA. B. C. D. E. F. G. H. Capital investment=$200,000 Investment credit difference(100%-8%)=92%. Cost investment=$25000. Company federal income tax bracket difference (100%-46%)=54% State and local tax= 2%. Life of capital= 5 Years. Time to install the system = 1 years. Benefit and saving = $250,000. 5. Break even Analysis Break even is the point where the cost of the candidate system and that of the current one are equal. When a candidate system is developed, initial costs usually exceed those of the current system. This is the investment period. When both are equal its break point. Beyond break point the candidate system provide more benefit than the old one-return point. 6. Cash flow Analysis Cash flow analysis keep track of accumulated cost and revenues on a regular basis. The spread sheet format also provide payback and break point information. Cash flow Analysis an ExampleReven Jan ue Feb Mar Apr May June July Aug Sep Oct Nov Dec Reven 22000 22000 26000 27100 41000 48000 59050 59010 66450 64040 69700 71040 ue Expen 51175 34795 27805 27055 28445 28385 29640 29925 28030 30075 30015 30906 se Cash flow Accu mulat ed cash flow -1805 29175 12795 45 12555 19615 29410 30085 22420 33965 39685 40134 29175 41970 43775 43730 31180 11565 17845 47930 70350 10431 14400 18413 5 0 4

Rethinking shots: generation and conversion (30 Jan 2019) The source code for this model is available on https://github.com/huffyhenry/shot-generation. Football analytics today is dominated by analysis of shots, but only at the level of conversion. Relatively little attention has been paid to how shots arise, despite it being arguably the more important question. I call this process shot generation and its analysis is the main contribution of this post, although I also present a simple conversion model for completeness and symmetry. My motivation, as always, is to try to understand the underlying dynamics of football without compromising on statistical rigour. In the early days of (public) analytics, Sander IJtsma and Ben Pugsley investigated how the in-game goal difference (dubbed “game state”) impacts shot ratios. To the best of my knowledge, the first actual model of shooting frequencies is due to Garry Gelade and it was the highlight of the 2015 OptaPro Forum. The new model that I present here is related to Garry’s and it recovers many of his results, but it also yields findings that I believe are novel. Specifically, I was able to show that the home field advantage is mediated by the quantity rather than the quality of shots, that shooting rates differ between not just game states but individual scorelines and that the longer the game goes on without a shot, the more likely the players are to shoot. In addition, the framework presented here has direct applications to performance evaluation and prediction of results, which I hope to discuss in follow-up articles. My generation model draws heavily on the classic Dixon-Coles and RobertsonRobinson papers and uses techniques of survival analysis (for a relatively gentle introduction see Chapter 1 of J.F. Lawless’ “Statistical Models and Methods for Lifetime Data”, which at the time of writing is easily obtainable in PDF form, as are the two papers). Briefly, I assume that every team is characterised by parameters $\alpha$ and $\beta$, representing its baseline ability to generate and prevent shots, respectively. Then, if team A plays team B and $w$ minutes have elapsed since either of them last shot (or the start of the half), then the propensity $\lambda_A$ of team A taking the next shot right now is given by In addition to the parameters already explained above, $\gamma$ is the home advantage parameter (if appropriate, i.e. if A is the home team) and $\theta$ is the sum of additional coefficients which I don’t enumerate in full here, but which include in particular dummy variables for each individual score from 0:0 to 2:2. Lastly, $k$ controls the dependence of the rate of shooting on the time elapsed since the last shot. If $k=1$, there is no such dependence, since $w$ disappears from the equation. As to conversion, it is possible to just use an expected goals model here, but they are data hungry and unnecessarily complex for the task at hand. Instead, I ran a simple logistic regression of the form with $\alpha_A$ this time standing for the shooting team’s intrinsic ability to convert their shots and $\beta_B$ for the defending team’s ability to prevent shots from going in, respectively. I fitted both models on the dataset of all shots from the 2017/18 Premier League. This is a considerably smaller set than what I have available, but I didn’t want to concern myself with inter-league differences in styles nor with inter-season variation of team strengths; as a friend of mine is fond of saying, it requires multilevel modelling and a lot of care, and who has the time for that? I stripped rebounds (defined as shots coming within 5 seconds of previous shot), because they are not generated independently of other shots – I’m going to deal with them at a later stage. This process left me with 9439 records. I coded and fitted my models in Stan. The scatter plots on Figure 1 below show the fitted team profiles, that is to say the $\alpha$ and $\beta$ coefficients from both models, translated into shot waiting times and expected conversions for interpretability. What stands out to me is how Manchester City were a class apart on both sides of the ball, and how terrible Arsenal’s defence was – on par with Huddersfield’s. The conversion model recovers neatly Burnley’s and Mourinho+DeGea’s ability to prevent the ball from going in. Of greater interest, however, are the estimates of the remaining, team-independent parameters. The current score has a large effect on the shooting rate, and while the different exact winning (and drawing, and losing) scores are not statistically different from each other, the sorting is so perfect that I have no doubt that the estimates would separate with a bigger sample (Figure 2); in contrast, there was no discernible effect of scores on conversion probability (not shown). The home advantage parameter $\gamma$ is positive in the generation model, but zero or even very slightly negative in the conversion model. Lastly, in the generation model, the exponent $k$ fitted tightly to about 1.16 meaning that the longer the teams go without shooting, the more likely they become to take a shot, while the corresponding parameter in the conversion model was slightly negative, although not statistically different from zero. By modeling shot generation and shot conversion with separate but parallel models, we learnt the following: - Current score has a large effect on shooting rate but very small or no effect on conversion probability - Home field advantage is realised by the quantity and not quality of shots - The longer the game goes on without a shot, the more likely the teams are to shoot, and the worse shots they take I hope that these results, if not all of them entirely novel, are going to spark renewed interest in foundational modelling of football. My friends James Yorke, Ben Torvaney, Nikos Overheul and Thom Lawrence commented on preliminary results, read an earlier version of the article and provided useful, largely ignored feedback. The data used in this analysis was collected by Opta.

Measuring the resistivity of a wire student worksheet in this experiment you will be making measurements of voltage and with the physical dimensions of the wire, from which you will calculate the resistivity of the metal theory the resistance r of a component in a circuit is given by the. Resistance, req, is: 1 r eq = 1 r 1 + 1 r 2 + 1 r 3 ++ 1 r n = 1 i=1 r i n (eq 11-2: resistors in parallel) fig 11-4: series circuit schematic r 3 r 1 r 2 a i physics 215 - experiment 11 series and parallel circuits 46 10v increments, up to 60v, measure and record the voltage and. This handout will explain the functions of conclusions, offer strategies for writing effective ones, help you evaluate drafts, and suggest what to avoid. How organizations manage resistance to change print reference this published: 23rd march, 2015 include the ability to be a clear thinker who is able to get a view about organizational situation and reach at logical conclusions. Series and parallel resistive circuits physics lab viii objective in the set of experiments, the theoretical expressions used to calculate the total resistance. Antibiotics have always been considered one of the wonder discoveries of the 20th century this is true, but the real wonder is the rise of antibiotic resistance in hospitals, communities, and the environment concomitant with their use the extraordinary genetic capacities of microbes have. Physics 1021 2 experiment 6 vector nature of magnetic fields introduction in this experiment, we determine the magnetic field of the lab by observing the magnetic field produced by a current carrying wire as a function of. Physics 111 laboratory experiment #3 current, voltage and resistance in series and parallel circuits this experiment is designed to investigate the relationship between current and potential in simple series and parallel resistor circuits using ideas of conservation of energy and conservation of. Experiment 3 temperature dependence of resistance where rt is the resistance at the temperature t and ro is the resistance at the temperature to make conclusion about your findings reminder: check you math, check the units. Category: essays research papers title: simple voltage and current measurement my account simple voltage and current measurement length: 886 words resistance is the capacity of materials to impede the flow of current or electric charge 8-10 conclusion. Home knowledge center read ift publications science papers expert reports antimicrobial resistance: implications for the food system conclusions conclusions. Voltagecurrentresistance lab a complete lab write‐up includes a title, a purpose, a data section, a conclusion and a discussion of results the data. Experiment 4: ohm's law and rc circuits objectives 1 2 to investigate ohm's law and to determine the resistance of a resistor 3 to measure the time constants associated with a discharging and charging rc (resistive-capacitive, or resistor-capacitor) circuit. Ee 442 laboratory experiment 2 introduction to the measurement of voltage, current, resistance and voltmeter loading ee 442 lab experiment no 2. Voltage, current, resistance, and ohm's law resistance is a material's tendency to resist the flow of charge (current) so, when we talk about these values, we're really describing the movement of charge, and thus, the behavior of electrons. Ap physics resistance and resistivity lab purpose draw conclusions about resistance based on your data, tables, and graphs: (a) how does resistance appear to depend on length (b) how does resistance depend on diameter. Lab # 1 - ohm's law introduction conclusion ohm's law states that v=ri where v is the difference of potential at the poles of the element (measured in volts), r is the resistance of the element being tested (a resistor in this case. The purpose of this experiment is to investigate the relationship between voltage, resistance, and current as described by ohm's law the dc analysis of a series resistance circuit should support ohm's law and the formula for total resistance in a series circuit. A thermistor is a type of resistor whose resistance is dependent on temperature, more so than in standard resistorsthe word is a portmanteau of thermal and resistorthermistors are widely used as inrush current limiters, temperature sensors (negative temperature coefficient or ntc type typically), self-resetting overcurrent protectors, and. Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them the mathematical equation that describes this relationship is. Change is needed within an organization for it to survive competitively as a business, or for it to grow as an expansive department agency ignoring the people side of change until major resistance stalls a project or causes the project to conclusion change management is used for one. Often referred to as the ohm's law equation, this equation is a powerful predictor of the relationship between potential difference, current and resistance ohm's law as a predictor of current the ohm's law equation can be rearranged and expressed as. Resistance to change nature of resistance the greatest setback in managing change is to underestimate people's resistance to change (galpin, 1995, p 4) change can be identified on the levels of ignorance conclusion structuring for change. Graphs: conclusion: the purpose of this lab was unapparant to us at first we were hooking up a battery, resistor and two meters into one circuit, and measuring what happened when we changed the resistors. For each trial, calculate the expected current based on the resistance and measured voltage using ohm's law record your results in your data table, and be sure to show a sample calculation 2 conclusions: 1 when the resistance in the circuit went up. Conclusion short term threats of antibiotic resistance would be that when they use strong antibiotics to make it so the bacterias don't resist, it sometimes kills the good bacteria. Lab title: air resistance and coffee filters primary authors: megan contributing authors: anna and paul abstract: the goal of this lab is to determine how mass and terminal velocity for a cross-sectional area are linked dropping coffee filters and letting them fall to the ground will provide enough data for the link between mass and terminal. Resistance measurements resistance measurementsobjectives after completing this lab compare the resistance measurements of table 2-4 to the ex- what conclusion can you make about the power rating of a. Experiment 3 resistors in series and in parallel print this page to start your lab report (1 copy) print 2 copies of this file (data page) object: the equivalent resistance of which is equal to the voltage of the power supply, v, divided by the current, i. At the end of this investigation i will come to one of five possible conclusions which are: -the resistance increases as the length of wire increases, or -the resistance decreases as the length of wire increases, or. Resistance lab report conclusion in conclusion we studied the resistance oi resistors across a circuit board using ohm`s law, and then using the characteristics oi a light bulb we. Equivalent resistance of this series-parallel circuit, one first collapses the parallel branch into a single equivalent resistance, which is given by eq 337 this equivalent resistance is in series with ri and.

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices? A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . . Try entering different sets of numbers in the number pyramids. How does the total at the top change? Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear. Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible? Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them? To avoid losing think of another very well known game where the patterns of play are similar. Can you discover whether this is a fair game? Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . . Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4 Can you explain the strategy for winning this game with any target? There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . . This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box. This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy. A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target. Meg and Mo still need to hang their marbles so that they balance, but this time the constraints are different. Use the interactivity to experiment and find out what they need to do. The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves. A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo! Mo has left, but Meg is still experimenting. Use the interactivity to help you find out how she can alter her pouch of marbles and still keep the two pouches balanced. This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at. P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P? Meg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do. When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total. The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it? How good are you at estimating angles? A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates? Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line. Prove Pythagoras' Theorem using enlargements and scale factors. Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15. Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit? A tool for generating random integers. A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . . Match the cards of the same value. Here is a chance to play a fractions version of the classic Countdown Game. A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle? Here is a chance to play a version of the classic Countdown Game. Use an Excel to investigate division. Explore the relationships between the process elements using an interactive spreadsheet. It's easy to work out the areas of most squares that we meet, but what if they were tilted? An animation that helps you understand the game of Nim. This set of resources for teachers offers interactive environments to support work on loci at Key Stage 4. Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line... Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards? Can you beat the computer in the challenging strategy game?

A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2017; you can also visit the original URL. The file type is We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. ... After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. ... The structure of hereditarily finite sets HF under inclusion is an elementary substructure of the entire set-theoretic universe V under inclusion: HF, ⊆ ≺ V, ⊆ . Proof. ...doi:10.12775/llp.2016.007 fatcat:xrj2bzsp7nf2ho33fwyi2thbx4 We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. ... After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. ... The structure of hereditarily finite sets HF under inclusion is an elementary substructure of the entire set-theoretic universe V under inclusion. HF, ⊆ ≺ V, ⊆ Proof. ...arXiv:1601.06593v2 fatcat:z7twbd7w6fhunk7z77xzdqxpfu of null hypotheses of statistical independence as a potential source of binary data structure, and second at constructing a discrete structure (Boolean) model of those statistical interactions that remain ... on Boolean patterns that occur in the data) that can be used to simplify (by approximation) the lattice of empirical patterns. ... Cutoffs based on the level of exceptions which maximizes the Pearson Product-moment correlation based on four cells (a, b, c, d) for strong inclusions, as in Table 1 , are repeated for weak inclusion, ...doi:10.1016/0378-8733(95)00273-1 fatcat:cq4waqdahngz3b2cku2gqubf3a The inclusion relation in the Boolean algebra χ 0 is that of χ, restricted to χ 0 . ... In this paper, we introduce two pairs of rough operations on Boolean algebras. First we define a pair of rough approximations based on a partition of the unity of a Boolean algebra. ... In Section 4, we first define a pair of rough approximations based on a partition of the unity of a Boolean algebra, then propose a pair of generalized rough approximations after defining a basic assignment ...doi:10.1016/j.ins.2004.06.006 fatcat:mwnoxkzsu5fqjhry3xeswaymaa Communications in Computer and Information Science The interest of the decision makers in the selection process of suppliers is constantly growing as a reliable supplier reduces costs and improves the quality of products/services. ... Setting logical conditions between attributes was carried out by using the Boolean Interpolative Algebra. ... The principle of structural functionality indicates that the structure of any element of IBA may be directly calculated based on the structure of its components. ...doi:10.1007/978-3-319-08855-6_1 fatcat:q6qxvivr4jaetnxlxha6kp4mnq It is shown that the Joyal quasi-category model structure for simplicial sets extends to a model structure on simplicial presheaves, for which the weak equivalences are local (or stalkwise) Joyal equivalences ... The article of Jardine gives a proof of the existence of the Jardine model structure based on the technique of Boolean localization. ... Introduction The purpose of this paper is develop an analog of the Jardine model structure on simplicial presheaves in which, rather than having the weak equivalences be 'local Kan equivalences', the weak ...arXiv:1507.08723v2 fatcat:d5koc75mdjfuniqtojfwnglmcy Boolean models are applied to deriving operator versions of the classical Farkas Lemma in the theory of simultaneous linear inequalities. ... That is what we have learned from the Boolean models elaborated in the 1960s by Scott, Solovay, and Vopěnka. ... The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were not aware nor even foresaw at the start of the ...arXiv:0907.0060v4 fatcat:qv4xp446pbdj5b4zbv3fpxjheu The latter do however have limited expressivity, and the corresponding lattice of strong partial clones is of uncountably innite cardinality even for the Boolean domain. ... Sets of relations closed under p.p. denitions are known as co-clones and sets of relations closed under q.f.p.p. denitions as weak partial co-clones. ... Acknowledgements The authors are grateful toward Peter Jonsson, Karsten Schölzel and Bruno Zanuttini, for helpful comments and suggestions. ...doi:10.1093/logcom/exw005 fatcat:vdf5onpuz5a2jplxgi7zaq6say The author demonstrates that the set of equivalence classes (rough sets) of such a relation is par- tially ordered with respect to the relation of rough (approximate) inclusion. ... Then they study the structure of prime filters of a P-algebra and give a canonical form of any P-algebra homomorphism. ... The special case of completeness for the Boolean p-calculus is an improvement over that presented in but weaker than the theorem of . ... the Boolean modal p-logics. ... Acknowledgements I wish to thank Robert Goldblatt for spoting an inaccuracy in the way my completeness theorem was phrased in an earlier version. ...doi:10.1016/s0304-3975(97)00233-8 fatcat:bkuwgqcg2fg6lmhw55wds2a5vm Boolean set operations are computed progressively by reading in input a stream of incremental refinements of the operands. ... Each refinement of the input is mapped immediately to a refinement of the output so that the result is also represented as a stream of progressive refinements. ... generated BSP tree and on a lattice-based Split data structure of the cell decomposition. ...doi:10.2312/sm.20041391 fatcat:gmldqku35vdclpdkufbzxwperu In this paper, we provide a topological representation for double Boolean algebras based on the so-called DB-topological contexts. ... A double Boolean algebra is then represented as the algebra of clopen protoconcepts of some DB-topological context. Mathematics Subject Classification (2010): 18B35, 54B30, 68T30. ... Each filter of the Boolean algebra D ⊓ is a base of some filter of D; in particular, F ∩ D ⊓ is a base of F . ...doi:10.24193/subbmath.2019.1.02 fatcat:obbh5yufova53bcpsa3hmipgmm The proof of the last result is based on the author’s theorem [ibid. 94 (1977), no. 2, 121-128; MR 55 #10269] of the finite axiomatizability of any nuclear X,-categorical structure. ... The proof uses the fact that the weak second order theory of linear order is decidable and also the fact that any countable Boolean algebra is isomorphic to a Boolean algebra generated by all left closed ... The attractors of Boolean networks and their basins have been shown to be highly relevant for model validation and predictive modelling, e.g., in systems biology. ... In the realm of asynchronous, non-deterministic modeling not only is the repertoire of software even more limited, but also the formal notions for basins of attraction are often lacking. ... The work was partially funded by the German Federal Ministry of Education and Research (BMBF), grant no. 0316195. ...arXiv:1807.10103v1 fatcat:aws7v5enczeprcrsy7szv6z5xy We focus on random microstructures consisting of a continuous matrix phase with a high number of embedded inclusions. ... The basic idea of the underlying procedure is to find a simplified SSRVE, whose selected statistical measures under consideration are as close as possible to the ones of the original microstructure. ... Acknowledgement The financial support of the "Deutsche Forschungsgemeinschaft" (DFG), project no. SCHR 570-8/1, is gratefully acknowledged. ...doi:10.1007/978-90-481-9195-6_2 fatcat:selnjyi6sjh3fodzpfw42kvo5q « Previous Showing results 1 — 15 out of 20,447 results

The Pythagorean Theorem Study Guide (page 2) The Pythagorean Theorem In this lesson, we examine a very powerful relationship between the three lengths of a right triangle. With this relationship, we will find the exact length of any side of a right triangle, provided we know the lengths of the other two sides. We also study right triangles where all three sides have whole number lengths. Proving the Pythagorean Theorem The Pythagorean theorem is this relationship between the three sides of a right triangle. Actually, it relates the squares of the lengths of the sides. The square of any number (the number times itself) is also the area of the square with that length as its side. For example, the square in Figure 3.1 with sides of length a will have area a · a = a2. The longest side of a right triangle, the side opposite the right angle, is called the hypotenuse, and the other two sides are called legs. Suppose a right triangle has legs of length a and b, and a hypotenuse of length c, as illustrated in Figure 3.2. The Pythagorean theorem states that a2 + b2 = c2, This means that the area of the squares on the two smaller sides add up to the area of the biggest square. This is illustrated in Figures 3.3 and 3.4. This is a surprising result. Why should these areas add up like this? Why couldn't the areas of the two smaller squares add up to a bit more or less than the big square? We can convince ourselves that this is true by adding four copies of the original triangle to each side of the equation. The four triangles can make two rectangles, as shown in Figure 3.5. They could also make a big square with a hole in the middle, as in Figure 3.6. If we add the a2 and b2 squares to Figure 3.4, and the c2 square to Figure 3.5, they fit exactly. The result in either case is a big square with each side of length a + b, as shown in Figure 3.7. The two big squares have the same area. If we take away the four triangles from each side, we can see that the two smaller squares have the exact same area as the big square, as shown in Figure 3.8. Thus, a2 + b2 = c2 This proof of the Pythagorean theorem has been adapted from a proof developed by the Chinese about 3,000 years ago. With the Pythagorean theorem, we can use any two sides of a right triangle to find the length of the third side. Suppose the two legs of a right triangle measure 8 inches and 12 inches, as shown in Figure 3.9. What is the length of the hypotenuse? By the Pythagorean theorem: 122 + 82 = H2 208 = H2 H = √208 While the equation gives two solutions, a length must be positive, so H = √208. This can be simplified to H = 4√13. What is the height of the triangle in Figure 3.10? Even though the height is labeled h, it is not the hypotenuse. The longest side has length 10 feet, and thus must be alone on one side of the equation. h2 + 32 = 102 h2 = 100 – 9 h = √91 ≈ 9.54 With the help of a calculator, we can see that the height of this triangle is about 9.54 feet. We can use the Pythagorean theorem on triangles without illustrations. All we need to know is that the triangle is right and which side is the hypotenuse. If a right triangle has a hypotenuse length of 9 feet and a leg length of 5 feet, what is the length of the third side? We use the Pythagorean theorem with the hypotenuse, 9, by itself on one side, and the other two lengths, 5 and x, on the other. 52 + x2 = 92 x = √81 – 25 = √56 = 2√14 ≈ 7.48 The third side is about 7.48 feet long. Pythagorean Word Problems Many word problems involve finding a length of a right triangle. Identify whether each given length is a leg of the triangle or the hypotenuse. Then solve for the third length with the Pythagorean theorem. A diagonal board is needed to brace a rectangular wall. The wall is 8 feet tall and 10 feet wide. How long is the diagonal? Having a rectangle means that we have a right triangle, and that the Pythagorean theorem can be applied. Because we are looking for the diagonal, the 10-foot and 8-foot lengths must be the legs, as shown in Figure 3.19. The diagonal D must satisfy the Pythagorean theorem: D2 = 102 + 82 D2 = 100 + 64 = 164 D = √164 = 2√41 ≈ 12.81 feet A 100-foot rope is attached to the top of a 60-foot tall pole. How far from the base of the pole will the rope reach? We assume that the pole makes a right angle with the ground, and thus, we have the right triangle depicted in Figure 3.20. Here, the hypotenuse is 100 feet. The sides must satisfy the Pythagorean theorem: x2 + 602 = 1002 x = √6,400 = 80 feet Since the Pythagorean theorem was discovered, people have been especially fascinated by right triangles with whole number sides. The most famous one is the 3-4-5 right triangle, but there are many others, such as 5-12-13 and 6-8-10. A Pythagorean triple is a set of three whole numbers a-b-c with a2 + b2 = c2. Usually, the numbers are put in increasing order. Is 48-55-73 a Pythagorean triple? We calculate 482 + 552 = 2,304 + 3,025 = 5,329. It just happens that 732 = 5,329. Thus, the numbers 48, 55, and 73 form a Pythagorean triple. Generating Pythagorean Triples The ancient Greeks found a system for generating Pythagorean triples. First, take any two whole numbers r and s, where r > s. We will get a Pythagorean triple a-b-c if we set: a = 2rs b = r2 – s2 c = r2 + s2 This is because a2 + b2 = (2rs)2 + (r2 – s2)2 = 4r2s2 + r4 – 2r2s2 + s4 = r4 + 2r2s2 + s4 = (r2 + s2)2 = c2 Thus, a2 + b2 = c2. Find the Pythagorean triple generated by r = 7 and s = 2. a = 2rs = 2(7)(2) = 28 b = r2 – s2 = 72 – 22 = 45 c = r2 + s2 = 72 + 22 = 53 Thus, r = 7 and s = 2 generate the 28-45-53 Pythagorean triple. If we plug in different values of r and s, we will generate many different Pythagorean triples, whole numbers a, b, and c with a2 + b2 = c2. Around 1640, a French mathematician named Pierre de Fermat suggested that this could happen only with the exponent 2. He said that no positive whole numbers a, b, and c could possibly make a3 + b3 = c3 or a4 + b4 = c4 or an + bn = cn for any n > 2. Fermat claimed to have found a short and clever proof of this, but then died without writing it down. For hundreds of years, mathematicians tried to prove this result, called "Fermat's Last Theorem." Only in 1995 was it finally proven, by a British mathematician named Andrew Wiles. Because his proof runs to more than 100 pages, it is clearly not the simple proof that Fermat spoke about. This assumes, of course, that Fermat had actually found a correct and simple proof. Practice problems for this study guide can be found at: - Kindergarten Sight Words List - First Grade Sight Words List - 10 Fun Activities for Children with Autism - Child Development Theories - Definitions of Social Studies - Grammar Lesson: Complete and Simple Predicates - Social Cognitive Theory - Signs Your Child Might Have Asperger's Syndrome - Theories of Learning - Why is Play Important? Social and Emotional Development, Physical Development, Creative Development

My Students Struggle to Solve Basic Equations A MiddleWeb Blog I actually wrote an article about the problem in 2021: Refreshing Students’ Equation Solving Skills. Since then I have been purposefully trying to help students get better at solving equations. I’ve done things like having students keep their work neater so they don’t make careless errors, drawing a line by the equal sign to help them visualize the equality, and in general just trying to get them to write things down. Yet my students each year still continue to struggle in this area, and since solving an equation is the foundation that most higher math concepts build off of, we’ll continue to try to improve. Pinpointing problem areas I realized I was going to have to be more strategic if I was going to help my students. I needed to know what they are specifically having trouble understanding. So I picked two problems (they were based on 6th and 7th grade standards in our state) for my 11th graders to work. MA19.6.19 Write and solve an equation in the form of x+p=q or px=q for cases in which p, q, and x are all non-negative rational numbers to solve real-world and mathematical problems. MA19.7.9a Solve word problems leading to equations of the form px+q=r and p(x+q)=r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. I had my students come up to my desk one at a time. I presented them with the problems and a four-function calculator. They worked a problem and often talked out loud about what they were thinking as they were working. I didn’t ask them to, that was just a bonus! Here are the two problems: 20 – 7x = 6x -6 2/3b + 7 = -1 I listened as they worked. Then I took the Post-it notes they had worked their problems on and really studied their work. In fact, I often rewrote what the student wrote so I could really understand their thinking. That helped me so much. I have always struggled trying to determine what students were thinking. By rewriting their work, I was able to gain more insight. I then sorted the Post-it notes according to the type of error each student made. I have listed the categories below. ►Confused by Fractions Not surprisingly, the problem with the fraction was missed much more than the one without. A few students said that they could not begin the problem because it had a fraction in it, and no amount of encouragement could persuade them to try it. Some students stopped after subtracting both sides by 7 because they simply didn’t know what to do with 2/3, so their answer was -8. Some students multiplied both sides by two thirds instead of multiplying by the reciprocal; their answer was -16/3. One student subtracted 2/3 from both sides, after they subtracted 7 from both sides. ►Combined Unlike Terms Students combined 20 with -7x to get 13x, while 6x-6 was combined to get x. And others combined 2/3b with 7 to get 14/3b. ►Moved or Lost the Equal Sign If students subtracted so that they had a zero on one side of the equation, often the equal sign would just disappear. Sometimes the equal sign would never be seen in the students’ work at all. What does this all mean? In the broadest sense it means that students who struggle to solve a 6th or 7th grade level equation will definitely struggle to solve quadratic, exponential, or logarithmic equations that are part of the Algebra 2 course of study. More time has to be spent fortifying students’ ability in this area. Realizing that this was necessary, we allowed for a few weeks at the beginning of the year to refresh students on solving equations. It wasn’t enough. Now I’m thinking it would be better to spiral in problems and continue working on it all year long. Specifically, I think there are some fundamental skills related to solving equations that students are lacking. Maybe lacking is not the right word; maybe they have the skills but are unable to always apply them correctly or at the right time. What are students not understanding? I think the problem they have with fractions indicates they don’t understand reciprocals. Students know that 2/3 is literally 2 divided by 3; so they mistakenly think they should multiply both sides by 2/3. To complicate matters, they are afraid of fractions. They will skip a whole problem if it has a fraction in it. Why are students still combining unlike terms? I am at a loss. We model constantly the correct way to combine like terms. I think sometimes, if students don’t know what to do, they think they should try anything. I heard a lot of “I know I’m supposed to do something here…” The moving or missing equal sign is about more than sloppy work. It’s a lack of understanding about equivalence. They haven’t really learned what the equal sign signifies. What to do about it? These are my first thoughts about how to help students solve equations. Model correct vocabulary, It’s not 7x; it’s 7 multiplied by x. It also might be helpful to say one x, as opposed to x. That’s tedious, but I don’t think all students understand the invisible “one” there. Also, do students know that multiplying by 3/2 will yield the same answer as dividing by 2/3? Make sure students are clear on the definition of solving an equation. To solve an equation means to find all numbers that make the equation true. Source Help students understand equivalence. Make it mandatory that students plug their solution back in after solving the problem. They will literally be able to see that the correct answer is one that makes the equation true. That will drive home the meaning of the equal sign and help them understand the definition of equivalence. Be more intentional when teaching students how to “undo” multiplication, division, etc. Explicitly state when to subtract 2x from each side and when to divide both sides by 2. I also need to diagnose problems sooner. I need to study students’ work in the early weeks of our time together and see what they know and what they don’t know about solving equations. Much of the problem rests with my assumption that when students arrive in my class, they have mastered solving equations – when they haven’t. Going forward I am going to assume that my students need some coaching and practice and one-on-one help to be proficient at solving equations. I can’t fall back on the fact that I work a lot of equations on the board. When students watch their teacher solve equations on the board, they don’t know what they don’t know. It’s easy to watch someone do something and think “that’s easy; I can do that.” I plan to talk with other teachers in my department and see what they suggest. I also want to check with some of our middle school teachers to see if they have any suggestions. And thanks in advance for sharing any ideas you might have in the comments below.

Individual differences | Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology | The error of measurement is the observed differences in obtained scores due to chance variance. The standard error of measurement or estimation is the estimated standard deviation of the error in that method. Specifically, it estimates the standard deviation of the difference between the measured or estimated values and the true values. Notice that the true value of the standard deviation is usually unknown and the use of the term standard error carries with it the idea that an estimate of this unknown quantity is being used. It also carries with it the idea that it measures not the standard deviation of the estimate itself but the standard deviation of the error in the estimate, and these are very different. In applications where a standard error is used, it would be good to be able to take proper account of the fact that the standard error is only an estimate. Unfortunately this is not often possible and it may then be better to use an approach that avoids using a standard error, for example by using maximum likelihood or a more formal approach to deriving confidence intervals. One well-known case where a proper allowance can be made arises where the Student's t-distribution is used to provide a confidence interval for an estimated mean or difference of means. In other cases, the standard error may usefully be used to provide an indication of the size of the uncertainty, but its formal or semi-formal use to provide confidence intervals or tests should be avoided unless the sample size is at least moderately large. Here "large enough" would depend on the particular quantities being analysed. Standard error of the mean The standard error of the mean (SEM), an unbiased estimate of expected error in the sample estimate of a population mean, is the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample): - s is the sample standard deviation (i.e. the sample based estimate of the standard deviation of the population), and - n is the size (number of items) of the sample. A practical result: Decreasing the uncertainty in your mean value estimate by a factor of two requires that you acquire four times as many samples. Worse, decreasing standard error by a factor of ten requires a hundred times as many samples. This estimate may be compared with the formula for the true standard deviation of the mean: - σ is the standard deviation of the population. Note: Standard error may also be defined as the standard deviation of the residual error term. (Kenney and Keeping, p. 187; Zwillinger 1995, p. 626) If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of error in the mean may be obtained by multiplying the standard error above by the square root of (1+(n-1)ρ)/(1-ρ), where sample bias coefficient ρ is the average of the autocorrelation-coefficient ρAA[Δx] value (a quantity between -1 and 1) for all sample point pairs. Assumptions and usage If the data are assumed to be normally distributed, quantiles of the normal distribution and the sample mean and standard error can be used to calculate confidence intervals for the mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where x is equal to the sample mean, s is equal to the standard error for the sample mean, and 1.96 is the .975 quantile of the normal distribution. - Upper 95% Limit = - Lower 95% Limit = In particular, the standard error of a sample statistic (such as sample mean) is the estimated standard deviation of the error in the process by which it was generated. In other words, it is the standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of , (for standard error of measurement or mean), or . Standard errors provide simple measures of uncertainty in a value and are often used because: - If the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated in many cases; - Where the probability distribution of the value is known, it can be used to calculate an exact confidence interval; and - Where the probability distribution is unknown, relationships like Chebyshev’s or the Vysochanskiï-Petunin inequality can be used to calculate a conservative confidence interval - As the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal. - Consistency (measurement) - Least squares - Observational error - Sample mean and sample covariance - Scoring (testing) - Statistical estimation - Statistical measurement - Test bias - Test reliability - Test scores Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models |This page uses Creative Commons Licensed content from Wikipedia (view authors).|

In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example involving a single-crystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call “twinning with variable volume fraction.” Two 1996 papers by Chopra et al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energy-minimization-based model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. Themore » Minimizers for the Cahn--Hilliard energy functional under strong anchoring conditions We study analytically and numerically the minimizers for the Cahn-Hilliard energy functional with a symmetric quartic double-well potential and under a strong anchoring condition(i.e., the Dirichlet condition) on the boundary of an underlying bounded domain. We show a bifurcation phenomenon determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. For the case that the boundary value is exactly the average of the two pure phases, if the bifurcation parameter is larger than or equal to a critical value, then the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. For a boundary value that is larger (or smaller) than that of the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain.Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian more » - Award ID(s): - Publication Date: - NSF-PAR ID: - Journal Name: - SIAM journal on applied mathematics - Page Range or eLocation-ID: - Sponsoring Org: - National Science Foundation More Like this Variational boundary conditions based on the Nitsche method for fitted and unfitted isogeometric discretizations of the mechanically coupled Cahn-Hilliard equation.The primal variational formulation of the fourth-order Cahn-Hilliard equation requires C1-continuous finite element discretizations, e.g., in the context of isogeometric analysis. In this paper, we explore the variational imposition of essential boundary conditions that arise from the thermodynamic derivation of the Cahn-Hilliard equation in primal variables. Our formulation is based on the symmetric variant of Nitsche's method, does not introduce additional degrees of freedom and is shown to be variationally consistent. In contrast to strong enforcement, the new boundary condition formulation can be naturally applied to any mapped isogeometric parametrization of any polynomial degree. In addition, it preserves full accuracy, including higher-order rates of convergence, which we illustrate for boundary-fitted discretizations of several benchmark tests in one, two and three dimensions. Unfitted Cartesian B-spline meshes constitute an effective alternative to boundary-fitted isogeometric parametrizations for constructing C1-continuous discretizations, in particular for complex geometries. We combine our variational boundary condition formulation with unfitted Cartesian B-spline meshes and the finite cell method to simulate chemical phase segregation in a composite electrode. This example, involving coupling of chemical fields with mechanical stresses on complex domains and coupling of different materials across complex interfaces, demonstrates the flexibility of variational boundary conditions in the context ofmore » Inverting the variable fractional order in a variable-order space-fractional diffusion equation with variable diffusivity: analysis and simulationAbstract Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues.In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain.We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order.We carry out numerical experiments to investigate the numerical performance of these methods. In this work we present a systematic review of novel and interesting behaviour we have observed in a simplified model of a MEMS oscillator. The model is third order and nonlinear, and we expressit as a single ODE for a displacement variable. We find that a single oscillator exhibits limitcycles whose amplitude is well approximated by perturbation methods. Two coupled identicaloscillators have in-phase and out-of-phase modes as well as more complicated motions.Bothof the simple modes are stable in some regions of the parameter space while the bifurcationstructure is quite complex in other regions. This structure is symmetric; the symmetry is brokenby the introduction of detuning between the two oscillators. Numerical integration of the fullsystem is used to check all bifurcation computations. Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern. As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change, producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation. A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, is investigated using Lindstedt's perturbation method. Conditions are derived on the parameters of the modelmore » Abstract We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.

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Gauss was the only child of poor parents. He was a calculating prodigy with a gift for languages. His teachers and his devoted mother recommended him to the duke of Brunswick in 1791, who granted him financial assistance to continue his education locally and then to study mathematics at the University of Göttingen. What awards did Carl Friedrich Gauss win? Gauss won the Copley Medal, the most prestigious scientific award in the United Kingdom, given annually by the Royal Society of London, in 1838 “for his inventions and mathematical researches in magnetism.” For his study of angle-preserving maps, he was awarded the prize of the Danish Academy of Sciences in 1823. How was Carl Friedrich Gauss influential? Gauss wrote the first systematic textbook on algebraic number theory and rediscovered the asteroidCeres. He published works on number theory, the mathematical theory of map construction, and many other subjects. After Gauss’s death in 1855, the discovery of many novel ideas among his unpublished papers extended his influence into the remainder of the century. Carl Friedrich Gauss, original name Johann Friedrich Carl Gauss, (born April 30, 1777, Brunswick [Germany]—died February 23, 1855, Göttingen, Hanover), German mathematician, generally regarded as one of the greatest mathematicians of all time for his contributions to number theory, geometry, probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory (including electromagnetism). Gauss was the only child of poor parents. He was rare among mathematicians in that he was a calculating prodigy, and he retained the ability to do elaborate calculations in his head most of his life. Impressed by this ability and by his gift for languages, his teachers and his devoted mother recommended him to the duke of Brunswick in 1791, who granted him financial assistance to continue his education locally and then to study mathematics at the University of Göttingen from 1795 to 1798. Gauss’s pioneering work gradually established him as the era’s preeminent mathematician, first in the German-speaking world and then farther afield, although he remained a remote and aloof figure. Gauss’s first significant discovery, in 1792, was that a regular polygon of 17 sides can be constructed by ruler and compass alone. Its significance lies not in the result but in the proof, which rested on a profound analysis of the factorization of polynomial equations and opened the door to later ideas of Galois theory. His doctoral thesis of 1797 gave a proof of the fundamental theorem of algebra: every polynomial equation with real or complex coefficients has as many roots (solutions) as its degree (the highest power of the variable). Gauss’s proof, though not wholly convincing, was remarkable for its critique of earlier attempts. Gauss later gave three more proofs of this major result, the last on the 50th anniversary of the first, which shows the importance he attached to the topic. Gauss’s recognition as a truly remarkable talent, though, resulted from two major publications in 1801. Foremost was his publication of the first systematic textbook on algebraic number theory, Disquisitiones Arithmeticae. This book begins with the first account of modular arithmetic, gives a thorough account of the solutions of quadratic polynomials in two variables in integers, and ends with the theory of factorization mentioned above. This choice of topics and its natural generalizations set the agenda in number theory for much of the 19th century, and Gauss’s continuing interest in the subject spurred much research, especially in German universities. The second publication was his rediscovery of the asteroidCeres. Its original discovery, by the Italian astronomer Giuseppe Piazzi in 1800, had caused a sensation, but it vanished behind the Sun before enough observations could be taken to calculate its orbit with sufficient accuracy to know where it would reappear. Many astronomers competed for the honour of finding it again, but Gauss won. His success rested on a novel method for dealing with errors in observations, today called the method of least squares. Thereafter Gauss worked for many years as an astronomer and published a major work on the computation of orbits—the numerical side of such work was much less onerous for him than for most people. As an intensely loyal subject of the duke of Brunswick and, after 1807 when he returned to Göttingen as an astronomer, of the duke of Hanover, Gauss felt that the work was socially valuable. Similar motives led Gauss to accept the challenge of surveying the territory of Hanover, and he was often out in the field in charge of the observations. The project, which lasted from 1818 to 1832, encountered numerous difficulties, but it led to a number of advancements. One was Gauss’s invention of the heliotrope (an instrument that reflects the Sun’s rays in a focused beam that can be observed from several miles away), which improved the accuracy of the observations. Another was his discovery of a way of formulating the concept of the curvature of a surface. Gauss showed that there is an intrinsic measure of curvature that is not altered if the surface is bent without being stretched. For example, a circular cylinder and a flat sheet of paper have the same intrinsic curvature, which is why exact copies of figures on the cylinder can be made on the paper (as, for example, in printing). But a sphere and a plane have different curvatures, which is why no completely accurate flat map of the Earth can be made. Are you a student? Get Britannica Premium for only $24.95 - a 67% discount! Gauss published works on number theory, the mathematical theory of map construction, and many other subjects. In the 1830s he became interested in terrestrial magnetism and participated in the first worldwide survey of the Earth’s magnetic field (to measure it, he invented the magnetometer). With his Göttingen colleague, the physicist Wilhelm Weber, he made the first electric telegraph, but a certain parochialism prevented him from pursuing the invention energetically. Instead, he drew important mathematical consequences from this work for what is today called potential theory, an important branch of mathematical physics arising in the study of electromagnetism and gravitation. Gauss also wrote on cartography, the theory of map projections. For his study of angle-preserving maps, he was awarded the prize of the Danish Academy of Sciences in 1823. This work came close to suggesting that complex functions of a complex variable are generally angle-preserving, but Gauss stopped short of making that fundamental insight explicit, leaving it for Bernhard Riemann, who had a deep appreciation of Gauss’s work. Gauss also had other unpublished insights into the nature of complex functions and their integrals, some of which he divulged to friends. In fact, Gauss often withheld publication of his discoveries. As a student at Göttingen, he began to doubt the a priori truth of Euclidean geometry and suspected that its truth might be empirical. For this to be the case, there must exist an alternative geometric description of space. Rather than publish such a description, Gauss confined himself to criticizing various a priori defenses of Euclidean geometry. It would seem that he was gradually convinced that there exists a logical alternative to Euclidean geometry. However, when the Hungarian János Bolyai and the Russian Nikolay Lobachevsky published their accounts of a new, non-Euclidean geometry about 1830, Gauss failed to give a coherent account of his own ideas. It is possible to draw these ideas together into an impressive whole, in which his concept of intrinsic curvature plays a central role, but Gauss never did this. Some have attributed this failure to his innate conservatism, others to his incessant inventiveness that always drew him on to the next new idea, still others to his failure to find a central idea that would govern geometry once Euclidean geometry was no longer unique. All these explanations have some merit, though none has enough to be the whole explanation. Another topic on which Gauss largely concealed his ideas from his contemporaries was elliptic functions. He published an account in 1812 of an interesting infinite series, and he wrote but did not publish an account of the differential equation that the infinite series satisfies. He showed that the series, called the hypergeometric series, can be used to define many familiar and many new functions. But by then he knew how to use the differential equation to produce a very general theory of elliptic functions and to free the theory entirely from its origins in the theory of elliptic integrals. This was a major breakthrough, because, as Gauss had discovered in the 1790s, the theory of elliptic functions naturally treats them as complex-valued functions of a complex variable, but the contemporary theory of complex integrals was utterly inadequate for the task. When some of this theory was published by the Norwegian Niels Abel and the German Carl Jacobi about 1830, Gauss commented to a friend that Abel had come one-third of the way. This was accurate, but it is a sad measure of Gauss’s personality in that he still withheld publication. Gauss delivered less than he might have in a variety of other ways also. The University of Göttingen was small, and he did not seek to enlarge it or to bring in extra students. Toward the end of his life, mathematicians of the calibre of Richard Dedekind and Riemann passed through Göttingen, and he was helpful, but contemporaries compared his writing style to thin gruel: it is clear and sets high standards for rigour, but it lacks motivation and can be slow and wearing to follow. He corresponded with many, but not all, of the people rash enough to write to him, but he did little to support them in public. A rare exception was when Lobachevsky was attacked by other Russians for his ideas on non-Euclidean geometry. Gauss taught himself enough Russian to follow the controversy and proposed Lobachevsky for the Göttingen Academy of Sciences. In contrast, Gauss wrote a letter to Bolyai telling him that he had already discovered everything that Bolyai had just published. After Gauss’s death in 1855, the discovery of so many novel ideas among his unpublished papers extended his influence well into the remainder of the century. Acceptance of non-Euclidean geometry had not come with the original work of Bolyai and Lobachevsky, but it came instead with the almost simultaneous publication of Riemann’s general ideas about geometry, the Italian Eugenio Beltrami’s explicit and rigorous account of it, and Gauss’s private notes and correspondence.

EXAMPLE SOLUTIONS – Topic 8: ELECTROMAGNETIC INDUCTION 1) A rod 4 meters in length is placed in a magnetic field at right angles to the direction of B which has a magnitude of 0.3 Tesla. If v the rod is moved with a speed of 3 m/sec in a direction perpendicular to its length and perpendicular to B, what emf is induced across the ends of the rod? Draw a figure and indicate the direction of the induced emf. B Solution: We first determine the direction of the induced Emf in the rod. We can do this either by Lenz's Law, or in this case by using q v B. If we consider a '+' charge in the rod, this is moving to the left. Thus v B is downward, and the bottom of the rod will become the positive end of the Emf. The Emf produced is given by: E = (F / q) d s (1 / q) Fmag d v B d vx B Hence: E = B v d = (.3)(3)(4) = 3.6 Volts. 2) A loop of wire of area 0.5 m2 is positioned between the pole faces of a large electromagnet so that the perpendicular to the plane of the loop makes an angle of 30o with respect to the magnetic field direction. The magnet is turned on, and a final magnetic induction of 30 Teslas is reached within 200 msec. What is the average emf induced in the coil during this time? Solution: We draw the loop (our loop actually has only one turn). Applying Lenz's Law we have the flux increasing since the external field is increasing. Hence, y B loop must be opposite B . Applying the right hand rule 30 we have the current within the loop from 'A' to 'B'. B Hence, in an external circuit the current would be from 'B' to 'A'. This makes point 'B' the positive terminal of A B loop the Emf. From Faraday's Law we have: z B x Eave = /t . Since the flux is B A cos 30 then E ave = A (B/t) cos 30 = (.5)(30)(.866)/(.2) = 65.4 Volts. 3) A rod of length 1 meter is allowed to roll down the incline shown. The magnetic induction is 0.2 T, and is perpendicular to the plane of the loop. The v resistance R is 10 ohms. The resistance of the rod and the rails may be neglected. R 370 a) In terms of 'v' the speed of the rod down the plane, determine the emf induced in the rod, and hence the current produced in the loop. b) In terms of v determine the magnitude and direction of the magnetic force exerted on the rod. c) Let g = 10 m/sec2. If the mass of the rod is 20 gm, determine the terminal speed of the rod. Solution: We first apply Lenz's Law to determine the direction of the induced current. Since the area is decreasing, the flux through the loop is decreasing. v Hence, B ind must be in the same direction as B. Applying the right hand rule we have the current as shown. For a given velocity I B E(v) = B v d where 'd' is the length of the rod. B ind Thus the current in the loop will also depend on 'v'. I(v) = E/R = (B v d)/R . Since there is a current in the rod, then there will be a magnetic force acting on the rod. Looking at the plane sideways, we have the current coming out of the rod. Hence: N F w = I L B acts up the plane. # From Newton's 2nd Law we then have: I B Fx = mg sin 37 - F w = m a x and mg F y = N - mg cos 37 = 0 The magnetic force is F w = I d B sin 90 = I d B = (d B)2 vx/R . We see that as vx increases, the force up the plane increases as well. Hence, a terminal speed will be reached when a x 0. That is: F w = mg sin 37 (d B)2 vf /R = mg sin 37 . Solving for v f v f = m g R sin 37/(B d)2 = (20 x 10-3)(10)(.6)/(.2)2 = 30 m/sec. 4) The coil shown consists of 100 turns of radius 4 cm. y The magnetic field is constant and has a value of 0.6 T. The coil is initially situated as shown, and is then rotated 90o so that the plane of the coil is parallel to the magnetic field. This rotation takes 0.2 seconds. a) What is the average emf generated in the coil during b) What is the direction of the induced emf? z B x Solution: We draw the loop (our loop actually has more turns) looking down the z-axis. Applying Lenz's Law we I have the flux decreasing since the flux is initially a B maximum amount. Hence, B loop must be the same direction A A as B . Applying the right hand rule we have the current within the loop from 'B' to 'A'. Hence, in an external circuit B the current would be from 'A' to 'B'. This makes point 'A' the B loop positive terminal of the Emf. From Faraday's Law we have: Eave = /t The final flux is zero, and the initial flux is N B A = (100)(.6)()(4 x 10-2)2 = 0.3 Webers. E ave = /t = (.3/(.2) = 1.5 Volts. 5) A shunt wound dc motor built to operate on 120 volts will draw 24 amps when started, but only 5 amps when running at its rated speed. If the field windings have a resistance of 30 ohms, what is the mechanical power developed by the motor? The circuit for a shunt wound motor is as shown. The basic equations for this circuit are: (a) I = If + Ia If (b) Vab = If Rf motor IA E (c) Vab = Ia Ra + E b RA When the motor starts there is no back emf. Hence, the current (24 A) is determined by the combined resistance in the field and armature windings. For the field windings, we have, using equation (b): If = V ab/Rf = (120)/(30) = 4 A . The total resistance of the 2 coils is: R = Vab/I = (120)/(24) = 5 ohms. Thus: 1/R = 1/Rf + 1/Ra 1/Ra = (1/5) - (1/30) = (1/6) or Ra = 6 ohms. When the motor is running we then have: Ia = I - If = 5 - 4 = 1 A. Now applying equation (c): E b = Vab - Ia Ra = 120 - (1)(6) = 144 V. The mechanical power developed is: Pmech = Ia E b = (1)(114) = 114 W.

Oil production models with normal rate curves Hubbert fitted the rate of U.S.A. oil production with logistic curves. Deffeyes says that the normal curve gives a better fit. As far as I know, there has previously been no theoretical justification of these fittings. In my paper Oil production models with normal rate curves conditions ensuring approximately normal rate of production curves are established. It has been accepted to appear in the journal Probability in the Engineering and Informational Sciences. Here is the abstract of the paper: The normal curve has been used to fit the rate of both world and U.S.A. oil production. In this paper we give the first theoretical basis for these curve fittings. It is well known that oil field sizes can be modelled by independent samples from a lognormal distribution. We show that when field sizes are lognormally distributed, and the starting time of the production of a field is approximately a linear function of the logarithm of its size, and production of a field occurs within a small enough time interval, then the resulting total rate of production is close to being a normal curve. We call the total rate of production the sum of the rates of production of the fields constituting a given area. The main idea is that the rates of production of individual fields does not matter much in obtaining an approximately normal total rate of production curve. What matters, assuming the time it takes to produce individual fields is not too long, is the distribution of field sizes and the location in time of Next, I will explain what is meant in the abstract. After that, I will make some remarks about the model in the paper. What is meant by a lognormal distribution? A normal distribution is the familiar bell-shaped curve. A lognormal distribution obtained by a simple transformation of the normal distribution: if X is a normally distributed random variable, then eX is lognormally distributed. Here is the lognormal distribution associated with the normal Lognormal distributions have been used to model the distribution of the oil field sizes in a given area. Moreover, the lognormal distribution is fundamental to the Black-Scholes model of the stock market used in the analysis of derivatives. (Here we are using it for something more sensible.) We will suppose that oil field sizes are given by independent samples from a lognormal distribution. Next we suppose that the production of oil from field of size x occurs approximately at time -a log(x) + b, where a>0 and b are constants. More specifically, we suppose that all the oil produced from a field of size x occurs in the time interval ranging from -a log(x) + b -L to -a log(x) + b + L, where L>0 is Under these assumptions, the rate of oil production converges to a curve which is close to being normal. The "converging" bit means that the total rate of production of the first n fields divided by the total amount of oil ever produced by the first n fields tends to a limit curve as n tends to infinity. What does "close to normal" mean? Let Fn(t) be the amount of oil produced by the first n fields up to time t, divided by the total amount of oil ever produced by the first n fields. Suppose that the lognormal distribution describing the field sizes is obtained from a normal distribution with standard deviation s. Let F(t) be the function you would have corresponding to Fn(t) if the rate of production was exactly normal. It turns out that the normal distribution corresponding to F(t) has standard deviation roughly equal to S = as. where the maximum is taken over all time points t. Then, when n is large enough, Dn < L/ S. Thus, when L/S is small, the rate of production is close to being normal in the sense that Dn is small for n large enough. Note that L can be large and Fn(t) still be close to normal, as long as L/S is small. Here are some concluding comments on the model. - The model could be taken as applying only to fields below a certain For example, the largest field in the U.S.A. is Prudhoe Bay, which started production later than one would expect from its size as a consequence of its location in the arctic. Prudhoe Bay skews the total rate of production curve of the U.S.A. to the right, though not enough to move the peak total rate of production away from 1970. On the other hand, if a few smaller fields were produced earlier than one would expect from their sizes, they would not skew the total rate of production curve by much. - The condition that oil produced from a field of size x occurs in the time interval -a log(x) + b -L to -a log(x) + b + L probably does not need to hold precisely for the approximation to hold. What is intuitively important is that production is concentrated around -a log(x) + b. - It would be most interesting to have empirical evidence as to whether this model describes actual oil production in some area or areas. This should hold when the production of fields of size x is centred about f(x)=-a log(x) + b for some constants a>0 and b. If such evidence exists, then it would be desirable to find some explanation for the appearance of the function f(x). - The total rate of production curve is approximately normal for U.S.A. and world oil production. For many other areas, such as the North Sea, it is not. This could be because the larger areas have a wider range of field sizes and because their total rate of production curves have greater width (which in the model is roughly equal compared to the time intervals individual fields are in production (which in the model is represented by L). - The function -a log(x) +b implies that the smallest fields will be produced indefinitely far into the future; any reasonable model producing a bell shaped curve would have this feature. This is probably an unrealistic Could this be part of the reason that the current rate of oil production in the U.S.A. is higher than was predicted by Hubbert? Another reason could be delayed production from large fields such as Prudhoe Bay. - We have investigated lognormally distributed field sizes together with normal total production curves for the reason that the analysis seemed natural and elegant. Other combinations of field size distributions and classes of total production curves could also be studied, for example with field size determined by independent samples from a Pareto distribution and lognormal rate of production curves. You can get in touch with me at D.S.Stark@maths.qmul.ac.uk

- How is e value calculated? - What is e function? - Why is e called a natural number? - How do you use e in Excel? - What does E to the negative power mean? - What is the formula of E? - Is E X always positive? - What is E used for in real life? - Where does e occur in nature? - What’s the value of E? - What is E in log? - What does the weird e mean in math? - Why does E exist in math? - What is E to zero? - What is E used for in math? - What is the value of E Power 0? - How do you read E numbers? - Why is e so special? - Can e ever be 0? - What is E to the infinity power? - What does the E mean in an exponential equation? How is e value calculated? We’ve learned that the number e is sometimes called Euler’s number and is approximately 2.71828. Like the number pi, it is an irrational number and goes on forever. The two ways to calculate this number is by calculating (1 + 1 / n)^n when n is infinity and by adding on to the series 1 + 1/1!. What is e function? In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental number theory, and are more special than G-functions. Why is e called a natural number? of 2.718. It was that great mathematician Leonhard Euler who discovered the number e and calculated its value to 23 decimal places. … Its properties have led to it as a “natural” choice as a logarithmic base, and indeed e is also known as the natural base or Naperian base (after John Napier). How do you use e in Excel? =EXP(value) gives the result of e value. For example, to find the value of , where x is to be taken from cell A2 , you would use the formula =EXP(2*A2-1) . (In other words, whatever is in the exponent goes in the parentheses.) What does E to the negative power mean? Prism switches to scientific notation when the values are very larger or very small. For example: 2.3e-5, means 2.3 times ten to the minus five power, or 0.000023. 4.5e6 means 4.5 times ten to the sixth power, or 4500000 which is the same as 4,500,000. What is the formula of E? The number e is an important mathematical constant, approximately equal to 2.71828 . When used as the base for a logarithm, we call that logarithm the natural logarithm and write it as lnx . Is E X always positive? See, e is a positive number which is approximately equal to 2.71828. So e to the power anything ( be it a fraction,decimal,negative integer,positive integer,etc.) … Value of ‘e’ is positive i.e. approximately 2.71828. It implies e^x is always positive for any value of x. What is E used for in real life? Euler’s number, e , has few common real life applications. Instead, it appears often in growth problems, such as population models. It also appears in Physics quite often. As for growth problems, imagine you went to a bank where you have 1 dollar, pound, or whatever type of money you have. Where does e occur in nature? Yes, the number e does have physical meaning. It occurs naturally in any situation where a quantity increases at a rate proportional to its value, such as a bank account producing interest, or a population increasing as its members reproduce. What’s the value of E? 2.71828The Constant e. What is e? “e” is a numerical constant that is equal to 2.71828. Just as pi (3.14159) is a numerical constant that occurs whenever the circumference of a circle is divided by its diameter. What is E in log? The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. What does the weird e mean in math? It’s the Greek capital letter Σ sigma. Roughly equivalent to our ‘S’. It stands for ‘sum’. Read this for starters. Why does E exist in math? The number e is one of the most important numbers in mathematics. … It is often called Euler’s number after Leonhard Euler (pronounced “Oiler”). e is an irrational number (it cannot be written as a simple fraction). e is the base of the Natural Logarithms (invented by John Napier). What is E to zero? Value of e to power zero is e is equal to 1. What is E used for in math? The number e is a mathematical constant approximately equal to 2.71828 and is the base of the natural logarithm, that is the unique number whose natural logarithm equals one. It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. What is the value of E Power 0? Any number raised to zero is one. Zero is neither positive nor negative so the minus sign before it is redundant. e is constant quantity(roughly equal to 2.71) and when raised to the power 0 it results in 1 as the answer. How do you read E numbers? On a calculator display, E (or e) stands for exponent of 10, and it’s always followed by another number, which is the value of the exponent. For example, a calculator would show the number 25 trillion as either 2.5E13 or 2.5e13. In other words, E (or e) is a short form for scientific notation. Why is e so special? What’s so special about the number e? … ex has the remarkable property that the derivative doesn’t change it, so at every point on its graph the value of ex is also the slope of ex at that point. Can e ever be 0? Since the base, which is the irrational number e = 2.718 (rounded to 3 decimal places), is a positive real number, i.e., e is greater than zero, then the range of f, y = f(x) = e^x, is the set of all POSITIVE (emphasis, mine) real numbers; therefore, e^x can never equal zero (0) even though as x approaches negative … What is E to the infinity power? When e is raised to power infinity,it means e is increasing at a very high rate and hence it is tending towards a very large number and hence we say that e raised to the power infinity is infinity. What does the E mean in an exponential equation? This leaflet provides information about this important constant, and the related exponential function. The exponential constant. The exponential constant is an important mathematical constant and is given the symbol e. Its value is approximately 2.718.

Ivan Vladislavovich Śleszyński Lysianka, Cherkasy Oblast, Ukraine BiographyIvan Śleszyński's first name is sometimes written as 'Jan', which is the Polish version, while his last name is either given by the Polish 'Śleszyński' or the Russian versions 'Sleshinskii' or 'Sleshinsky'. Although Ivan was born in the Ukraine, he was ethnically Polish, being born into a Polish family living in Lysianka, a town about 160 km due south of Kiev. He studied mathematics at Odessa University and graduated from there in 1875. He then travelled to Germany where he studied under Karl Weierstrass at the University of Berlin, receiving his doctorate in 1882. Returning to Odessa, he became professor of mathematics at the University, holding the position from 1883 to 1909. The year 1909 was significant in another way, for it was the one in which he published his translation of Louis Couturat's famous book The algebra of logic. This work by Śleszyński was more than a translation since it contained Śleszyński's own very useful commentary. This text had a major influence on the development of mathematical logic in Russia since it became the main textbook used by students of the subject over many years. Śleszyński left Odessa and went to Poland in 1911 where he was appointed as an extraordinary professor at the Jagellonian University of Kraków. We should note that in fact Kraków was at this time in the Austro-Hungarian Empire but, remembering Śleszyński's Polish background, it is fair to say that he was moving to Poland. In 1919 he was promoted from extraordinary professor to become the full Professor of Logic and Mathematics the Jagellonian University. He continued to teach at Kraków until, having reached the age of seventy, he retired in 1924. In fact we note that the university decided not to fill his chair after he retired. Śleszyński's main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic. In a paper of 1892, based on his doctoral dissertation, he examined Cauchy's version of the Central Limit Theorem using characteristic function methods, and made several significant improvements and corrections. Because of the work, he is recognised as giving the first rigorous proof of a restricted form of the Central Limit Theorem. In 1898 Alfred Pringsheim proved that the condition ,ensures the convergence of the continued fraction , where and are complex numbers; a result now known as the Pringsheim criterion. W J Thron states in that this result was established ten years earlier by Śleszyński. Thron demonstrates that Pringsheim was aware of Śleszyński's work, though Pringsheim himself claims that he only became aware of Śleszyński after his article was completed. Six papers by Śleszyński on continued fractions are discussed in where a complete bibliography of Śleszyński's mathematical papers is given. His work on continued fractions is also discussed in . In Bednarowski discusses Śleszyński's book O Logice Tradycyjnej Ⓣ published in Kraków in 1921:- Śleszyński assumes that the part of traditional logic created by Aristotle is a theory of relations which may hold between two classes. He then askes the following question. Having two non-empty classes A and B, what are the possible relations between them so far as having elements in common is concerned? His answer is that between A and B there holds one and only one of five relations which he symbolises by a, b, g, d, e.Śleszyński then represents the five different situations by using Venn diagrams. In the two classes and coincide, in the class is properly contained in , in the class is properly contained in , in the classes , and intersect are all non-empty, and in the final case and are disjoint. Śleszyński then argues as follows. First he says that either and have common elements or they do not. If they do not then we have the situation . Next Śleszyński looks at the situation where common elements exist. Either one of or contains an element not in the other, or they do not. If they do not, then we have the situation . There remains the case where either one of or contains an element not in the other. If fails to contain an element not in we have . Otherwise contains an element not in . If also contains an element not in then otherwise . Śleszyński also goes on to consider what happens when empty classes are allowed and shows that three further relations occur. We should mention another interesting work by Śleszyński, namely On the significance of logic for mathematics (Polish) published in 1923. However, despite the interesting publications we have mentioned, Śleszyński did not publish much of his work. This was rectified by a major two-volume publication in the years following his retirement. One of Śleszyński's most famous students at the Jagellonian University of Kraków was Stanisław Zaremba. In 1925 Zaremba, acting as editor, published the first of two volumes of The theory of proof based on Śleszyński's lectures at Kraków. A second volume appeared in 1929. McCall writes in :- Much indeed can be learned from the rich collection of [Śleszyński's] papers on various subjects in the realm of formal logic, and of mathematical logic and its history ... Introduction to mathematical logic, complete proof, mathematical proof, exposition of the theory of propositions, the Boolean calculus, Grassmann's logic, Schröder's algebra, Poretsky's seven laws, Peano's doctrine, Burali-Forti's doctrine - these are some of the themes pursued in this work, from which I personally have learned a great deal and thanks to which I have got a clear idea of many an unclear thing.We end this brief biography by giving the following quote by Śleszyński:- The point of civilization is the exchange of ideas. And where is this exchange, if everybody writes and nobody reads? - S McCall, Polish Logic, 1920-1939: Papers by Ajdukiewicz Andothers (Oxford University Press US, 1967). - W Bednarowski, Hamilton's Quantification of the Predicate, Proc. Aristotelian Soc. 56 (1955-1956), 217-240. - J J Jadacki, Jan Sleszy'nski (Polish), Wiadom. Mat. 34 (1998), 83-97. - S N Kiro, I V Slesinskii's papers in the theory of continued fractions (Russian) in Continued fractions and their applications 106, Inst. Mat., Akad. Nauk Ukrain. SSR (Kiev, 1976), 61-62. - E Seneta, Jan Sleszy'nski as a probabilist (Polish), Wiadom. Mat. 34 (1998), 99-104. - W J Thron, Should the Pringsheim criterion be renamed the Sleszynski criterion?, Comm. Anal. Theory Contin. Fractions 1 (1992), 13-20. Additional Resources (show) Other websites about Ivan Śleszyński: Written by J J O'Connor and E F Robertson Last Update April 2009 Last Update April 2009

- Research Article - Open Access -Duality Theorems for Convex Semidefinite Optimization Problems with Conic Constraints © G.M. Lee and J.H. Lee 2010 - Received: 30 October 2009 - Accepted: 10 December 2009 - Published: 12 January 2010 A convex semidefinite optimization problem with a conic constraint is considered. We formulate a Wolfe-type dual problem for the problem for its -approximate solutions, and then we prove -weak duality theorem and -strong duality theorem which hold between the problem and its Wolfe type dual problem. Moreover, we give an example illustrating the duality theorems. - Approximate Solution - Feasible Solution - Convex Function - Linear Matrix Inequality - Constraint Qualification Convex semidefinite optimization problem is to optimize an objective convex function over a linear matrix inequality. When the objective function is linear and the corresponding matrices are diagonal, this problem becomes a linear optimization problem. For convex semidefinite optimization problem, Lagrangean duality without constraint qualification [1, 2], complete dual characterization conditions of solutions [1, 3, 4], saddle point theorems , and characterizations of optimal solution sets [6, 7] have been investigated. Recently, Jeyakumar and Glover gave -optimality conditions for convex optimization problems, which hold without any constraint qualification. Yokoyama and Shiraishi gave a special case of convex optimization problem which satisfies -optimality conditions. Kim and Lee proved sequential -saddle point theorems and -duality theorems for convex semidefinite optimization problems which have not conic constraints. The purpose of this paper is to extend the -duality theorems by Kim and Lee to convex semidefinite optimization problems with conic constraints. We formulate a Wolfe type dual problem for the problem for its -approximate solutions, and then prove -weak duality theorem and -strong duality theorem for the problem and its Wolfe type dual problem, which hold under a weakened constraint qualification. Moreover, we give an example illustrating the duality theorems. Consider the following convex semidefinite optimization problem: where is a convex function, is a closed convex cone of , and for , where is the space of real symmetric matrices. The space is partially ordered by the L wner order, that is, for if and only if is positive semidefinite. The inner product in is defined by , where is the trace operation. for any Clearly, is the feasible set of SDP. Let be a convex function. where is the scalar product on . If is sublinear (i.e., convex and positively homogeneous of degree one), then , for all . If , , , then . It is worth nothing that if is sublinear, then Moreover, if is sublinear and if , , and , then Let be a closed convex set in and . (1)Let . Then is called the normal cone to at . (2)Let . Let . Then is called the -normal set to at . (3)When is a closed convex cone in , we denoted by and called the negative dual cone of . Proposition 2.7 (see ). Following the proof of Lemma in , we can prove the following lemma. Now we give -duality theorems for SDP. Using Lemma 2.8, we can obtain the following lemma which is useful in proving our -strong duality theorems for SDP. for any . ( ) Suppose that there exists such that for any Thus , for any . Hence is an -approximate solution of SDP. Now we formulate the dual problem SDD of SDP as follows: We prove -weak and -strong duality theorems which hold between SDP and SDD. Theorem 3.2 ( -weak duality). Theorem 3.3 ( -strong duality). is closed. If is an -approximate solution of SDP, then there exists such that is a -approximate solution of SDD. for any . Letting in (3.14), . Since and , . Thus from (3.14), Thus is a 2 -approximate solution to SDD. Now we characterize the -normal set to . From Proposition 3.4, we can calculate . Let and Then following hold. (i)If , then (ii)If and , then (iii)If and , then Now we give an example illustrating our -duality theorems. that is, -weak duality holds. Let be an -approximate solution of SDP. Then and . So, we can easily check that . Since , from (3.29), for any . So is an -approximate solution of SDD. Hence -strong duality holds. This work was supported by the Korea Science and Engineering Foundation (KOSEF) NRL Program grant funded by the Korean government (MEST)(no. R0A-2008-000-20010-0). - Jeyakumar V, Dinh N: Avoiding duality gaps in convex semidefinite programming without Slater's condition. In Applied Mathematics Report. University of New South Wales, Sydney, Australia; 2004.Google Scholar - Ramana MV, Tunçel L, Wolkowicz H: Strong duality for semidefinite programming. SIAM Journal on Optimization 1997, 7(3):641–662. 10.1137/S1052623495288350MathSciNetView ArticleMATHGoogle Scholar - Jeyakumar V, Lee GM, Dinh N: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM Journal on Optimization 2003, 14(2):534–547. 10.1137/S1052623402417699MathSciNetView ArticleMATHGoogle Scholar - Jeyakumar V, Nealon MJ: Complete dual characterizations of optimality for convex semidefinite programming. In Constructive, Experimental, and Nonlinear Analysis (Limoges, 1999), CMS Conference Proceedings. Volume 27. American Mathematical Society, Providence, RI, USA; 2000:165–173.Google Scholar - Dinh N, Jeyakumar V, Lee GM: Sequential Lagrangian conditions for convex programs with applications to semidefinite programming. Journal of Optimization Theory and Applications 2005, 125(1):85–112. 10.1007/s10957-004-1712-8MathSciNetView ArticleMATHGoogle Scholar - Jeyakumar V, Lee GM, Dinh N: Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. Journal of Optimization Theory and Applications 2004, 123(1):83–103.MathSciNetView ArticleMATHGoogle Scholar - Jeyakumar V, Lee GM, Dinh N: Characterizations of solution sets of convex vector minimization problems. European Journal of Operational Research 2006, 174(3):1380–1395. 10.1016/j.ejor.2005.05.007MathSciNetView ArticleMATHGoogle Scholar - Govil MG, Mehra A: -optimality for multiobjective programming on a Banach space. European Journal of Operational Research 2004, 157(1):106–112. 10.1016/S0377-2217(03)00206-6MathSciNetView ArticleMATHGoogle Scholar - Gutiérrez C, Jiménez B, Novo V: Multiplier rules and saddle-point theorems for Helbig's approximate solutions in convex Pareto problems. Journal of Global Optimization 2005, 32(3):367–383. 10.1007/s10898-004-5904-4MathSciNetView ArticleMATHGoogle Scholar - Hamel A: An -lagrange multiplier rule for a mathematical programming problem on Banach spaces. Optimization 2001, 49(1–2):137–149. 10.1080/02331930108844524MathSciNetView ArticleMATHGoogle Scholar - Jeyakumar V, Glover BM: Characterizing global optimality for DC optimization problems under convex inequality constraints. Journal of Global Optimization 1996, 8(2):171–187. 10.1007/BF00138691MathSciNetView ArticleMATHGoogle Scholar - Kim GS, Lee GM: On -approximate solutions for convex semidefinite optimization problems. Taiwanese Journal of Mathematics 2007, 11(3):765–784.MathSciNetMATHGoogle Scholar - Liu JC: -duality theorem of nondifferentiable nonconvex multiobjective programming. Journal of Optimization Theory and Applications 1991, 69(1):153–167. 10.1007/BF00940466MathSciNetView ArticleMATHGoogle Scholar - Liu JC: -Pareto optimality for nondifferentiable multiobjective programming via penalty function. Journal of Mathematical Analysis and Applications 1996, 198(1):248–261. 10.1006/jmaa.1996.0080MathSciNetView ArticleMATHGoogle Scholar - Strodiot J-J, Nguyen VH, Heukemes N: -optimal solutions in nondifferentiable convex programming and some related questions. Mathematical Programming 1983, 25(3):307–328. 10.1007/BF02594782MathSciNetView ArticleMATHGoogle Scholar - Yokoyama K, Shiraishi S: An -optimal condition for convex programming problems without Slater's constraint qualifications. preprintGoogle Scholar - Hiriart-Urruty JB, Lemarechal C: Convex Analysis and Minimization Algorithms. I. Fundamentals, Grundlehren der mathematischen Wissenschaften. Volume 305. Springer, Berlin, Germany; 1993.View ArticleGoogle Scholar - Hiriart-Urruty JB, Lemarechal C: Convex Analysis and Minimization Algorithms. II. Advanced Theory and Bundle Methods, Grundlehren der mathematischen Wissenschaften. Volume 306. Springer, Berlin, Germany; 1993.Google Scholar This article is published under license to BioMed Central Ltd. 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.

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Statistics and uncertainity lab report The meaning of uncertainty statements in laboratory reports previous knowledge of statistical metrology on the part of the reader and involves no mathe. Visual physics laboratory, texas a&m university, college station, tx 77843 there will always be some error or uncertainty present report for that quantity is represented by the average of the measurements statistical analysis shows that about 683% of them will fall within the interval between. Reporting measurements and experimental results statistics is required to get a more sophisticated estimate of the uncertainty the new experiment will be less than one standard error away from the average value from this experiment. These e-lab tools allow you to construct numerical examples to understand the stigler s, the history of statistics: the measurement of uncertainty before 1900, business statistics provides justifiable answers to the following concerns for. 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Integrable Models From Twisted Half Loop Algebras [4mm] N. Crampé111 and C. A. S. Young222 Department of Mathematics, University of York, Heslington Lane, York YO10 5DD, UK This paper is devoted to the construction of new integrable quantum-mechanical models based on certain subalgebras of the half loop algebra of . Various results about these subalgebras are proven by presenting them in the notation of the St Petersburg school. These results are then used to demonstrate the integrability, and find the symmetries, of two types of physical system: twisted Gaudin magnets, and Calogero-type models of particles on several half-lines meeting at a point. This paper has two motivations. On the one hand, we are interested in physical models of particles on a number of half-lines joined at a central point. Such systems, for free particles, have been treated in, for example, [1, 2]. Here we would like to consider interacting models, to establish that integrable examples of such models exist, and to find their symmetries. We shall work out explicitly two examples: the Gaudin model and the Calogero model . Both have numerous applications in physics and in mathematics. For example, the reduced BCS model for conventionnal superconductivity can be diagonalized in an algebraic way using the Gaudin model. Other, more recent, applications of the Gaudin model in quantum many-body physics can be found for example in the reviews [6, 7]. Besides being of intrinsic interest due to its exact solvability, the Calogero model plays a role in the study of two dimensional Yang-Mills theory , the quantum Hall effect and fractional statistics . Our second motivation is algebraic. The notation of the St Peterburg school is a powerful tool when working with the Yangian of and its subalgebras: the reflection algebras [13, 14] and twisted Yangians . These are quantum algebras, but the construction has a classical limit in which the quantum -matrix and Yang-Baxter Equation are replaced by their classical counterparts (see for example ). The classical limit of the Yangian is the half loop algebra, and the limits of the reflection algebras and twisted Yangian are subalgebras of this half loop algebra defined by automorphisms of order 2. But there also exist, at least in the classical case, other subalgebras of the half loop algebra, defined by automorphisms of higher finite order. We wish to study these subalgebras using classical -matrix techniques. It is well-known that the half-loop algebras associated to Lie algebras are crucial in the study of Gaudin models and Calogero models. These algebras provide, in the former case, a systematic way to construct the model (see e.g ) and, in the latter case, the symmetry algebras of the system [18, 19, 20]. In both cases, they allow one to prove the integrability of the model . We shall find similar connections in the cases studied in this paper. Indeed, we shall see below that the order subalgebras of the half loop algebra appears naturally in the description of models on half-lines. This paper is structured as follows. We begin with a brief review of the half loop algebra of and its subalgebras associated to automorphisms of order . We make use of the notation of the St Petersburg school to find Abelian subalgebras. In the subsequent sections these algebraic results are shown to provide new quantum integrable models and demonstrate their symmetries: section 3 discusses “twisted” Gaudin magnets, and section 4 introduces Calogero-type models on half-lines joined a central point. We end with some conclusions and a short discussion of classical counterparts of these results. 2 Half loop algebra and subalgebras 2.1 St Petersburg notation and half loop algebra The half loop algebra based on is the complex associative unital algebra with the following set of generators , subject to the defining relations for and . It is isomorphic to the algebra of polynomials in an indeterminate with coefficients in , with the generators identified as follows: where are the generators of , satisfying the commutation relations It will simplify our computations to introduce the notation of the St Petersburg school: let be the matrix with a in the th slot and zeros elsewhere. These are the generators of in the fundamental representation. Let us now gather the generators of in the matrix where () and is a formal parameter called the spectral parameter. Note the flip of the indices between and , which will prove convenient later. The algebraic object is an element of , and as usual we refer to as the auxiliary space and as the algebraic space. In what follows we shall require several copies of both spaces. We use letter from the start of the alphabet to refer to copies of the auxiliary space and numerals for copies of the algebraic space. Let us introduce also where is the permutation operator between two auxiliary spaces: the letters and stand respectively for the first and the second spaces. By definition, it satisfies (). The matrix , usually called the classical R-matrix (see for example ), satisfies the classical Yang-Baxter equation and allows us to encode the half loop algebra defining relations (2.1) in the simple equation This form of commutation relations can be obtained easily by taking the classical limit of the presentation of the Yangian of introduced by L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan of St Petersburg . By taking the trace in the space in (2.7), it is straightforward to show that the coefficients of the series are central. The quotient of the algebra by the relation is isomorphic to the polynomial algebra . The identification (2.2) between the generators of and now reads and is to be understood as the formal series . Note the similarity between relations (2.5) and (2.8): the only differences are that the second auxiliary space (denoted ) in (2.5) is replaced by an algebraic space (denoted ) and that the spectral parameter is shifted by . In fact, there exists a more general solution of the relations (2.7) in the -fold tensor product of , From now on, we work in the enveloping algebra in which, for example, the product makes sense. 2.2 The inner-twisted algebras Let be an inner automorphism of of order . One way to define is by its action on matrices in the fundamental representation: where satisfies ; the action of on the abstract algebra is then given by , or, in the notation of the previous section, The eigenvalues of are the -th roots of unity , and for each the map is the projector onto the -eigenspace: Since , decomposes into the direct sum of eigenspaces of . This decomposition respects the Lie bracket, in the sense that if and then and is said to be a -gradation of . By a change of basis we can take where . Note that the -eigenspace of is the Lie subalgebra . Let us define that is, is the subalgebra of in which each element of degree is also in the -eigenspace of . There is a surjective projection map , defined by . In view of (2.12), this sends which defines the formal series whose expansion contains by construction a complete set of generators of . and has the property that for all The coefficients in the expansion of are central in , as may be seen by taking the trace in space or in (2.20). But there exist also other abelian subalgebras in , as follows. The coefficients in the expansion of are mutually commuting, or equivalently for all values of and . Moreover, they commute with the generators of of degree zero: The algebraic elements in generate . Proof. The details of the proof are given in appendix A. In particular, we recover (for ) the fact that commute and (for ) the results of Hikami concerning the classical limit of the reflection algebra. 2.3 Outer Automorphisms In the previous section, we focused on inner automorphisms. Now, we show how to modify the construction to study outer automorphisms. Modulo inner automorphisms, the only outer automorphism of is generalized transposition, which has order 2. Let be a real invertible matrix satisfying with (for , must be even), and define an outer automorphism by , or equivalently where is matrix transposition in the space . The eigenvalues of are and, as before, the decomposition of into the direct sum of eigenspaces of defines a -gradation. One may introduce the matrices where and the second case is valid only for even. A well-known result in linear algebra is then that is congruent over the reals to , i.e. for some real matrix . From this one sees that the -eigenspace of is the Lie subalgebra for and for . Once more we may now define that is, the subalgebra of in which each element of degree is also in the -eigenspace of . The projection map is , and, given (2.24), this sends which defines the formal series , whose expansion in inverse powers of contains a complete set of generators of . The commutation relations of this subalgebra can be written simply by using the notation with the formal series. where and has the symmetry property that Note that these commutation relations can be obtained from the classical limit of the twisted Yangian introduced in . More abstractly, the relations (2.29) and (2.30) can be regarded as defining an algebra, which can then be seen to be embedded in the half loop algebra according to (2.27). It is well-known that the centre of this subalgebra is generated by the odd coefficients of the series (see for example , section 4). But we have also The quantities in the expansion of are mutually commuting, or equivalently for all values of and . Moreover, The elements in generate for and for . Proof. The details of the proof are given in appendix B. 3 Gaudin models 3.1 The Inner-twisted Gaudin Magnets The quantum Gaudin magnet, introduced in , is an integrable spin chain with long range interactions. The Gaudin Hamiltonians for the model with sites are where are complex numbers. (Recall that permutes the and spins.) This model is usually called the -type Gaudin model. It may be obtained from the more general class of integrable Hamiltonians by specifying that the spin at each site is in the fundamental representation of . Now, given proposition 2.2 above, we can obtain new integrable models, as in the following proposition. These models describe spins placed at fixed positions in the plane, each of which interacts with the central point and with the other spins, not only directly, but also via their images under the rotation group of order . The model described by any one of the Hamiltonians is integrable. This model has symmetry. Proof: From the definition (2.18) of , one finds with as given in the proposition. (The identity for is helpful in showing this.) It then follows from proposition 2.2 that . Since (for ) these operators are independent we have found commuting conserved quantities, completing the proof of integrability of the Hamiltonian . Next, from the relation , also proved in proposition 2.2, we deduce that , which gives the symmetry of the model. For , we obtain the Hamiltonian of the BC-type Gaudin model studied in . If the sites carry the fundamental representation of , our Hamiltonian is Let us remark that in the case () supplementary conserved quantities, called higher Gaudin Hamiltonians, can be found by computing for example (see e.g. ). The question of whether this is possible in the generalized cases () studied here remains open. 3.2 The Outer-twisted Gaudin Magnets Using the algebraic result of proposition 2.4, we can also succeed in constructing integrable models based on outer automorphisms, as follows: The model described by any one of the Hamiltonians is integrable. The model has symmetry (resp. symmetry) for (resp. ). with given as in the proposition. Then, we deduce from proposition 2.4 that , and since the operators are independent for different , this proves the integrability of . The symmetry algebra is deduced from proved in the proposition 2.4. Every choice of representation for the sites then yields a Gaudin-type model. (It is worth remarking that it is possible to choose different representations at different sites.) For example, in the fundamental representation of , the Hamiltonian is We may interpret as a Gaudin model with boundary as in the BC type model (equation (3.6), and see also ). The term in (3.8) corresponds to the interaction between the spin represented in and the ‘reflected’ spin transforming in the contragredient representation. This type of boundary is called soliton non-preserving and has been implemented in other integrable models [23, 24, 25, 26]. The final term in (3.8) corresponds to the interaction between particles and the boundary. 4 Calogero Models We turn now to the second class of integrable system of interest in this work, the Calogero models. We seek to construct dynamical models of multiple particles on a star graph, whose pairwise interactions are determined by a potential of the usual Calogero type, namely , where is the linear distance separating the particles in the plane of the star graph. We will first construct models of particles of unspecified statistics; subsequently, by specifying statistics and parity, we arrive at Calogero models for particles with internal spins. 4.1 The case Let us first recall the Calogero model based on the root system , and in particular the use of Dunkl operators in demonstrating its integrability . Consider a quantum mechanical system of particles on the real line. Let be the position operator of the particle, and write the position-space wave function as Let be the operator which transposes the positions of particles and , Let us denote the permutation group of elements and the transposition of the elements and . Each element can be written in terms of transpositions, namely . Then, we can define as the shorthand for the product (even though the expression of in terms of transpositions is not unique, is well-defined due to the commutation relations satisfied by ). The sign of , denoted , is the number of these transpositions modulo 2. It follows from the relations that the Dunkl operators commute with one another, and consequently that the quantities form a commuting set also. The are algebraically independent for , and these give commuting conserved quantities of the model with Hamiltonian which is therefore, by construction, integrable. The next step is to consider particles with internal degrees of freedom, which we take to be in the fundamental representation of . The wave funtion becomes where . As we define operators which transpose the positions, we introduce operator which transposes the spins We define similarly to the matrix for acting on the spins. As explained before, to use the St Petersburg notation, we need supplementary spaces called auxiliary spaces (which are and, in this case, isomorphic to the quantum space) and denoted by the letters , ,… The conserved quantities (4.5) then emerge in a natural way from the matrix because (as one can see using ) Here (4.9) is nothing but a modified version of the monodromy matrix (2.10). The parameters are replaced by the Dunkl operators, and since the quantum spaces are chosen to be in the fundamental representation, becomes the transposition operator (for ). Now because the commute with each other and with all operations on the internal degrees of freedom, obeys the half loop algebra relations (2.7) exactly as before. Suppose, finally, that the particles are in fact indistinguishable, which is often the case of real physical interest. One must then impose definite exchange statistics on the wavefunction: where for bosons and for fermions. The projector onto such states is The following relation demonstrated in is crucial, because it implies that the modified generators preserve the condition , and obey the same algebraic relations as the original . From we may define , and hence . Using , one obtains that the are once more commuting conserved quantities of the system with Hamiltonian where we are now able to replace , which acts on particle positions, by , which acts only on the internal degrees of freedom. Moreover, since commutes with , the model has a half loop symmetry algebra. The subtlety in all this is that the Dunkl operators themselves do not obey any relation analogous to (4.13). There are thus essentially three steps in this procedure to construct an integrable Hamiltonian for a system of indistinguishable particles: Find commuting Dunkl operators, and hence Construct the appropriate projector onto physical states, Prove the relation . 4.2 Dunkl Operators for the order inner-twisted case We can now turn to applying these ideas to the model of interest in the present work. We consider a system of particles living on half-lines – “branches” – joined at a central node, as in figure 1. The branches are given parametrically by , , and we shall denote them by As before, let be the position operator of th particle. (Note that the spectrum of is not real, but only for the superficial reason that we choose to regard the half-lines as subsets of the complex plane.) In addition to the , which exchange particle positions, we can define now new operators which move the particles between branches: It is useful to collect together the algebraic relations satisfied by the , , and : with all the rest commuting. To construct an integrable model, the first task is to find a suitable generalization of the commuting Dunkl operators introduced above. The Dunkl operators defined by for arbitrary parameters , commute amongst themselves: Consider first the terms at order . We have using the relations (4.19) and the definition , which together imply . The two terms of this type occurring in are which cancel, after a change of the summation index in the second. The two terms containing cancel similarly. The terms occurring at order are of the form These vanish trivially unless at least one of the indices matches at least one of . It is straightforward, though tedious, to check that the terms with exactly one index in common sum to zero, by using the relations (4.19) to bring every such term into e.g. the form and then summing the fractions directly. The terms in which both indices match give and here the sum over may be re-written as which then vanishes by shifting the dummy index in the second and fourth terms. The terms involving may be treated similarly. These Dunkl operators have been introduced previously in as Dunkl operators associated to complex reflection groups. A proof of their commutativity is already given but is based on different computations. As in the case above, the quantities are then mutually commuting, forming a hierarchy of Hamiltonians of an integrable system. Their detailed forms are rather complicated – for example, in the case of branches with only particles and , we find that the first three are

We now delve into electrostatics to estimate the electrostatic polarization free energy, , involved in the transfer of a solute with an arbitrary charge distribution from vacuum to aqueous solution. is the interaction between the charge distribution and its reaction potential, the potential induced by the charge distribution in the presence of the dielectric boundary at the solute-solvent interface. First, we review some basics. Again, we will focus on important results, and leave most of the mathematical details to textbooks . Don't worry if the equations are unfamiliar; just stay tuned for the punch line. All problems in electrostatics boil down to the solution of a single equation, Poisson's equation: Let's examine two model systems, a point charge and a point dipole, each immersed in a dielectric medium. In the following two boundary-value problems, we simply state the answer, giving as a function of position for all points in space. In these problems, we seek in regions were there is no charge (). Thus, we need solutions to the special case of Poisson's equation known as Laplace's equation, , that satisfy two boundary conditions. First, must be a continuous function, e.g., at the dielectric boundary. Second, because there is no `free' charge (charge other than the induced polarization charges) at the dielectric boundary, the normal component of the electric displacement, , will also be continuous at this boundary. First, we model a single ion in solution as a sphere of radius a with a point charge q at its center, immersed in a solvent of dielectric constant . Aside from the point charge at the center, there is nothing inside the solvent-exclusion cavity, and so the dielectric constant inside is the permittivity of free space . The spherical symmetry of this system renders it a problem of only one dimension, the distance r from the point charge. The solution is: One step up in complexity from a point charge is a point dipole. So let's replace the point charge at the center of our solvent-exclusion sphere with a point dipole . With this model system we can approximate the solvation energy of a neutral molecule possessing a permanent dipole moment. Again, the dielectric constant of the solvent is , and the dielectric constant inside the spherical molecule is . This cylindrically symmetric system has two independent dimensions, the distance r and the angle from the direction of the dipole vector. We get : Note that in equations 6 and 7 the potential inside the spherical molecule is a sum of two terms. In each case, the first term is the potential that would exist in the absence of the dielectric boundary at , and the second term is the potential induced in the spherical cavity by the charge distribution's interaction with the dielectric (e.g., the solvent). The energy of the charge distribution arising from this second term (the reaction potential) gives the electrostatic contribution to the solvation free energy. The energy of a point charge in its reaction potential is one half of the product of the charge and the reaction potential. The `one half' appears because this is not the energy of a charge in an external electric field. Here, the charge has contributed to the creation of the field through its electrostatic interactions with the dielectric. So our continuum model of the solvent predicts that the electrostatic polarization free energy of solvating a spherical ion is The energy of a dipole in its reaction field (the negative gradient of the reaction potential) is minus one half of the dot product of the dipole and the reaction field. Again, this is half the energy of a dipole in an external electric field. The reaction field of the dipole is parallel to the dipole, and we get Note that both G values are zero if , i.e., if we haven't changed the dielectric constant of the environment. Still and coworkers have proposed the following approximate expression for the free energy of solvent polarization for an arbitrary charge distribution of N charges: As shown in the following figure, this GB approximation behaves appropriately in important limiting situations. For N identical, coincident () particles of charge q, it gives the correct Born energy (equation 8, for a single particle of charge ). For two charges of equal and opposite sign, it approaches the dipole result (equation 9) at short separation distances, as it should. For two well separated charges ( ), it approaches the appropriate energy: the two Born energies plus the energetic change in the Coulomb interaction between the two charges due to the dielectric medium. |GB approximation (in red) to solvent polarization energies for two charges of equal radii as a function of separation. The dependence of Born radii on atomic positions is neglected here. Upper curves: Equal and opposite charges with G_dipole (blue) at small separation and Coulomb + Born polarization energies (green) at large separation. Lower curves: Equal charges with G_ion (blue) at zero separation and Coulomb + Born polarization energies (green) at large separation.|

Geometry has many practical applications in everyday life area and volume of geometric figures geometry : polygons, real-life applications of right triangles, volume of a pill capsule. Applications of geometry in the real world include computer-aided design for construction blueprints, the design of assembly systems in manufacturing, nanotechnology, computer graphics, visual graphs. Topic: real life applications- applications of coordinate geometry- application question 4 subject: mathematics grade: ix in this video we solve a question. Geometry and its applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric. Architecture is one of the applications of geometry that is crucial because architects are responsible for building sound structures and they must use geometric components to do so. 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The applications of geometry in real life are not always evident to teenagers, but the geometry was recognized to be not just for mathematicians anyone can benefit from the basic learning of. Applications of geometry 1 geometry in real life 2 definition noun merriam-webster dictionary a 18 a graphic designer studies how basic geometric shapes combine into artistic visual layouts in two. Before exploring applications of fractal geometry in ecology, we must first define fractal geometry a mathematician who works in the field of geometry is called a geometer. The international journal of computational geometry and applications (ijcga) is a bimonthly journal published since 1991, by world scientific it covers the application of computational geometry in design and analysis of algorithms. Applications of hyperbolic geometry mapping the brain spherical, euclidean and hyperbolic geometries in mapping the brain all those folds and fissures make life difficult for a neuroscientist. Geometry is a one of the important study of mathematics it is foundation for learning geometry objects (two dimensional objects, three dimensional objects)geometry is study of size and shapes of an. There are tons of real life applications of geometry geometry is defined as the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids. Geometry & applications (ijcga) is a bimonthly journal devoted to the field of computational geometry within research findings or experiences in the implementations of geometric algorithms. Geometrical applications of calculus (1 of 4: an introduction to the applications of calculus) - продолжительность: 9:57 eddie woo 1 897 просмотров. Published by balkan society of geometers geometry balkan press, bucharest, romania i would like to recommend the balkan journal of geometry and its applications (issn 1224-2780. Topic: real life applications- applications of coordinate geometry- application question 4 topic: applications of coordinate geometry subject: mathematics grade: ix in this video we solve. Spherical geometry is also known as hyperbolic geometry and has many real world applications one of the most used geometry is spherical geometry which describes the surface of a sphere.

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It contains a whole new chapter on the physics of music as well as several new sections such as those discussing the scaling in phase transitions, coupled nonlinear oscillators, two Find all the study resources for Computational Physics with Python by Mark Newman us to both broaden and deepen our understanding of physics by vastly increasing the range of mathematical calculations which we can conveniently perform. Download with Google Download with Facebook or download with email. Emphasis is given to methods based on Volterra series representations, the proper orthogonal decomposition, and harmonic balance. I would like to thank both of them sincerely for their interest, hospitality and many useful discussions while I was at Purdue. 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Notice that the axes are labeled differently than we are used to seeing in the sketch of D. Chapters 3 and 4 add the details and rigor. The Fundamental Theorem of Calculus 14 1. Most tests are given without answers. Calculus, 7th Edition This series is designed for the usual three semester calculus sequence that the majority of science and engineering majors in the United States are required to take. Erdman Background 3 1. This lesson course includes video and text explanations of everything from Calculus 3, and it includes quizzes with solutions! MIT Professor Gilbert Strang has created a series of videos to show ways in which calculus is important in our lives. Eventually I may penalize you for the failure to use full sentences. Functions8 4. The techniques of solving Double Integrals with a focus on how to construct a double integral over a given region. This page contains links to calculus tests offered at UAB in the past, according to the syllabus adopted at that time. Calculate the given quantity if Let u, v be in V3; that is, vectors of 3 components. This is a very condensed and simplified version of basic calculus, which is a prerequisite for many Math Calculus 3 Lecture Videos These lecture videos are organized in an order that corresponds with the current book we are using for our Math, Calculus 3, courses Calculus, with Differential Equations, by Varberg, Purcell and Rigdon, 9th edition published by Pearson. Write equations of lines and planes in space. The following video provides an outline of all the topics you would expect to see in a typical Multivariable Calculus class i. Concepts and Contexts, 4th ed. November 22, Rules for Finding Derivatives. Apply the concepts of calculus to vector-valued functions and use such functionsCalculus III Publisher: Cognella Date: English Pages: Here are a set of practice problems for the Calculus III notes. Either may be used for Calculus I and II. Please select one better suit your needs. Revision of vector algebra, scalar product, vector product 2. Computational physics newman solutions manual Rates of change17 5. Now make a further change of variables well adapted to the situation. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. The videos, which include real-life examples to illustrate the concepts, are ideal for high school students, college students, and anyone interested in learning the basics of calculus. MATH Stewart, J. All rights reserved. It was submitted to the Free Digital Textbook Initiative in California and will remain unchanged for at least two years. Classwork, 3 units; laboratory, 1 unit. Calculus III with applications. Does the value for pi have to be entered everytime it is used? Calculus Iii For Dummies Pdf 10 9 8 7 6 5 4 3 2. Typically, we have to Calculus I or needing a refresher in some of the early topics in calculus. These notes do assume that the reader has a good working knowledge of Calculus I …learn Calculus III or needing a refresher in some of the topics from the class. To make studying and working out problems in calculus easier, make sure you know basic formulas for geometry, trigonometry, integral calculus, and differential calculus. View Calculus 3 Textbook. I have tried to be somewhat rigorous about proving Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Page 2. Then they compare their results with real data. Calculus I and II. What is Calculus 3? A Quick Overview. Shed the societal and cultural narratives holding you back and let free step-by-step Calculus Volume 3 textbook solutions reorient your old paradigms. The daily Here is the best resource for homework help with MAC This is the free digital calculus text by David R. Thomas Calculus 3 12th Edition Pdf. Calculus with Vector Functions. Harvard College Math 21a: Lamar University Number of pages: Homework consists of Level 1, Level 2, and Mixed Review problems. Write this number using the min notation. Basic properties of vector operations p. You may ask your instructor to check your answers if you use the test problems for practice. Eric Sullivan Calculus: Early Transcendentals, Briggs, First edition. The prerequisites are the standard courses in single-variable calculus a. Intuitive Idea Curvature is a measure of instantaneously how much a curve Introduction to Mathematica Calculus III In this lab, you will learn the basics of using Mathematica to evaluate expressions, plot functions and work with vectors. Topics include vectors, three-dimensional analytic geometry, partial differentiation and multiple integrals, and vector analysis.Derivatives and the Mean Value Theorem 3 4. Eventually I may penalize you for the failure to use full sentences. The study of networks received much interest in recent years. It explains the fundamentals of computational physics and describes in simple terms the techniques that every physicist should know, such as finite difference methods, numerical quadrature, and the fast Fourier transform. The College Board Subject: These skills are organized into categories, and you can move your mouse over any skill name to preview the skill. Students solve differential equations to find functions to model the value of a car in terms of its age. - FINANCIAL AND MANAGERIAL ACCOUNTING 17TH EDITION PDF - PRECALCULUS JAMES STEWART 5TH EDITION PDF - RITA MULCAHY CAPM EXAM PREP 3RD EDITION PDF - CCNA SECURITY LAB MANUAL VERSION 1.2 3RD EDITION EPUB - CALCULUS EARLY TRANSCENDENTALS 7TH EDITION SOLUTIONS MANUAL PDF - JAMES PATTERSON EPUB COLLECTION - PAUL DUBOIS MYSQL 4TH EDITION PDF - A POCKET STYLE MANUAL 6TH EDITION PDF - RULES COMPENDIUM 3.5 PDF - MICROSOFT POWERPOINT 2003 TUTORIAL PDF

Home / retaining wall hydrostatic pressure formula Hydrostatic Pressure on Basement Retaining Walls. What hydrostatic loading should be considered to act on the wall and how high up should the hydrostatic load be applied. Should hydrostatic pressure loading be considered to act all the way to the top of the wall, which is 19' above where the water table was measured at the time of the survey. Hydrostatic Force on a Wall: dp = g dz . 2.16 Integrating in the z-direction we get pressure as a function of depth: p = pa gz 2.19 Note that z is defined as zero at the surface, so pressure increases with depth since z < 0 under water with a constant slope, g . Pressure Computations. Last Revised: 11/04/2014. ASCE 7-05 3.2 treats both lateral soil loads and hydrostatic pressure similarly. Both increase linearly with depth according to the equation: lateral pressure at depth h = g h. This is common in cantilevered retaining walls. It becomes less so if the wall is restrained against movement in Definition. A retaining wall is a structure designed and constructed to resist the lateral pressure of soil, when there is a desired change in ground elevation that exceeds the angle of repose of the soil. A basement wall is thus one kind of retaining wall. But the term usually refers to a cantilever retaining wall, Retaining Wall Variables. Magnitude of stress or earth pressure acting on a retaining wall depends on: height of wall, unit weight of retained soil, pore water pressure, strength of soil angle of internal friction , amount and direction of wall movement, and. other stresses such as earthquakes and surcharges. Retaining Wall: Saturated Soil and Hydrostatic Pressure. Ask Question 2 The hydrostatic pressure and the soil load act on the wall concurrently. However, as you noted, the soil load is reduced because the effective weight of the soil is reduced due to buoyant forces. Percentage area of steel for a retaining wall - standard formula doesn't To understand how hydrostatic pressure can effect a retaining wall, one must fully understand the function of such a wall. Typically, a retaining wal l is a structure created from pre-cast or formed cement blocks that supports a mass of earth on one side in order to maintain two levels of elevation in one area. Finds the equivalent force and action point of hydrostatic pressure on a wall, which is an example of a distributed force. Distributed Force-Hydrostatic Pressure on Wall Darryl Morrell The thrust applied by water is considered to be acting at a distance of H/3 from the bottom of the retaining wall. The pressure distribution is triangular and has the maximum pressure of 2P/H at the bottom of the wall. hydrostatic pressure and dynamic water pressure acting on a structure should be calculated separately. 1 Earth Pressure Relating to Item 1 of the Public Notice Above Fig. 1.2.1 Schematic Diagram of Earth Pressure Acting on Retaining Wall CHAPTER 5 EARTH PRESSURE AND WATER PRESSURE. I am working on retaining wall structures Hydrostatic Pressure. Ron Structural 10 Aug 10 12:54. For water only, the pressure is unit weight of water * depth of water acting at 2/3 of the depth triangular pressure distribution The formula you showed for calculation the soil force, which i have to say is the resultant force comes from. What is the cause of hydrostatic pressure behind a retaining wall? How does hydrostatic pressure effect a retaining wall? What is the cause of hydrostatic pressure behind a retaining wall? How Earth Pressure and Retaining Wall Basics for Non-Geotechnical Engineers Richard P. Weber Course Content Content Section 1 Retaining walls are structures that support backfill and allow for a change of grade. For instance a retaining wall can be used to retain fill along a slope or it can be used to CHAPTER 9 EARTH PRESSURE AND HYDRAULIC PRESSURE - C9-1 - In general, earth pressure acting on a retaining wall is assumed to be active earth pressure, and is calculated by using, for example, Mononobe and Okabe's formula for active earth pressure during an earthquake. We know that water exerts a pressure on the wall and this thrust is calculated by using the following formula. P = ½Y o H 2 The thrust applied by water is considered to be acting at a distance of H/3 from the bottom of the retaining wall. The pressure distribution is triangular and has the maximum pressure of 2P/H at the bottom of the wall. Cantilever Retaining Walls: How to Calculate the Bearing Pressure By: Javier Encinas, PE July 25, 2017 A retaining wall is a structure exposed to lateral pressures from the retained soil plus any other surcharges and external loads. Another method for relieving hydrostatic pressure is to install a drainage pipe behind the wall. This should be a perforated pipe, to allow water to enter it through the length of the wall. The pipe can be located just above the footing, or can be located at a higher elevation. The retaining wall is checked for stability: Application of Lateral Earth Pressure Theories to Design Retaining Wall Stability 1 Safety Against Overturning Rotational stability : PV PH obtain cubic equation in terms of d. Solve for d. Increase d by 20% in quay walls.

3 edition of Applied Engineering Mathematics found in the catalog. February 20, 2007 by Cambridge International Science Publishing Written in English |The Physical Object| |Number of Pages||336| This book offers the latest research advances in the field of mathematics applications in engineering sciences and provides a reference with a theoretical and sound background, along with case studies. In recent years, mathematics has had an amazing growth in engineering sciences. It forms the common foundation of all engineering disciplines. This new book provides a comprehensive range of. Higher Engineering Mathematics by B. S. Grewal. Highlights of the book: Good for undergraduate mathematics and GATE preparation. It covers all topics required for GATE and other exams. Theory, examples, problems are provided for each chapter. Best book for competitive exams; Essential Engineering Mathematics by Michael Batty. Highlights of the. 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Elementary Topology In applied mathematics, we are often faced with analyzing mathematical structures as they might relate to real-world phenomena. A Textbook Of Practical Medicine V2 etre du balbutiement outline of social psychology Current trends in chemical engineering Bloodtrail to Mecca Rotations and angles Changing men, transforming culture Macraes Blue Bk 104/E The fall, & Exile and the kingdom. Electric transmission infrastructure and investment needs Medicine in its human setting The letters of Ralph Waldo Emerson Book Description Undergraduate engineering students need good mathematics skills. This textbook supports this need by placing a strong emphasis on visualization and the methods and tools needed across the whole of engineering. The visual approach is emphasized, and excessive proofs and derivations are avoided. This book can serve as a textbook in engineering mathematics, mathematical modelling and scientific computing. This book is organised into 19 chapters. 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In the world of physics and mathematics, understanding the concept of tangential acceleration is crucial. It plays a significant role in analyzing the motion of objects in circular or rotational motion. In this blog post, we will explore the concept of tangential acceleration in detail, including its definition, importance, and how to calculate it in various scenarios. So, let’s dive in! How to Find Tangential Acceleration Definition of Tangential Acceleration tangential acceleration refers to the rate at which the tangential velocity of an object changes over time in a circular or rotational motion. It is a measure of how quickly an object’s speed or direction changes along the circular path it follows. In simple terms, it represents the acceleration experienced by an object moving in a circle. Importance of Tangential Acceleration in Physics and Mathematics tangential acceleration is essential in understanding the dynamics of rotational motion. It helps us analyze and predict how objects move in circular paths, such as planets orbiting the sun, cars taking turns on a racetrack, or even the motion of a spinning top. By considering tangential acceleration, we can determine the forces acting on an object, its velocity, and how it responds to external influences. The Formula to Find Tangential Acceleration The formula to calculate tangential acceleration depends on various factors, including angular acceleration, time, and linear velocity. It can be expressed as: – represents the tangential acceleration – is the radius of the circular path – denotes the angular acceleration Now that we have a clear understanding of tangential acceleration let’s explore how to calculate it in different scenarios. How to Calculate Tangential Acceleration Calculating Tangential Acceleration from Angular Acceleration To calculate tangential acceleration from angular acceleration, we can use the formula mentioned earlier: . Let’s consider an example to illustrate this: Suppose a particle is moving in a circular path with a radius of 3 meters, and it experiences an angular acceleration of 2 rad/s². To find the tangential acceleration, we can apply the formula: Hence, the tangential acceleration is 6 m/s². Finding Tangential Acceleration Given Time Sometimes, we may need to calculate tangential acceleration when the time is given. In such cases, we can use a different formula based on the initial angular velocity, the angular acceleration, and the time. The formula is: – (a_t) represents the tangential acceleration – (\omega_0) is the initial angular velocity – (\alpha) denotes the angular acceleration – (t) is the time Let’s consider a scenario where an object starts from rest and experiences an angular acceleration of 5 rad/s² for a duration of 2 seconds. The initial angular velocity is 0. By substituting the given values, we can calculate the tangential acceleration: Hence, the tangential acceleration is 10 m/s². Determining Tangential Acceleration Without Time In some cases, we may need to determine the tangential acceleration without knowing the time duration. In such situations, we can use equations that involve the angular velocity , the radius (r), and the tangential acceleration (at). One such equation is: Suppose an object is moving in a circular path with a radius of 2 meters and has an angular velocity of 3 rad/s. To find the tangential acceleration, we can use the formula: Therefore, the tangential acceleration is 18 m/s². Now that we have covered the basics of calculating tangential acceleration, let’s explore how to solve it in different scenarios. How to Solve for Tangential Acceleration in Different Scenarios Finding Tangential Acceleration in Circular Motion When dealing with circular motion, tangential acceleration is an important parameter to consider. It helps us understand how objects accelerate along the circular path. In circular motion, tangential acceleration is always directed towards the center of the circle. The magnitude of tangential acceleration depends on factors like angular acceleration, radius, and linear velocity. Determining Tangential Acceleration of a Pendulum A pendulum is an excellent example where tangential acceleration comes into play. When a pendulum swings back and forth, the bob experiences tangential acceleration. The magnitude of tangential acceleration is determined by the length of the pendulum, the angle it swings, and the gravitational acceleration. Calculating Tangential Acceleration in Vertical Circular Motion In vertical circular motion, the tangential acceleration helps us understand how objects accelerate or decelerate as they move up or down along the circular path. The tangential acceleration in vertical circular motion varies depending on the location of the object in the circular path. At the topmost point, the tangential acceleration is directed downward, while at the bottommost point, it is directed upwards. How to Find Tangential Velocity and Speed with Centripetal Acceleration Finding Tangential Velocity with Centripetal Acceleration and Radius tangential velocity represents the linear velocity of an object moving along a circular path. It is related to centripetal acceleration (the acceleration towards the center of the circle) and the radius of the circular path. The formula to calculate tangential velocity is: – represents the tangential velocity – is the centripetal acceleration – denotes the radius Calculating Tangential Speed with Centripetal Acceleration tangential speed refers to the magnitude of the tangential velocity. It represents how fast an object is moving along a circular path. To calculate tangential speed, we need to know the tangential acceleration and the time it takes for the object to complete one revolution around the circle. The formula for tangential speed is: – represents the tangential speed – is the tangential acceleration – denotes the time How to Find Tangential Component of Linear Acceleration Finding Tangential Acceleration from Radial Acceleration In certain cases, we may need to determine the tangential acceleration from the radial acceleration. Radial acceleration is the component of acceleration directed towards or away from the center of the circle. It is perpendicular to the tangential acceleration. To find the tangential acceleration from radial acceleration, we can use the following formula: – represents the tangential acceleration – is the radial acceleration Calculating Tangential Acceleration from Tangential Velocity In some scenarios, we may need to find the tangential acceleration using the tangential velocity and the time taken to change the velocity. The formula to calculate tangential acceleration in such cases is: – represents the tangential acceleration – is the final tangential velocity – denotes the initial tangential velocity – is the time Determining Tangential Acceleration from Velocity Sometimes, we may need to find the tangential acceleration when only the velocity of the object is known. In such cases, we can use the following formula: – represents the tangential acceleration – is the tangential velocity – denotes the radius How to Find Acceleration Tangential and Normal When an object moves in a circular path, it experiences two types of acceleration: tangential acceleration and radial or centripetal acceleration. tangential acceleration is responsible for the change in the object’s speed or direction along the circular path, while radial acceleration keeps the object moving towards the center of the circle. The sum of these two accelerations gives the total acceleration of the object. How to Find Direction of Tangential Acceleration The direction of tangential acceleration is determined by the change in the object’s velocity along the circular path. It always points tangent to the circular path, either in the same direction as the motion or in the opposite direction, depending on whether the object is accelerating or decelerating. Multivariable Questions on Tangential Acceleration How to Find Tangential Acceleration with Multiple Variables In more complex scenarios, we may come across questions that involve multiple variables to find the tangential acceleration. To solve these problems, we need to carefully analyze the given information, identify the relevant formulas, and apply them step by step. Let’s consider an example: Suppose an object is moving along a circular path with a radius of 5 meters. The object’s tangential velocity is 10 m/s, and the time taken to complete one revolution is 4 seconds. To find the tangential acceleration, we can use the formula: Substituting the given values: Hence, the tangential acceleration is 2.5 m/s². Quick Facts : Q: What is the concept of tangential acceleration? A: The concept of tangential acceleration is related to the acceleration of an object moving in a circular path. It can be understood as the rate of change in the speed of the object along its tangential direction. It is known as tangential acceleration because the direction of the acceleration vector is tangential to the direction of the velocity vector at any given point. Q: What is the formula for tangential acceleration? A: The formula for tangential acceleration is a = r * α, where ‘a’ represents the tangential acceleration, ‘r’ is the radius, and ‘α’ represents the angular acceleration of the object. It is the product of the radius of the motion and the angular acceleration. Q: How does tangential acceleration relate to uniform circular motion? A: In uniform circular motion, the magnitude of the velocity remains constant but the direction of the velocity changes continuously. Hence, there is an additional acceleration acting along the radius towards the center, known as centripetal acceleration. If the object executing circular motion has uniform acceleration, then the tangential acceleration is zero. |Attribute Of Tangential Acceleration |Characteristic in Uniform Circular Motion |None (tangential acceleration is zero) |Not applicable (since speed is constant) |No direction (as there is no tangential acceleration) |0 m/s² (no change in the magnitude of velocity) |Effect on Speed |No effect (speed is constant) |Effect on Trajectory |No effect (trajectory remains circular at constant radius) |Resulting Motion Type |Uniform circular motion (constant speed, constant radius) |No net force in the tangential direction Q: What’s the difference between radial and tangential acceleration? |Radial (Centripetal) Acceleration |Always points radially inward regardless of the object’s motion direction. |Aligned with the instantaneous direction of velocity change, either forward or backward along the path. |Dependence on Velocity |Depends on the square of the tangential velocity (speed) and inversely on the radius of curvature. |Directly related to the rate of change of the object’s speed, irrespective of its path curvature. |Role in Circular Motion |Provides the necessary force component to keep an object in a circular path without influencing the object’s speed. |Responsible for the change in speed of an object in circular motion, without affecting the radius of the path. |Independence from Speed |Independent of changes in the object’s speed; an object in uniform circular motion has constant radial acceleration. |Directly dependent on changes in speed; without a change in speed, tangential acceleration is nonexistent. |Represented in Equations |Prominently features in Newton’s second law for rotational motion (F=ma_r) when considering the force necessary for circular motion. |Featured in the kinematic equations of motion when an object’s speed is changing. |Measured in terms of centripetal force required per unit mass to maintain the circular path (N/kg or m/s²). |Measured as the rate of change of speed, indicating how quickly an object accelerates or decelerates (m/s²). |In Rotational Dynamics |Analogous to force in linear dynamics, but for rotating systems, it represents the radial force per mass needed to maintain rotation. |Analogous to the force component in linear dynamics that causes a change in kinetic energy due to speed variation. |Does no work because the radial acceleration is perpendicular to the displacement of the object in circular motion. |Does work as it is in the direction of displacement, contributing to a change in the kinetic energy of the object. |Effect on Angular Momentum |Does not change the angular momentum of an object in a closed system since there is no torque involved. |Can change the angular momentum if it is associated with a torque, affecting the rotational speed. |Since it doesn’t change the speed, it doesn’t directly contribute to a change in kinetic energy; it affects potential energy in a gravitational field. |Directly affects kinetic energy as it changes the speed; in a gravitational field, it can also affect potential energy. Q: What does tangential acceleration tell us? A: Tangential acceleration gives us an idea about how rapidly the speed of an object is changing with time in the tangential direction. If tangential acceleration is positive, the object is speeding up. If it is negative, the object is slowing down. Q: How does the tangential acceleration formula apply to solving problems? A: The tangential acceleration formula is particularly useful in cases where an object moves in a circular path and its speed changes at a uniform rate. It helps calculate the change in speed at any given point of time. The formula can be applied directly or by integrating the equation if the angular acceleration is not constant. Q: Could you provide a solved example using the tangential acceleration formula? A: Sure. Suppose an object is moving on a circular path of radius 4 meters with an angular acceleration of 2 rad/s². The tangential acceleration (a) would be a = r * α = 4 m * 2 rad/s² = 8 m/s². Here, we’ve used the formula for tangential acceleration to calculate the acceleration of the object. Q: What is the relationship between total acceleration, centripetal and tangential acceleration? A: The total acceleration of an object moving in a circular path is the vector sum of the centripetal and tangential acceleration. Mathematically, total acceleration = √((centripetal acceleration)² + (tangential acceleration)²). The centripetal acceleration is directed towards the center of the circle, whereas the tangential acceleration is in the tangent direction to the circle at that point. Q: How are the tangential acceleration and the velocity vector related? A: The velocity vector of an object executing circular motion has two components: the radial and the tangential. And tangential acceleration has an effect on the magnitude of the velocity vector along the tangential direction. If there is any tangential acceleration, it means that the magnitude of the velocity vector is changing. How can tangential acceleration and angular acceleration be related? To understand the relationship between tangential acceleration and angular acceleration, it is important to consider the concept of Finding Angular Acceleration of a Wheel. Angular acceleration refers to the rate at which the angular velocity of a rotating object changes over time. On the other hand, tangential acceleration refers to the linear acceleration experienced by an object moving in a circular path. These two concepts are interconnected because the tangential acceleration of a point on a rotating object is related to the angular acceleration of the object. By understanding how tangential acceleration and angular acceleration are connected, we can gain insights into the dynamics of rotational motion. Q: What are the applications of tangential acceleration in real life? A: Tangential acceleration has many practical applications in real-life situations such as turning of vehicles where the speed changes due to tangential acceleration. It’s used in the dynamics of rotational motions such as gears, pulleys, and wheels. It’s also applicable in the field of astronomy for studying the planetary motion of celestial objects. - How to find mass and acceleration with force - How to find tension force with acceleration - How to find angular acceleration without time - How to find acceleration projectile motion - Torque and angular acceleration - Centripetal acceleration and mass - How to find acceleration with mass and radius - How to calculate force without acceleration - How to find total acceleration - How to find total acceleration in circular motion Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess.

The path of a particle moving in a plane need not trace out the graph of a function, hence we cannot describe the path by expressing y directly in terms of x. An alternate way to describe the path of the particle is to express the coordinates of its points as functions of a third variable using a pair of equations Equations of this form are called parametric equations for x and y, and the unknown t is called a parameter. The parameter t may represent time in some instances, an angle in other situations, or the distance a particle has traveled along the path from a designated starting point. x = f(t), y = g(t). An easy example of a parametric representation of a curve is obtained by using basic trigonometry to obtain parametric equations of a circle of radius 1 centered at the origin. We have the relationships between a point (x,y) on the circle and an angle t as shown in the following figure. By elementary trigonometry we have the parametric As t goes from 0 to 2p the corresponding points trace out the circle in a counter clockwise direction. The following animation illustrates the process. x = cos(t), y = sin(t). (For a related demonstration for generation of sine and cosine curves see the Circular Functions demo. ) Generating a circle based on circles. A famous curve that was named by Galileo in 1599 is called a cycloid. A cycloid is the path traced out by a point on the circumference of a circle as the circle rolls (without slipping) along a straight line. A cycloid can be drawn by a pencil (chalk or marker) attached to a circular lid which is rolled along a ruler. The following animation illustrates the generation of a cycloid. If the circle that is rolled has radius then the parametric equations of the cycloid are x = a(t - sin(t)), y = a(1 - cos(t)) where parameter t is the angle through which the circle was rolled. As in the case of the circle, these parametric equations can be derived using elementary trigonometry. To see the basics of the derivation click on the following: The Equations of a Cycloid. For some history related to cycloids click on the following: St.Andrews-cycloid Take a large circle centered at the origin. Place a smaller circle tangent to the original circle at the point where it crosses the positive x-axis and outside of the original circle. Identify the point of tangency. See the next figure. Next we roll the smaller circle around the larger circle and follow the path of the point of tangency. The resulting curve is called an epicycloid. The shape of the curve generated in this manner depends on the relationship between the radius of the large circle and the small circle. The following animations illustrate three With a careful analysis we can show that the parametric equations of an epicycloid using a large circle of radius and a small circle of radius b, where a > b, are , y=(a+b)sin(t)-bsin((a+b)t/b) . The epicycloid has been studied by such luminaries as Leibniz, Euler, Halley, Newton and the Bernoullis. The epicycloid curve is of special interest to astronomers and the design of cog-wheels with minimum friction. To experiment with epicycloids see the files available at the end of this module. For more information and an on-line animator click on the following link: animator for epicycloid. Here take a large circle centered at the origin. Place a smaller circle tangent to the original circle at the point where it crosses the positive x-axis and inside the original circle. Identify the point of tangency. See the next figure. Roll the smaller circle around the larger circle and follow the path of the point of tangency. The resulting curve is called an hypocycloid. The shape of the curve generated in this manner depends on the relationship between the radius of the large circle and the small circle. The following two animations illustrate the generation of hypocycloids. (Also see the animation at the beginning of this demo.) Again with a careful analysis we can show that the parametric equations of an epicycloid using a large circle of radius a and a small circle of radius b, where a > b, , y=(a-b)sin(t)-bsin((a-b)t/b) . I have used demos of this type in several Each semester for calculus we have four computer labs. Our In calculus class we define parametric equations and have the students plot a few by hand before we do the demonstration. The demonstration then wows them as they realize how difficult it would be to plot these objects by hand. For the epicycloid I usually use two rolls of tape with different radii, identify a point on the outer one with a magic marker, and then roll it around the other and ask the students what they think the path would look like. Then using DERIVE (see downloads available below) we plot epicycloids for various radii and ask questions about what they would expect and (as I already pointed out in discussion) the numbers of times we need to rotate (i.e. what the parameters should be) in order to obtain a closed epicycloid curve. We try to get them to state a little theorem about this. students must go to the lab, perform the exercises or experiments on the computer using DERIVE, and then write up their results. One of these labs requires students to plot a collection of functions using parametric equations and polar coordinates. The main purpose of the lab is to familiarize students with the parametric plotting capabilities of DERIVE. This lab is assigned after students have seen some of the demo material shown above. I have also used this demo or a variation thereof during various admission (student recruiting) days. Generally, here, the audience consists of prospective students and their parents. It is relative easy to explain what the cycloids are and then it is exciting and informative to see them plotted. For DERIVE the following file was supplied by Anthony Berard and can be downloaded by clicking on (See the imbedded instructions concerning change of scale.) For Matlab the following files were written for this demo and can be downloaded by clicking on the file name: epicycloid.m , hypocycloid.m . This demo was submitted by Department of Mathematics and is included in Demos with Positive Impact with his permission.

Question 1: How many non-negative integers are there? In other words, p is a real number without a natural number partner—an apple without an orange. The tl;dr version is: Assume the numbers between 0 and 1 not including 1 itself are countable aleph-zero cardinality then you can order them 1, 2, 3, 4. I then took out my clothes as I put them in my backpack before I shifted into my Grey Wolf. When you receive the information, if you think any of it is wrong or out of date, you can ask us to change or delete it for you. The weird thing is that it seems like this definition should be obvious that no matter how many things there are, of course you can list all of them. I suppose this is largely a matter of taste. So they're the same size. He is the soon to be Alpha of the Dark Shadows Pack. The natural numbers look nothing like the rational numbers, but both are countably infinite, for example. Coly walked over to me as we gave a wolf hug, I'm sorry it has to be like this, I don't think Aaron realises what he is doing by sending his mate off to be a rogue I know what you are thinking, How does he know? I was walking around taking a breather as I realized how dehydrated I really was. We've gone beyond aleph null. Then you know that the set of men in the room is the same size as the set of women there. It is continuous and flowing, never sharp, never pointy. Can we arrange this into a countable list? This is the first counter intuitive point of infinite numbers. It's kind of independent of standard axioms. This entry was posted in on by. Hazel and Augustus are both smart, thoughtful kids who are coping with terrible circumstances, but they also have that combination of naivet? The most common challenge to mathematical platonism argues that mathematical platonism requires an impenetrable metaphysical gap between mathematical entities and human beings. The basic idea is to assume that you have such an association and then construct a number between 0 and 1 that isn't associated to any integer. Let's say it starts like this: 1 0,1,1,0,1,0,. Let's call the number of positive integers Aleph Zero, because that's what it's called. So it therefore cannot be on the list. Even after a sustained effort lasting more than half a century, no renormalized quantum field theory of gravity has ever been produced. The question is, is there a function that maps every real number or even just the real numbers between zero and one to a unique counting number and vice versa. It means first of all that gravity is infinite at the center of a black hole. Zero is one of the original stumpers. Certainly we can say that some infinite sets are bigger than others, as mathematics nowadays routinely does. Same thing with time: will it go on for all eternity, and does it stretch back infinitely far into the past? After even more torture with the Red Moon Pack, I finally escape to stumble stupidly across another pack territory. Furthermore, if S is a set, then the power set of S always has cardinality strictly greater than that of S, so there cannot be a largest cardinal number. Pick any ordering, write out the numbers in their decimal expansions, then from the first number, take the first digit after the decimal point, second number pick second digit after the decimal point, etc. But what does Aleph Zero + Aleph Zero equal? I hope it is clear, my english is not that good. But we can also say that the set of numbers is only finite, as I have suggested there would be some rationale for doing. Unified field theory Excerpt: Gravity has yet to be successfully included in a theory of everything. What we do know is that if life has infinite moments or infinite love or infinite being then a life twice as long still has exactly the same amount. Edit: If the columns of this matrix : consist of the digits of each number in my list and the rows for each number, what number is not in my list? If you take the numbers in the diagonal in a sequence, and then invert them, the sequence formed will never be found in the set. Now ask, are there more natural numbers than even numbers? Specifically, the fact that Jesus Christ dealt with both general relativity and quantum mechanics in His resurrection from the dead is made evident by the Shroud of Turin. And in fact, with one crucial qualification that we shall come back to, this argument can be applied to anything whatsoever: there are more sets of bananas than there are bananas, more sets of stars than there are stars, more sets of points in space than there are points in space, more sets of sets of bananas than there are sets of bananas, and so on. As soon as we cross the territory boundry line. How many permutations can this number have? You can't find any number that I haven't used. Why can't X exist on the list somewhere else? He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption. We have no control over, and assume no responsibility for, the conduct, practices or privacy policies of MailChimp. Going all the way to Aleph Zero, you are still limiting your set, essentially, to an identity matrix sort of number: with 1's only at ii, and with the rest of the row being 0's. This is the whole point behind reductio ad absurdum arguments - you follow logical steps and if you come to a contradiction an absurd claim then you know your initial assumption was wrong. A non renormalizable theory has no predictive value because it contains an infinite number of singular coefficients. And what does that mean? The proof of this is known as. Thus you know that the set of older twins that have ever been born is the same size as the set of younger twins that have ever been born. This is true, there is a contradiction here. For instance, the set of real numbers is larger than the set of integers. If the previous statement, some infinities are larger than others is true then we can also say some infinities are smaller than others and also some infinities are equal in size to others. What the two theories have in common — and what they clash over — is zero. Of supplemental note is this quote from Newton: The Supreme God is a Being eternal, infinite, absolutely perfect;,,, from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. We have taken reasonable measures to protect information about you from loss, theft, misuse or unauthorised access, disclosure, alteration and destruction. If I'm incorrectly paraphrasing you, correct me. How can there be more sets of anything than there are sets altogether? We use MailChimp to issue our newsletters, donation requests and reader surveys. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. I can post it myself, but probably not until later.

Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not! (I think) (If you'd rather just know what 1+2+3+4+ . . .actually is equal to, check out our next post in this series) Brief Summary: 1+2+3+4+ . . . is not equal to -1/12, but both the infinite series and the negative number are associated with each other in a way that can be seen in this graph The area of the little region below the horizontal axis equals -1/12, and the infinite area under the curve on the right gives you 1+2+3+4+. . . , which goes to infinity as you add terms, not to -1/12. (Update 2-6-14: of course, this graph shows what it looks like if you can only see a finite region of the graph. So 1+2+3+4+...+m is going to infinity, provided m is some finite number. This doesn't say anything about what happens if you could see all the way to infinity - I'd need a much bigger screen for that. For this reason, I am crossing out all the things that I am not so sure about in this post.) For a longer explanation, read on . . . Thanks in large part to the patience and persistence of people like Bernd Jantzen, who commented extensively on my previous post on this subject, I have found a way to simply, and graphically explain how the series 1+2+3+4+ . . . is associated with (and not equal to) the number -1/12, (update 2-6-14: provided you are actually taking the limit as you add terms and not looking at all the infinite terms at the same time as the Numberphiles did). I usually try to avoid writing equations on this blog, so you're going to have to bear with me. But at least there will be pictures, and those are the most important parts. Here we go . . . I find it easier to understand things visually, so it occurred to me to plot out the series 1+2+3+4+ . . . at various points as you add the numbers. These points are called partial sums. This is what you get if you stop the sum after each step for n=1 the partial sum is 1 for n=2 the partial sum is 1+2 = 3 for n=3 the partial sum is 1+2+3=6 for n=4 the partial sum is 1+2+3+4=10 for n=5 the partial sum is 1+2+3+4+5=15 If you draw the first five terms on a piece of paper, it looks like this But you don't have to add all those numbers to calculate value at each point. The numbers, it so happens, are a sequence (1,3,6,10,15, . . .) that you can calculate with a simple formula called a generating function. In this case the generating function is To get a number at any point in the sequence, like say the 5th spot, just plug in 5 for n, and you get the answer for n=5, G(5)=5(5+1)/2=15 As it turns out, you can also use the generating function to figure out what the values would be between whole numbers. Essentially, you replace the number n with an x, which can have any numerical value you like. If you plot the result, you get this, for positive x. The curve you see here goes up to infinity as x goes to infinity. So far, so good. But if we're going to plot the graph between the various values of n, we might as well look at negative values of x too. When you do that, you get a graph like this There are three interesting areas in this graph. The area above the horizontal axis and to the right of the curve (let's call it A), the area above the horizontal axis and to the left of the curve (call it B), and the area trapped between the curve and the horizontal axis (which I call C). A and B are large areas - infinite actually, as you extend the curves to infinity, but C is small. In fact, if you use calculus to determine it's size, it's 1/12. And because it's below the axis, it's conventionally considered negative, so it's -1/12. It's an easy integral to do, but in case you're feeling lazy, I did it on Wolfram Alpha for you, just click here. Interesting, isn't it? The curve generated using the partial values of the series 1+2+3+4+ . . . gives you a graph with a little region in it that has an area of -1/12. Hmmmmm. This is how 1+2+3+4+ . . . and -1/12 are associated. They aren't equal (update 2-6-14: provided you are taking limits), but -1/12, in the form of area C, is a characteristic of the curve While 1+2+3+4+ . . is the series that generates the curve in the first place. So, despite my previous, non-mathematical argument to the contrary . . . Update 2-6-14: The limit as m goes to infinity of 1+2+3+4+ . . .+m does not equal -1/12. As I see it, -1/12 is a kind of label for the curve that you can generate using partial sums of 1+2+3+4+ . . . The same thing works for 1+2^3+3^3+4^3+ . . . , and 1+2^5+3^5+4^5+ . . . and so on for any odd power (i.e., zeta(-3), zeta (-5), etc). I used this method to calculate the associated values of the zeta function for powers up to 13. In each case, you get a specific value the area C that's associated with the zeta function that creates the curve. Here's a list of the C areas I calculated for curves generated by several series zeta(-1) = 1+2+3+4+ . . . ---> -1/12 zeta(-3) =1+2^3+3^3+4^3+ . . . ---> 1/120 zeta(-5) =1+2^5+^5+4^5+ . . . ---> -1/252 zeta(-7) =1+2^7+3^7+4^7+ . . . ---> 1/240 zeta(-9) =1+2^9+3^9+4^9+ . . . ---> -1/132 zeta(-11) =1+2^11+3^11+4^11+ . . . ---> 691/32760 zeta(-13) =1+2^13+3^13+4^13+ . . . ---> -1/12 They agree with the published values of the zeta function for negative integers listed on Wikipedia. (Although Wikipedia stops at zeta(-7) and I go to zeta(-13). The fact that the number associated with zeta(-1) and zeta(-13) are the same looks like potential trouble, BTW - after all, how would you know if your -1/12 is associated with zeta(-1) or zeta(-13)? It also suggests that the Numberphiles could have shown that -1/12 = 1+2^13+3^13+4^13+ . . ., or that 1+2^13+3^13+4^13+ . . .= 1+2+3+4+ . . ., if they felt like it.) This little procedure works for even powers too, except the answer is always zero. Here's what the C area looks for for the curve generated from zeta(-2)=1+2^2+3^2+4^2+ . . . The parts above and below the axis cancel for all series of this type with even powers, and as a result the total area doesn't give you any information. As you can see for the integral of the curve that comes from1+2^2+3^2+4^2+ . . . How Could I Have Screwed Up So Badly? I'm going to blame my misadventure on the trouble with using words to describe mathematical ideas. Before working out this problem, there was no way I could understand what it means when someone says that the value of -1/12 "can be assigned to an infinite series." They sounded like gibberish, Ramanujan is to blame a bit too. After all, how are we supposed to understand what he was trying to say here? "I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal." -S. Ramanujan in a letter to G.H. Hardy Are we supposed to realize that "under my theory" means that "=" doesn't mean equal? I haven't found his original work, but several people have reproduced a calculation by Euler that uses an equal sign in the same way. If the two sides aren't equal then, as I recall from second grade math, you can't use an equal sign. At least one person I spoke to said that in order to understand it, you have to know what the ". . ." in the expression 1+2+3+4+. . . means. As you can see from the example above, the dots mean exactly what they always mean. If they didn't, then we'd be in almost as much trouble as having equal signs that don't mean equal. Finally, there are the physicists that say they need the relation 1+2+3+4+ . . . = -1/12. Some of them seem to believe the equation is what it is, and that our failure to understand it shouldn't stand in the way of using it. That struck me as the most romantic view, and it was the one I latched onto at the end of the day. Now, I realize that this is a far too mysterious view. It's interesting, but I hope this post shows While 1+2+3+4+ . . . = -1/12 is clearly not true I hope there are other, comparably bizarre mathematical controversies out there. One thing is for sure, though, I'm not going to rely on words, even from the most decorated experts in their fields, to try to understand things like this. I'm going to get out a pencil, fire up Wolfram Alpha, and just do the math. Andy Holland said... In order to use negative numbers for the integral, doesn't that require the problem to be the sum of all natural numbers including the negative ones? The left hand integral area cannot cancel the right hand area otherwise. The sum of all natural numbers from -x to x in the limit as x approaches infinity is -1/12? Monday, July 18, 2016 at 10:44 PM alan doak said... Every term of Zeta(-1) is larger than Zeta(1), except the first which are both 1. Since Zeta(1) equals infinity (proven by the comparison test) -> Zeta(-1) is also infinity, due to the comparison test. QED. Thursday, June 23, 2016 at 9:38 PM Ronnie Johansson said... Please delete "Dr Ebute"'s fraudish nonsense. Tuesday, May 3, 2016 at 6:05 AM Merychris Calib-og said... How can I find the following cardinalities? Sunday, January 31, 2016 at 7:29 AM The term above -143n^10 should be -143n^10/60. Inadvertently the denominator 60 was omitted.Also the term n^c should be +n^c with a plus sign inserted. Integrating this corrected function from 0 to -1 will now render the result of -1/12 = RZ(-13) = RZ(-1). Alan Walter,Sydney. Friday, January 29, 2016 at 9:00 PM Thanks for your youtube email reference. RZ(-c) is Riemann's Zeta value at -c = I(n=0to -1)[1^c+2^c+3^c+...+n^c]dn Where I(n=0to-1) is the definite integral from 0 to -1(lower limit) RZ(-13)=I(n=0to-1)[1^13 +2^13 +3^13 +...+n^13]dn +715n^4/132 -691n^2/420]dn= -1/12 on evaluation= -B(14)/14= -(7/6)(1/14) Note1^c+2^c+3^c+...n^c for odd c values has factors of n(n+1)with zeros at 0,-1. Anyone interested in an expression for the yet unsolved sum of the positive ODD Zeta series Z(2n+1) in terms of π^(2n+1)? Alan Walter, Sydney. Wednesday, January 27, 2016 at 8:39 PM The author of this article is absolutely correct to point out that these supposedly mysterious values can be found by calculating the definite integral between 0 and -1 of the expression for the respective sum to the nth term (a.k.a. their partial sum expressions). This is also mentioned in this interesting 'response' video (it responds to the claims made in the video that is the subject of this discussion): https://www.youtube.com/watch?v=BpfY8m2VLtc It shows what happens if you do the series manipulations with rigorously (hint - you do not get -1/12) and it explains why other methods get this -1/12 result. The response video claims this -1/12 result is the result of a mistake. The mistake is one of taking a function that applies to just positive whole numbers, manipulating it in ways that bring decimal numbers and negative numbers into play, and then interpreting the result as though it still relates to positive whole numbers. Tuesday, January 26, 2016 at 7:02 PM Let's assume I gave you zero fucks. How many fucks have I given you? Friday, May 15, 2015 at 11:28 PM Interesting, except removing thingZ from bagA doesn't leave you with -thingZ in bagA. to do that you would have to remove 2*thingZ. Otherwise, taking thingZ out of bagA, and then replacing it would leave you with nothing in bagA. But clearly removing thingZ from bagA and replacing it shouldn't make thingZ dissappear. Friday, April 3, 2015 at 9:39 AM I understand how puzzling it can be. Is zero a quantity or not? Is a negative number a quantity or not? Talking about apples and pears is a little worn out, I think. There are two different bags; let's call them bagA and bagB. They are freshly made, and nothing has ever been put in them before. Each bag has nothing in it. The bags are in an enclosed vacuum, so there is no air in them and no air outside of them. There are two different things; let's call them thingZ and thingY. You put thingZ into bagA, and thingY into bagB. You then have a bag - bagA - containing thingZ and a bag - bagB - containing thingY. If you remove thingZ from bagA and thingY from bagB, the bags will then be empty because the things that were in them have been removed. Because there used to be a thingZ in bagA, and one was taken out, there is a shortage of one thingZ. BagA therefore contains -1 thingZ. Likewise, bagB therefore contains -1 thingY. But all the time there was a thingZ in bagA, bagA also had no thingY. Similarly, while there was a thingY in bagB, bagB also had no thingZ. In fact, ever since the bags were made, there was also none of thingX, none of thingW, no tomatoes, onions, apples, bicycles, orang-utans or saxophones in them. None of any other thing, in fact, in either bag. And the same goes for what is outside of each bag either, because it was a vacuum. If it is true that zero is something that does exist, then zero is a complete lack of stuff. But we know that it wasn't always like that, because there used to be a thingZ and thingY in the bags. Regarding anything and everything else, there always was a complete lack of anything else in the bags, and everything outside the bags. Because you cannot remove something that isn't there, bagA still contains zero thingY and bagB still contains zero thingZ - and zero everything else except as was stated in the paragraph before this one. And outside of the bags is still zero everything. But the bags don't contain zero of everything, they contain -1 of one thing. Zero is more than -1. The bags therefore collapse under the pressure of absolutely nothing. Friday, April 3, 2015 at 2:30 AM do you wants....idea from amateur...from me or not... Easy...! Open Mind... Zeta Functions not importtant ....Harmonic important more! Zeta function that important Zeta(-1) and Zeta(-3) I show you....develop zeta in term cartoon.....and blinding set and anti and i predict somethings in Higher Dimension...You can help me proof Wednesday, December 3, 2014 at 11:13 PM To be more specific, irrational mathematics('mathematics'), is not sustainable. Monday, November 3, 2014 at 11:53 PM I have posted a message explaining the dilemma. What has happened here was inevitable, mathematics is not sustainable. Here is the link to my message: http://marques.co.za/duke/news_win.htm It will not surprise me if this comment is censored (Moderated) Friday, October 31, 2014 at 5:13 AM 1+2+3+4+... = -1/12 (R) where (R) is the Rumanujan Summation. This is not a normal -1/12. It basically is a categorization of the series in question. It should be read, "the sum of one plus two plus . . . has a Rumanujan summantion of -1/12" (as opposed to "equals -1/12"). Tuesday, October 14, 2014 at 3:12 PM Johnson Briney said... Contact a Great man on email@example.com or call him on +2347060552255 who help me to solve my problems when my ex boyfriend was blackmailing me after trying many ways to stop him but it didn't work for me until i met this man who help me cast a spell that stop him for blackmailing me and now he is pleading forgiveness and i believe he can solve any problems you are having because he just solve mine. Sunday, August 31, 2014 at 9:38 AM way2 college said... Valuable liveliness is also saved which you can set aside to your family or to manually. Completing an Online Distance Learning course gives you more flexibility while studying over conformist classroom set up. Saturday, August 9, 2014 at 1:31 AM No, zero is not in the real world. Imagine that you have two bags, in one there is one apple and in the other you have one pear. Then we remove the apple and the pear out of the bags, now in one bag you have zero apples and in the other zero pears, but zero apples and zero pears have the very same properties, therefore they must be the same thing, as we know, pears and apples are not equal to each other, so it must be that zero is nothing but a concept that does not exist in the real world. Monday, June 2, 2014 at 1:58 AM Area A = Area B since they both are infinitely large areas. That series diverges, you learned that in Cal2 or Math Physics of DiffEq. C'mon. Thursday, May 29, 2014 at 10:47 AM Richard Smart said... This article is superb, I love it. That infinite series thing really perturbed me when I first saw it but I couldn't see how to examine it more effectively such that I could get to the point that I wasn't perturbed by it any more. Watching you do it above now makes me annoyed I didn't have the idea of doing that myself. But the fact is I didn't. So simple, so insightful, so satisfying. Thanks for that and if you could now just solve every other annoying problem for me I would be most grateful ;o) Thursday, May 22, 2014 at 3:41 PM Bernd Jantzen said... Of course, the partial sums of 1+2+3+4+... are tending to infinity. Nobody claimed anything different. The whole discussion here is about how to assign a meaningful finite value even to a divergent sum like this one. And the discussion is about the question whether one may call this finite number the value of the divergent series or just a meaningful number that one may use in replacement of the series under some conditions. Sunday, April 6, 2014 at 6:01 PM Saturday, April 5, 2014 at 10:22 AM Saturday, April 5, 2014 at 10:20 AM i think 1+2+3+4+.........=infinity Saturday, April 5, 2014 at 10:18 AM I possess no sheep, therefore I possess 0 sheep. 0 is very much in the real world. Wednesday, March 19, 2014 at 11:08 AM Imre Fabian said... For me, not a mathematician, this is a perfect example for that you can prove anything with infinity mathematics. In my view, infinity does not exist (in the real world), nor does 0. Infinity = anything / 0 0 = anything / infinity. Physical argument: the smallest thing (measure) is the planck dimension (planck lenght, planck time etc.) so there cannot be an infinite number of since the beginning of time (the big bang). Wednesday, March 5, 2014 at 4:58 AM

Thanks! Can’t I start it with the value closer to my lowest data? The class upper limits then permit the use of the FREQUENCY function in spreadsheets such as Excel, LibreOffice.org, OpenOffice.org, Google Sheets and Gnumeric. To make the histogram for the above data, follow these steps: You should now see a histogram on your worksheet. I’d been a google sheets skeptic until I read through some of your posts. You do not have *smoothing* anymore in Google Sheets. comes up when I try to use the normdist formula for some values. Relative Frequency Graph Maker. Then open the Google Sheets with the data you want to use to make a histogram. If pie chart is made right click and select Change Chart Type, Bar Chart Click on a red bar and right click select Change Series Chart Type to Line Chart. We now need to calculate the distribution of the 1,000 exam scores for our histogram chart. But now, you can make one in a matter of seconds. On the Chart Editor pane, select the Histogram⦠For example, a shop might have a goal of selling 5% of their total items in the $41 â $50 price range. it has a longer tail on the left, more spread on the left. The new chart editor opens in a side pane, but the steps and options are essentially the same. We showed you why and how you can use a histogram. This was a great exploration for me, both in learning more advanced charts (advanced for me, at least!) I get a big tail of zeros by your method in Histogram 1 but everything else seems to work. On the Insert drop-down, click Chart. Highlight the entire data and click the + (Insert) sign on top of Google Sheets. The “Legend” category, as its name suggests, lets you provide settings and formatting for the histogram legend. Hi! Tally up the number of values in the data set that fall into each group (in other words, make a frequency table). Simply start with a blank chart or a histogram templates. As a starting point, you can take you max value (99.2 in this example) and min value (9.7 in this example), calculate the range between them (89.5) and then divide by how many bins you want to show (e.g. Make a title for the OY axis in a similar way. Create a named range from these raw data scores, called scores, to make our life easier. Adjusting the min and max inputs really helps you provide context to your histogram. Thank you, that was very helpful. Or, you can choose the smooth option in the customization menu: Hi Ben, Do you know how to make a histogram when I have a theoretical ‘Engagement Score’, a continuous variable, in Col A and counts of a given score in Col B? This thread is outdated. Your email address will not be published. How do array formulas work in Google Sheets? Should I always start my bins with 0? Google Sheets will insert a Column Chart. how did you come up with increments of 5 with the example that you used in the tutorial above? Formula to calculate relative frequency. ; From the add-on description page, click the "+Free" in the top right corner to add it ⦠Other Google Sheets tutorials you may like: Save my name, email, and website in this browser for the next time I comment. Your email address will not be published. you can easily understand the dynamics, trends, and relationships among data items and draw important inferences. When you click the + (Insert) sign, a drop-down will be displayed. Select the bins column and the Normdist column then Insert > Chart and select line chart, and make it smooth: That’s a normal distribution curve, around our mean of 56.9. This category lets you provide the text and formatting for the chart title and subtitle as well as the titles for both x and y axes. Make it count Google Sheets makes your data pop with colorful charts and graphs. To make a histogram, you first divide your data into a reasonable number of groups of equal length. This tool will create a histogram representing the frequency distribution of your data. So we can set the legend position to none. However, after the creation of the ND, I see false input in the Advanced settings’ Chart types’ first box where is only one column, while I expect there to be two columns. Bit of a mouthful, but in essence, the data converges around the mean (average) with no skew to the left or right. If you want to create histograms in Excel, youâll need to use Excel 2016 or later. Select the data you want to visualize in your histogram. The last bin gives the total number of datapoints. In column I, let’s use the FREQUENCY formula to assign our 1000 scores to the frequency bins. You should see an ellipsis (or hamburger icon) on the top right corner of the box containing the graph. Google Sheets Developer & Data Analytics Instructor. Series: Change bar colors. You can choose the smooth line chart option in the chart choose menu: 2. This is super helpful, not simple. How can I make it work? The third column is for the count or frequency of data in each class. Histograms are a useful tool in frequency data analysis, offering users the ability to sort data into groupings (called bin numbers) in a visual graph, similar to a bar chart. Select the Smooth option: Select the vertical axis. To plot the Histogram chart, first, select the whole data in column A and go to the menu Insert > Chart. So you plot how data of a single category is distributed. A survey – Zubair Lutfullah Kakakhel, http://datapigtechnologies.com/blog/index.php/understanding-standard-deviation-2/, Does GPA matter for my salary? Double-click the chart you want to change. I myself have a similar data but I choose graphically those two columns with their titles (like you do in Excel) and do Chart > Line > …. Copy the raw data scores from here into your own blank Google Sheet. A histogram is the best chart you can use to illustrate the frequency distribution of your data. It’s advisable for them to be whole numbers too, both aesthetically and to ease understanding. Let me help you with Google Sheets and Apps Script. It allows you to see the proportion or percentage that one value is repeated among all the elements in the sample. What if you are trying to compare your data not to the normal distribution of your data, but say the district’s average? Is the Scale Factor 0.39 (78 * 0.005)? If we look closely, it’s skewed very, very slightly to the left, i.e. Then, edit the chart data through the spreadsheet editor - Just replace the values by typing in your own data set. (ie. Let’s set up the normal distribution curve values. After selecting a combo chart, I am not getting the “Smooth” option. Thank you for a tutorial that was clear and concise. However you should not truncate the y-axis (vertical axis) because the height of the bars is measured from zero and this prevents the data being distorted. However, sometimes the Chart editor goes away after your histogram has been created. This could sometimes help make the histogram easier to read and understand. Enter âRelative Frequencyâ in cell C1. The data I have is at an aggregate level and instead of Engagement Score of 18.8 listed out in 1,000 rows, I simply have a2 = 18.8 | b2 = 1000. This is a great article that goes into more detail about standard deviations: http://datapigtechnologies.com/blog/index.php/understanding-standard-deviation-2/. ), 1) The P at the end of STDEVP stands for population and should be used for calculating standard deviations on whole populations, as opposed to when you’re looking at samples (when you’d use the regular STDEV function). To understand how to create a histogram, we are going to use the data shown in the image below: This dataset contains scores of students in an exam. For example, if you had to compare the distribution of marks for two different classes, you could use one color for grade 6 and another for grade 7 (say). Meaning, when I multiply the normal distribution values by 5,000, they’ll be comparable to the histogram values on the same axis. Can you please be precise and say how you choose the data in the step six? A relative frequency histogram is a graph that displays the relative frequencies of values in a dataset. The #NUM! Some other settings available under these categories include: Finally, you can format the histogram to contain major and/or minor gridlines. It says “…the averages of random variables independently drawn from independent distributions converge in distribution to the normal…” What does this mean? A histogram is one such helpful visualization tool that helps you understand the distribution of your data. (Or just click the link here). It divides the range of your data into intervals, displaying how many of the data values fall into each interval. Can you please be more precise on the step 6 because I cannot reproduce your method. In our example, we changed the background color to “light green 3”, and allowed the other settings to remain the same. The kind of data plotted by histograms and Bar Graphs is also different. Its calculations, however, are usually far from perfect. Raw data. To make the histogram for the above data, follow these steps: We want to create a histogram to understand how the student scores in the exams were distributed. For example, you can use it to give a title for the vertical axis, by selecting the “Vertical axis title” option from the dropdown menu and then set the title as “Student Count”. If a data point falls on the boundary, make a decision as to which group to put it into, making sure you stay consistent (always put it in the higher of the two, or always put it in the lower of the two). The normal distribution curve is a graphical representation of the normal distribution theorem stating that “…the averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large”. A survey | cloud5, How to Create a Scatter Plot in Google Sheets. Hope you have the data to plot the Histogram Chart in Google Sheets now. Paste the frequency distribution into cell A1 of Google Sheets so the values are in column A and the frequencies are in column B. Last updated: October 12, 2017 [Download PDF]What are Histograms? You can use it to count the frequency of values in a range. This range is actually called a one column array. Before Excel 2016, making a histogram is a bit tedious. At the right, click Customize. We need to scale our normal distribution curve so that it’ll show on the same scale as the histogram. Creating a Histogram in Google Sheets. Type into the field the number for the bucket size that you would like represented in the histogram. FREQUENCY Function in Google Sheets. As such, you will usually feel the need to customize the histogram to give it the look and functionality you want. In this tutorial, I will show you how to make a histogram in Google Sheets and how to customize it. We expect our exam scores will be pretty close to the normal distribution, but let’s confirm that graphically (it’s difficult to see from the data alone!). Begin with a column that lists the classes in increasing order. To make it appear again and to customize your histogram, do the following: The Chart style category in the Chart editor lets you set the background color, border color, font style, and size of your chart. What’s the average score? You have two options: 1. With the help of visualization tools like charts, graphs, maps, etc. We expect 68% of values to fall within one standard deviation of the mean, and 95% to fall within two standard deviations. How clustered around the average are the student scores? Highlight all the data in column A, i.e. Seems the new chart editor does not have the “smooth” option for the combo charts, but you can click “Use the old chart editor” at the bottom of the sidebar to go back to the old editor which does still have the smooth option. Itâs a list of 1,000 exam scores between 0 and 100, and weâre going to look at the distribution of those scores. One quick note – I think the text for normal distribution theorem should be slightly different? On your computer, open a spreadsheet in Google Sheets. 3) Good question. Fantastic tutorial. While in Excel and LibreOffice.org the existence of a gap width or spacing setting allows the columns to touch, Google Docs did not provide this option, hence the chart seen below. Insert Histogram Chart in Google Sheets. Usually, the Chart editor has a ‘Customize’ tab that lets you enter all your specifications. In our example, it would make sense to distribute the scores between 0 and 100. Create a new Google Spreadsheet (or open an existing one) From the menu bar, choose: Add-ons -> Get Add-ons. Google Sheets performs its own calculations on your data and displays what it believes to be the optimal number of bins for your histogram. Distributions were in intervals of 10 let ’ s set up the normal distribution theorem should be different., youâll need to use for the bucket size that you would represented!: October 12, 2017 [ Download PDF ] what are histograms ( an Easy Guide ) represented in sample... Alongside the normal distribution curve and to ease understanding normal distribution or choose to not have * smoothing anymore... Edit or format title text been a Google Sheets performs its own calculations on your.! You able to do this using the native histogram charts in Google Sheets,... Understand the distribution of data points per bin to distribute the scores between 0 and 100, if! Api, run kstest on the x-axis to be a range, like 0-9 %, %... Sample, with respect to the frequency, so it would be better if the distributions were intervals... Could sometimes help make the histogram for the tutorial above a normal sheet table of frequencies a! When youâre interested in percentage values precise and say how you can ’ t I start it with native., http: //datapigtechnologies.com/blog/index.php/understanding-standard-deviation-2/ cover your whole dataset you might want to create a new Google spreadsheet ( hamburger. Is one such helpful visualization tool that helps you understand the distribution of data... Or change bucket size that you used in the chart choose menu: 2 above... Get Add-ons the cell range A1: A12 skewed or just all over the place of bins... That plots frequency distribution shows how a variable is distributed available under these categories include:,! Actually called a one column Array dashboards and reports own blank Google sheet Apps. Just checked now and the normal distribution function for a property for your histogram one quick note – think! Clustered around the average are the “ legend ” category, as name... To draw the histogram chart usually feel the need to customize the histogram frequencies is particularly good displayed in tutorial... Can be interpreted as probabilities ticks at how to make a relative frequency histogram in google sheets, to make a histogram a! To reflect every little change you made alongside the normal distribution called scores, bins ) ) ` not! But the steps and options are essentially the same size and cover your dataset... The mean, our scale factor is 5,000 copy this column of frequency instead of COUNTIFS fill... Enter all your specifications PDF ] what are histograms a survey | cloud5, how much I! Apps Script determining if your data is normally distributed, skewed or just all over the place frequencies values. It easier in perceiving, analyzing the results and presenting them to be whole numbers too both. You plot how data of a bar graph life easier to relative frequency histogram is when! 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And Apps Script of COUNTIFS to fill up the histogram easier to read and understand change bucket that. Chooses the number of bins for your histogram line chart option in the data you want bins! Chooses the number of data in each class your requirement size you get the range perceiving, analyzing the and... Editor has a longer tail on the left, more spread on the same scale as the and! Contain major and/or minor ticks on your data is very close to lowest!, https: //productforums.google.com/forum/ #! topicsearchin/docs/category $ 3Aspreadsheets, Does GPA matter for my salary s for. These categories include: Finally, you can format the histogram from the menu >... For this, you will usually feel the need to change the scale factor 0.39 ( 78 0.005. Have * smoothing * anymore in Google Sheets and Apps Script Google spreadsheet ( open... Get a big tail of zeros by your method in histogram 1 but everything else seems work. Determine what increments to use for the “ charts ” tab of the normal distribution columns: show dividers... Horizontal axis category to change the min and max to do this in shortly ) let! Right corner of the Google Sheets API, run kstest on the left, more on. Event or category of data in each class ) how do you determine what increments to use the histogram the! Optimal number of times each category how to make a relative frequency histogram in google sheets in the chart choose menu: 2 are marked *, Learn about. Values for the bucket size that you used in the histogram for the above data, while bar! Usually far from perfect without seeing your data that category at least ). They ’ ll show on the other hand, plot categorical data by counting the of! YouâLl need to use to make our life easier Google spreadsheet ( or hamburger icon on! This for our histogram chart in Google Sheets the number for the OY axis in a.. Size that you how to make a relative frequency histogram in google sheets like represented in the histogram to understand how the chart editor goes away after histogram. Determining if your data option: chart style: change how the student scores the! Survey – Zubair Lutfullah Kakakhel, http: //datapigtechnologies.com/blog/index.php/understanding-standard-deviation-2/, Does GPA matter my... One column Array values for the tutorial ) equal length us the frequency bins are histograms to cell.! Statistics add-on from the Add-ons gallery and select it I think the text for distribution. Bins, displaying how many values occurred close to the menu Insert >.. Scale our normal distribution theorem should be slightly different: type category histogram Google! Data to be how to make a relative frequency histogram in google sheets optimal number of bins for you now the basics and done do following. Tutorial above step 6 because I can not reproduce your method in histogram 1 but everything else seems to.. Why and how you choose the Smooth option is definitely still available your.! On your worksheet steps and a few seconds then you could have different colors different! See a histogram in Google Sheets skeptic until I read through some of posts... Sheets ( an Easy Guide ) axis in a Microsoft spreadsheet in Google Sheets by calculating the distribution! Were in intervals of 10, more spread on the left, i.e an Easy Guide ) min max... ( not too narrow, not too narrow, not too wide ) increments to use for above. And displays what it believes to be, or choose to not have them at all made a... Classes in increasing order Polygon '' box to show the old chart editor ] histogram Maker normal distribution columns histogram... Replace the values by typing in your histogram has been created the vertical axis and bar graphs is also.! From a bar graph the general distribution of your posts item in the histogram from the how to make a relative frequency histogram in google sheets example we. Dividers checkbox lets you provide context to your liking or ‘ buckets ’ will be updated instantly to every. A kind of data, follow these steps: create the histogram legend position to.... Values into the field the number for the histogram: show item,... A category is distributed ( advanced for me, at least! draw the histogram of. Me become way more productive charts ” tab of the 1,000 exam scores for our histogram.! Set and format major and/or minor ticks on your data pop with charts! Along the x-axis have very arbitrary sizes me to get the range of your.... The use of frequency instead of COUNTIFS to fill up the normal distribution theorem should slightly! Editor opens in a range of values in a sample, with respect to the mean else seems to.. Of this template > > like charts, graphs, maps,.... Which you want to visualize in your histogram ’ s helped me become more! Up the frequency of data points per bin #! topicsearchin/docs/category $ 3Aspreadsheets, GPA! And understand updated: October 12, 2017 [ Download PDF ] what are histograms that! Learned how to make a title for the Horizontal axis category to change the and. 12, 2017 [ Download PDF ] what are histograms and select it like bars of a of... Allows you to compare how often values occur relative to the lowest provided! Frequency, and change the range named range from these raw data scores here. A level of automation [ ⦠] histogram Maker displaying how many values occurred to... How did you come up with increments of 5 we showed you why how... Visualizations of your data you add a line between each item in chart. Up when I try to use the function ArrayFormula together with frequency use to make a,! That lists the classes in increasing order m not getting the “ legend ” category as!

A guide to calculating the capacitance of capacitors when used how to calculate capacitors in series and parallel how to calculate capacitors in series and parallel. I have a 125v 2ah battery and i'm trying to calculate a equivalent capacitance with rated voltage of 27v for each of those batteries this is what i did: work of battery = \$125v \cdot 2a. Capacitors in series and in parallel next: the equivalent capacitance of two capacitors connected in parallel is the sum of the individual capacitances. Multiple connections of capacitors act like a single equivalent capacitor the total capacitance of this equivalent of capacitors in series and parallel. Capacitors are passive devices used in electronic circuits to store energy in the form of an electric field they are the compliment of inductors, which store energy in the form of a. Three capacitors are connected in series the equivalent capacitance of this combination is 318 f two of the individual capacitances are 654 f and 819 f. This is a series and parallel capacitor calculator it computes the total capacitance value of a circuit, either of capacitors in series or in parallel. It's all about charge, voltage, and capacitance in exactly the opposite way as resistors in series and parallel are all about voltage, current, and resistan. Answer to capacitors in series: find (a) the equivalent capacitance, (b) the charge on each capacitor, (c) the voltage across each. Combine capacitors in parallel calculating the total capacitance of two or more capacitors in electronics components: capacitors in the circuits are equivalent. Ap physics practice test: capacitance, resistance capacitor with capacitance c the equivalent capacitance of the three capacitors is. Types of capacitor, definition of capacitors, storing charge, charging and discharging, dielectric constant, capacitors series and parallel. G12: capacitance revision study play how is the equivalent capacitance of a parallel combination in a parallel-plate capacitor capacitance is directly. This video guides you through the steps in finding the equivalent capacitance of a basic circuit the circuit contains 11 capacitors in total like us on f. Capacitors in series and parallel let us see how to calculate the equivalent capacitance of capacitors when connected in parallelconsider two capacitors. Esr equivalent series resistance the esr rating of a capacitor is a rating of quality a theoretically perfect capacitor would be lossless and have an esr of zero. Electronics tutorial about connecting capacitors in series including how to calculate the total capacitance of series connected capacitors. Capacitors experiment we can find the equivalent capacitance value capacitor and determine its capacitance by measuring v 2 and using equations 1 and 3. For this final equivalent capacitor with capacitance c eq2, the voltage across it is, of course, 120 v the charge on this capacitor, q eq2, is. Question: a and a capacitor are connected in parallel, and this pair of capacitors is then connected in series with a capacitor, as shown in the diagram what is the equivalent capacitance. When capacitors are connected in series, the total capacitance is less than any one of the series capacitors’ individual capacitances if two or more capacitors are connected in series, the. Capacitance, insulators, dielectrics, capacitors, capacitors in series, capacitors in parralel, energy stored in a capacitor. The canonical definition for capacitance between two nodes, whether total, equivalent or other, is the charge in coulombs required to change the potential difference by one volt. The capacitance of a capacitor is proportional to the surface area of the plates the earliest unit of capacitance was the jar, equivalent to about 111 nanofarads. Chapter 5 capacitance and dielectrics figure 57: (a) series capacitors in previous figure have been combined as a single equivalent capacitor (b. Capacitors in series and parallel systems including capacitors more than one has equivalent capacitance capacitors can be connected to each other in two ways. First i calculated equivalent capacitance between r and m the two 2μf capacitors are in series their combination is in parallel with the 1μf capacitor their equivalent capacitance is. Capacitors and rc circuits when capacitors are arranged in parallel when capacitors are arranged in series, the equivalent capacitance is. Formula of equivalent capacitance in series series combination when capacitors are connected in series,the total capacitance is less than the smallest capacitance value because the. Capacitors in parallel capacitors can be connected in parallel: the equivalent capacitance for parallel-connected capacitors can be calculated as. Practice problems: capacitors solutions 1 (easy) determine the amount of charge stored on either plate of a capacitor (4x10-6 f) when connected across a 12 volt battery c = q/v. Answer to: how to find equivalent capacitance of a network of capacitors by signing up, you'll get thousands of step-by-step solutions to your. For an arbitrary number of series-connected capacitors the charge is the same on each capacitor and the equivalent capacitance is determined from. All Rights Saved.

Well, Melaina, the Earth’s mass does decrease when fossil fuels are burnt. But not in the sense you were probably imagining, and only to a very, very small degree. There is no decrease in chemical mass. Burning fossil fuels rearranges atoms into different molecules, in the process releasing energy from chemical bonds, but in the end, the same particles — protons, neutrons, and electrons — remain, so there is no decrease in mass there. But energy is released, and some of that energy is radiated out into space, escaping from the Earth entirely. Einstein's Theory of Relativity tells us that energy does have mass: E=mc^2, or m=E/c^2. When a chemical bond that stores energy is formed, the resulting molecule has a very tiny bit more mass than the sum of the masses of the atoms from which it was formed, so a net gain. Wait, what? Again, this is an exceedingly tiny bit. In very rough numbers, worldwide energy consumption is about 160,000 terawatt-hours per year, and about 80% of that comes from fossil fuels. That is about 450,000,000 TJ/year (tera-joules/year). The speed of light is 300,000,000 meter/s; dividing 450,000,000 TJ by (300,000,000 m/s)^2 gives a decrease in mass of 5000 kilograms per year. That is an exceedingly small fraction — 50 billionths of one percent — of the approximately 10,000,000,000,000 kilograms of fossil fuels consumed per year. And as far as making the Earth lighter, it’s a tenth of a billionth of a billionth of a percent of the Earth’s mass. Of course, the energy in fossil fuels originally came from the Sun, and in absorbing that sunlight the Earth’s mass increases slightly. I picture the Earth expanding and contracting, taking a deep breath, then exhaling. We don't see this when we look, but it is a great visual for imaging this never-ending give and take process. I'm not sure how we'd measure the small changes to the Earth's net mass on any given day. The mass of the Earth may be determined using Newton's law of gravitation. It is given as the force (F), which is equal to the Gravitational constant multiplied by the mass of the planet and the mass of the object, divided by the square of the radius of the planet. Newton's insight on the inverse-square property of gravitational force was from an intuition about the motion of the earth and the moon. The mathematical formula for gravitational force is F=GMmr2 F = G Mm r 2 where G is the gravitational constant. I know, Newton’s law could use some curb appeal but it is super useful when understanding what keeps the Earth and other planets in our solar system in orbit around the Sun and why the Moon orbits the Earth. We have Newton to thank for his formulas on the gravitational potential of water when we build hydroelectricity dams. Newton’s ideas work in most but not all scenarios. When things get very, very small, or cosmic, gravity gets weird... and we head on back to Einstein to make sense of it all. There was a very cool paper published yesterday by King Yan Fong et al. in the journal Nature that looked at heat transferring in a previously unknown way — heat transferred across a vacuum by phonons — tiny, atomic vibrations. The effect joins conduction, convection and radiation as ways for heating to occur — but only across tiny distances. The heat is transferred by phonons — the energy-carrying particles of acoustic waves, taking advantage of the Casimir effect, in which the quantum fluctuations in the space between two objects that are really, really close together result in physical effects not predicted by classical physics. This is another excellent example of the universe not playing by conventional rules when things get small. Weird, but very cool! But the question was specifically about the mass of the Earth and the burning of fossil fuels, and that process does decrease the mass. So it is mostly true that the Earth’s mass does not decrease due to fossil fuel burning because the numbers are so low, but not entirely true. The fuel combines with oxygen from the atmosphere to produce carbon dioxide, water vapour, and soot or ash. The carbon dioxide and water vapour go back into the atmosphere along with some of the soot or ash, the rest of which is left as a solid residue. The weight of the carbon dioxide plus the water vapour and soot is exactly the same as the weight of the original fuel plus the weight of the oxygen consumed. In general, the products of any chemical reaction whatsoever weigh the same as the reactants. There is only one known mechanism by which Earth’s mass decreases to any significant degree: molecules of gas in the upper atmosphere (primarily hydrogen and helium, because they are the lightest) escape from Earth’s gravity at a steady rate due to thermal energy. This is counterbalanced by a steady rain of meteors hitting Earth from outer space (if you ever want to hunt them, fly a helicopter over the frozen arctic, they really stand out), containing mostly rock, water, and nickel-iron. These two processes are happening all the time and will continue at a steady rate unchanged by anything we humans do. So, the net/net is about the same. So, the answer is that the Earth's mass is variable, subject to both gain and loss due to the accretion of in-falling material (micrometeorites and cosmic dust), and the loss of hydrogen and helium gas, respectively. But, drumroll please, the end result is a net loss of material, roughly 5.5×107 kg (5.4×104 long tons) per year. The burning of fossil fuels has an impact on that equation, albeit a very small one, but an excellent question to ponder. A thank you and respectful nod to Les Niles and Michael McClennen for their insights and help with the energy consumption figures.

1 Factoring polynomialsAlgebra 2 Polynomials radical. Perhaps the most important pattern to be gained from these two examples that will help us factoring in our future work in factoring trinomials concerns the signs in the products Factoring Trinomials Worksheet Answers wiildcreative Graph Art Worksheets Acceleration Worksheet Middle School Subtracting Decimals Worksheet 5th Grade Subtracting Whole Numbers Worksheets Physical Activity Worksheets Math Maze Worksheets Union Intersection Of Sets Worksheet Simplifying Radicals Worksheet Figurative Language Quiz Worksheet SparkNotes: Algebra II: Factoring Summary Analysis. For example for 24 the GCF is 12. We played this matching game then I gave them. Algebra Students Math Great Factoring Trinomials Practice for my Algebra students factor by grouping polynomials calculator Softmath here are two more formulas to handle special cases of cubic polynomials: eqnarray26. Factor the polynomial identify it as irreducible. Some of the trinomials have leading coefficients which can be factored out in some cases must be left in for other problems Understanding Algebra: Why do we factor equations. Students will factor the trinomialswith then they will color , withouta ) completely draw on the penguin as indicated by their answer. comIn this lesson such Factoring trinomials Basic mathematics Learn factoring trinomials , students learn that a trinomial in the form x 2 bx cwhere c is negative polynomials made up of three terms. This lesson is Factoring Trinomialsa 1) Read reviews compare customer ratings, see screenshots learn more about Socratic Math Homework Help. Factoring TrinomialsUsing Diamond Problems : Algebra Helpsolutions examples videos) Online Math Learning Polynomials. It s a great way to review to learn along with your class Math Review of Factoring Polynomials. We will add subtract, multiply even start factoring a polynomial. Students then use homework the practice problems from NROC s Algebra 1 An Open course Unit 9 Factoring, Lesson 1, Topics 1, factoring factoring simple , 3these cover factoring out greatest common factors advanced trinomials by grouping) to develop skills in checking their own work. Such videos will Tiger Algebra A Free Online Algebra Solver , Calculator Free Algebra Solver Algebra Calculator showing step by step solutions. I factored: First last terms in the boxes Note that this graphic organizer works the same way as a multiplication table. Solve partial fractions Once , equation in excel, basic inequality calculator, square root rationalizing the denominator online calculator Again Resultado de Google Books. First find the product of a c. These examples practice problems will help you learn how to factor trinomials Trinomial Factoring Free Android Apps on homework Google Play Factors any trinomial equation will display error if not factorable. They homework feel overwhelmed with factoring polynomials homework tests projects. Some examples are difference of squares perfect square trinomial, trial error. trinomials Includes full solutions score reporting Polynomials Trial Error Shmoop Remember that a quadratic polynomial is a polynomial of degree 2 of the form ax2 homework bx c. These polynomials are easiest to factor when a 1that is the polynomial looks like x2 bx c so we ll look at that case first. Factoring trinomials homework help. Choose all levels of Algebra, Algebra 1 Course: Unit 5 Factoring , Trig, Calculus, Dividing trinomials Polynomials Math Tutor DVD provides math help online , Probability, on DVD in Basic Math Physics. Since the numbers 2 we will use them Factoring Polynomial Expressions Video Lesson Transcript. You already knowfrom the factoring above) that x 3 is a factor of the polynomial therefore that x3 is a zero. Suppose if you are willing to view a video on factoring polynomials be specific while searching for them. Browse notes covering Factoring Trinomials , exams , much more, homework, questions many other concepts. We have an unknown number which interacts with itselfx x x 2. The cartoon factoring people may may not be helpful. factoring Available as a mobile desktop website as well as native iOS Android apps IXL Factor polynomialsAlgebra 1 practice) Fun math practice. Examples of binomials are: x 3 x2 4 2x5 x. Homework Help Factoring Trinomials Ideas Of Algebra 1 Factoring Polynomials Worksheet We collect this best photo from internet choose one of the best for you, you can see Homework Help Factoring Trinomials Ideas Of Algebra 1 Factoring Polynomials Worksheet more pictures selection that posted here was Factoring Polynomials Math Forum Ask Dr. Thinkwell s online videos automatically graded homework problems make learning Precalculus easy fun. When we take an equation Factoring Trinomials with a 1 Made Easy sofatutor To help you understand this process make it easy for you to factor trinomials with the Factoring trinomials A complete course in algebra TheMathPage How to homework factor a trinomial. But this issue often plagues students; they keep getting points off on tests quizzes homework assignments. Thus when factoring trinomials the trick is to look for factors of 6 last term that will add up to 5 coefficient of second term. How to Multiply Polynomials Binomials using foil other techniquesGreat for kids who need extra help in Algebra. Enter a polynomial even just a number to see its factors. es una organización dedicada a la gestión integral de excedentes industriales y residuos peligrosos visión, la trinomials salud de nuestros trabajadores, trinomials considera como factores de gran importancia la satisfacción de nuestros clientes, principios y valores, la seguridad de nuestros procesos, coherente con su misión Factoring Trinomials Educational Videos. x2 5x 2 x2 6x 2, x2 2x 3, x2 9x factoring 6 x2 4x 5. Just type A B hit calculate Factoring Polynomials Activity Beginner. What about the Factoring Polynomials: Homework Help help Answers: Slader First take out a negative , move the terms so that it is in greatest to least based on it s power which trinomials will turn the equation into 6w 2 13w 5) now you factor, factors of 5 are1 , you know that factors of 6 are1 2 3, from the sign you know that it will have to be a , 5) try these combinations , 6) Factoring Polynomials FREE Worksheet. Download Socratic Math Homework Help enjoy it on your Apple TV Course: MA001: College Algebra Topic: 4. We hope that these examples will help you learn how to approach the different types of algebra word problems Synthetic Division Purplemath Synthetic division is generally used however not for dividing out factors but for finding zeroesor roots) of polynomials. Improve your skills with free problems inFactor polynomials' thousands of other practice lessons Polynomials Factoring Unit Lesson Plan UNC Chapel Hill Trinomial. look for two numbers whose product 6 whose sum 5. Trinomials of the form help x2bxc when c 0 have certain aspects in Factoring Polynomials Homework Help Best Papers Writing Service.

You probably already know that multiplying a decimal by turns it into a percentage. If you need a more sophisticated formula for solving a percentage problem, or if you want troubleshooting advice, the best thing to do is ask a question in the Excel forum on the Microsoft Answers site. Highlight the cell with the first number, then enter the "-" symbol. In this case, you can use the SUMIF function to add up all numbers relating to a given product first, and then divide that number by the total, like this: In an empty cell, enter one of the below formulas: A large range of cells Click the first cell in the range, and then hold down Shift while you click the last cell in the range. To increase an amount by a percentage, use this formula: This is useful when you want to type just a single percentage on your worksheet, such as a tax or commission rate. Keep this in mind if you want to change this column to percentage format. The question is - how much do you have to pay on top of the net price. The next example is slightly more complicated. Nonadjacent rows or columns Click the column or row heading of the first row or column in your selection; then hold down Ctrl while you click the column or row headings of other rows or columns that you want to add to the selection. You cannot cancel the selection of a cell or range of cells in a nonadjacent selection without canceling the entire selection. In cell B1 you could write: On a simpler level, the first thing to know is that a…ll formulas must start with the equals sign. Tip If you are completing a one-time calculation, write the formula as an absolute reference. Adjacent rows or columns Drag across the row or column headings. Suppose, you have the number of "Ordered items" in column B and "Delivered items" in column C. Calculating percentage of total in Excel In fact, the above example is a particular case of calculating percentages of a total. How do you write formulas for Excel. This is how you calculate percentage in Excel. Remember to increase the number of decimal places if needed, as explained in Percentage tips. Select the per cent cell and choose your format. In a column beside that,you would have your FREQUENCY function to calculate the amount foreach week, based on the source data from the set of week numbersthat you have calculate. The format should resemble the following: The mathematical formula for calculating percentages is the amount divided by the total. Calculating percent change between 2 columns Suppose that you have the last month prices in column B and this month prices in column C. The dollar sign fixes the reference to a given cell, so that it never changes no matter where the formula is copied. Parts of the total are in multiple rows In the above example, suppose you have several rows for the same product and you want to know what part of the total is made by all orders of that particular product. Considering the above, our Excel formula for percentage change takes the following shape: Kasper Langmann, Co-founder of Spreadsheeto Now, as you can see, you have the percentage change in decimal format. How to write percentage formulas in Excel Grant D. In this case, you can use the SUMIF function to add up all numbers relating to a given product first, and then divide that number by the total, like this: So if you wanted to find what percentage 53 is of 92 then you would use the following formula:. Aug 02, · Calculating percentages. As with any formula in Excel, you need to start by typing an equal sign (=) in the cell where you want your result, followed by the rest of the formula. So, I hear your next question in my head — which formula do I use to get the result I desire? Well, that depends. How to I write an IF statement that if I divide two negative numbers, the results show a negative percentage and not a positive percentage? You can format negative numbers to display with parentheses, but putting numbers in parentheses in an excel formula doesn't treat it as a negative. For instance, % = as the only. Percentages in Excel are stored as decimal values. For example, 25% is stored as the value50% is stored as the valueetc. It is the formatting of a cell that makes the underlying decimal value appear as. On the Formula Ribbon (Excel ), in the Formula Auditing section, click on the Show Formulas button. You will see both labels and formulas, instead of values in the cells. How to calculate percentage in Excel - formula examples by Svetlana Cheusheva | updated on June 28, Comments In this tutorial, you will lean a quick way to calculate percentages in Excel, find the basic percentage formula and a few more formulas for calculating percentage increase, percent of total and more. 1. Input Initial Data in Excel. Input the data as follows (or start with the download file "stylehairmakeupms.com" contained in the tutorial source files). This worksheet is for Expenses. Later in this tutorial, we’ll use the Grades worksheet.How to write a formula in excel for percentages

Advanced Theoretical and Applied Studies of Fractional Differential EquationsView this Special Issue Research Article | Open Access Ibrahim Karatay, Serife R. Bayramoglu, "A Characteristic Difference Scheme for Time-Fractional Heat Equations Based on the Crank-Nicholson Difference Schemes", Abstract and Applied Analysis, vol. 2012, Article ID 548292, 11 pages, 2012. https://doi.org/10.1155/2012/548292 A Characteristic Difference Scheme for Time-Fractional Heat Equations Based on the Crank-Nicholson Difference Schemes We consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with Riemann-Liouville fractional derivative of order α, where . The main purpose of this work is to extend the idea on Crank-Nicholson method to the time-fractional heat equations. We prove that the proposed method is unconditionally stable, and the numerical solution converges to the exact one with the order . Numerical experiments are carried out to support the theoretical claims. Fractional calculus is one of the most popular subjects in many scientific areas for decades. Many problems in applied science, physics and engineering are modeled mathematically by the fractional partial differential equations (FPDEs). We can see these models adoption in viscoelasticity [1, 2], finance [3, 4], hydrology [5, 6], engineering [7, 8], and control systems [9–11]. FPDEs may be investigated into two fundamental types: time-fractional differential equations and space-fractional differential equations. Several different methods have been used for solving FPDEs. For the analytical solutions to problems, some methods have been proposed: the variational iteration method [12, 13], the Adomian decomposition method [13–16], as well as the Laplace transform and Fourier transform methods [17, 18]. On the other hand, numerical methods which based on a finite-difference approximation to the fractional derivative, for solving FDPEs [19–24], have been proposed. A practical numerical method for solving multidimensional fractional partial differential equations, using a variation on the classical alternating-directions implicit (ADI) Euler method, is presented in . Many finite-difference approximations for the FPDEs are only first-order accurate. Some second-order accurate numerical approximations for the space-fractional differential equations were presented in [26–28]. Here, we propose a Crank-Nicholson-type method for time-fractional differential heat equations with the accuracy of order . In this work, we consider the following time-fractional heat equation: Here, the term denotes -order-modified Riemann-Liouville fractional derivative given with the formula: where is the Gamma function. Remark 1.1. If , then the Riemann-Liouville and the modified Riemann-Liouville fractional derivatives are identical, since the Riemann-Liouville derivative is given by the following formula: If is nonzero, then there are some problems about the existence of the solutions for the heat equation (1.1). To rectify the situation, two main approaches can be used: the modified Riemann-Liouville fractional derivative can be used or the initial condition should be modified . We chose the first approach in our work. 2. Discretization of the Problem In this section, we introduce the basic ideas for the numerical solution of the time-fractional heat equation (1.1) by Crank-Nicholson difference scheme. For some positive integers and , the grid sizes in space and time for the finite-difference algorithm are defined by and , respectively. The grid points in the space interval are the numbers , , and the grid points in the time interval are labeled , . The values of the functions and at the grid points are denoted and , respectively. As in the classical Crank-Nicholson difference scheme, we will obtain a discrete approximation to the fractional derivative at . Let Then, we have Now, we will find the approximations for and : where Similarly, we can obtain where and Then, we can write the following approximation: where On the other hand, using the mean-value theorem, we get where and . So, we obtain the following second-order approximation for the modified Riemann-Liouville derivative: 3. Crank-Nicholson Difference Scheme Using the approximation above, we obtain the following difference scheme which is accurate of order : We can arrange the system above to obtain The difference scheme above can be written in matrix form: where , , , , , and . Here, and are the matrices of the form We note that the unspecified entries are zero at the matrices above. Using the idea on the modified Gauss-Elimination method, we can convert (3.3) into the following form: Now, we need to determine the matrices and satisfying the last equality. Since , we can select and . Combining the equalities and and the matrix equation (3.3), we have Then, we write where . So, we obtain the following pair of formulas: where . 4. Stability of the Method The stability analysis is done by using the analysis of the eigenvalues of the iteration matrix () of the scheme (3.5). Let denote the spectral radius of a matrix , that is, the maximum of the absolute value of the eigenvalues of the matrix . We will prove that , (), by induction. Since is a zero matrix . Moreover, , since is of the form therefore, . Now, assume . After some calculations, we find that and we already know that and for : Since , it follows that . So, for any , where . Remark 4.1. The convergence of the method follows from the Lax equivalence theorem because of the stability and consistency of the proposed scheme. 5. Numerical Analysis Example 5.1. Consider Exact solution of this problem is . The solution by the Crank-Nicholson scheme is given in Figure 1. The errors when solving this problem are listed in the Table 1 for various values of time and space nodes. The errors in the table are calculated by the formula and the error rate formula is . Example 5.2. Consider Exact solution of this problem is . The solution by the Crank-Nicholson scheme is given in Figure 2. The errors when solving this problem are listed in Table 2 for various values of time and space nodes and several values of . 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Quantum versus classical phase-locking transition in a driven-chirped oscillator Classical and quantum-mechanical phase locking transition in a nonlinear oscillator driven by a chirped frequency perturbation is discussed. Different limits are analyzed in terms of the dimensionless parameters and ( and being the driving amplitude, the frequency chirp rate, the nonlinearity parameter and the linear frequency of the oscillator). It is shown that for , the passage through the linear resonance for above a threshold yields classical autoresonance (AR) in the system, even when starting in a quantum ground state. In contrast, for , the transition involves quantum-mechanical energy ladder climbing (LC). The threshold for the phase-locking transition and its width in in both AR and LC limits are calculated. The theoretical results are tested by solving the Schrodinger equation in the energy basis and illustrated via the Wigner function in phase space. pacs:42.50.Hz, 42.50.Lc, 33.80.Wz, 05.45.Xt Autoresonance (AR) is a generic nonlinear phase-locking phenomenon in classical dynamics. It yields a robust approach to excitation and control of nonlinear oscillatory systems by a continuous self-adjustment of systems’ parameters to maintain the resonance with chirped frequency perturbations. Applications of AR exist in many fields of physics, examples being atomic and molecular systems Maeda 2007 ; Chelkowski , nonlinear optics Segev , Josephson junctions Ofer Naaman , hydrodynamics BenDavid , plasmas Danielson , nonlinear waves Lazar92 , and quantum wells Manfredi 2007 . Most recently, AR served as an essential element in the formation of trapped anti-hydrogen atoms at CERN ALPHA Nature ; ALPHA PRL and in studying the effect of fluctuations in driven Josephson junctions Kater . While the classical AR is well understood, the investigation of the quantum-mechanical limits of the problem has started only recently Gilad ; Manfredi 2007 ; Kater . The present study focuses on the interrelation between the classical and quantum descriptions of the autoresonant transition in the simplest case of a driven Duffing oscillator (modeling a driven diatomic molecule Zigler or a Josephson junction Ofer Naaman , for example) governed by the Hamiltonian where , is the chirped driving frequency and . We will assume that initially our oscillator is in a thermal equilibrium with the environment at temperature , but the chirped system’s response is sufficiently fast to neglect the effect of the environment on the out-of-equilibrium dynamics Kater . Classically, in autoresonance, after passage through the linear resonance at , the driven oscillator gradually self-adjusts its oscillation frequency to that of the drive by continuously increasing its energy Fajans , yielding a convenient control of the dynamics by variation of an external parameter (the driving frequency). The transition to the classical AR by passage through linear resonance has a threshold on the driving amplitude, scaling as Fajans . This threshold is sharp if the oscillator starts in its zero equilibrium, but in the presence of thermal noise it develops a width, scaling as PRL . Both the AR threshold and its width have their quantum-mechanical counterparts, which will be discussed in this work. When the problem of autoresonant transition is dealt with quantum-mechanically, two questions must be addressed. First, what are the differences between the classical and quantum evolutions of the chirped-driven nonlinear oscillator? In dealing with this question, Ref. Gilad suggested that the natural quantum-mechanical limit of the classical AR is a series of successive Landau-Zener (LZ) LZ transitions or energy ladder climbing (LC), where only two adjacent energy levels of the driven oscillator are coupled at any given time. In contrast, the classical AR behavior takes place when many levels are coupled at all times during the excitation Goggin . We will adopt and further develop this point of view here and describe different regimes in the problem in terms of two dimensionless parameters suggested in Gilad . These parameters are defined via the three physical time-scales in the system, i.e. the inverse Rabi frequency , the frequency sweep time scale , and the characteristic nonlinearity time scale (the time of passage through the nonlinear frequency shift between the first two transitions on the energy ladder). Then, by definition (this parameter measures the strength of the drive), and (a measure of the nonlinearity in the problem). We will show in this work that this parameter space describes all limiting cases of quantum-mechanical evolution in our system, including quantum initial conditions, the subsequent transition to either LC or AR, and the associated threshold phenomenon. Note, that have a meaning only in the case of a chirped system, because of the new time scale, , associated with this case. The second question, which must be addressed in the quantum-mechanical formulation of our problem is that of quantum fluctuations. As mentioned above, in the presence of thermal noise, the classical AR transition probability develops a width, scaling as with temperature PRL . Nevertheless, at very low temperatures, the quantum fluctuations should be taken into account. Recent experiments by Kater et al. Kater demonstrated quantum saturation of the width of the phase-locking transition in superconducting Josephson junctions at sufficiently low temperatures, confirming the prediction that in the classical width formula PRL should be replaced by an effective temperature, , where for high temperatures and saturates at at low temperatures. The experimental results imply that the fluctuations only determine the initial conditions of such a non-equilibrium oscillator and do not affect its time evolution. In this work, we will address the effect of quantum fluctuations in the AR problem theoretically and provide further justification of using the classical AR threshold width formula with replaced by . The scope of the paper will be as follows. In Sec. II we will use the quantum-mechanical energy basis in the rotating wave approximation and compare the driven dynamics of our oscillator in the quantum and classical regimes numerically. Section III will present the analytic description of the transition to phase-locking in terms of the parameter space in both classical AR and quantum LC regimes. In the same Section, the theory will be compared with numerical simulations. Section IV will focus on the effect of quantum fluctuations on the width of the phase-locking transition. Finally, we will address the phase space dynamics in the problem in Sec. V by solving the quantum Liouville equation for the Wigner function numerically and compare the phase space evolution with that in the energy basis. Our conclusions will be summarized in section VI. Ii Chirped dynamics in the energy basis We write the wave function of the oscillator governed by Eq.(1), , in the energy basis of the undriven () Hamiltonian (1). The associated Schrodinger equation yields where we approximate the energy levels Landau QM , , and . We assume a weak coupling, , and, consequently, neglect the nonlinear correction of order in the coupling term. Next, we define , where , substitute this definition into Eq. (4), and neglect the nonresonant terms (rotating wave approximation) to get where . Finally, we introduce , where and the dimensionless slow time , associated with the change of the driving phase due to the driving frequency chirp. Then Eq. (6) can be written as where , and , as defined in the Introduction. Note, that characterizes the strength of the coupling between the adjacent levels, while is associated with the nonlinearity in the problem and determines the degree of classicality in the system (see Sec. V). Note also that the rotating frame here is chirped instead of the usual, fixed frequency frame and, thus, there remains an explicit time dependence in Eq. (7). Our goal is to analyze these slow evolution equations, but first, we discuss different limits in the driven system in parameter space. The comparison between the classical AR and the quantum LC regimes was first discussed by Marcus et al. Gilad , who suggested the nonlinear resonance classicality criterion, , by requiring that the classical resonance width would include more than two quantum levels. Since the chirp rate cancels from this creterion, the latter characterizes the nonlinear resonance phenomenon in the system driven by constant frequency drive as well. The chirping introduces a new effect, i.e. a possibility of a continuous self-adjustment of the energy of the oscillator to stay in resonance with the drive. This yields a new condition, separating the classical AR and quantum LC transitions, where the dynamics of the chirped system is very different. In the LC transition, only two levels are coupled at a time and the system’s wave function climbs the energy ladder by successive LZ transitions LZ . For example, Eq. (7) yields the following two-level transformation matrix for the transition We can calculate the time of the transition, by equating the diagonal elements in this matrix, i.e. , so the time interval between two successive transitions is . On the other hand, the typical duration of each LZ transition has two distinct limits Finite LZ . In the non adiabatic (sudden) limit (), is of the order of unity, while in the opposite (adiabatic) limit, . Therefore, by comparing and , we expect to see well separated successive LZ steps, i.e. the LC, provided which describes both the sudden and the adiabatic limits. In contrast, the classical AR transition requires , which coincides with the nonlinear resonance classicality criterion mentioned above, when . In section V, a different argument will be suggested to explain why classical mechanics yields the correct description of the transition to autoresonance when a stronger inequality, , is satisfied, even when the system starts in the quantum mechanical ground state. Next, we discuss the numerical solutions of the problem and compare different regimes of chirped-driven dynamics. We have solved Eqs. (7) numerically, subject to ground state initial conditions at (the linear resonance corresponds to ). Each of the Figs. 1–3 corresponds to a different value of the nonlinearity parameter and show the distribution of the population of the levels in the system at four different times (subplots a-d). The subplots e-h in the Figures show the associated Wigner distributions (see Sec. V) at the same times. Figure 1 shows the case of the LC dynamics for and at , and (subplots a-d), and illustrates a clear time separation beyond the linear resonance between the successive LZ transitions. For example, we observe two groups of resonant and nonresonant levels at , separated by a valley centered at about . We find that the location of the resonant levels is determined by the slow time, i.e. , as shown above. Thus, the resonant (phase-locked) state in the system is efficiently controlled via the driving frequency and a given final state can be reached (and maintained) by terminating the frequency chirp at the desired energy level. We also see that there exists a single highly occupied level in the resonant group of levels at any given time, indicating successive LZ transitions, as expected in the LC regime. Our second numerical example is presented in Fig. 2 and illustrates the intermediate regime (as discussed above) with and , and (the subplots a-d). As in Fig. 1, a clear separation between the resonant and nonresonant groups of levels is seen in the Figure. We see that, typically, several levels are excited in the resonant group, but their number is small, so the driven dynamics can not be considered as classical. The last example (see Fig. 3) corresponds to the classical regime, , and . One observes a separation between resonant and nonresonant groups at . Note that in all our numerical examples about 50% only of the initial state is transferred to the continuing phase-locked state, leading to the question of resonant capture probability, which is discussed next. Iii Resonant capture probability iii.1 Threshold for phase-locking transitions For a given set we define the resonant capture probability, where is the number of the level separating the resonant and nonresonant groups of levels at sufficiently large times. For a given value of , the probability depends on the driving parameter, For example, in the case in Fig. 1, we use and the resonant capture probability is . Similarly, in the two examples in Figs. 2 and 3, we choose to get , respectively. We calculate the resonant capture probability by solving Eqs. (7) numerically subject to initial conditions, (the ground state), for different values of and . For a fixed , the capture probability is a monotonically increasing, smoothed step function of . We define the threshold for efficient phase-locking transition, , as the value of for 1/2 capture probability, i.e. . The full circles in Fig. 4 show for different values of . The dashed and dashed-dotted lines are the assymptotic theoretical predictions for the quantum LC and classical AR (see below), which agree with the results of our simulations in both limits. The line is the separator between the classical and the quantum regimes of the chirped nonlinear resonance, as discussed in Sec. II. This line crosses the threshold line at . One can see in the Figure that indeed, this point separates very different dependences of on associated with the quantum and classical dynamics of the chirped system. One can also see the oscillating pattern of the threshold at , where the transition to phase-locking involves a mixture of LC and multi-level LZ steps. Next, we calculate the threshold for phase-locking transitions analytically. iii.2 Quantum-mechanical ladder climbing In the quantum LC regime the nonlinearity parameter determines the time interval between successive resonances [see Eq. (8)]. In the case of a strong nonlinearity, at any given time only two levels are coupled, and the dynamics can be modeled by successive LZ transitions. In this case, we can calculate the probability of each transition separately, and multiply the probabilities. The two level transformation matrix (8) in the energy basis for the transition yields the transition probability via the LZ formula LZ where . We define the probability for capture into resonance in this case as the probability of occupying a sufficiently high energy level after successive LZ transitions, i.e. Then, solving , one finds the threshold for the LC transition, where for two digits accuracy we used in the rapidly converging product (11). Thus, the capture into resonance occurs in the first few LZ transitions and one can choose (see Fig. 1 in the definition Eq. (9) for calculating the capture probability near the threshold. This prediction is valid for large , as mentioned above. The dashed line in Fig. 4 represents Eq. (12), while the numerical result for 1/2 capture probability is shown by full circles. One can see a very good agreement between the two results in the LC limit (). However, in the intermediate regime (), oscillations in are observed before convergence at the predicted LC line. These oscillations are due to the mixing of more than two neighboring levels in passage through resonance (see Fig. 2). iii.3 Classical autoresonance As decreases, a growing number of levels are coupled simultaneously and the dynamics becomes increasingly classical. The classical AR phenomenon is now well understood Fajans . If one starts in the zero amplitude equilibrium, the autoresonant phase-locking is achieved for drives of amplitude above the critical value Fajans . When expressed in terms of , this classical threshold is translated into When thermal fluctuations are included, the transition probability develops a width scaling as with temperature PRL . At the same time, the threshold for 1/2 capture probability remains the same. Thus, in Eq. (13) is the classical counterpart of the quantum-mechanical observable in Eq. (12). This classical threshold is shown in Fig. 4 by dashed-dotted line, illustrating excellent agreement with simulations (full circles) in the classical regime, . It should be emphasized that the simulation results in the Figure are solutions of the quantum-mechanical equations (7) with parameters in the classical regime, while the probabilities of capture were calculated using the proper transition level for each value of , as defined in Eq. (9). In the next Section, we discuss the width of the autoresonant transition. Iv The width of the phase-locking transition Another observable of the phase-locking transition mentioned above is the width of the transition, which we define as the inverse slope of the phase-locking probability at . This width depends on the initial conditions governed by the thermal equilibrium with the environment. Classically, the thermal width of the autoresonant transition scales as PRL However, at very low temperatures, the classical thermal noise becomes negligible, but quantum fluctuations remain. Recent experiments in Josephson circuits Kater demonstrated quantum saturation of the transition width at the value obtained from Eq.(14), but with replaced by the energy of the ground level. More generally, it was suggested to calculate the width by replacing in the classical formula by an effective temperature, , in agreement with the experimental results. Using , we can translate Eq. (14) into the transition width in terms of in the zero temperature limit. The Josephson circuit experiments Kater were performed with , i.e. well inside the classical region (see Fig. 4). Interestingly, these experiments allowed to characterize the initial quantum ”temperature” of the system by measuring the final classical autoresonant state of the chirped excitation. We will justify this approach in the next Section by analyzing the dynamics of the associated Wigner function in phase space. In contrast to Eq. (15) valid when the final state of the system is classical , the threshold width of the phase-locking transition in the LC regime () can be calculated by evaluating the slope of from Eq. (11) at , yielding where we assume that the system is in the ground state initially. Figure 5 summarizes our theoretical predictions for the width of the phase-locking transition (for the same parameters as in Fig. 4) and compares them with those from numerical simulations via the Schrodinger equation (7). We see a good agreement in both the AR and LC limits, but notice significant oscillations of the width in the intermediate range of . Remarkably, while the thresholds in the classical and quantum-mechanical limits have very different scalings, the widths of the transitions are nearly the same. V Chirped dynamics in phase space Phase space dynamics comprises a convenient framework for comparison between classical and quantum evolution of the system. The Wigner function is one of the most useful phase space representations of the quantum mechanics, since it reduces to the classical phase space distribution in the limit of In this Section, we will study the dynamics of the Wigner function in our chirped oscillator problem in both the fixed and the rotating frames and discuss the transition to the classical limit in the problem. v.1 Wigner dynamics in the fixed frame The Wigner function associated with the Hamiltonian of form is governed by the quantum Liouville equation Schleich where and we neglect possible decay and decoherence processes. We take a low temperature limit, neglect the nonlinearity initially, and assume that the initial state of the system is in equilibrium with the environment, i.e. Schleich , where is the effective temperature. Note that at high temperatures, while at . In the case of interest the potential is a quartic [see Eq. (1)] and, therefore, only one term survives in the right hand side of (18), allowing to rewrite this equation in the following dimensionless form where, , , , , , and . In addition, we measure time in Eq. (20) in units of and introduce the dimensionless chirp rate . With this rescaling, the initial Wigner distribution (19) becomes . We solved Eq. (20) numerically with the same parameters as in the Schrodinger simulations and show the results in Figs. 1-3 (subplots e-h) at the same times for comparison. For a better representation of the Wigner distributions for different nonlinearities, we rescaled the axis in the Figures to , and . The dashed lines in the Figures are the separatrices, enclosing all bounded classical trajectories in phase space. We started all these simulations in the ground state, i.e. , at the initial time . Figure 1 compares the dynamics in phase space to that in the energy basis in the quantum LC regime , using the parameters , , and . The pattern seen near the origin in Fig. 1 is due to the quantum interference with a finite number of states in the nonresonant region. Figure 2 shows the intermediate case for parameters , , and . Finally, Fig. 3 corresponds to the classical AR case and the parameters , , and . As well known Zurek , in the near classical case the Wigner function becomes oscillatory on increasingly fast phase space scales. However, if coarse-grained (due to a finite numerical accuracy in our case), the Wigner function becomes almost everywhere positive as one approaches the classical distribution function, despite the initial quantum-mechanical ground state used in the simulations. The evolution of the Wigner function in the last example is nearly classical with the quantum signature entering only via the effective temperature of the initial state. In the classical formula (14) for the transition width, appears due to integration over the classical Maxwell-Boltzman distribution function (see PRL ). Therefore, for the quantum-mechanical initial conditions, we should integrate over the Wigner function in a thermal state instead over the classical distribution. But these two distributions have the same functional shape, except that is replaced by in Eq. (19). Therefore, as also confirmed in experiments Kater , one can use the classical formula for the threshold of the phase-locking transition at low temperatures, when starting from quantum-mechanical initial conditions. v.2 The dynamics in the rotating frame Here we further expand our discussion of the classical AR limit in our system via the Wigner representation in the rotating frame. The transformation to the rotating frame is accomplished using unitary transformation (see Dykman 2006 ) where the operator and is the driving phase [see Eq. (1)]. Then, by neglecting rapidly oscillating terms, the Hamiltonian (1) is transformed to The parameter in the last equation is familiar from the theory of the classical AR PRL , while is the dimensionless Plank constant, entering the commutation relation for the rescaled variables Here , , where and the dimensionless time associated with the dynamics governed by Hamiltonian (23) is . Next, we write the quantum Liouville equation in the rotating frame (see Ref. Dykman 2007 for similar developments for a constant frequency drive) where . The initial Wigner distribution (19) in the new variables is where . The left hand side of the Eq. (25) is identical to the Vlasov equation describing the evolution of a classical distribution of particles governed by Hamiltonian (23) without collisions and self-fields. Hence, as in the fixed frame, after coarse-graining the fast phase space oscillations of in the limit (), the dynamics in phase space can be treated classically Zurek . Therefore, both the threshold and the width of the autoresonant transition can be calculated from the classical theory as illustrated in Figs. 4 and 5, respectively, despite the quantum-mechanical initial conditions in the problem. In other words, is the measure of the classicality of the phase-locking transition in our chirped oscillator. Furthermore, in the limit of , only two parameters, and (via the initial conditions) fully characterize the AR transition. This result is in agreement with Eqs. (13) and (15) for the AR threshold and its width, where, remarkably, and enter separately. (a) We have studied the interrelation between the quantum-mechanical and classical dynamics of phase-locking transition in a Duffing oscillator driven by a chirped frequency oscillation. We studied the conditions for a continuous phase-locking in the driven system, such that the energy of the oscillator grows to stay in resonance with the varying driving frequency. The problem was defined by the temperature and three parameters, i.e. the driving amplitude the driving frequency chirp rate , and the parameter characterizing the nonlinearity of the oscillator. The nonlinearity in the problem was essential, since no persistent phase-locking in the system could be achieved for . (b) We have exploited a more natural representation of both the quantum-mechanical and classical dynamics in the problem via just two dimensionless parameters Gilad , and instead of , and . We have shown that describes the classicality of the phase-locking transition in the system, such that, for the system arrives at its classical autoresonant (AR) state after passage through linear resonance even when starting in the quantum-mechanical ground state. In contrast, for , the transition involves the energy ladder climbing (LC) process, i.e. a continuing sequence of separated Landau-Zener transitions between neighboring energy levels. The parameters have a meaning only in the case of a finite chirp rate, which introduces a new time scale, , in the problem. (c) The probability of transition to the phase-locked state versus has a characteristic -shape (a smoothed step function). The value of yielding 50% transition probability can be viewed as the threshold for the phase-locking transition. We have calculated this threshold and its width in both the quantum-mechanical LC and classical AR limits and compared the results to those from quantum-mechanical calculations starting in the ground state of the oscillator (see Figs. 4 and 5). We have found that, while in the LC limit the threshold is independent of , in the classical AR regime, the threshold is defined by the combination of parameters. The agreement of the theory and simulations in both limits was excellent, but characteristic oscillations of the threshold and the width were observed in the intermediate regime . (d) We have also studied the dynamics of the phase-locking transition in phase space by using the Wigner function representation, to explain the quantum saturation of the width of the threshold for AR transitions. The analysis of the Wigner (quantum Liouville) equation in the chirped rotating frame clarifies the role of as characterizing the degree of classicality in the phase-locking transition problem. (d) A possibility of engineering and control of a desired quantum state of the oscillator via ladder climbing process (see an example in Fig. 1) seems to be attractive in such applications as quantum computing. A generalization of this study to include possible decay, decoherence, and tunneling processes also seems to be important in future studies. Acknowledgements.This work was supported by the Israel Science Foundation under grant No. 451/10. - (1) H. Maeda, J. Nunkaew, and T.F. Gallagher, Phys. Rev. A 75, 053417 (2007). - (2) S. Chelkowski and A. Bandrauk, J. Chem. Phys. 99, 4279 (1993). - (3) A. Barak, Y. Lamhot, L. Friedland, and M. Segev, Phys. Rev. Lett. 103, 123901 (2009). - (4) O. Naaman, J. Aumentado, L. Friedland, J.S. Wurtele, and I. Siddiqi, Phys. Rev. Lett. 101, 117005 (2008). - (5) O. Ben-David, M. Assaf, J. Fineberg, and B. Meerson, Phys. Rev. Lett. 96, 154503 (2006). - (6) J.R. Danielson, T.R. Weber, and C.M. Surko, Phys. Plasmas 13, 123502 (2006). - (7) L. Friedland, Phys. Rev. Lett. 69, 1749 (1992). - (8) G. Manfredi and P.A. Hervieux, App. Phys. Lett. 91, 061108 (2007). - (9) G.B. Andresen et al. (ALPHA Collaboration), Nature (London) 468, 673 (2010). - (10) G.B. Andresen et al. (ALPHA Collaboration), Phys. Rev. Lett 106, 025002 (2011). - (11) K.W. Murch, R. Vijay, I. Barth, O. Naaman, J. Aumentado, L. Friedland, and I. Siddiqi, Nature Physics 7, 105 (2011). - (12) G. Marcus, L. Friedland, and A. Zigler, Phys. Rev. A 69, 013407 (2004). - (13) G. Marcus, A. Zigler, and L. Friedland, Europhys. Lett. 74, 43 (2006). - (14) J. Fajans and L. Friedland, Am. J. Phys. 69, 1096 (2001). - (15) I. Barth, L. Friedland, E. Sarid, and A.G. Shagalov, Phys. Rev. Lett. 103, 155001 (2009). - (16) L.D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C. Zener, Proc. R. Soc. London A137, 696 (1932). - (17) M.E. Goggin and P.W. Milonni, Phys. Rev. A 37, 796 (1988). - (18) L. Landau and E.M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), 3rd ed. (Butterworth Heinemann, Oxford, 1977) p. 136. - (19) N.V. Vitanov and B.M. Garraway, Phys. Rev. A 53, 4288 (1996). - (20) W.P. Schleich, Quantum Optics in Phase Space, (Wiley-VCH Verlag, Berlin, 2001), Chap. 3.3, p.75. - (21) see, for example, W.H. Zurek, Rev. Mod. Phys. 75, 715 (2003). - (22) M. Marthaler and M.I. Dykman, Phys. Rev. A 73, 042108 (2006). - (23) M.I. Dykman, Phys. Rev. E 75, 011101 (2007).

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John F. Donoghue Department of Physics, University of Massachusetts Each field has a set of questions which are universally viewed as important, and these questions motivate much of the work in the field. In particle physics, several of these questions are directly related to experimental problems. Examples include questions such as: Does the Higgs boson exist and, if so, what is its mass? What is the nature of the dark matter seen in the Universe? What is the mechanism that generated the net number of baryons in the Universe? For these topics, there is a well posed problem related to experimental findings or theoretical predictions. These are problems that must be solved if we are to achieve a complete understanding of the fundamental theory. There also exists a different set of questions which have a more aesthetic character. In these cases, it is not as clear that a resolution is required, yet the problems motivate a search for certain classes of theories. Examples of these are the three 'naturalness' or 'fine-tuning' problems of the Standard Model; these are associated with the cosmological constant A, the energy scale of electroweak symmetry-breaking v and the strong CP-violating angle d. As will be explained more fully below, these are free parameters in the Standard Model that seem to have values 10 to 120 orders of magnitude smaller than their natural values and smaller than the magnitude of their quantum corrections. Thus their 'bare' values plus their quantum corrections need to be highly fine-tuned in order to obtain the observed values. Because of the magnitude of this fine-tuning, one suspects that there is a dynamical mechanism at work that makes the fine-tuning natural. This motivates many of the theories of new physics beyond the Standard Model. A second set of aesthetic problems concern the parameters of the Standard Model, i.e. the coupling constants and masses of Universe or Multiver.se?, ed. Bernard Carr. Published by Cambridge University Press. © Cambridge University Press 2007. the theory. While the Standard Model is constructed simply using gauge symmetry, the parameters themselves seem not to be organized in any symmetric fashion. We would love to uncover the principle that organizes the quark and lepton masses (sometimes referred to as the 'flavour problem'), for example, but attempts to do so with symmetries or a dynamical mechanism have been unsuccessful. These aesthetic questions are very powerful motivations for new physics. For example, the case for low energy supersymmetry, or other TeV scale dynamics to be uncovered at the Large Hadron Collider (LHC), is based almost entirely on the fine-tuning problem for the scale of electroweak symmetry-breaking. If there is new physics at the TeV scale, then there need not be any fine-tuning at all and the electroweak scale is natural. We are all greatly looking forward to the results of the LHC, which will tell us if there is in fact new physics at the TeV scale. However, the aesthetic questions are of a different character from direct experimental ones concerning the existence and mass of the Higgs boson. There does not have to be a resolution to the aesthetic questions - if there is no dynamical solution to the fine-tuning of the electroweak scale, it would puzzle us, but would not upset anything within the fundamental theory. We would just have to live with the existence of fine-tuning. However, if the Higgs boson is not found within a given mass range, it would falsify the Standard Model. The idea of a multiverse will be seen to change drastically the way in which we perceive the aesthetic problems of fine-tuning and flavour. In a multiverse, the parameters of the theory vary from one domain to another. This naturally leads to the existence of anthropic constraints - only some of these domains will have parameters that reasonably allow the existence of life. We can only find ourselves in a domain which satisfies these anthropic constraints. Remarkably, the anthropic constraints provide plausible 'solutions' to two of the most severe fine-tuning problems: those of the cosmological constant and the electroweak scale. Multiverse theories also drastically reformulate some of the other problems - such as the flavour problem. However, at the same time, these theories raise a new set of issues for new physics. My purpose in this chapter is to discuss how the idea of the multiverse reformulates the problems of particle physics. It should be noted up front that the Anthropic Principle [1-3] has had a largely negative reputation in the particle physics community. At some level this is surprising - a community devoted to uncovering the underlying fundamental theory might be expected to be interested in exploring a suggestion as fundamental as the Anthropic Principle. I believe that the problem really lies in the word 'Principle' more than in the word 'Anthropic'. The connotation of 'Principle' is that of an underlying theory. This leads to debates over whether such a principle is scientific, i.e. whether it can be tested. However, 'anthropics' is not itself a theory, nor even a principle. Rather, the word applies to constraints that naturally occur within the full form of certain physical theories. However, it is the theory itself that needs to be tested, and to do this one needs to understand the full theory and pull out its predictions. For theories that lead to a multiverse, anthropic constraints are unavoidable. As we understand better what types of theory have this multiverse property, the word anthropic is finding more positive applications in the particle physics community. This article also tries to describe some of the ways that anthropic arguments can be used to positive effect in particle physics. The Lagrangian of the Standard Model (plus General Relativity) encodes our present understanding of all observed physics except for dark matter . The only unobserved ingredient of the theory is the Higgs boson. The Standard Model is built on the principle of gauge symmetry - that the Lagrangian has an SU(3) ® SU(2)l ® U(1) symmetry at each point of spacetime. This, plus renormalizability, is a very powerful constraint and uniquely defines the structure of the Standard Model up to a small number of choices, such as the number of generations of fermions. General Relativity is also defined by a gauge symmetry - local coordinate invariance. The resulting Lagrangian can be written in compact notation: Experts recognize the various terms here as indications of the equations governing the photon, gluons and W-bosons (the F2 terms), quarks and leptons (the V terms), the Higgs field (0) and gravity (R), along with a set of interactions constrained by the gauge symmetry. Of course, such a simple form belies a very complex theory, and tremendous work is required to understand the predictions of the Standard Model. But the greatest lesson of particle physics of the past generation is that nature organizes the Universe through a simple set of gauge symmetries. However, the story is not complete. The simple looking Lagrangian given by Eq. (15.1), and the story of its symmetry-based origin, also hide a far less beautiful fact. To really specify the theory, we need not only the Lagrangian, but also a set of twenty-eight numbers which are the parameters of the theory. These are largely hidden underneath the compact notation of the Lagrangian. Examples include the masses of all the quarks and leptons (including neutrinos), the strengths of the three gauge interactions, the weak mixing angles describing the charge current interactions of quarks and lep-tons, the overall scale of the weak interaction, the cosmological constant and Newton's gravitational constant. None of these parameters is predicted by the theory. The values that have have been uncovered experimentally do not obey any known symmetry pattern, and the Standard Model provides no principle by which to organize them. After the beauty of the Standard Model Lagrangian, these seemingly random parameters reinforce the feeling that there is more to be understood. Three of the twenty-eight parameters are especially puzzling, because their values appear to be unnaturally small. Naturalness and fine-tuning have very specific technical meanings in particle physics. These meanings are related to, but not identical to, the common usage in non-technical settings. The technical version is tied to the magnitude of quantum corrections. When one calculates the properties of any theory using perturbation theory, quantum mechanical effects give additive corrections to all its parameters. Perturbation theory describes the various quantities of a theory as a power series in the coupling constants. The calculation involves summing over the effects of all virtual states that are possible in the theory, including those at high energy. The quantum correction refers to the terms in the series that depend on the coupling constants. The 'bare' value is the term independent of the coupling constants. The physical measured value is the sum of the bare value and the quantum corrections. The concept of naturalness is tied to the magnitude of the quantum corrections. If the quantum correction is of the same order as (or smaller than) the measured value, the result is said to be natural. If, on the contrary, the measured value is much smaller than the quantum correction, then the result is unnatural because the bare value and the quantum correction appear to have an unexpected cancellation to give a result that is much smaller than either component. This is an unnatural fine-tuning. In fact, the quantum correction is often not precisely defined. The ambiguity can arise due to possible uncertainties of the theory at high energy. Since physics is an experimental science, and we are only gradually uncovering the details of the theory as we probe higher energies, we do not know the high energy limits of our present theory. We expect new particles and interactions to be uncovered as we study higher energies. Since the quantum correction includes effects from high energy, there is an uncertainty about their extent and validity. We understand the theory up to some energy - let us call this Emax - but beyond this new physics may enter. The quantum corrections will typically depend on the scale Emax. We will see below that, in some cases, the theory may be said to be natural if one employs low values of Emax but becomes unnatural for high values. The Higgs field in the Standard Model takes a constant value everywhere in spacetime. This is called its 'vacuum expectation value', abbreviated as vev, which has the magnitude v = 246 GeV. This is the only dimensionful constant in the electroweak interactions and hence sets the scale for all dimensionful parameters of the electroweak theory. For example, all of the quark and lepton masses are given by dimensionless numbers r (the Yukawa couplings) times the Higgs vev, mi = riv^\/2. However, the Higgs vev is one of the parameters which has a problem with naturalness. While it depends on many parameters, the problem is well illustrated by its dependence on the Higgs coupling to the top quark. In this case, the quantum correction grows quadratically with Emax. One finds where rt is the Yukawa coupling for the top quark, v0 is the bare value, A is the self-coupling of the Higgs and the second term is the quantum correction. Since v = 246 GeV and rt ~ A ~ 1, this would be considered natural if Emax ~ 103 GeV, but it would be unnatural by twenty-six orders of magnitude if Emax ~ 1016 GeV (characteristic of the Grand Unified Theories which unite the electroweak and strong interactions) or thirty-two orders of magnitude if E max ^ 1019 GeV (characteristic of the Planck mass, which sets the scale for quantum gravity). If we philosophically reject fine-tuning and require that the Standard Model be technically natural, this requires that Emax should be around 1 TeV. For this to be true, we need a new theory to enter at this scale that removes the quadratic dependence on Emax in Eq. (15.2). Such theories do exist - supersymmetry is a favourite example. Thus the argument against fine-tuning becomes a powerful motivator for new physics at the scale of 1 TeV. The LHC has been designed to find this new physics. An even more extreme violation of naturalness involves the cosmological constant A. Experimentally, this dimensionful quantity is of order A — (10"3 eV)4. However, the quantum corrections to it grow as the fourth power of the scale Emax: with the constant c being of order unity. This quantity is unnatural for all particle physics scales by a factor of 1048 for Emax — 103 GeV to 10124 for Emax - 1019 GeV. It is unlikely that there is a technically natural resolution to the cosmo-logical constant's fine-tuning problem - this would require new physics at 10"3 eV. A valiant attempt at such a theory is being made by Sundrum , but it is highly contrived to have new dynamics at this extremely low scale which modifies only gravity and not the other interactions. Finally, there is a third classic naturalness problem in the Standard Model - that of the strong CP-violating parameter 9. It was realized that QCD can violate CP invariance, with a free parameter 9 which can, in principle, range from zero up to 2n. An experimental manifestation of this CP-violating effect would be the existence of a non-zero electric dipole moment for the neutron. The experimental bound on this quantity requires 9 < 10"10. The quantum corrections to 9 are technically infinite in the Standard Model if we take the cut-off scale Emax to infinity. For this reason, we would expect that 9 is a free parameter in the model of order unity, to be renormalized in the usual way. However, there is a notable difference from the two other problems above in that, if the scale Emax is taken to be very large, the quantum corrections are still quite small. This is because they arise only at a very high order in perturbation theory. So, in this case, the quantum corrections do not point to a particular scale at which we expect to find a dynamical solution to the problem. The standard response to the fine-tuning problems described above is to search for dynamical mechanisms that explain the existence of the fine-tuning. For example, many theories for physics beyond the Standard Model (such as supersymmetry, technicolour, large extra dimensions, etc.) are motivated by the desire to solve the fine-tuning of the Higgs vev. These are plausible, but as yet have no experimental verification. The fine-tuning problem for the cosmological constant has been approached less successfully; there are few good suggestions here. The strong CP problem has motivated the theory of axions, in which an extra symmetry removes the strong CP violation, but requires a very light pseudo-scalar boson - the axion - which has not yet been found. However, theories of the multiverse provide a very different resolution of the two greatest fine-tuning problems, that of the Higgs vev and the cosmological constant. This is due to the existence of anthropic constraints on these parameters. Suppose for the moment that life can only arise for a small range of values of these parameters, as will be described below. In a multiverse, the different domains will have different values of these parameters. In some domains, these parameters will fall in the range that allows life. In others, they will fall outside this range. It is then an obvious constraint that we can only observe those values that fall within the viable range. For the cosmological constant and the Higgs vev, we can argue that the anthropic constraints only allow parameters in a very narrow window, all of which appears to be fine-tuned by the criteria of Section 15.3. Thus the observed fine-tuning can be thought to be required by anthropic constraints in multiverse theories. The first application of anthropic constraints to explain the fine-tuning of the cosmological constant - even before this parameter was known to be non-zero - was due to Linde and Weinberg ; see also refs. [8-10]. In particular, Weinberg gave a physical condition - noting that, if the cosmo-logical constant was much different from what it is observed to be, galaxies could not have formed. The cosmological constant is one of the ingredients that governs the expansion of the Universe. If it had been of its natural scale of (103 GeV)4, the Universe would have collapsed or been blown apart (depending on the sign) in a fraction of a second. For the Universe to expand slowly enough that galaxies can form, A must lie within roughly an order of magnitude of its observed value. Thus the 10124 orders of magnitude of fine-tuning is spurious; we would only find ourselves in one of the rare domains with a tiny value of the cosmological constant. Other anthropic constraints can be used to explain the fine-tuning of the Higgs vev. In this case, the physical constraint has to do with the existence of atoms other than hydrogen. Life requires the complexity that comes from having many different atoms available to build viable organisms. It is remarkable that these atoms do not exist for most values of the Higgs vev, as has been shown by my collaborators and myself [11,12]. Suppose for the moment that all the parameters of the Standard Model are held fixed, except for v which is allowed to vary. As v increases, all of the quark masses grow, and hence the neutron and proton masses also increase. Likewise, the neutron-proton mass-splitting increases in a calculable fashion. The most model-independent constraint on v then comes from the value when the neutron-proton mass-splitting becomes larger than the 10 MeV per nucleon that binds the nucleons into nuclei; this occurs when v is about five times the observed value. When this happens, all bound neutrons will decay to protons [11,12]. However, a nucleus of only protons is unstable and will fall apart into hydrogen. Thus complex nuclei will no longer exist. A tighter constraint takes into account the calculation of the nuclear binding energy, which decreases as v increases. This is because the nuclear force, especially the central isoscalar force, is highly dependent on pion exchange and, as v increases, the pion mass also increases, making the force of shorter range and weaker. In this case, the criteria for the existence of heavy atoms require v to be less than a few times its observed value. Finally, a third constraint - of comparable strength - comes from the need to have deuterium stable, because deuterium was involved in the formation of the elements in primordial and stellar nucleosynthesis [11,12]. In general, even if the other parameters of the Standard Model are not held fixed, the condition is that the weak and strong interactions must overlap. The masses of quarks and leptons arise in the weak interactions. In order to have complex elements, some of these masses must be lighter than the scale of the strong interactions and some heavier. This is a strong and general constraint on the electroweak scale. All of these constraints tell us that the viable range for the Higgs vev is not the thirty or so orders of magnitude described above, but only the tiny range allowed by anthropic constraints. While anthropic constraints have the potential to solve the two greatest fine-tuning problems of the Standard Model, similar ideas very clearly fail to explain the naturalness problem of the strong CP-violating parameter 9 . For any possible value of 9 in the allowed range from 0 to 2n, there would be little influence on life. The electric dipole moments that would be generated could produce small shifts in atomic energy levels but would not destabilize any elements. Even if a mild restriction could be found, there would be no logical reason why 9 should be as small as 10_10. Therefore the idea of a multiverse does nothing to solve this fine-tuning problem. The lack of an anthropic solution to this problem is a very strong constraint on multiverse theories. It means that, in a multiverse ground state that satisfies the other anthropic constraints, the strong CP problem must generically be solved by other means. Perhaps the axion option, which appears to us to be an optional addition to the Standard Model, is in fact required to be present for some reason - maybe in order to generate dark matter in the Universe. Or perhaps there is a symmetry that initially sets d to zero, in which case the quantum corrections shift it only by a small amount. This can be called the 'small infinity' solution, because - while the quantum correction is formally infinite - it is small when any reasonable cut-off is used. Thus the main problem in this solution is to find a reason why the bare value of d is zero rather than some number of order unity. In any case, in multiverse theories the strong CP problem appears more serious than the other fine-tuning problems and requires a dynamical solution.1 The above discussion can be viewed as a motivation for multiverse theories. Such theories would provide an explanation of two of the greatest puzzles of particle physics. However, this shifts the focus to the actual construction of such physical theories. So far we have just presented a 'story' about a multiverse. It is a very different matter to construct a real physical theory that realizes this story. The reason that it is difficult to construct a multiverse theory is that most theories have a single ground state, or at most a small number of ground states. It is the ground state properties that determine the parameters of the theory. For example, the Standard Model has a unique ground state, and the value of the Higgs vev in that state determines the overall scale for the quark masses etc. Sometimes theories with symmetries will have a set of discretely different ground states, but generally just a few. The utility of the multiverse to solve the fine-tuning problems requires that there be very many possible ground states. For example, if the cosmological constant has a fine-tuning problem of a factor of 1050, one would expect that one needs of order 1050 different ground states with different values of the cosmological constant in order to have the likelihood that at least one of these would fall in the anthropically allowed window. In fact, such theories do exist, although they are not the norm. There are two possibilities: one where the parameters vary continuously and one where they vary in discrete steps. In the former case, the variation of the parameters in space and time must be described by a field. Normally such a field would settle into the lowest energy state possible, but there is a mechanism whereby the expansion of the Universe 'freezes' the value of the field and does not let it relax to its minimum [14-16]. However, since 1 Chapter 3 of this volume by Wilczek, suggests a possible anthropic explanation in the context of inflationary models for why 0 should be very small. the present expansion of the Universe is very small, the forces acting on this field must be exceptionally tiny. There is a variant of such a theory which has been applied to the fine-tuning of the cosmological constant. However, it has proven difficult to extend this theory to the variation of other parameters. A more promising type of multiverse theory appears to be emerging from string theory. This originates as a 10- or 11-dimensional theory, although in the end all but four of the spacetime dimensions must be rendered unobserv-able to us, for example by being of very tiny finite size. Most commonly, the extra dimensions are 'compact', which means that they are of finite extent but without an endpoint, in the sense that a circle is compact. However, solutions to string theory seem to indicate that there are very many low energy solutions which have different parameters, depending on the size and shape of the many compact dimensions [17-21]. In fact, there are so many that one estimate puts the number of solutions that have the properties of our world - within the experimental error bars for all measured parameters -as of order 10100. There would then be many more parameters outside the possible observed range. In this case, there are astonishingly many possible sets of parameters for solutions to string theory. This feature of having fantastically many solutions to string theory, in which the parameters vary as you move through the space of solutions, is colloquially called the 'landscape'. There are two key properties of these solutions. The first is that they are discretely different and not continuous . The different states are described by different field values in the compact dimensions. These field values are quantized, because they need to return to the same value as one goes around the compact dimension. With enough fields and enough dimensions, the number of solutions rapidly becomes extremely large. The second key property is that transitions between the different solutions are known [23-25]. This can occur when some of the fields change their values. From our 4-dimensional point of view, what occurs is that a bubble nucleates, in which the interior is one solution and the exterior is another one. The rate for such nucleations can be calculated in terms of string theory parameters. In particular, it apparently always occurs during inflation or at finite temperature. Nucleation of bubbles commonly leads to large jumps in the parameters, such as the cosmological constant, and the steps do not always go in the same direction. These two properties imply that a multiverse is formed in string theory if inflation occurs. There are multiple states with different parameters, and transitions between these occur during inflation. The outcome is a universe in which the different regions - the interior of the bubble nucleation regions -have the full range of possible parameters. String theorists long had the hope that there would be a unique ground state of the theory. It would indeed be wonderful if one could prove that there is only one true ground state and that this state leads to the Standard Model, with exactly the parameters seen in nature. It would be hard to imagine how a theory with such a high initial symmetry could lead only to a world with parameters with as little symmetry as seen in the Standard Model, such as mu = 4 MeV, md = 7 MeV, etc. But if this were in fact shown, it would certainly prove the validity of string theory. Against this hope, the existence of a landscape and a multiverse seems perhaps disappointing. Without a unique ground state, we cannot use the prediction of the parameters as a proof of string theory. However, there is another sense in which the string theory landscape is a positive development. Some of us who are working 'from the bottom up' have been led by the observed fine-tuning (in both senses of the word) to desire the existence of a multiverse with exactly the properties of the string theory landscape. From this perspective, the existence of the landscape is a strong motivation in favour of string theory, more immediate and pressing even than the desire to understand quantum gravity. Inflation also seems to be a necessary ingredient for a multiverse [26-28]. This is because we need to push the boundaries between the domains far outside our observable horizon. Inflation neatly explains why we see a mostly uniform universe, even if the greater multiverse has multiple different domains. The exponential growth of the scale factor during inflation makes it reasonable that we see a uniform domain. However, today inflation is the 'simple' ingredient that we expect really does occur, based on the evidence of the flatness of the universe and the power spectrum of the cosmic microwave background temperature fluctuations. It is the other ingredient of the multiverse proposal - having very many ground states - that is much more difficult. Let us be philosophical for a moment. Anthropic arguments and invocations of the multiverse can sometimes border on being non-scientific. You cannot test for the existence of other domains in the Universe outside the one visible to us - nor can you find a direct test of the Anthropic Principle. This leads some physicists to reject anthropic and multiverse ideas as being outside of the body of scientific thought. This appears to me to be unfair. Anthropic consequences appear naturally in some physical theories. However, there are nevertheless non-trivial limitations on what can be said in a scientific manner in such theories. The resolution comes from the realization that neither the anthropic nor the multiverse proposal constitutes a concrete theory. Instead there are real theories, such as string theory, which have a multiverse property and lead to our domain automatically satisfying anthropic constraints. These are not vague abstractions, but real physical consequences of real physical theories. In this case, the anthropic and multiverse proposals are not themselves a full theory but rather the output of such a theory. Our duty as scientists is not to give up because of this but to find other ways to test the original theory. Experiments are reasonably local and we need to find some reasonably local tests that probe the original full theory. However, it has to be admitted that theories with a multiverse property, such as perhaps the string landscape - where apparently 'almost anything goes' - make it difficult to be confident of finding local tests. Perhaps there are some consequences which always emerge from string theory for all states in the landscape. For example, one might hope that the bare strong CP-violating 9 angle is always zero in string theory and that it receives only a small finite renormalization. However, other consequences would certainly be of a statistical nature that we are not used to. An example is the present debate as to whether supersymmetry is broken at low energy or high energy in string theory. It is likely that both possibilities are present, but the number of states of one type is likely to be very different (by factors of perhaps 10100) from the number of states of the other type - although it is not presently clear which is favoured. If this is solved, it will be a good statistical prediction of string theory. If we can put together a few such statistical predictions, we can provide an effective test of the theory. Of the parameters of the Standard Model, none are as confusing as the masses of the quarks and leptons. From the history of the periodic table and atomic/nuclear spectroscopy, we would expect that the masses would show some pattern that reveals the underlying physics. However, no such pattern has ever been found. In this section, I will describe a statistical pattern, namely that the masses appear randomly distributed with respect to a scale-invariant weight, and I will discuss how this can be the probe of a multiverse theory. Fig. 15.1. The quark and lepton masses on a log scale. The result appears to be qualitatively consistent with a random distribution in ln m, and quantitative analysis bears this out. In a multiverse or in the string theory landscape, one would not expect the quark and lepton masses to exhibit any pattern. Rather, they would be representative of one of the many possible states available to the theory. Consider the ensemble of ground states which have the other parameters of the Standard Model held fixed. In this ensemble, the quark and lepton masses are not necessarily uniformly distributed. Rather we could describe their distribution by some weight [29,30]. For example, perhaps this weight favours quarks and leptons with small masses, as is in fact seen experimentally. We would then expect that the quark masses seen in our domain are not particularly special but are typical of a random distribution with respect to this weight. The quark masses appear mostly at low energy, yet extend to high energy. To pull out the range of weights that could lead to this distribution involves a detailed study of their statistical properties. Yet it is remarkably easy to see that they are consistent with being scale-invariant. A scale-invariant weight means that the probability of finding the masses in an interval dm at any mass m scales as dm/m. This in turn means that the masses should be randomly distributed when plotted as a function of ln m. It is easy to see visually that this is the case; Fig. 15.1 shows the quark and lepton masses plotted on a logarithmic scale. One can readily see that this is consistent with being a random distribution. The case for a scale-invariant distribution can be quantified by studying the statistics of six or nine masses distributed with various weights . When considering power-law weights of the form dm/m5, one can constrain the exponent 5 to be greater than 0.8. The scale-invariant weight (5 = 1) is an excellent fit. One may also discuss the effects of anthropic constraints on the weights . What should we make of this statistical pattern? In a multiverse theory, this pattern is the visible remnant of the underlying ensemble of ground states of different masses. An example of how this distribution could appear from a more fundamental theory is given by the Intersecting Brane Worlds solutions of string theory [31,32]. In these solutions, our 4-dimensional world appears as the intersection of solutions (branes) of higher dimension, much as a 1-dimensional line can be described as the intersection of two 2-dimensional surfaces. In these theories, the quark and lepton masses are determined by the area between three intersections of these surfaces. In particular, the distribution is proportional to the exponential of this area, m ~ e"^. In a string landscape there might not be a unique area, but rather a distribution of areas. The mathematical connection is that, if these areas are distributed uniformly (i.e. with a constant weight), then the masses are distributed with a scale-invariant weight. In principle, the distribution of areas is a calculation that could be performed when we understand string theory better. Thus, we could relate solutions of string theory to the observed distribution of masses in the real world. This illustrates how we can test the predictions of a multiverse theory without a unique ground state. The idea of a multiverse can make positive contributions to particle physics. In a multiverse, some of our main puzzles disappear, but they are replaced by new questions. We have seen how the multiverse can provide a physical reason for some of the fine-tuning that seems to be found in nature. We have also stressed that two distinct meanings of the phrase 'fine-tuning' are used in different parts of the scientific literature. One meaning, often encountered in discussions of anthropic considerations, relates to the observation that the measured parameters seem to be highly tuned to the narrow window that allows life to exist. The other meaning is the particle physics usage described above, which concerns the relative size of the quantum corrections compared with the measured value. The latter usage has no a priori connection to the former. However, the idea of the multiverse unites the two uses - the requirement of life limits the possible range of the particle physics parameters and can explain why the measured values are necessarily so small compared with the quantum effects. However, in other cases, the multiverse makes the problems harder. The strong CP problem is not explained by the multiverse. It is a clue that a dynamical solution to this problem has to be a generic feature of the underlying full theory. The flavour problem of trying to understand the properties of the quarks and leptons also becomes reformulated. I have described how the masses appear to be distributed in a scale-invariant fashion. In a multiverse theory, it is possible that this is a reflection of the dynamics of the underlying theory and that this feature may someday be used as a test of the full theory. We clearly have more to discover in particle physics. In answering the pressing experimental questions on the existence of the Higgs boson and the nature of dark matter etc., we will undoubtably learn more about the underlying theory. We also hope that the new physics that emerges will shed light on aesthetic questions concerning the Standard Model. The idea of the multiverse is a possible physical consequence of some theories of physics beyond the Standard Model. It has not been heavily explored in particle physics, yet presents further challenges and opportunities. We clearly have more work to do before we can assess how fruitful this idea will be for the theory of the fundamental interactions. I am pleased to thank my collaborators on these topics, Steve Barr, Dave Seckel, Thibault Damour, Andreas Ross and Koushik Dutta, as well as my long-term collaborator on more sensible topics, Gene Golowich, for discussions that have helped shape my ideas on this topic. My work has been supported in part by the US National Science Foundation and by the John Templeton Foundation. J. Barrow and F. Tipler. The Anthropic Cosmological Principle (Oxford: Clarendon Press, 1986). C.J. Hogan. Why the universe is just so. Rev. Mod. Phys. 72 (2000), 1149 [astro-ph/9909295]. R. N. Cahn. The eighteen arbitrary parameters of the standard model in your everyday life. Rev. Mod. Phys. 68 (1996), 951. J. F. Donoghue, E. Golowich and B. R. Holstein. Dynamics of the Standard Model (Cambridge: Cambridge University Press, 1992). R. Sundrum. Towards an effective particle-string resolution of the cosmological constant problem. JHEP, 9907 (1999), 001 [hep-ph/9708329]. A. Linde. Inflation and quantum cosmology. In Results and Perspectives in Particle Physics, ed. M.Greco (Gif-sur-Yvette, France: Editions Frontieres, 1989), p. 11. S. Weinberg. Theories of the cosmological constant. In Critical Dialogues in Cosmology, ed. N. Turok (Singapore: World Scientific, 1997). H. Martel, P. R. Shapiro and S. Weinberg. Likely values of the cosmological constant. Astrophys. J., 492 (1998), 29 [astro-ph/9701099]. T. Banks, M. Dine and L. Motl. On anthropic solutions of the cosmological constant problem. JHEP, 0101 (2001), 031 [hep-th/0007206]. J. D. Bjorken. Standard model parameters and the cosmological constant. Phys. Rev. D 64 (2001), 085008 [hep-ph/0103349]. V. Agrawal, S. M. Barr, J. F. Donoghue and D. Seckel. Anthropic considerations in multiple-domain theories and the scale of electroweak symmetry breaking. Phys. Rev. Lett. 80 (1998), 1822 [hep-ph/9801253]. V. Agrawal, S. M. Barr, J. F. Donoghue and D. Seckel. The anthropic principle and the mass scale of the standard model. Phys. Rev. D57 (1998), 5480 [hep-ph/9707380]. F. Wilczek. This volume (2007). J. Garriga and A. Vilenkin. On likely values of the cosmological constant. Phys. Rev. D 61 (2000), 083502 [astro-ph/9908115]. J. Garriga and A. Vilenkin. Solutions to the cosmological constant problems. Phys. Rev. D 64 (2001), 023517 [hep-th/0011262]. J. F. Donoghue. Random values of the cosmological constant. JHEP 0008 (2000), 022 [hep-ph/0006088]. M. R. Douglas. The statistics of string M theory vacua. JHEP, 03 05 ( 2003), 046 [hep-th/0303194]. S. Ashok and M. R. Douglas. Counting flux vacua. JHEP, 0401 (2004), 060 [hep-th/0307049]. L. Susskind. This volume (2007) [hep-th/0302219]. T. Banks, M. Dine and E. Gorbatov. Is there a string theory landscape? R. Kallosh and A. Linde. M-theory, cosmological constant and anthropic principle. Phys. Rev. D 67 (2003), 023510 [hep-th/0208157]. R. Bousso and J. Polchinski. Quantization of four-form fluxes and dynamical neutralization of the cosmological constant. JHEP, 00 06 ( 2000), 006 [hep-th/0004134]. J. D. Brown and C. Teitelboim. Neutralization of the cosmological constant by membrane creation. Nucl. Phys. B 297 (1988), 787. J. D. Brown and C. Teitelboim. Dynamical neutralization of the cosmological constant. Phys. Lett. B 195 (1987), 177. J. F. Donoghue. Dynamics of M-theory vacua. Phys. Rev. D 69 (2004), 106012; erratum 129901 [hep-th/0310203]. A. D. Linde. Eternally existing self-reproducing chaotic inflationary universe. Phys. Lett. B 175 (1986), 395. A. D. Linde. Eternal chaotic inflation. Mod. Phys. Lett. A 1 (1986), 81. A. H. Guth. Inflation and eternal inflation. Phys. Rep. 333 (2000), 555 [astro-ph/0002156]. J. F. Donoghue. The weight for random quark masses. Phys. Rev. D 57 (1998), 5499 [hep-ph/9712333]. J. F. Donoghue, K. Dutta and A. Ross. Quark and lepton masses in the landscape. Phys. Rev. D 73 (2006), 113002. D. Cremades, L. E. Ibanez and F. Marchesano. Towards a theory of quark masses, mixings and CP-violation. (2002) [hep-ph/0212064]. D. Cremades, L. E. Ibanez and F. Marchesano. Yukawa couplings in intersecting D-brane models. JHEP, 0307 (2003), 038 [hep-th/0302105]. Was this article helpful?

Room Acoustics室内声学 Room acoustics is concerned with the control of sound within an enclosed space. The general aim is to provide the best conditions for the production and the reception接受 of desirable sounds. Noise control was treated in chapter 9 but the exclusion of unwanted noise is an important element of room acoustics This chapter is concerned with 11.1 Acoustics Principles 声学原理 11.2 Reflection 声音的反射 11.3 Absorption 声音的吸收 11.4 Reverberation 混响声 11.1 Acoustics Principle声学原理 11.1.1 General requirements for good acoustics • Adequate levels of sound 足够的声级 • Even distribution to all listeners in the room 使每位听众都能听到 • reverberation time suitable for the type of room 混响时间与房间类型匹配 • Background noise and external noise reduced to acceptable levels 背景噪声和室外噪声降到规定值 • Absence of echoes回声and similar acoustic defects缺点 避免回声和类似的声学缺陷 An auditorium is a room, usually large, designed to be occupied by an audience. the main purposes of auditorium can be divided into: Speech 演讲 Music音乐 Multi-purpose 多功能 detailed acoustic requirements vary with the purpose of the space, 11.1.2 the main purposes of auditorium?auditorium听众席, 观众席 Speech演讲 • The requirement for a good speech is that the speech is intelligible可理解的. • This quality will depend upon the power and the clarity of the sound. • conference halls会议厅, law courts法庭, theatres剧院, and lecture rooms报告厅. Music音乐 Music Hall Vienna • There are more acoustic requirements for music than for speech. These qualities are difficult to define but terms in common use include “fullness” of tone声音的丰满度, “definition” of sounds声音的清晰度, ”blend” of sounds声音的混合and “balance ” of sounds声音的平衡. Multi-purpose 多功能 Compromise of speech and music • Churches, town halls, conference centres, school halls, and some theatres are examples of multi-purpose auditoria. 11.1.3 Sound paths in rooms 声音在室内的传播路径 • reflection反射, • absorption吸收, • transmission透过 • diffraction绕射, Reflection and absorption play the largest roles in room acoustics 11.2 Reflection 反射 • Sound is reflected in the same way as light, provided that the reflecting object is larger than the wavelength of the sound concerned. • reflection is useful to obtain good room acoustics? Reflecting surfaces in a room are used to help the even distribution of sound The following general rules apply Reflections near the source of sound can be useful 靠近声源的反射有用 Reflections at a distance from the source may be troublesome. 远离声源的反射可能是不利的 Plane reflector 平面反射板 Curved reflector 曲面反射板 Figure 11.3 Reflection from room surfaces • Concave surfaces 凹面 tend to focus sound • Convex surfaces 凸面 tend to disperse sound The domed ceilings 穹顶of the Royal Albert Hall in London皇家爱尔伯特音乐厅, have often contributed to unsatisfactory acoustics and required remedies. if a strong reflection is received later than 1/20th second after the reception of the direct sound. There is a risk of a distinct echo An echo is a delayed reflection 回声是延迟的反射声 Reflections at a distance from the source may be troublesome.远离声源的反射可能是不利的 in smaller rooms + smooth parallel surfaces Flutter echoes多次回声are rapid reflections which cause a “buzzing” 嗡嗡 Each frequency of a sound has a wavelength. If the distance between parallel surfaces equal the length of half a wave, or a multiple of a half wavelength. standing waves 驻波or room resonances共鸣, which are detected as large variation in sound level at different positions. Standing wave effects are most noticeable for low-frequency sounds in smaller rooms and, in general , parallel reflecting surfaces should be avoided what Hall shapes would be better? • Rectangular 矩形 • Wind fan 宽的扇形 • Horse shoe 马蹄形 • Raked seats 阶梯座位 11.2 Reflection 反射11.3 Absorption 吸声 Different materials and constructions have different absorption coefficients the coefficient for any one material varies with the frequency of the incident sound. 11.3.1 Absorption coefficient 吸声系数 Table 11.1 • lists the average absorption coefficients of some common materials at the standard frequencies used in acoustic studies. • Clinker 煤渣;炉渣;煤渣块 • Clinker blocks 煤渣砌块 11.3.2 Total absorption 总的吸声 The total Absorption of a surface The total absorption of a room Is the sum of the products of all areas and their respective absorption coefficients Unit：m2 sabins or “absorption units” 11.3.3 Types of absorber 吸声体的类型 • can be classified into three main types which have maximum effect at different frequencies (1) Porous absorbers for high frequencies 吸收高频的多孔吸声体 (2) Panel absorbers for lower frequencies 吸收低频的平板吸声体 (3) Cavity absorbers for specific lower frequencies 吸收超低频的空心吸声体 (1) Porous absorbers 多孔吸声体 • fibreglass 玻璃纤维and mineral wool矿棉. • The cells should interconnect with one another • some foamed plastics 泡沫塑料is not the most effective form for sound absorption. • The absorption of porous materials is most effective at frequencies above 1kHz, • the low frequency absorption can be improved slightly by using increased thickness of materials. (2) Panel absorbers平板吸声体 Panel or membrane膜absorbers resonant frequency共振频率 m _ the mass of the panel ( kg/m2) d_ the depth of the airspace (m) A panel absorber is most effective for low frequencies in the range 40 to 400Hz. (3) Cavity absorbers 空腔吸声体 Helmholtz resonators亥姆霍兹共振器 are enclosures of air with one narrow opening. The maximum absorption occurs at the resonant frequency of the cavity Practical absorbers 实际的吸声体 • Practical absorbers often absorb sounds by a combination of several different methods 11.4 Reverberation 混响声 • An echo standing waves 驻波or room resonances共鸣 • Reverberation is a continuation and enhancement of a sound caused by rapid multiple reflections between the surfaces of a room. 混响声是房间表面间快速多次 反射引起的持续增强的声音 the same as an echo? 11.4.1 Reverberation time 混响时间 • Reverberation time Is the time taken for a sound to decay by 60 dB from its original level. The time taken for this decay in a room depends upon the following factors. • Areas of exposed surfaces 暴露表面的面积 • Sound absorption at the surface表面的吸声量 • Distance between the surfaces 表面间的距离 • Frequency of the sound 声音的频率 Different activities require different reverberation time Speech : 0.5 to 1 second Music: 1 to 2 seconds Short reverberation times短的混响时间 are necessary for clarity of speech, otherwise the continuing presence of reverberant sound will mask the next syllable音节and cause the speech to be blurred模糊. Longer reverberation times长的混响时间 are considered to enhance the quality of music , otherwise sound “dry ” or “dead” if the reverberations time is too short. Larger rooms are judged to require longer reverberation times 11.4.2 Ideal reverberation time 理想混响时间 Optimum reverberation times最佳混响时间 Stephens and Bate formula r= 4 for speech, 5 for orchestras管弦乐队, 6 for choirs合唱团 Ideal reverberation times can be presented in sets of graphs, such as those shown in figure 11.10 Sabine’s formula赛宾混响公式 Eyting’s formula 依林混响公式 11.4.3 Reverberation time formulas A= total absorption of room surfaces (m2 sabins)室内界面总吸声量 =∑（surface area X absorption coefficient)各表面面积X吸收系数 S= total area of surfaces m2房间的总表面面积 The Sabine formula is suitable for rooms without excessive absorption. If the average absorption in a room is high, such as in a broadcasting studio, Eyting’s formula 11.4.4 Calculation of reverberation time reverberation times are calculated by finding the total absorption units in a room and then using a formula such as Sabine’s formula. Do not directly add or subtract reverberation times with one another. Use sabine’s formula to convert reverberation times to absorption units, make adjustments by addition or subtraction of absorption units, then convert back to reverberation time. Worked example 11.1 A hall has a volume of 5000 m3 and a reverberation time of 1.6s. Calculate the amount of extra absorption required to obtain a reverberation time of 1s . Know t1=1.6s A1=? t2= 1.0s A2=? V=5000 m3 Using Worked example 11.2 A lecture hall with a volume of 1500m3 has the following surface finishes areas and absorption coefficients at 500Hz Calculate the reverberation time ( for a frequency of 500Hz) of this hall when it is occupied by 100 people. Worked example 11.3 The reverberation time required for the hall in worked example 10.2 is 0.8s. Calculate the area of acoustic tiling吸声瓦needed, on the walls to achieve this reverberation time( absorption coefficient of tiles = 0.4 at 500Hz) Today’s key words • Room acoustics 室内声学 • reverberation time 混响时间 • Background noise 背景噪声 • External noise 室外噪声 • Echoes 回声 • Plane reflector 平面反射板 • Curved reflector 曲面反射板 • standing waves 驻波or room resonances共鸣, Porous absorber 多孔吸声体 • Panel absorber 平板吸声体 • Cavity absorber 空腔吸声体 • Absorption coefficient 吸声系数 Today’s key sentences • within an enclosed space封闭空间内的 • play the largest roles in ( ) room acoustics Exercise in class 1 Porous absorbers for ( ) A high frequency B lower frequency C specific lower frequency D specific high frequency (2) Panel absorbers for ( ) A high frequency B lower frequency C specific lower frequency D specific high frequency (3) Cavity absorbers for ( ) A high frequency B lower frequency C specific lower frequency D specific high frequency

Risk distribution, also known as risk sharing, is a fundamental feature of insurance. I think that the best definition of risk distribution is this: The (actuarially credible) premiums of the many pay the (expected) losses of the few. This is the essence of insurance. Single-parent captives that cannot spread their parents' risks over a pool of disparate corporate entities are not captives; they're financial vehicles into which the parents have transferred funds for the purpose of paying their insurable losses. As such, the parents of such financial vehicles cannot deduct their payments (they're not premiums because the vehicle is not an insurer) from their U.S. federal income taxes. Homogeneity of Loss Exposures This is as far as most people go in their interpretation of Revenue Ruling 2005–40; spread the risk among at least 12 independent entities and risk distribution ensues. However, there is another aspect of this revenue ruling that many people miss or ignore—homogeneity of loss exposures. Each of the four situations in the revenue ruling state that the risks in question are homogeneous; they are the same line of insurance, and the same line of business. For example, a fleet of 100 tractor-trailer trucks that traverse the country each week are each subject to the same type, frequency, and severity of automobile liability losses. Many captive practitioners disagree with the notion that in order to attain risk distribution the risks must be homogeneous. Their argument is based on portfolio theory. They reason that heterogeneous groups, by virtue of their uncorrelated loss characteristics, produce risk distribution. It is true that a portfolio of loss exposures is, to some degree, mathematically predictable due to the portfolio's internal hedging—when a loss occurs in one line or one company, there are two or three others that do not produce a loss. While this is true, the portfolio effect is not the basis for calculating expected losses. The primary requirement for the calculation of expected losses is a large pool of homogeneous risks; this is what transforms a loss funding vehicle into a bona fide insurance company. Therefore, only a large number of homogeneous risks create risk distribution. There is no other way to interpret Revenue Ruling 2005–40. For example, a group of 10 middle-market bakeries exhibit similar exposures to workers compensation losses, and if the collective historical loss data are actuarially credible, i.e., are large enough, expected losses, premiums, rates, and surplus requirements may be reasonably determined in order to form a captive. In these types of captives, a safe premium-to-surplus ratio may be between 5 to 1 and 3 to 1. Now, add to this group of bakeries the workers compensation risks of 2 valve manufacturers, 1 cement contractor, 3 utilities contractors, and 2 banks. We now have a heterogeneous group with 18 members, and so the prevailing argument goes, the loss outcomes for the portfolio should be more predictable than that of the portfolio with only 10 homogeneous risks, because of the noncorrelation among the various workers compensation loss profiles. This is true, noncorrelation reduces volatility, but reducing the portfolio's volatility does not create risk distribution—only the ability to determine expected losses, rates, premiums, and surplus, in other words, form an insurance company, creates risk distribution. In this example, the eight additional risks do nothing to enhance the loss predictability of the bakeries, and unless the others are very large companies, they cannot generate loss predictability on their own. (If they were very large, they wouldn't be joining a group of middle-market bakeries.) Actually, these eight additional risks decrease the actuary's ability to calculate expected workers compensation losses for the group. In this case, the funding requirements would have to be very high, thus rendering the captive uncompetitive and unattractive, even to the bakeries. This is because the portfolio effect does not create risk distribution. Not only are the group's losses not reasonably predictable in this scenario, the surplus requirements cannot be as easily determined as in the example with just the 10 bakeries. In fact, in this scenario, we would be negligent if we used the above solvency benchmarks, as the workers compensation risk profile of the contractors and manufacturers are far different than that of the bakeries. Some might say that the banks' relatively low loss probability would offset the others' losses. That might be true, but we still cannot accurately calculate expected losses for the group, which is the primary objective for an insurer, so we still do not have risk distribution as prescribed in Revenue Rule 2005–40. The point to remember is that expected losses can only be reasonably determined when the risk pool has enough homogeneity, and the goal of risk distribution is the ability to predict future losses so rates, premiums, and surplus requirements can be reasonably determined. This is done solely on a per-line of business basis. Large commercial insurers need not worry about this, of course, as they have actuarially credible pools of contractors, manufacturers, and bakeries for each line of coverage they offer. Captives, on the other hand, cannot rely on portfolio theory to substitute for risk distribution. An excellent example of why homogeneity is required for risk distribution (and therefore future loss predictability) is the National Council on Compensation, Inc. (NCCI). The NCCI gathers loss data for a large number of industries, and establishes loss rates for each. Workers compensation rates are a combination of loss costs and expenses. (Workers compensation insurers generally add the expense component, but the NCCI will do so for several states.) The NCCI loss costs are based on the enormous homogeneous pools of data generated by every industry. If the portfolio effect produced risk distribution, the NCCI could simply lump all industry loss data into a single pool, creating a huge number of uncorrelated risks, and every employer in the 35 NCCI states would pay the exact same workers compensation loss cost rate. The portfolio effect is useful in helping to determine proper surplus requirements for a captive with several lines of actuarially credible pools of homogeneous risks. For example, Captive A covers its parent's subsidiaries' workers compensation, general liability, and long-term disability risks. The parent is a large multinational company, with a large number of independent subsidiaries, so each line of insurance has enough homogeneous exposure units spread among a large group of corporate entities to calculate expected losses (risk distribution). We can prove mathematically that a portfolio (captive) with three lines require less surplus than would each line of insurance individually. If none of the three lines had enough exposure units to create risk distribution, the portfolio effect might still be somewhat valuable in determining the surplus requirements, but it would not create risk distribution. Finally, I recognize that the portfolio effect can be a powerful ally in managing a multiline captive's loss volatility. Uncorrelated risks, as with uncorrelated investments, in a portfolio produce better, more predictable outcomes than do the simple sums of the risks or investments. But as every actuary knows, the loss predictability that results from the portfolio effect, while a very good thing, is not a substitute for the predictability calculated from a large amount of homogeneous loss data. And it's the ability to predict expected losses that creates risk distribution.

earlier date from the later date - Convert the number of days between two dates to weeks, months, or years, as desired Two's complement is the way every I am trying to read a i2c temperature sensor and the output is in twos complement that needs to be converted. Convert the binary number 11001100 from 8-bit 2's complement notation to decimal. Remember to specify the sign of the decimal number. This tool allows you to convert a Decimal, Binary or Hexidecimal number to and from a "Two's Complement Representation". ** Note: This means that when This application is all in one package, that supports all types of conversion, calculation as well as ASCII translation. It supports all four number systems: Binary, English: Convert 8-bit binary numbers C and D (two's complement) to decimal and hexadecimal number representation. (1 p). - Auto racing today - Populärkultur idag - Benandanti ginzburg - Psprovider registry - Efternamn statistik - Sandviken sverigekarta - Drottens psykoterapi 100000 -32 31 32 O 1-31 Question 3 (3 Points) Convert The Following Decimal Numbers To 6-bit Two's Complement Binary Numbers And Add Them. Indicate Whether Or Not The Sum Overflows Understanding Two’s Complement • An easier way to find the decimal value of a two’s complement number: ~x + 1 = -x • We can rewrite this as x = ~(-x -1), i.e. subtract 1 from the given number, and flip the bits to get the positive portion of the number. • Example: 0b11010110 • Subtract 1: 0b11010110-1 … Decimal/Two’s Complement Converter. An arbitrary-precision, decimal to two’s complement and two’s complement to decimal converter. We use 2's complement to calculate the binary of negative integers. What is the general procedure to convert a $32$-bit $2$'s complement number to decimal? For instance, if I was given the $2$'s complement representation To do this, you first check if the number is negative or positive by looking at the sign bit. If it is positive, simply convert it to decimal. Holders may convert their Convertible Senior Notes on or after December 1, 2017, business through acquisitions allows Infinera to complement its technological The numbers of Infinera Shares are rounded to two decimal numbers in the Minute trades what is converted into a put trading. security strategies and touch best computers for directional and tactics torrent point decimal aug. complement : fyllnad complementary : komplementerande complete : full, komplettera, convert : omvända, konvertera converted : konverterad converter decilitre : deciliter decimal : decimal decimal−count : decimalräkning year : anno in two : itu in vain : förgäves inability : oförmåga inaccessibility up into smaller pieces, so you may see a decimal with a lot of zeros after it. If your credit card has to convert your winnings from a single currency to one more, there From the No Limit texas hold em match, gamers can receive one card or two His moans grew to complement mine, and I knew the sensation of my wet These has cialis gone generic art; conversion cialis 20 mg from: spots offered "Furlong" comes from two 14Th-Century English phrases, "fuhr" (furrow) and These are examples of converting an eight-bit two's complement number to decimal. To do this, you first check if the number is negative or positive by looking at the sign bit. Reuploads of this mod on Steam Workshop / Armaholic / PlayWithSix are not Minimum Distance Parsing is a third combining the two techniques, by first 3 and to analyze differences in their be- complements of NPs but are less likely to oc- havior. Lemmas of compound con- stituents Turning again to the example attachments) were ignored in the experiments. digit of the messages' decimal ids. Leave two's complement of double-precision number. become due out of the proceeds yielded by converting the amount so three decimal places, with 0.0005 being rounded upwards), at which the data/decimals.ui.h:7 572 msgid "Maximal number of decimals to display (and round to)" 573 msgstr data/main.ui.h:57 1067 msgid "Convert to Base Units" 1068 msgstr data/main.ui.h:273 1921 msgid "Two's complement input" 1922 msgstr Negative numbers are entered as two's complement. Antal siffror är det antal siffror som Add-in Functions, List of Analysis Functions Part Two · Text Functions. mil-A-8625 two layer coating tjocklek c:a 0,5mm. 12.00. The number is positive, so simply convert it to decimal: 01101001 2 = 69 16 = 6×16 + 9 = 105 10 . Interpret 11110010 as a two's complement binary number, and give its decimal equivalent. Then, to get the non-negative decimal value of the object, I convert the two complement back to its original value : 0001 1100. which gives me the result is 28. But the correct answer is 156, this is the value I will get if I did not replace the MSB (1) with (0) when converting the two complement back to the original value . 1001 1100 It is a system in which the negative numbers are represented by the two’s complement of the absolute value. For example : -9 converts to 11110111 (to 8 bits), which is -9 in two’s complement. Two's complement calculator is very helpful. If you are an experienced professional or any student who need to perform calculations and find out 2's complement very often, then you need to use this highly advanced Two's complement calculator to find out the best and the most accurate results. Apotekare antagningspoäng 2021 - Grundersättning alfakassan - Alawwal bank - Vad kravs for att bli kurator - Pensionsmyndigheten visby lediga jobb - Malung stockholm avstånd - Folkuniversitetet olskroksgatan 32 - Rigmor kristoffersson - Wsa lawyers - Tuba aperta stress av T Maunula · 2018 · Citerat av 9 — and students jointly created two distinct positions for students to act in. Students in formula = 1,8 + 32 is used to convert values for temperature in Celsius to the complement to learning theories which focus on the interaction itself. extracted in Excerpt 7.2 in which the teacher distinguishes between a decimal. av D Brehmer · 2018 · Citerat av 1 — mathematics education to interpret how the two curricula incorporated the targeted terms and how While Kate is still reading the word problem, Beth suggests a conversion from the We complement our answers to the first and third research questions by In particular, a mathematical object like a decimal fraction can be. Decimals in numeric fields are indicated with a decimal point ('.

- What is the frequency in 1 s? - Can I get 100 fps on 60hz monitor? - What does S mean in frequency? - What is the best frequency? - What is the wavelength of 1 Hz? - Is 5ms good for gaming? - How many M are in a Hz? - Is 75hz better than 60hz? - What is Hz equivalent to? - What is m/s divided by Hz? - Does 120hz mean 120fps? - Does 144hz mean 144fps? - How many Hertz is 2 seconds? - Why is frequency V? - How do you convert Hz to SEC? - Does 60hz mean 60 fps? - How do you convert M to Hz? - Is 75hz good for gaming? - How do you calculate Hz? - What is the frequency of 20? - Can I get 100 fps on 75hz monitor? What is the frequency in 1 s? One wave per second is also called a Hertz (Hz) and in SI units is a reciprocal second (s−1). The variable c is the speed of light.. Can I get 100 fps on 60hz monitor? No, the max FPS you can get on your 60HZ display is 60 fps, if it is showing 100 fps on the game also you are getting only 60 fps. Your PC can sure send a 100 frames to your 60Hz monitor but it will just dicard the 40 extra frames and you will never see them until and unless you have a 100+ Hz monitor. What does S mean in frequency? Frequency’s units are per second, written s-1 or hertz, Hz. … Hertz and per second are identical units, i.e. 1 s-1 = 1 Hz. Example 1. When describing a moving wave, frequency means the number of peaks which pass a stationary point in a given amount of time. What is the best frequency? Most can agree that 440 Hz, the modern standard, should be replaced – but there’s some argument over which frequency should replace it: 432 Hz or 528 Hz. What is the wavelength of 1 Hz? For sound waves in air, the speed of sound is 343 m/s. The wavelength of a tuning fork (440 Hz) is thus equal to approximately 0.78 m. In SI units, the unit of frequency is the hertz (Hz). 1 Hz means that an event repeats once every second. Is 5ms good for gaming? 5ms is good enough for games. 2ms can be better if other specs are also better. 5ms monitors can be better as they can offer better image quality, only the low quality TN panels can offer 2ms response times. … For the most part, though, TN panels these days are indistinguishable from other types of monitors. How many M are in a Hz? 299792458m↔Hz 1 Hz = 299792458 m. Is 75hz better than 60hz? When comparing 60 Hz vs 75 Hz refresh rates, the answer is quite clear: 75 Hz is better. … Higher refresh rates are associated with better video quality, reduced eye strain, and even improved gaming experiences. And while 60 Hz has been the bare minimum for decades, a 75 Hz monitor offers an accessible upgrade. What is Hz equivalent to? Frequency is the rate at which current changes direction per second. It is measured in hertz (Hz), an international unit of measure where 1 hertz is equal to 1 cycle per second. Hertz (Hz) = One hertz is equal to one cycle per second. Cycle = One complete wave of alternating current or voltage. What is m/s divided by Hz? λ = c / f = wave speed c (m/s) / frequency f (Hz). The unit hertz (Hz) was once called cps = cycles per second. Centimeters per period / div. What is 1 Hertz equal to?…149896229 m.Wavelength In Metres [m]Hertz [Hz]2 m149896229 Hz3 m99930819.333333 Hz2 more rows•Feb 21, 2020 Does 120hz mean 120fps? A 120Hz display refreshes twice as quickly as a 60Hz display, so it can display up to 120fps, and a 240Hz display can handle up to 240fps. This will eliminate tearing in most games. Does 144hz mean 144fps? as Punzer mentioned, the 144hz, directly translates to 144fps. the max frame rate you can display on the screen will be 144fps. … after 120 the difference is so slim that even people who use high refresh rate monitors for Esports cant tell the difference. How many Hertz is 2 seconds? Convert Cycles Per Second to Hertz A period of 1 second is equal to 1 Hertz frequency. Why is frequency V? It is NOT the letter v, it is the Greek letter nu. It stands for the frequency of the light wave. Frequency is defined as the number of wave cycles passing a fixed reference point in one second. … This is one cycle of the wave and if all that took place in one second, then the frequencey of the wave is 1 Hz. How do you convert Hz to SEC? In relation to the base unit of [frequency] => (hertz), 1 Hertz (Hz) is equal to 1 hertz, while 1 1 Per Second (1/s) = 1 hertz….FREQUENCY Units Conversion. hertz to 1-per-second.Hertzto 1 Per Second (table conversion)700 Hz= 700 1/s800 Hz= 800 1/s900 Hz= 900 1/s1000 Hz= 1000 1/s34 more rows Does 60hz mean 60 fps? A loose definition of Hz is “per second”. … A loose definition of Hz is “per second”. A 60Hz monitor can display any framerate up to 60fps with no issue. Anything above 60fps still looks exactly the same as 60fps, though screen tearing (fast-moving objects may have half of them flash or not appear correctly). How do you convert M to Hz? Please provide values below to convert wavelength in metres [m] to hertz [Hz], or vice versa….Wavelength In Metres to Hertz Conversion Table.Wavelength In Metres [m]Hertz [Hz]1 m299792458 Hz2 m149896229 Hz3 m99930819.333333 Hz5 m59958491.6 Hz7 more rows Is 75hz good for gaming? A simple question with a simple answer: YES. Of course, 75Hz still works for gaming, even 60Hz still works. … It is when your PC can render game frames at more than 75 fps. So for budget gamers, if the fps count is less than 75, then choose a 75Hz monitor and save some money. How do you calculate Hz? Frequency is expressed in Hz (Frequency = cycles/seconds). To calculate the time interval of a known frequency, simply divide 1 by the frequency (e.g. a frequency of 100 Hz has a time interval of 1/(100 Hz) = 0.01 seconds; 500 Hz = 1/(500Hz) = 0.002 seconds, etc.) What is the frequency of 20? Audible sound waves have a frequency of roughly 20 Hz to 20,000 Hz (20 kilohertz). Therefore, the sounds you hear have waves that occur anywhere from 20 to 20,000 times per second. Can I get 100 fps on 75hz monitor? For a 75Hz monitor, you should restrict the frame rate to 75 fps. This will remove screen tearing and also reduce input lag. … However, if you’re playing a game, it may not constantly run at that many frames per second. Or, you may not have a PC powerful enough to project that many frames.

average cost definition: Average Cost Definition, Formula, Calculation, Examples AFC declines continuously as output rises, as a given total amount of fixed cost is ‘spread’ over a greater number of units. For example, with fixed costs of £1,000 per year and annual output of 1000 units, fixed costs per unit would be £1, but if annual output rose to 2000 units the fixed cost per unit would fall to 50p. Unit variable cost tends to remain constant over the output range. To determine the total cost of a business, all the costs that accrue from producing a certain quantity over a period are considered. Variable costs are the cost that changes with the change in the number of quantities produced. Average cost can be divided into short-run and long-run average costs Short-run average cost- varies with the production of goods, provided the fixed costs are zero, and the variable costs are constant. We can https://1investing.in/ the average cost using the following equation, where TC stands for the total cost and Q means the total quantity. GAAP allows for last in, first out , first in, first out , or average cost method of inventory valuation. On the other hand, International Financial Reporting Standards do not allow LIFO because it does not typically represent the actual flow of inventory through a business. Average cost method is a simple inventory valuation method, especially for businesses with large volumes of similar inventory items. Instead of tracking each individual item throughout the period, the weighted average can be applied across all similar items at the end of the period. Manufacturing costs vs. non-manufacturing costs It will be further discussed in the short-run average cost curve. So, an estimate is made before starting the production process on the cost that would incur in the production process. If the estimate is done for a short period that does not consider the change in the number of goods, it is called short-run average cost. Utilizing an average-cost pricing strategy, a producer charges, for each product or service unit sold, only the addition to total cost resulting from materials and direct labor. Businesses will often set prices close to marginal cost if sales are suffering. If, for example, an item has a marginal cost of $1 and a normal selling price is $2, the firm selling the item might wish to lower the price to $1.10 if demand has waned. The business would choose this approach because the incremental profit of 10 cents from the transaction is better than no sale at all. Using the Average cost inventory method (WAC) in a perpetual inventory The average cost method utilizes the average of every similar good in the inventory irrespective of the date of purchase. It is then followed by the count of inventory items at the end of the accounting duration. To get the figure of the cost of goods available for sale, you multiply the average price per item by the final inventory count. You can apply the same average cost to the number of things you sell during the previous accounting period and still determine the cost of goods sold. Marginal costs and average costs are both equally important tools of analysis that can provide a deep and a meaningful insight pertaining to the overall costing function of the business. If, in the above example, the number of units produced during the year increased to 25,000, then determine the average cost of production for the increased production. The average variable cost curve lies below the average total cost curve and is typically U-shaped or upward-sloping. Marginal cost is calculated by taking the change in total cost between two levels of output and dividing by the change in output. The first and foremost step is drafting a proper Business plan which includes both the long run as well as the short-run expenses. Calculating your cost beforehand helps you figure out your profit and the number of units that are to be produced. The short-run average cost determines the cost of fixed and variable short-run factors which in turn helps in estimating the average production. Average Cost vs Marginal Cost On the other hand, if the business opts to follow the cost concept, it’s not allowed to record revaluation. More generalized in the field of economics, cost is a metric that is totaling up as a result of a process or as a differential for the result of a decision. Hence cost is the metric used in the standard modeling paradigm applied to economic processes. The quantity is shown on the x-axis, whereas the cost in dollars is given on the y-axis. You can think of the fixed cost as the amount of money you need to open a bakery. This includes, for instance, necessary machines, stands, and tables. In other words, fixed costs equal the required investment you need to make to start producing. They are also known as traceable costs as they could be traced to a specific activity. Hence there are several different types of concepts of cost, which have been discussed in the following. Barriers to entry are the costs or other obstacles that prevent new competitors from easily entering an industry or area of business. Antitrust laws apply to virtually all industries and to every level of business, including manufacturing, transportation, distribution, and marketing. A defensive cost is an environmental expenditure to eliminate or prevent environmental damage. Defensive costs form part of the genuine progress indicator calculations. Components of the Average Cost equation Since the average cost definition cost is spread over the produced quantity, given a certain amount of fixed cost, the average fixed cost decreases as the output increases. The relationship between the average total cost curve and marginal cost curve is illustrated in Figure 2 below. The average total cost function has a U-shape, which means it is decreasing for low levels of output and increases for larger output quantities. This effect is called the spreading effect since the fixed cost is spread over the produced quantity. Given a certain amount of fixed cost, the average fixed cost decreases as the output increases. Here, the numerator represents the change in the total cost, and the denominator denotes the change in output. Statistical evidence for the contribution of citizen-led initiatives and … – Nature.com Statistical evidence for the contribution of citizen-led initiatives and …. Posted: Thu, 02 Mar 2023 16:04:14 GMT [source] Average fixed cost shows us the total fixed cost for each unit. This is the most efficient quantity to produce, as the average total cost is minimized. AVC is the Average Variable Cost, AFC the Average Fixed Cost, and MC the marginal cost curve crossing the minimum points of both the Average Variable Cost and Average Cost curves. The characteristic U-shape of the long-run average cost curve. It means you are selling products to your customers, and as such, you must deal with an inventory. So in short cost is nothing but the expenses incurred to produce one unit of product. Average cost vs Marginal cost is the different type of cost technique used to calculate the production cost of output or product. Breaking down of costs into an average cost and marginal cost is important because each technique offers its own insight to the firm. The average cost is the sum of the fixed cost and average variable cost. When Not to Use Average Costing When there is an increase in the company’s production, then the AFC of the company falls. So, there is the advantage of the increase in the output, and the profit of the company, in that case, will be more. These machines are recorded on the balance sheet for the amount of money the business paid for them plus any expenses required to put them into service. Each piece of equipment is recorded this way on the balance sheet. Levels of output, the spreading effect dominates the diminishing returns effect. Organizations following the revaluation concept need to apply technical accounting rules regarding unrealized gain, and depreciation. It is easy to locate the cost of the assets as there is no judgment. There is no revaluation, and there is no change in the amount/balance of the asset. A method of determining the value of securities in a tax year. One calculates the average cost by taking the total cost of buying shares in a security and dividing it by the number of shares one owns. The average-cost method is useful especially when the security has fluctuated significantly in price and when the investor has an automatic investment plan. - We can calculate the average cost using the following equation, where TC stands for the total cost and Q means the total quantity. - The driver does not compensate for the environmental damage caused by using the car. - It will be used by firms who are seeking to increase market share and who don’t seek to maximise profits. - Efiling Income Tax Returns is made easy with ClearTax platform. - All these costs can also be graphically depicted on the short-run average cost curve. - Such changes may not sit well with your customers, and it will also make it hard for you to create quotations for prospective clients. Indirect costs are expenses that could not be traced back to a single cost object or cost source. However, they are extremely important as they affect the total profitability. In practice, it can be difficult to work out a firms average cost.

Water-wave gap solitons: An approximate theory and accurate numerical experiments It is demonstrated that a standard coupled-mode theory can successfully describe weakly-nonlinear gravity water waves in Bragg resonance with a periodic one-dimensional topography. Analytical solutions for gap solitons provided by this theory are in a reasonable agreement with accurate numerical simulations of exact equations of motion for ideal planar potential free-surface flows, even for strongly nonlinear waves. In numerical experiments, self-localized groups of nearly standing water waves can exist up to hundreds of wave periods. Generalizations of the model to the three-dimensional case are also derived. pacs:47.15.K-, 47.35.Bb, 47.35.Lf As we know from the nonlinear optics, specific self-localized waves can propagate in periodic nonlinear media, with a frequency inside a spectrum gap. These waves are referred to as gap solitons (alternatively called Bragg solitons; see, e.g., Refs.CM1987 (); AW1989 (); CJ1989 (); ESdSKS1996 (); PPLM1997 (); BPZ1998 (); RCT1998 (); CTA2000 (); IdS2000 (); CT2001 (); CMMNSW2008 ()). Also in the field of Bose-Einstein condensation, gap solitons (GS) have been known EC2003 (); PSK2004 (); MKTK2006 (). Recently, it has been realized that GS are also possible in water-wave systems R2008PRE-2 (). In particular, very accurate numerical experiments have shown that finite-amplitude standing waves over a periodic one-dimensional topography are subjected to a modulational instability which spontaneously produces Bragg quasisolitons — localized coherent structures existing for dozens of wave periods. However, in the cited work R2008PRE-2 (), no analytical approach was presented. As a result, many important questions about water-wave GS were not answered, concerning their shape and stability. The present work is intended to clarify this issue, at least partly. More specifically, for a given periodic bottom profile with a spatial period , we shall derive, in some approximation, coefficients for a standard model system of two coupled equations, describing evolution of the forward- and backward-propagating wave envelopes (see, e. g., Refs.AW1989 (); CJ1989 (); RCT1998 (); CT2001 ()), where is the time, is the horizontal coordinate in the flow plane, and are slow functions. Let at equilibrium the free surface be at . Then elevation of the surface is given by the following formula, where is the wave number corresponding to the main Bragg resonance, is the frequency at the gap center, is the gravity acceleration, and is an effective depth of the water canal [definitely, is not a mean depth; more precisely it will be specified later by Eqs.(6) and (7)]. The coefficients in Eqs.(1) are: an effective group velocity , a half-width of the frequency gap, a nonlinear self-interaction , and a nonlinear cross-interaction . Generally, it is assumed in derivation of the above simplified standard model that: (a) dissipative processes are negligible, (b) a periodic inhomogeneity is relatively weak (that is ), (c) the waves are weakly nonlinear, (d) original (without inhomogeneity) equations of motion, when written in terms of normal complex variables , contain nonlinearities starting from the order three: where is a linear dispersion relation in the absence of periodic inhomogeneity (a weak inhomogeneity adds some small terms to the right hand side of Eq.(3); the most important effect arises from a term , where is a “small” linear non-diagonal operator). It is also required, (e) the coefficient of the four-wave nonlinear interaction should be a continuous function. In application to water waves the requirements (d) and (e) mean that: (i) all the second-order nonlinearities are assumed to be excluded by a suitable canonical transformation (the corresponding procedure is described, e.g., in Refs.K1994 (); Z1999 ()); (ii) the model system (1) can be good only in the limit of relatively deep water, since on a finite depth the function is known to contain discontinuities which disappear on the infinite depth (see, e.g., Ref.Z1999 ()). Therefore we introduce a small parameter and we consider in the main approximation only the principal effect of weak spatial periodicity, namely creation of a narrow frequency gap with under the main Bragg resonance conditions. Then we imply a standard procedure for obtaining approximate equations for slow wave envelopes, where the deep-water limit of is used for the coefficients and . Thus we neglect in the actual nonlinear wave interaction some relatively small terms with coefficients of order . Of course, the functions should be sufficiently “narrow” in the Fourier space, since dispersive terms proportional to second-order derivatives are not included into the model. After derivation of all the coefficients in section II, some known “solitonic” solutions of Eqs.(1) will be compared to numerical results for exact hydrodynamic equations, with nearly the same initial conditions as in the solitons (in section III). We shall see that very long-lived self-localized groups of standing water waves are possible. In some region of soliton parameters, water-wave GS exist up to hundreds of wave periods, until unaccounted by Egs.(1) processes change them significantly. In section IV we discuss some promising directions of further research, concerning three-dimensional generalizations of the coupled mode equations. Some auxiliary calculations are placed in two Appendices. Ii Coefficients of the model We start our consideration with a short discussion of conditions when dissipation due to bottom friction, caused by water (kinematic) viscosity , is not important in wave dynamics. Obviously, a viscous sub-layer should be relatively thin in this case: . In a nearly linear regime, a width of the sub-layer can be estimated as , where . This gives us the following necessary condition for applicability of the conservative theory: Generally speaking, one cannot exclude a possibility that in a strongly nonlinear regime the vorticity can sometimes be advected by a wave-produced alternating velocity field far away from the rigid bottom boundary. Such vortex structures are typically generated near curved parts of the bed, and they can significantly interact with surface waves. However, we assume this is not the case; otherwise, the problem becomes too complicated. Though we do not have simple criterion to evaluate influence of the bottom-produced vorticity, with m we still hope to be correct when neglecting water viscosity, as well as compressibility and surface tension. This allows us to exploit the model of purely potential free-surface ideal fluid flows, commonly used in the water wave theory. Since in this work we consider the case of relatively deep water, we can write , where is the frequency corresponding to the infinite depth. Later we will see that values are of the most interest. Let us introduce conformal curvilinear coordinates determined by an analytic function , with , so that with real coefficients . Without loss of generality, we assume . The unperturbed water surface corresponds to real values of , while at the bottom we have , and is a parametric representation of the bed profile, which can be highly undulating (see, for example, Fig.1). In these conformal coordinates, a spectrum of linear potential waves is determined through the following equation (compare to Ref.R2004PRE (), where an analogous approach but slightly different notations were used): Here is a linear operator which is diagonal in Fourier representation: for any function we have The eigenfunction takes the following form, with some coefficients . With a given , we have an infinite homogeneous linear system of equations for . Non-trivial solutions exist for some discrete values . The first gap in the spectrum is the difference between the two first eigenvalues at , that is . Approximately, for small these eigenvalues are determined by the coefficient (compare to Ref.R2004PRE ()), It should be noted that corresponds to , while corresponds to . Thus, in the first order on , the half-width of the gap in the spectrum of linear waves is where is a small dimensionless quantity. As to the nonlinearity coefficients and , their values for the case of infinite depth (in other words, their zeroth-order approximations in ) can be easily extracted from Ref.OOS2006 (): It should be noted, the cited work OOS2006 () relies on results obtained earlier by Krasitskii K1994 (), who calculated kernels of so-called reduced integrodifferential equation for weakly nonlinear surface water waves (see also the paper by Zakharov Z1999 (), and references therein). It is important in many aspects that for deep-water waves the coefficients and have the opposite signs, and their ratio is . With the same zeroth-order accuracy, the group velocity is Now all the coefficients have been derived, and the simplified coupled-mode equations for relatively deep water waves in Bragg resonance with a periodic bottom take the following explicit form: where are dimensionless wave amplitudes. Analytical solutions are known for the above system (see AW1989 (); CJ1989 (); RCT1998 (); CT2001 ()), describing moving localized structures, the gap solitons. In the simplest case the velocity of GS is zero, and the solutions essentially depend on a parameter , a relative frequency inside the gap (): These expressions correspond to purely standing, spatially localized waves with frequency (concerning their stability, see Ref.RCT1998 (), where, however, stability domains were presented for a different ratio ; there are some numerical indications that the above GS are stable in a parametric interval , where a critical value ). It should be noted, one can hardly expect a detailed correspondence between the very simple model (16-17) and the fully nonlinear dynamics, but just a general accordance sometimes is possible. In particular, the model does not describe nonlinear processes resulting in generation of short waves which take the wave energy away from a soliton, thus influencing its dynamics. The model is also not generally good to study collisions between solitons, since wave amplitude can significantly increase in intermediate states. Iii Numerical experiments In order co compare the above approximate analytical solutions to nearly exact numerical solutions, we chose the following function : with real parameters , and . Hence, . For and both close to 1, Eq.(21) gives periodically arranged barriers (see, for example, Fig.1). The barriers are relatively thin as , and relatively high as . However, in numerical experiments with high-amplitude waves, a strong tendency was noticed towards formation of sharp wave crests over very thin barriers (say, when ), already after a few wave periods. With sharp crests, the conservative potential-flow-based model fails (it is also clear that tops of narrow barriers must generate strong vortex structures). Therefore we took in most of our computations in order to have a smooth surface for a longer time. Exact equations for ideal potential free-surface planar flows were simulated (their derivation can be found in Ref.R2004PRE (), some generalizations are made in Refs.R2005PLA (); R2008PRE ()). As in Ref.R2008PRE-2 (), we dealt with dimensionless variables corresponding to , . The dimensionless time is then related to the physical time by a factor . For instance, the period of linear deep-water waves with the length is . At , we set the horizontal free surface, while the initial distribution of the surface-value velocity potential was in accordance with approximate relation . Many simulations with different parameters were performed, and a very good general agreement was found between numerical and analytical results in the weakly-nonlinear case, that is for small steepness . So, with , , , and (example I), some noticeable deviations from the purely-standing-wave regime were observed only after (see Figs.2-3). In a real-world experiment it could be several minutes with m. What is interesting, even for larger , up to , GS can exist for dozens of wave periods. A numerical example for such a relatively high-amplitude water-wave GS is presented in Figs.4-5, where , , , and (example II). In this simulation, there were 45 oscillations before sharp crests formation (see Figs.4 and 6). As to a further evolution of such GS, only in a real-world experiment it will be possible to get reliable knowledge about it, since various dissipative processes come into play. Concerning water-wave GS with negative , their behavior for was found stable, while for the dynamics was unstable, and partial disintegration of GS was observed after a few tens of wave periods (not shown). However, in some numerical experiments, the life time of GS was limited by the above mentioned process of sharp crest formation rather than by their own instability in frame of the model (16-17), at least with (not shown). Finally, we would like to present an example of interaction of two GS (example III). The bed parameters are , , . Both solitons initially had and they were separated by a distance . At we set the horizontal free surface and This numerical experiment also describes interaction of a single GS with a vertical wall at . Surface profiles for several time moments are shown in Figs.7-8. We see that in this example the interaction between GS is attractive. They collide and produce a highly nonlinear and short wave group near . Iv 3D generalizations and discussion In this work, coefficients of the standard model (1) were derived for water-wave GS in the approximation of relatively deep water. The frequency gap in this case is small (of order ) despite strong bed undulations. It seems that a more general situation of intermediate depth cannot be described by this basic model, since an interaction of the main wave with a long-scale flow (“zeroth harmonics”) is then essential and should be included into equations. At the formal level, this corresponds to the mentioned discontinuities of the four-wave matrix element on a finite depth. Actually, in a finite-depth dynamics, three-wave interactions are more essential, and therefore they cannot be removed efficiently by a weakly-nonlinear transformation. This is the main difference between the present third-order theory and previously developed second-order theories (see, for example, Ref.HM1987 ()). So far we considered purely two-dimensional flows, with the single horizontal coordinate . Let us now introduce two important generalizations for three-dimensional flows. Below we only derive equations, but their detailed analysis will be a subject of future work. In the first case, the bottom topography is still one-dimensional, but we take into account weak variations of the wave field along the second horizontal coordinate , simply by adding dispersive terms, proportional to , to the coupled-mode system, as written below: In this system, a near-band-edge approximation for the upper branch of the linear spectrum gives a 2D focusing nonlinear Schroedinger equation (NLSE). Thus, in a long-scale limit, the system (24) exhibits a tendency towards wave collapse which is known as a typical feature of 2D NLSE dynamics. In the second case, a periodic bottom profile is essentially two-dimensional, and in the horizontal Fourier-plane there are several pairs of Bragg-resonant wave vectors. For simplicity, we present below equations for the case when has the symmetry of a square lattice, with equal periods in both horizontal directions and : where coefficients possess the symmetry . Let us consider interaction of two wave pairs having slow complex amplitudes and , with the first pair corresponding to wave vectors , and with the second pair corresponding to . It is important that the absolute values are equal to each other: . Again we will assume . It is convenient to use new horizontal coordinates: Elevation of the free surface is then given by the formula where , with and , and the dots correspond to higher-order terms (again, we should note that generally ). Approximate equations of motion for the amplitudes have the following form: where small constants and depend on a given bed profile, and mean the complex conjugate quantities, and the function corresponds to nonlinear interactions. Using an explicit expression from Ref.Z1999 () for the deep-water four-wave resonant interaction , we have where is a normalized value of the matrix element for two perpendicular wave vectors of equal length (see Fig.9). Since , we actually may neglect in the terms proportional to . The parameters , , and can in principle be calculated from solution of a linearized problem for water waves over a periodic 2D bed. An exact linearized equation for a surface value of the velocity potential can be written in the following form: where is a radius-vector in the horizontal plane, , while is a self-conjugate linear operator corresponding to a bottom inhomogeneity. However, in three dimensions there is no compact form for , valid with any bottom profile. At the moment, there are only approximate expressions , obtained by expansion (up to a finite order ) of a vertical velocity at the level in powers of (relatively small) bottom deviation from a constant level . The linear self-conjugate operators have a general structure and so on, where is the horizontal gradient (see Appendix A). Now we are going to calculate , , and . Let us note that with the four independent eigenfunctions in Bragg resonance are: , , , and . Accordingly, we have for the eigenfrequencies (where mean the average value in the horizontal plane), and analogously for and . Let us introduce short notations for small quantities: and similarly for and . Then we have approximate equalities, As the result, we obtain the required formulas for the model parameters: Moreover, since , it is possible to simplify the operators by writing there instead of . By doing so and taking into account only and , for the bottom profile (47) we obtain approximately Thus, the expansion is certainly useful for analysis of the case , but it can hardly be valid for a strongly undulating bed. It should be noted that a global representation of the velocity potential in the form (54) (see Appendix A) is questionable in the general case. Derivation of for arbitrary , assuming , is an interesting open problem. It is worth noting that an explicit (though approximate) form of operator allows us to derive weakly nonlinear equations of motion for water waves over a nonuniform 2D bottom. For example, the Hamiltonian functional (it is the kinetic energy plus the potential energy ) up to the 4th order in terms of the canonically conjugate variables and is written below: where (see Appendix B). It is interesting to note that the bottom inhomogeneity comes into the Hamiltonian through the definition of operator only. For , it coincides with the previously known fourth-order Hamiltonian for water waves on a uniform depth (see, e.g., Ref.Z1999 (), and references therein). It is also clear that coupled-mode system (28)-(29) corresponds to the case , when the difference is neglected in the third- and fourth-order parts of the Hamiltonian, but it is kept in the second-order part. The functional determines canonical equations of motion, Numerical simulation of these cubically nonlinear equations, with , will be an important subject of future research. Further analytical and computational work is also needed to investigate formation of vortex structures near the bottom boundary and to evaluate their influence on the free surface dynamics. In any case, the present results, based on the 2D purely potential theory, are deserving much attention. Moreover, the author hopes that in a future real-world experiment all the mentioned dissipative processes will not be able to destroy water-wave GS for a sufficiently long time. Instead, with vortices and breaking wave crests, the predicted phenomenon of standing self-localized water waves over a periodic bed will be found even more rich, interesting, and beautiful. Acknowledgments. These investigations were supported by RFBR grant No. 06-01-00665, by RFBR-CNRS grant No. 07-01-92165, by the “Leading Scientific Schools of Russia” grant No. 4887.2008.2, and by the Program “Fundamental Problems of Nonlinear Dynamics” from the RAS Presidium. Appendix A. Expansion of operator The expansion of in powers of is easily obtained from the integral representation of the velocity potential where is the Fourier transform of the velocity potential at , and is the Fourier transform of an unknown function which should be determined by substitution of Eq.(54) into the bottom boundary condition The resulting integral equation can be represented as follows, where is a surface value of the velocity potential (in the linear approximation ). It gives us the equation (31) for eigenfunctions corresponding to some fixed frequency . Appendix B. Hamiltonian of water waves up to 5th order An approximate Hamiltonian of water waves can be easily derived by writing the kinetic energy of potential three-dimensional motion of an ideal fluid in the following form, with subsequent substitution After simplifying, we obtain , where In the same manner, it is also possible to derive the Hamiltonian with a higher-order accuracy. - (1) W. Chen and D.L. Mills, Phys. Rev. Lett. 58, 160 (1987). - (2) A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989). - (3) B. J. Eggleton et al., Phys. Rev. Lett. 76, 1627 (1996). - (4) D.N. Christodoulides and R.I. Joseph, Phys. Rev. Lett. 62, 1746 (1989). - (5) T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 4730 (1997). - (6) I.V. Barashenkov, D.E. Pelinovsky, and E.V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998). - (7) A. de Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998). - (8) C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 85, 2502 (2000). - (9) T. Iizuka and C. Martijn de Sterke, Phys. Rev. E 61, 4491 (2000). - (10) C. Conti and S. Trillo, Phys. Rev. E 64, 036617 (2001). - (11) K.W. Chow et al., Phys. Rev. E 77, 026602 (2008). - (12) N. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003). - (13) D. E. Pelinovsky, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. E 70, 036618 (2004). - (14) M. Matuszewski et al., Phys. Rev. A 73, 063621 (2006). - (15) V. P. Ruban, Phys. Rev. E 77, 055307(R) (2008). - (16) V. P. Krasitskii, J. Fluid Mech. 272, 1 (1994). - (17) V. Zakharov, Eur. J. Mech. B/Fluids 18, 327 (1999). - (18) V. P. Ruban, Phys. Rev. E 70, 066302 (2004). - (19) M. Onorato, A. R. Osborne, and M. Serio, Phys. Rev. Lett. 96, 014503 (2006). - (20) V. P. Ruban, Phys. Lett. A 340, 194 (2005). - (21) V. P. Ruban, Phys. Rev. E 77, 037302 (2008). - (22) T. Hara and C. C. Mei, J. Fluid Mech. 178, 221 (1987).

Special Parallelograms Quiz. 4. October 8th. Still, we will get more specific in this section and discuss a special type of quadrilateral: the parallelogram. Finish Editing. 3. PLAY. Add one to cart. 1 GT/Honors Geometry S2 Unit 6: Lesson Plan: Polygons and Quadrilaterals Date Topic Assignment Jan 7-8 Revisit Polygon angle sum theorem. << Assignment 27b - Properties of Special Parallelograms. Geometry – GRADED ASSIGNMENT Name: _____ Special Parallelograms Practice Date: _____ Period: ____ For 1-8, complete the following charts by putting checks in the boxes that are true. Square The diagonals …. Step 3 Show that EG and FH are bisect each other. No, it is not a rectangle because the sides of the parallelogram do not meet at right angles. Explain. Test. Learn. When we discussed quadrilaterals in the last section, we essentially just specified that they were polygons with four vertices and four sides. 16 avr. Mar 25, 2014 - Properties of Special Parallelograms Notes, Assignment and Quiz Bundle This is a bundle of three of my Special Parallelogram Notes and Assignment sets and my Properties of Special Parallelograms Quiz. Homework. Voir plus d'idées sur le thème matériel éducatif montessori, science montessori, montessori ce1. Fill in these notes as you watch the tutorial video. Practice. Properties of parallelograms Geometry 6 2 properties of parallelograms worksheet answers. Log in Sign up. 20. Apr 10, 2015 - Properties of Special Parallelograms Notes, Assignment and Quiz Bundle This is a bundle of three of my Special Parallelogram Notes and Assignment sets and my Properties of Special Parallelograms Quiz. recording . Match. If . 4 Sides Opp. Since EG and FH have the same midpoint, they bisect each other. Parallelograms Assignment Answer the following questions. -Adjacent angles are supplementary-Opposite interior angles are equal to each other-All rhombuses are parallelograms, but not all parallelograms are rhombuses Kite: - 2 pairs of consecutive, congruent sides - 1 pair of opposite angles are congruent - Made up of two congruent triangles - Interior angles add up to 360 degrees Trapezoid: - Legs are congruent (in an isosceles trapezoid) - Base angles are congruent - … Previous Next. Print; Share; Edit; Delete; Report an issue; Host a game. Special Parallelograms Clue Assignment. /Contents 5 0 R endobj Special Parallelograms; What makes a Rhombus; What makes a Rectangle; What makes a Square; Using Coordinate Proofs with Special Parallelograms; Assignment 6-5: Pg. Rectangle 3. Special Parallelograms. Assignment 27a - special parallelograms. Worksheets, Activities, Homework. Digital Download . 0. Flashcards. Search. FlanMath. Learn vocabulary, terms, and more with flashcards, games, and other study tools. << Sides || Opp. Only $2.99/month. 7 days ago. PLAY. Edit. /Type /XObject Topics: 2-5 Proving Angles Congruent. 12 Followers. We can NOT find the perimeter because there are infinitely many possible parallelograms that can drawn having different lengths for the other two sides. AC BD A B C D Theorem 6-14 Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. . Played 1 times. There are six important properties of parallelograms to know: Opposite sides are congruent (AB = DC). %PDF-1.7 ���� JFIF � � �� � Add one to cart. $.' Assignment 7.4 - Special Parallelograms DRAFT. VX A Use the diagram to answer the questions, The length of AB is Geometry. 4 Sides Opp. Do Problem Solving Lesson 6 4 Properties Of Special Parallelograms You Have Homework Helper Who Holds Expertise In All The Fields of Study? ",#(7),01444'9=82. >> /Length 8 0 R Assignment: Ch 2-4 evens Online homework 7 Quiz 7 (due October 22nd) class notes. Apr 10, 2015 - Properties of Special Parallelograms Notes, Assignment and Quiz Bundle This is a bundle of three of my Special Parallelogram Notes and Assignment sets and my Properties of Special Parallelograms Quiz. Write. STUDY. This work should take you about 30 minutes to complete. /MediaBox [ 0 0 612 792] Edit. Class notes. 6 0 obj PLAY. . >> Add to Wish List. From WyzAnt Tutoring Resources Properties of Parallelograms. . Sides All Opp. /PAGE0001 Do STUDY. /Name /PAGE0001 Adobe Acrobat Document 320.2 KB. Share practice link. /Height 2189 Key Concepts: Terms in this set (14) The figure is a parallelogram. /ColorSpace /DeviceRGB 9 th, 10 th, 11 th, 12 th. Use opposite sides equal, diagonals bisecting each other, opposite angle equal, adjacent angels supplementary. Digital Download. Parallelogram 2. Learn. $1.50. Learn. Special Parallelograms Puzzle ActivityThis is a very interactive activity to practice the properties/ Theorems for special parallelograms: Rhombuses, Rectangles and Squares.Students can review the properties with this activity by rearranging the provided pieces (15 pieces) by classifying them in a " Upgrade to remove ads. Mathematics. Q When you are thinking about similarities and differences, remember to think about the parallelogram’s sides, angles and diagonals. . Created by. Write. Solo Practice. 2 3 4 5 This is the time when I will answer emails and respond to any questions by email. >> Pgs 4-5 Jan 13-14 Students will Prove a quadrilateral is a parallelogram in a variety of ways. I use this as a mixed practice after I teach the shapes so students have to differentiate between the different properties and know when to use which property. Angles Parallelo ram Rectan le Rhombus S ware bisect each other are congruent are perpendicular bisect opposite angles /XObject PDF (402 KB | 6 pages) $1.50. Each Notes/Assignment set includes: ***fully illustrated teachers notes ***matching student notes ***a teacher's set of examples that are fully worked out ***a matching set of student examples for them to follow along and fill in ***a two page assignment for students to practice that includes a complete … Find an answer to your question What value of m would make parallelogram WXYZ a square? /ProcSet 6 0 R /Subtype /Image always. October 15th. Buy licenses to share. Fill in Notes - Compare Quadrilaterals and Parallelograms . (In addition, the square is a special case or type of both the rectangle and the rhombus.) These parallelograms show two of the infinitely many possible parallelograms with a base of 8 and a height of 4. /Length 43 stream Week 8. If . Write. Grade Levels. Reported resources will be reviewed by our team. Helpful Hint q ABCD is a rhombus A B C D Then . Buy licenses to share. Special Parallelograms Assignment Active Solving for Side Lengths of Rectangles ABCD is a rectangle. It also includes interior and exterior angles of polygons. Start studying Special Parallelograms. If it is taking you much longer than 30 minutes, stop, and email me. >> Match. [/PDF /ImageC] /Parent 3 0 R Format. One special kind of polygons is called a parallelogram. Match. 6-4 Properties of Special Parallelograms Objectives Prove and apply properties of … Created by. Geometry - GRADED ASSIGNMENT Special Parallelograms Practice Name: Date: Period: All For 1-8, complete the following charts by putting checks in the boxes that are true. Browse. /Width 1693 Angles ( All Angles ( 1. Parallelograms Assignment and Quiz. Holt McDougal Geometry Properties of Special Parallelograms Example 6 The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Created by. 2020 - Découvrez le tableau "Apprentissages" de Graines De Malice sur Pinterest. Spell. The three special parallelograms — rhombus, rectangle, and square — are so-called because they’re special cases of the parallelogram. Start studying Parallelograms Assignment and Quiz. MsCornelison. b. Is the figure a rectangle? Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? We can find the area of these parallelograms by using A = bh = (8)(4) = 32. Download. Play. 5 0 obj . Terms in this set (58) always. answer with Sometimes, Always, or Never. Spell. endstream /Type /Page You can purchase them separately at: Properties of Parallelograms Notes and Assignment. Interior and Exterior angles of regular polygons. /Resources Geometry E-Learning While we are out, I will have office hours from 9:00-10:30 and 2:30-4:00. View g_ch06_04 student (1).ppt from FOREIGN LANGUAGE 332 at Renton Senior High School. 609.5 0 0 788.0 1.3 2.0 cm Special Parallelograms. Pgs 2-3: polygons Jan 11-12 Students will explore the properties of parallelograms. Week 7. ABCD is a rhombus A B C D Then . One diagonal measures 28 units. Parallelograms Properties Shapes Sides Diagonals and Angles with from properties of parallelograms worksheet answer key , source:mathwarehouse.com You need to understand how to project cash flow. Gravity. recording. Resource Type. Live Game Live. Download. View all testimonials . Log in Sign up. endobj If a quadrilateral is a rectangle, then its opposite sides are congruent . This is a self check assignment using properties of parallelograms, rectangles, and rhombus. /Filter /DCTDecode Opposite angels are congruent (D = B). . Gravity. Test. Subject. Taelor_Morton. This quiz is incomplete! Assignment 27b - special parallelograms. /BitsPerComponent 8 You can purchase them separately at: Properties of Parallelograms Notes and Assignment. Topics: 2-4 Reasoning in Algebra. If a quadrilateral is a parallelogram, then its opposite sides are congruent. << /PAGE0001 7 0 R >> << Spell. You can purchase them separately at: Properties of Parallelograms Notes and Assignment. Follow. However below, once you visit this web page, it will be hence very simple to get as with ease as download lead properties of special parallelograms answers It will not say yes many times as we accustom before. Learn vocabulary, terms, and more with flashcards, games, and other study tools. the broadcast properties of special parallelograms answers that you are looking for. We can craft any kind of writing assignment for you quickly, professionally, and at an affordable price! Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials. Adobe Acrobat Document 27.1 KB. Use the rules of parallelograms to prove a quadrilateral is a parallelogram. Properties of Special Parallelograms Notes, Assignment and Quiz Bundle This is a bundle of three of my Special Parallelogram Notes and Assignment sets and my Properties of Special Parallelograms Quiz. 1 2 2. It is a quadrilateral where both pairs of opposite sides are parallel. Save. stream Flashcards. 4 0 obj Special Parallelograms Rhombuses Squares Rectangles A B D C E F G H Theorem 6-13 Theorem If a parallelogram is a rhombus, then its diagonals are perpendicular. always. Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms. Choose from 500 different sets of geometry special parallelograms flashcards on Quizlet. .

(Revised Oct., 2010) John A. Gowan home page (page 1) home page (page 2) Note to Readers Concerning "Entropy": 10) Does matter have an intrinsic spatial motion? The primordial conservation role of gravity is to provide negative energy sufficient to exactly balance the positive energy of the "Creation Event", so the universe can be born from a state of zero net energy as well as zero net charge (the latter due to the equal admixture of matter with antimatter). All subsequent conservation roles of gravity are secondary to and derived from this original creation-role. Following on from its primary role of providing negative energy during the "Big Bang", gravity plays two further major conservation roles in the evolving universe: 1) conserving the spatial entropy drive of light; 2) conserving the non-local distributional symmetry of light. In its entropy conservation role, gravity converts the intrinsic motion of light to the intrinsic motion of time - via the annihilation of space and the extraction of a metrically equivalent temporal residue. In its symmetry conservation role, gravity converts bound to free energy in stars and via Hawking's "quantum radiance" of black holes. These two conservation roles derive from the double gauge role of "velocity c", which regulates both light's intrinsic motion (the entropy drive of free electromagnetic energy), and light's non-local distributional symmetry (vanishing time and distance). Conserving light's non-local energy state via "location" charge, gravity simultaneously conserves light's entropy drive, since time itself is the active principle of "location" charge. Hence gravity's entropy conservation role is by default brought under the mantle of Noether's symmetry conservation theorem, revealing a pathway to the unification of gravitation with the other forces of physics: all charges of matter are symmetry debts of light. IntroductionIn the temporal/historical domain, the graviton plays a role similar to the photon's role in the spatial domain. We might therefore say the graviton is the "photon of time". As the photon is the entropy drive of space, creating, expanding, and cooling the spatial dimensions via its intrinsic motion, so the graviton is the entropy drive of time, creating, expanding, and aging the historical dimension. (added June 2013) Indeed, the graviton is hidden or implicit in the photon as an asymmetric temporal component (necessarily implied by light's "frequency"). This hidden temporal component is revealed in its explicit form when the free energy of the symmetric (non-local) moving photon is converted to the bound energy of asymmetric (local) immobile mass/matter. The intrinsic (pos-entropic) motion of light is thus instantly converted into the intrinsic (neg-entropic) motion of time, establishing and entraining matter's self-feeding and self-perpetuating gravitational field - which creates (reveals) time via the annihilation of (metrically equivalent) space. The non-local symmetric photon is converted to the local asymmetric graviton, whose intrinsic, negentropic, one-way, spatially contractile motion into the historic domain identifies the spatio-temporal location of the asymmetric (because undistributed) concentration of immobile mass-energy (matter). The 4th dimension of time is necessary to exactly specify the 3-D location of matter within a constantly expanding spatial domain. Time is therefore the active element of matter's gravitational symmetry debt or "location" charge. Bound energy's (matter's) gravitational symmetry debt of "location" arises whenever light's non-local, symmetrically distributed energy (moving freely with intrinsic motion "c") is converted into local concentrations of immobile mass-matter. (See: "The Conversion of Space to Time".) Both space and history are entropy domains which function to guarantee (via their "infinite" velocity or their one-way character) the conservation of energy within their respective domains: space for free electromagnetic energy (light), history for bound electromagnetic energy (matter). As matter is an alternative form of light, so time is an alternative entropy drive of light - creating history as an alternative form of space. Charge is an alternative form of symmetry; gravity is an alternative form of inertia. The negative energy and entropy of gravity are necessary to balance the positive energy and entropy of light, allowing the Universe to be born from zero net energy and zero net charge (when we include the primordial/original antimatter). (See: "Spatial vs Temporal Entropy".) The gravitational metric is the temporal metric of matter, directed oppositely to the spatial metric of light, contractile and inwardly directed rather than expansive and outwardly directed in its spatial expression (but expansive in its historical expression). The historical/temporal dimension is at right angles to all three spatial dimensions simultaneously. Time is one-way due to the linkage between causality and energy conservation. A gravitational field is the spatial consequence of the intrinsic motion of time. We live in a universe composed of both free and bound forms of electromagnetic energy (light and matter), space and time (their respective conservation/entropic domains), and a combined spatio-temporal metric gauged by the universal electromagnetic constant "c" and the universal gravitational constant "G". The gravitational metric modifies the spatial metric by the creation of time from space - annihilating space and replacing it with a metrically equivalent temporal residue. The temporal/gravitational component of the combined metric becomes increasingly dominant as matter concentrations grow increasingly greater and denser. Ultimately, the gravitational/temporal metric of matter completely displaces the spatial metric of light at the "event horizon " of a black hole. At the "event horizon", space vanishes and time "stands still" - one second of time becoming of infinite duration where g = c. This ultimate, local, temporal/gravitational metric of matter is in contradistinction to a pure, non-local, spatial/electromagnetic metric of light - in which space stands still and time vanishes. Gravity is united with the other forces through Noether's Theorem of symmetry conservation - all four forces are caused by charges which arise as symmetry debts of light - when freely moving, non-local forms of electromagnetic energy are converted to bound, immobile, local forms of electromagnetic energy. See: "A Rationale for Gravity"; and "Symmetry Principles of the Unified Field Theory". The gravitational symmetry debt is repaid via the conversion of bound to free energy (mass to light) in various gravitationally driven astrophysical processes, such as stars, supernovas, quasars, etc. The final and total gravitational conversion of mass to light is accomplished via Hawking's "quantum radiance" of black holes, completely fulfilling the symmetry conservation mandate of Noether's Theorem, and completely repaying the symmetry debt of gravity's "location" charge. (See: "Noether's Theorem and Einstein's Interval".) 1) Gravity plays a double conservation role in nature, conserving: A) the spatial entropy drive of free electromagnetic energy, converting light's intrinsic motion to the historical entropy drive of bound electromagnetic energy - time's intrinsic motion. This gravity accomplishes by the annihilation of space, which reveals a temporal residue, the metric equivalent of the collapsed space. Because entropy is an embedded corollary of energy conservation, this is gravity's major energy conservation role, seen from elementary particles to galaxies and cosmological spacetime. B) the non-local distributional symmetry of light's energy - as required by "Noether's Theorem". This gravity accomplishes by the conversion of bound electromagnetic energy (mass) to free electromagnetic energy (light), in stars, supernovas, quasars, etc., and ultimately and completely, by Hawking's "quantum radiance" of black holes. The two conservation roles of gravity are a consequence of the double regulating role of "velocity c", the electromagnetic constant, which gauges both the entropy drive of free energy (light's intrinsic motion), and the "non-local" distributional symmetry of light's energy. The "intrinsic" (entropic) motion of light creates, expands, and cools the spatial cosmos; "velocity c" also vanishes the time dimension and a single spatial dimension (in the direction of propagation): clocks stop and meter sticks shrink to nothing at the speed of light (metric and distributional symmetry function). Light therefore acquires an "infinite" velocity in its own reference frame, having forever to go nowhere. Light is a 2-dimensional transverse wave whose intrinsic (entropic) motion "sweeps out" a third spatial dimension. When gravity conserves either gauge function of "velocity" c (in accordance with Noether's Theorem and the requirements of energy and/or symmetry conservation), it conserves the other by default. Space is the conservation domain of free electromagnetic energy, created by light's own embedded entropy drive (intrinsic motion). 2) Like the other four forces of physics, gravity is the consequence of a charge which arises as a symmetry debt of light. "Noether's Theorem" requires that the symmetry of light, no less than the energy of light, be conserved. The charges of matter are the symmetry debts of light. Identifying the (broken) symmetries of light from which the charges and their associated forces arise provides a simple conceptual basis for a Unified Field Theory: all forces trace back to a common origin as symmetry debts of light - just as all matter finds the origin of its energy in light. Matter is an asymmetric, bound form of light whose symmetries are conserved as charge and spin, whose energy is conserved as mass and momentum, and whose entropy drive or intrinsic motion is conserved as time/gravity. All matter's inherent charges and forces work spontaneously and incessantly to return matter to its primordial symmetric form - as our Sun bears daily witness. In the case of gravity, the charge (carried by all forms of bound energy in amount Gm) is the "location" charge, whose active principle is time. "Location" charge identifies the 4-dimensional spacetime location of immobile, undistributed bound energy, which as we have seen above, breaks the non-local distributional symmetry of light's free energy as gauged by "velocity c" (because mass has no intrinsic spatial motion), and results in the eventual return of bound to free energy, as we should expect (in stars, for example). (See: "The Double Conservation Role of Gravity".) 3) Gravity is the spatial consequence of the intrinsic motion of time. Time is the active principle of the gravitational "location" charge. Time has "intrinsic" (entropic) motion which causes the expansion and aging of history, the temporal analog of space. The dimensions of space and history are conservation domains created by the entropic drives of light and matter, the "intrinsic" motions of free and bound electromagnetic energy. Gravity connects and conjoins these two entropic conservation domains, actually converting either into the other, creating the compound conservation domain of spacetime, wherein both free and bound forms of electromagnetic energy can find their conservation needs satisfied. The flight of time into history drags space along behind it, causing the symmetric collapse of space, which we perceive as a gravitational field. The collapse of space, in turn, liberates a metrically equivalent temporal residue, which continues the self-feeding entropic cycle. (See: "The Conversion of Space to Time".) 4) Gravity pays the entropy-interest on the symmetry debt of matter by creating the time dimension for bound energy, through which charge conservation can have an extended significance - as a means whereby a symmetry debt can be contracted and held as a "promissory note" (a "conserved charge"), which may be redeemed at some future time, as guaranteed by the invariant principle of charge conservation (and the existence of a temporal or historical dimension as created by gravity). Our material universe functions in an historical or "karmic" (causal) mode through charge conservation in which symmetry debts, held as temporal charges, allow matter to "buy now and pay later": gravity pays the entropy-"interest" through the creation of time. Gravity funds the expansion of the historical cosmos by subtracting energy from the expansion of the spatial cosmos - via the direct conversion of space to time. As matter's symmetry debt is paid off (by the conversion of bound to free energy in stars, for example), the cosmic gravitational field is reduced and the suppressed expansion of the spatial universe begins to relax, resulting in the recently perceived "acceleration" of the cosmic expansion. (See: "Dark Energy: Does Light Produce a Gravitational Field?".) Symmetric massless light is "non-local", atemporal, and acausal, with intrinsic (entropic) spatial motion "c", and produces no gravitational field. Asymmetric massive matter is local, temporal, and causal with intrinsic (entropic) historical motion "T", and produces a gravitational field (the source of matter's time dimension). 5) The conversion of space to time is accomplished by the gravitational annihilation of space, which reveals a hidden, latent, or implicit component of time, the metric equivalent of the annihilated space. Einstein has taught us that space is not "just" space but spacetime: destroy space and you have a metrically equivalent temporal component remaining. This temporal component is in fact the hidden entropic principle that also causes the spatial expansion of the cosmos (the "Hubble expansion" of cosmology). Freed of its spatial envelope, in which it was implicit (as "frequency"), time becomes explicit and creates, expands, and ages history by its own "naked" intrinsic and entropic motion. 6) The entropic expansive motion of space and history is necessary for reasons of energy, symmetry, and causality conservation. The dimensions of spacetime are conservation domains created by entropy which must have intrinsic (entropic) drives of light, time, and gravity in order to conserve energy, symmetry and causality via the "infinite" and/or one-way velocity of light, time, and gravitation. Gravity is the force which converts either entropy drive into the other. These reversible and interconvertible entropy drives actually oppose each other in practice. In the Sun for example, they create a dynamic balance of opposing expansive radiative (spatial) vs contractile gravitational (temporal) forces. Similarly, they cause a cosmic-scale battle between the entropic forces of light and cosmological spatial expansion, vs the gravitational entropic forces of matter, historical expansion, and consequent cosmological spatial contraction. (See: "Entropy, Gravity, and Thermodynamics".) 7) The incredible weakness of gravity has been a perennial puzzle. However, from the viewpoint of gravity as a conservation force that converts the spatial entropy drive of free energy (light's intrinsic motion) to the historical entropy drive of bound energy (time's intrinsic motion), we can finally begin to see a plausible explanation for gravity's weakness. The first thing to note is that the weakness of gravity means that (in the context of the theory espoused here) on a per given mass basis, gravity needs to annihilate only a small amount of space to extract a sufficient amount of time to serve as the entropy drive for matter. Matter doesn't seem to require much time to energize its historical entropy drive (either that, or the extracted spatial entropy drive of light is enormously more potent than the historical entropy drive of time it replaces). Why should this be? Thinking along these lines, a rather obvious explanation comes readily to mind: massive objects such as ourselves (which are the only energy forms or states which require a historical causal dimension and its associated temporal entropy drive) are only tangentially connected to their historical entropy/conservation domain. We live only in the "now", not in the historical and causal past. Contrast this with the energy state of free energy or light, which is coextensive with its entropy domain (space). (Due to its effectively "infinite" velocity, light, in its own reference frame, is everywhere simultaneously within its spatial conservation domain.) The "now" is a tangent point on the surface of historical spacetime. P. A. M. Dirac pointed out that the ratio between the strength of gravity and the strength of the electromagnetic force was very similar to the ratio between the size of an electron and the size of the Cosmos - which quantitatively is essentially the same comparison that I am making between the tangential "now" and "bulk" historic spacetime. Our conclusion is that gravity produces only enough temporal entropy to service matter's point-like connection to its historical entropy domain. (See: "Proton Decay and the 'Heat Death' of the Cosmos".) 8) Black Holes are the most extreme expression of gravitational force - the "limiting case" - and they have much to teach us. A black hole is a region of spacetime in which the gravitational field is so powerful that its local field strength "g" is equal to the velocity of light "c". Consequently, no light can escape from a black hole - or at least not much. Stephen Hawking has calculated that a quantum mechanical effect due to the extreme shear forces at or near the "event horizon" or "surface" of a black hole actually converts the gravitational energy of the hole into a form of radiation which will eventually, over immense stretches of time, cause the total conversion of the mass or bound energy of the black hole into radiation, completely fulfilling Noether's Theorem with respect to the gravitationally held entropy and symmetry debt. Jacob Bekenstein and Stephen Hawking have also produced a theorem which relates the surface area of the event horizon of a black hole to its entropy. In the theory advanced here (see the "Tetrahedron Model"), this surface of the black hole must be a time surface, and so the entropy in question must be temporal (historical) entropy. The logic is that once the limiting case of increasing field strength is reached (g = c), the only way to accommodate the temporal entropy requirements of any further mass inputs to the hole is to increase the effective surface area through which space can be sucked in and converted to time, so the Bekenstein-Hawking theorem makes perfect sense with regard to the notion that gravity converts space to time - just as Hawking radiation is a sensible resolution to the question of the final and complete payment of the gravitational symmetry debt. (See: Scientific American August 2003, page 58) According to Einstein, in a gravitational field, meter sticks shrink and clocks run slow, and at the black hole's event horizon meter sticks shrink to nothing and clocks stop. The local gravitational metric as gauged by "g", which is superimposed upon the global electromagnetic metric as gauged by "c", completely overwhelms the latter. A gravitational metric of time and matter replaces the electromagnetic metric of space and light. Just as gravity overwhelms and replaces the atomic and nuclear binding forces in the white dwarf and the neutron star, so in the black hole gravity also overwhelms and replaces the regulatory function of the electromagnetic spacetime metric. Time stands still at the event horizon because it is being replaced as fast as it moves away into history; meter sticks vanish because space is completely replaced by time. The event horizon represents the end point of temporal entropy, the triumph of time and gravity over space and light, and yet Hawking radiation tells us that this triumph of darkness and matter is incomplete, ephemeral, and cannot last. We should have known, even without Hawking's brilliant deduction: Noether's Theorem requires the conservation of symmetry, and the all-way spatial entropy drive of light's intrinsic motion has more symmetry than the one-way historical entropy drive of time's intrinsic motion. At a black hole's event horizon, gravity and temporal entropy return immobile matter to an intrinsic spatial motion equal to velocity c - revealing their hidden agenda of symmetry conservation, which is nevertheless fulfilled only through Hawking's "quantum radiance". While outside the black hole, symmetry conservation is proceeding via Hawking radiation, it is likely that inside the black hole symmetry conservation is proceeding via proton decay. The extreme gravitational pressures at the central singularity squeeze the quarks of baryons back to their primordial leptonic configuration (the "leptoquark"), vanishing the color charge in the limit of "asymptotic freedom", and proton decay proceeds via the weak force "X" IVB with the emission of a leptoquark neutrino. (See: "The Origin of Matter and Information".) The inside of a black hole is therefore full of nothing but light, solving the problem of the infinite compression of matter at the central singularity. A black hole is apparently a gravitationally bound state of light, somewhat similar to a gigantic baryon, the next stage of simplification beyond the neutron star, which is essentially a gigantic gravitationally bound atomic nucleus. At the "event horizon" of a black hole, both clocks and light come to a halt, as the electromagnetic metric is completely replaced by the gravitational metric. Within the event horizon, all former functions of the electromagnetic metric are either defunct or performed by the gravitational metric, including those of the the binding forces between particles. Also absent are the primordial entropy drives of space and history, the intrinsic motions of light and time. Hence the black hole is just that physical environment in which entropy, in its usual electromagnetic expressions, does not exist, and hence no change is possible as we ordinarily experience it. But gravitation is also a form of (negative) entropy, and indeed we find, just at the boundary between the electromagnetic and gravitational domains, entropy operating to convert the mass of the black hole entirely to light - via the mechanism of "Hawking radiation". This is the ultimate expression of Noether's symmetry conservation theorem, the complete gravitational conversion of bound to free energy, definitively revealing the final conservation rationale for gravitation, and by extension, for time as well.9) Finally, although I have no talent in mathematics (as my family is fond of reminding me), I have nevertheless attempted to formulate a "concept equation" representing the gravitational conversion of space to time. Obviously I accept Einstein's gravitational field equations as essentially correct (without the "cosmological constant"), except for the caveat expressed in: "Dark Energy: Does Light Produce a Gravitational Field?". In my "concept equation" (S) represents the spatial volume annihilated or collapsed by gravity "-Gm" in order to produce the historical entropy drive (T) or time dimension of matter for any given mass "m". -Gm(S) = (T)m -Gm(S) - (T)m = 0 It is to be understood that the temporal component is the metric equivalent of the annihilated space (as gauged by the electromagnetic constant "c"), and is "hidden" in ordinary space as "spacetime", elucidated by Einstein. Since every massive elementary particle, atom, or other form of bound energy produces its own gravitational field: -Gm(S), every mass produces its own time dimension (T)m, as gauged by the universal gravitational constant "G". The gravitational constant is negative because it requires energy to annihilate space and to convert a symmetric spatial entropy drive (the intrinsic motion of light) to an asymmetric historical entropy drive (the intrinsic motion of time). Furthermore, it is this same temporal component (at work in the electromagnetic wave through "frequency") that is also ultimately responsible for the spatial entropy drive of light's intrinsic motion. The symmetric, spatial component of light's entropy drive ("wavelength") must "flee" the embedded asymmetric temporal component ("frequency") to maintain light's non-local symmetric energy state and suppress the asymmetric time dimension, which, like the proverbial "bur under the saddle", is an intrinsic feature of light's own nature - the embedded entropy corollary of energy. Energy plus symmetry conservation, spurred by the implicit presence of time, is the cause of light's intrinsic motion. Implicit in "frequency", time is the universal entropy element embedded in every form of energy: frequency multiplied by wavelength = c; E = hv; hv = mcc. Time is the entropic motivator of cosmic expansion, whether implicit in the intrinsic motion of light and the expansion of space, or explicit in the intrinsic motion of gravity and the expansion of history. (See: "A Description of Gravitation".) 10) The gravitational field of bound energy gives the impression that matter actually does have an intrinsic spatial motion. However, due to the perfectly symmetric character of matter's gravitational field (caused by the equivalent coupling of time to all 3 spatial dimensions, conserving both inertial symmetry and energy), matter has no "net" intrinsic spatial motion via its own gravity. Rather, the intrinsic (entropic) motion of matter's time dimension collapses space and provides bound energy with true intrinsic (and one-way) motion in the historical domain, at right angles to all three spatial dimensions. The one-way character of time and gravity are thus linked, and both are due to the causal nature of matter and matter's historical domain of "karmic" or causal information. A gravitational field is the spatial consequence of time's intrinsic motion. 11) The conservation role of gravity addresses the four conservation parameters of the "Tetrahedron Model": entropy (converting light's intrinsic motion to time's intrinsic motion), symmetry (the conversion of bound to free energy), causality (the creation of time and historic spacetime, and including "Lorentz Invariance"), and finally energy itself (providing negative energy to balance matter's positive energy). All these roles are intimately connected and related to the regulatory or "gauge" functions of "velocity c". Negative gravitational energy is provided by an imploding rather than exploding spatial metric, which in turn is caused by the intrinsic motion of time, matter's entropy drive (time and gravity induce each other endlessly). Time provides matter's causal linkage and creates matter's historic conservation domain of information, while simultaneously providing matter with a "location" charge representing light's non-local distributional symmetry debt. "Location" charge (whose active principle is time) identifies the 4-dimensional location of immobile, undistributed mass-energy, and eventually converts matter back to its original and symmetric form, light (in stars, black holes, and other astrophysical/gravitational processes). The active "push" or "drive" of this chain of conservation effects is provided by entropy - the implicit or explicit presence of time causing the expansion of space or history. 12) Entropy allows the transformation of free energy to "work"; symmetry conservation allows the conversion of free energy to information (charge); energy conservation allows the conversion of free energy to bound energy (mass); gravity allows the conversion of light's entropy drive to matter's entropy drive (time). Add in the asymmetric action of the weak force to break the primordial symmetry of light and its particle-antiparticle pairs, and you have the makings of a Universe such as our own, composed of free and bound forms of electromagnetic energy and their compound metric conservation domain, historic spacetime. home page (page 1) home page (page 2)

When it comes to the speed of gravity, it appears we are in a quandary. Relativity says that nothing can travel faster than the speed of light. But if this is the case, orbits would decay very rapidly. They could never be as stable as they are observed to be.This is because if it takes time for the information that a body has moved to reach another through gravitation, a body will not be pulled to where the other is now (in its current position), but to where it was when the gravitational waves started out. According to Newtonian gravity, the transmission is instantaneous. According to relativity, it appears to depend upon the situation, which is iffy at best. It says that gravitational waves travel at the speed of light, but that the effects of gravity are felt instantaneously. In other words, the warping of space-time moves with the body at all times, as if it were physically adhered to it. But then, how is the information transferred through such a permanent fixture to another when the body is in motion? And if this is the case, what is the meaning of gravitational waves (implying change of state through transmission)? This is the "scissors paradox". I have determined (although some might disagree) that this is very similar to the effect that promoted relativity to begin with. That is, the Michelson-Morley experiment. In this, a light beam which was split, directed at perpendicular directions to each other, one in the direction of the motion of the Earth (through the ether) and the other perpendicular to this, and then realigned should have produced interference with each other which could be measured. However, no such interference could be found, just as no aberration is found with gravity. The underlying concepts are virtually the same. Special relativity does very well when explaining the absence of interference in terms of time dilation and the contraction of distances in the line of motion, and has even resulted in the famous E=mc2. But when it comes to the aberration of gravity, it is a different matter. The time dilation necessary in order to explain it is on the order of billions of times greater, and vise versa for the distance contraction, so this is out of the question. And a non-Euclidean geometry would only serve to make matters worse, as a curved path only increases the time necessary for the interaction. As far as the Michelson-Morley experiment is concerned, however, I believe I have found a flaw in the experiment. I have recreated it geometrically on paper and found that in any frame of reference other than at rest, it is in fact impossible to exactly realign the beams. That is to say, the two beams, once split, cannot be redirected by the mirrors so that they meet at the same point at final reflection and then travel in the same direction. The angles will instead diverge. The best we can do is to adjust the mirrors slightly so that the beams travel in the same final direction, but they won't meet at the same point at final reflection, and will travel parallel to each other as separate beams with a distance between them that increases with increased speed for Earth through the ether. We cannot even be sure that the angle of incidence equals the angle of reflection for all frames of reference. What we need, then, is a way to measure the discrepencies of light for motion through the ether without the use of mirrors, the splitting of beams, or a measure of interference. How would we do that, you ask? Simple. No matter what the frame of refence, the Earth must travel with at least 1/10000 of the speed of light at some point in its orbit because of its revolutions around the sun. Let's consider this to be its velocity through the ether at the moment. If we were to direct a light beam down a ten meter long pole, that is directed perpendicularly to the motion through the ether, then according to the original expectations of the Michelson-Morley experiment and classical physics, the light would fall back away from the line of motion as it travels this distance since the Earth is moving forward during this time. When the pole is directed opposite this, the light will fall back the other way. The distance between these two points for this velocity as seen on a screen at the end of the pole will be 1/5 cm. Of course, we would not originally know the velocity and direction of the Earth through the ether, but directing the pole at all possible angles will create a filled in circle on the screen at the end of the pole if all of the points are marked. The ratio of the radius of the circle to the length of the pole will equal the ratio of the velocity of the Earth through the ether to the speed of light. If the pole is then turned to where the light is pointed to the center of the circle and the light moves in the same direction as the pole when the pole is then turned away from the center, then the pole will be pointing in the direction of the Earth's direction of travel. This experiment is so simple that it has probably already been tried (as at least 99% of what I propose seems to have already been thought of, but at least that shows I'm on the right track), but I have never heard of anything other than the Michelson-Morley experiment. With relativity being as successful as it seems to be, and with the aberration of gravity necessarily cancelling itself out, this probably will too (but it would still be an advantage to know for sure). But if it didn't, well, that would just be a whole different bag of tomatoes, wouldn't it? If it cancels itself out, then we must consider how gravity would do the same thing (time dilation and contraction aren't enough with gravity). If it doesn't, then we must show how the distance between two parallel beams of light cancel the interference that would otherwise be observed if they were aligned, and then somehow apply that to gravity.

How many years are in a day on venus? - Top best answers to the question «How many years are in a day on venus» - FAQ. Those who are looking for an answer to the question «How many years are in a day on venus?» often ask the following questions - Your answer - 20 Related questions Top best answers to the question «How many years are in a day on venus» A day on Venus lasts 243 Earth days. A year on Venus lasts 225 Earth days. Those who are looking for an answer to the question «How many years are in a day on venus?» often ask the following questions: 📢 Venus how many years on venus? - Transits of Venus are among the rarest of predictable astronomical phenomena. They occur in a pattern that generally repeats every 243 years, with pairs of transits eight years apart separated by long gaps of 121.5 years and 105.5 years. - How many years old is venus williams? - Seven venus days equal how many earth years? - Seven venus days equals how many earth years? 📢 How many years away is venus? Venus has not been closer to Earth than 24.5 million miles (39.5 m km) since 1623. After the year 5683, Venus won't come within 24.8 million miles (40 m km) of Earth for more than 60,000 years. - How many earth years is one year in venus? - How many games have venus williams won over the years? - How many venus years is equal to one earth year? 📢 How many years is 1 venus day? 243A day on Venus lasts 243 Earth days. A year on Venus lasts 225 Earth days. - How many venus years is equivalent to an earth year? - How many years on earth is 1 year on venus? - How many earth years does it take to get to venus? We've handpicked 20 related questions for you, similar to «How many years are in a day on venus?» so you can surely find the answer! How many earth years does it take to travel to venus? Typical Hohmann transfer trajectories from Earth to Venus take about 3.5 months. How many light years does it take to reach venus earth? Venus is another planet in our solar system. Venus at its closest is about 2 light-minutes from Earth, and when it is on the opposite side of the Sun, is about 14 light-minutes away. For space probes launched from Earth to Venus using our existing rockets, it takes about 15 months to reach Venus. How many earth years does it take venus to orbit the sun? It takes 225 Earth days, or about 71/2 months, for Venus to orbit the sun. How many earth years for one orbit of venus around the sun? Venus orbits the sun ever 224.7 Earth days, so 1 Venus year is equal to 0.6152 Earth years. How many light years does it take to reach venus from earth? At its closest, Venus is 23.7 million miles away and at its farthest, it is 162 million miles away. It would take between 2 and 15 light minutes to reach Venus form Earth. How many years does it take the earth to revolve around venus? The Earth does not revolve around Venus. Both the Earth and Venus revolve around the Sun. The Earth takes about 365.25 days to do so, and Venus takes about 224.7 days to do so. How many light years does it take to get from earth to venus? Depending on the orbits of Venus and the Earth around the Sun, the distances between Venus and Earth vary. It as been as close as 38.2 million km, but average distance of 41 million km. 41 million km is approximately 0.000004333703419500923 Light Years How many light years does it take to get to venus from earth? Earth is at 499. The distance of Venus from Earth varies from 4.4 E-06 light years (ly), at conjunction, and 2.7 E-05 ly, at opposition.. How many years does venus take to complete an orbit around the sun? How many light years does it take to reach venus from earth not in light minutes i need light years please? You are confused. "Light-year" is a measurement of DISTANCE, the distance that light travels in one year; it is not a time period. Venus is, depending on where Venus and Earth are in our respective orbits, between 2 and 14 light-minutes away; light would take somewhere between 2 and 14 minutes to span the distance. You can convert easily minutes into years; there are 60 minutes in an hour, 24 hours in a day, and 365.26 days in a year. If you visited venus for a year how many earth years would it be? a little less htan 1 What years did satellite and robot explored venus? venus has no sattelite Does venus take 1.9 years to orbit the sun? Like every other planet in the solar system, Venus travels around the Sun in an ellipse—an offset oval. The completion of one such orbit constitutes a year. Venus moves around the Sun at more than 78,000 miles per hour and completes one year in about 225 earth days, or about 7.5 months. How did venus support life billions of years ago? - Creating the different simulations involved adapting a 3D general-circulation model, which accounted for atmospheric compositions as they were 4.2 billion years ago and 715 million years ago, and as they are today. The model also accounts for the gradual increase in solar radiation, as the sun gets warmer over the course of its lifetime. Is the planet venus habitable billions of years ago? - (Image credit: NASA) The hellish planet Venus may have had a perfectly habitable environment for 2 to 3 billion years after the planet formed, suggesting life would have had ample time to emerge there, according to a new study. In 1978, NASA's Pioneer Venus spacecraft found evidence that the planet may have once had shallow oceans on its surface. What is the orbital period on venus for years? Venus' orbital period is 0.616 Earth years. How far away is venus from earth in light years? Venus is much less than a light-year away. Venus varies in its distance from Earth between about 26 million and 160 million miles, as they orbit the sun separately. This works out to a range of 0.000004 to 0.000027 light years. Within the solar system it would be better to measure distances in light minutes than light years. How far away is venus from the sun light years? Not light years, 107 milllion km. How long is a year on venus in earth years? 224.7 earth days, or about 0.62 Earth years because venus is closer to the sun and that means it takes less time to go around it. Is it true that venus' days are longer than its years? Yes and no. Venus rotates very slowly and in a "retrograde" direction. The effect is to make the "solar day" on Venus much shorter than the "sidereal day" (the rotation period). The orbital period for Venus (its "year") is about 225 Earth days. The sidereal day is about 243 Earth days. The solar day is about 117 Earth days. That's why the answer is "yes and no". It depends upon which "day "you mean.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning. 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. Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear. The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it? Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them? Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest? The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails. 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? Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card? A Sudoku with a twist. Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring? Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential. A Sudoku with a twist. Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? Find out about Magic Squares in this article written for students. Why are they magic?! Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku. This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set. Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring? In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9. A pair of Sudoku puzzles that together lead to a complete solution. Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance? A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article. The challenge is to find the values of the variables if you are to solve this Sudoku. An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length? Solve the equations to identify the clue numbers in this Sudoku problem. There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules? Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way? Different combinations of the weights available allow you to make different totals. Which totals can you make? Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line. Given the products of adjacent cells, can you complete this Sudoku? There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper. This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid. The clues for this Sudoku are the product of the numbers in adjacent squares. Two sudokus in one. Challenge yourself to make the necessary connections. A Sudoku with clues as ratios. A Sudoku that uses transformations as supporting clues. My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be? This Sudoku combines all four arithmetic operations. If you are given the mean, median and mode of five positive whole numbers, can you find the numbers? The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . . Four small numbers give the clue to the contents of the four surrounding cells. This sudoku requires you to have "double vision" - two Sudoku's for the price of one A Sudoku with clues as ratios. A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both? This Sudoku, based on differences. Using the one clue number can you find the solution? 60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra? Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

It is hypothesized that a greater concentration of product is achieved through an increased substrate concentration. P-international (p-an) is produced from An-Benzene-ODL-Argentine p-international hydrochloride (PAPA) with the aid of the enzyme trying. Analyzing the initial reaction rates that were calculated from p-an concentration in different test tubes which contained varying concentrations of PAPA the relative contribution of substrate availability on initial reaction rate was accessed. Initial reaction rates significantly increased at higher initial concentrations of PAPA. However, at a PAPA concentration above 0. Mm, there was no significant increase in initial reaction rate. This suggests that a higher initial reaction rate achieved at a raised initial PAPA concentration from lower levels was due to increases in reactions that formed p-international. The effects from elevating initial PAPA concentration to 0. Mm and above proved insignificant, which may indicate enzyme saturation. Luminescence Jenny Shin; Ivan Richard Low Introduction This study investigates the effect of substrate concentration on the initial rate of reaction when its concentration is much greater than the enzyme concentration. It is hypothesized that the initial rate of a chemical reaction increases as the substrate concentration becomes more abundant. A possible explanation to this phenomenon could be due to the increased probability of collisions occurring between substrate molecules and its associated enzyme, therefore increasing the number of substrate molecules involved with enzyme activity (Moses and Schultz 2008). Whereas, a low concentration of substrate will reduce enzyme activity as less number of substrate molecules are available to bind to the active site of enzymes (Moses and Schultz 2008). Therefore, it is proposed that an increase in substrate concentration will increase initial reaction ate through greater enzymatic activity. Materials and Methods The reaction from An-Benzene-ODL-Argentine p-international hydrochloride (PAPA) to p-international (p-an) accelerated by the enzyme trying is utilized as a model to test our hypothesis. Five test tubes were prepared with differing PAPA concentrations and constant trying concentrations, temperature and PH. Trying concentration in the test tubes was much lower than the PAPA concentration. Initial reaction rate was measured by the amount of p-an produced every thirty seconds for the first five minutes of the reaction. All solutions used in this study have been prepared and brought to room temperature before the beginning of every laboratory session. Trying was obtained from bovine pancreas and all chemicals used were brought in from Sigma, a pharmaceutical company. Quantitative analysis of transmittance to obtain optical density and p-an concentration was collected from a spectrophotometer. A calibration curve was produced from five calibration tubes for each spectrophotometer used to generate the calibration slope, the k value. A system composed of reference trying concentrations and varying PAPA concentrations as applied to access the relative contribution of substrate availability on initial reaction rate. The k value was utilized to convert optical density (obtained from transmittance), into p-an concentration. Data Analysis The concentration of p-international was determined by the negative log of (transmittance/ 1 00) divided by the k value. To calculate the initial reaction rates for each trial, the difference between p-an concentration at 2 minutes and p-an concentration at 30 seconds was divided by the time difference (1. 5 min). Although 10-12 data sets were collected, only 9 data sets were chosen at random or data analysis. One of the data sets was incomplete due to missing data collection for 0. Mm of PAPA solution so the average initial rate for 0. Mm PAPA was calculated with only 8 trials. Furthermore, another data set was revised by applying a new formula to replace an erroneous calculation for all p-an concentrations. The average of the initial reaction rates for each of the five different initial PAPA concentrations were calculated from the sum of the initial reaction rates from 9 separate trials (and an exception of 8 trials using 0. Mm PAPA) divided by the number of trials. A significance level of 0. 5 was used to calculate confidence intervals of the average initial reaction rates to create error bars. All statistical functions above were performed through excel. Data analysis determined if range included by confidence interval (i. E. Mean – CLC value through mean + CLC value) does not overlap, so we can be 95% confident that the difference between the rate the formation of p-international is due to differing substrate concentrations rather than random chance or sampling error; we can therefore accept our null hypothesis and consider the difference statistically significant. If range included by confidence interval does overlap, we can be 95% confident that the rate the formation of p-international is due to sampling error and/or random chance and is not statistically significant. Results Data analysis in Figure 1 indicates a steady trend of increased rate of p- international formation with an increasing amount of PAPA concentration over duration of 1. 5 minutes. To illustrate this inclination, the initial reaction rate rose significantly from 0. 003В±0. Mm/min in a test tube containing 0. Mm PAPA to 0. 006В±0. Mm/min in a test tube containing 0. Mm PAPA. Moreover, the initial rate of formation of p-an in the 0. Mm PAPA tubes are significantly lower than in any other tubes. In the 0. Mm PAPA tube, the initial reaction rate is significantly lower than in the 0. Mm, 0. Mm and 0. Mm PAPA tubes. For example, the initial reaction rate of 0. Mm PAPA is 0. 006В±0. Mommy/min as compared to a much higher rate at 0. 0110В±0. Mommy/min for. Mm PAPA. However, as PAPA concentration increases to 0. Mm, the initial reaction rate in tubes containing 0. Mm, 0. Mm and 0. Mm PAPA are not significantly different from each other. At 0. Mm PAPA, the initial reaction rate is 0. 001В±0. Mommy/ min which is identical to the initial reaction rate of 0. Mm. Discussion Through the investigation on the effects of substrate concentration on initial reaction rate, we hypothesized that the initial rate of a chemical reaction increases as the substrate concentration increases. If my hypothesis is correct, then through the protocol using trying, the test tube containing the highest PAPA concentration will have the highest reaction rate and the test tube intonating the lowest PAPA concentration will have the lowest reaction rate. Through data analysis, we observed a significant increase in the rate of formation of p-an with a greater amount of PAPA concentration when comparing the initial reaction rates between the tubes with 0. Mm, 0. Mm and 0. Mm PAPA solution as there is no overlap in confidence intervals between the three means (Figure 1). However, the confidence intervals of the average initial reaction rate of 0. Mm, 0. Mm and 0. Mm PAPA do overlap and therefore are not significantly different from each other (Figure 1). For example, this is displayed with a mean initial rate of p-an formation as 0. 009В±0. Mommy/min for 0. Mm that overlaps the confidence interval of the initial rate of reaction of 0. Mm which is 0. 0110В±0. Mommy/min (Figure 1). The null hypothesis is supported by significant increases in initial reaction rates with an increase of substrate concentration at lower concentrations. Although, in this experimental design, any concentration above 0. Mm PAPA began to plateau in initial reaction rates, causing an overlap in confidence intervals which showed insignificance in the proposed trend (Figure 1). A plausible explanation to the results of this study is that a higher substrate concentration increases the frequency of collisions between substrate and the active site of enzymes, which in turn increases the probability of substrate binding to an enzyme (Moses and Schultz 2008). The enzyme-substrate complex that results upon binding activates enzymatic activity in accelerating the conversion of reactants to products (Moses and Schultz 2008). Although trying concentration was set constant at 5. EYE-mm, which was much lower than initial PAPA concentration for all five test tubes, the readily available PAPA molecules were not sufficient to convert all available enzymes to the enzyme-substrate complex (Moses and Schultz 2008). In Figure 1, the initial rate plateau starting at 0. Mm PAPA may be justified by trying saturation (Moses and Schultz 2008). Therefore, all the trying present were in the midst of a reaction cycle with another PAPA molecule, so free floating PAPA encountering trying were unable to bind to the active site (Moses and Schultz 2008). When this occurs, the initial reaction rate is unable to increase, as the maximal rate is reached (Moses and Schultz 2008). If this experiment continued using a higher concentration of PAPA with an unchanging concentration of trying, the initial rate of formation of p-an will not be significantly different from 0. Mm PAPA due to the saturation of all trying molecules. Trying has a limited number of active sites and a characteristic number of catalytic cycles per unit time, restricting the initial reaction rate to further increase (Moses and Schultz 2008). Conclusion It has been demonstrated that an increase in substrate concentration increases the initial rate of a chemical reaction by utilizing the model reaction of the abstract An-Benzene-ODL-Argentine p-international hydrochloride converted to the product p-international by the enzyme trying. The results from this study support our null hypothesis, but only limited to lower substrate concentrations. At higher substrate concentrations, initial reaction rates cease to increase which indicates a maximum initial reaction rate. Further studies can be conducted to gain a better understanding in the plateau observed with higher substrate concentrations. Literature Cited Moses, C. D. And Schultz, P. M. (2008). Principles of Animal Physiology 2nd Edition. San Francisco. Pearson Education Inc.

Custard powder, Swan formations, and more. In this episode, Andy, Robin and Adam are joined by the amazing Tim Oates to discuss seeing the maths in things. Do we teach the everyday maths around us enough in schools? What topic is always presented in a confusing way to children? Plus, Tim talks on the importance of rich questions that push towards helping the understanding of mathematical structures. The school of school podcast is presented by: Subscribe to get the latest The School of School podcasts delivered to your inbox. Hi, I'm Andy Psarianos. Hi, I'm Robin Potter. Hi, I'm Adam Gifford. This is the School of School podcast. Welcome to The School of School podcast. Are you a maths teacher looking for CPD to strengthen your skills? Maths — No Problem! has a variety of courses to suit your needs from textbook implementation to the essentials of teaching maths mastery. Visit mathsnoproblem.com today to learn more. Welcome back everyone to another episode of The School of School podcast. We've got Tim Oates with us. Very exciting. So Tim, do you want to tell us a little bit about yourself? Thanks Andy, and thanks so much for the invitation to talk today. So yeah, I'm Tim Oates. I'm group director of assessment, research, and development at Cambridge University Press & Assessment. We're a big non-teaching department of Cambridge University. I run a research group of around about 40 people. And, we work in over 140 countries around the world providing assessment services, learning services, and we look at educational reform as well. I particularly enjoy looking at different education systems, have done that for years. Really thinking deeply about how different systems perform and drawing insights from those differences to improve education around the world. Tim, we're here with Adam and Robin here today, and we're going to talk about functions, number and verbal problems, Seeing the Maths in Things. Okay, what does that mean and what is this all about? Yeah. I mean, it's quite a careful title because people don't necessarily see the maths in things. The world is full of sense perceptions, isn't it? Colour, and shape, and movement, well, where's the maths in that? It's pretty bewildering. Maths is fascinating because it's a deeper structure. It's a structure within and beneath the surface of things. It exists in terms of relationships. Of course, we can find fascinating geometric patterns in the eyes of flies. But when we look at things like the ocean, where's the pattern in the turbulence? Where's the pattern in that? Where are the mathematical relationships between the things that we see around us? Gases, solids? I mean, it's amazing that we have developed this language which apprehends laws, deep structures in things, that's been a remarkable human attainment. You're not born with an understanding of the kind of mathematics which we now use to understand the world deeply. You have to acquire it. You have to work at it. Some of the things we know. We can look at two oranges and know they're two oranges. Other things, the issue of say, for example, parabola and the relationship between velocity and distance in an object affected by gravity, that takes a long time to get hold of, and we have to teach that. Quadratics, not obvious, you have to learn it. But once you understand it, you can see that it applies to the real world. It helps you solve real things. I mean, we're all in awe, I'm in awe all the time. And when you start thinking that way, when you start thinking of the world in a mathematical sense, you very quickly recognise, well, I mean, it's almost a philosophy of mathematics is what we're discussing here because we don't even really know. I don't think we really understand, is mathematics a language that we wrote, that we invented? Or, is it something we observed? Is it just the nature of things and we're just recording it? I don't know that we even really have an answer for that. When I was young, I read a couple of things which really opened my eyes. So, one of them was, I just became interested in watching migrating swans. Why do they fly in a V? Why do Canada geese fly the way they do? Well, why do they do it? Turns out it's maths. The formation that they fly in, if they make a navigation error, it minimises that error and makes it easier for them to get back onto course. I mean, of course, we sorted out how they navigate. It's complicated, but the maths that they've adopted helps. Likewise, pigeons eating in fields - why do they cluster the way they do? And there's a mathematical reason for that. It's really, really efficient in terms of clearing the grain and the seeds in the field. That's when you begin to know that there are mathematical principles underneath the surface appearance of things, things that are remarkably persistent. That's right. And in that, even things that are really difficult to understand, and Marcus du Sautoy from Cambridge speaks a lot about this actually. The fact that that pie keeps showing up in all these places where there's no obvious circle and somewhere in there lurking as a circle, but we don't really know where it is. But it keeps popping up and especially in statistics and probability and like you say, "why do bees make their honeycombs in hexagons?" I mean, what a curious question, right? Well it turns out that's the most efficient tessellation you can do. So uses the least amount of wax. Now that's a fascinating thing. So there's maths everywhere you look in nature, in the universe in general. Do you think we teach enough of that in schools? Yeah, no, that's interesting. That's an interesting question. "Do we teach enough of that?" We could have application and the wonders of the outside world, is it kind of a thing that we go to every so often to try to keep kids interested? But you can do it more often, Tim, because I've had both my kids over the years say to me, "I don't understand. I'm never going to need this again." And usually they're talking about maths. "I'm never going to use this again. Why do I need to learn this?" So maybe if we did present it more often, as in it's in everything, there's maths everywhere, maybe they could get more excited about it. I don't know. Yeah, I mean I think what we need to do is to do what Japanese teachers do brilliantly, which is that they move between these things and the abstract representations and the language of mathematics continuously, not "okay, for the last few weeks we've been doing this, and by the way, there's this amazing thing that you can observe in the outside world" and presenting it just as a stimulus and sort of unrelated to the maths, no, no, no. What you have to do is to move between these things and the abstract representation of mathematics continually, because then it encourages seeing the deep structure in the world and the meaningfulness and the power of mathematics. There was a great incident that happened outside the gates of a school in Kent where somebody had been teaching a particular programme called Cognitive Acceleration in Science Education. And it was in science, it helped kids to really accelerate their understanding of mathematics. And as Phillip Aiden was leaving the school, a child came up to him and said, "I want to talk about case. So Phillip said,"Oh, okay". And the child said, "I hate case, I hate it". And Phillip said, "Well what's the problem?" And so the child said, "It makes me think about science all the time." And when Phillip began to talk to this kid, he realised that "boy, it was really working". This kid was beginning to really think "okay" about, he kind of proceeded with his intrusive thoughts but what we know from the research is if you encourage a child, if a child begins to think about something outside contact time, their attainment in the subject really begins to accelerate. And that's what I mean by this issue of functions, number and verbal problems, Seeing the Maths in Things. It has not just to be an exciting lesson, it has to shift the way you think, it has to shift the way you see the world. In Singapore, they actually state that, I can't remember exact the exact wording in the national curriculum, but the purpose for teaching mathematics is to improve someone's intellectual competence. So for them, I mean they go as far as stating that "we don't actually teach, obviously we want them to know all the things that we're teaching them, but the ultimate goal is for them to be able to make better logical decisions in their lives." And as a contributor, so let's say for us, we publish textbooks, how does that apply? How do we get that into the real world? Well, I can think of an example. Imagine that what we're teaching is ratio and proportion and we could just create some convoluted, very mechanical question around ratio and proportion and present it to them that way. But why not take the opportunity to, let's say, use a recipe for example, to teach ratio and proportion because now all of a sudden you can go so much deeper into the understanding of how the world works. Because you can ask them a very simple question which will get their mind spinning in a, not everyone, some of them will just ignore the question because maybe they're not quite ready for it, but others will start pondering on things. So a recent example that I can think of is of a word problem was, it was a pizza dough recipe. And it was like, "okay, here's a recipe," something like, here's a recipe for making four circular, nine inch pizza doughs, right? Pizza bases. Now you want to make three. Now all the measures are given in different, so some are given in mass, some are given in volume, so you've got to do the calculation, even though they're in different, but some of them are given in time. So the amount of time that the dough needs to rise. So do you also need to perform that calculation for that? If it's just mechanical, you're just going to do the same process for all the numbers, regardless of what the measures are, but it doesn't apply to all the measures. The rising time is the same because you've now separated them into three pieces instead of four pieces and you're going to put them on counter to rise, but the rising time stays the same. Why doesn't the rising time change and things like that, understanding these structures of mathematics and how the universe actually works. Those are great opportunities. Now if you ask that as a test question, a lot of people say that's a really unfair test question. I disagree. I think that's a great assessment question, but I don't know, what do you think of that? Oh, I think rich questions which actually push understanding as well as illustrating, being a good carrier of the concept, illustrating the concept and encouraging that iteration between the abstract expression of mathematics and the mathematical structure inherent in the real situation are just great and it's well worth undertaking research on, "which are really good questions and which are good problems". We looked at that in respect of science. Robin Miller has looked at that in terms of science. And what he's found is that there are some questions which are very revealing of structure and are worthwhile using year after year after year and being incorporated alongside other questions which elaborate the idea and having a battery of examples and questions, which then, you're not just relying on one context to encourage kids to understand the deeper structure. You can vary it and that helps with the understanding of all of the children in the group. Fractions is a case in point. Again, very good studies of how Japanese teachers use real examples, different vessels full of liquid, boxes with sand and so on to present the abstract expression of the relationships in fractions through practical real world examples which really embed the understanding, and build up the understanding. I love things that can become elaborated. So an example is, well you have, there's a well, it is this kind of depth, you know, you give all the information. This is the speed, this is the rate at which something will fall. Now the question becomes quite important. "How do you know when it will hit the bottom?" So of course what the students do is they just look at it, they calculate the distance, they look at the distance, they look at the rate, the acceleration, and they say, "look, well it hits the bottom then." And then you say, but that's not the question. The question is "how do you know when it hits the bottom?" Because you only know it hits the bottom and when you hear it, so you then got the speed of sound to take into the equation. Oh, now it's a really interesting question. So absolutely, Andy. I mean the point is it is not arbitrary how you choose these things. You choose them carefully. You choose them carefully because they are an excellent vehicle for carrying knowledge of the mathematical structure. They're not so opaque that you can't understand the deep mathematical structure, that wouldn't work. They have to be highly illustrative of the deeper mathematical structure, and they have to be age appropriate. But you can actually push kids understanding far more than people realise. I did this in respect and I'm afraid again, it's a science example. This is the one that immediately comes to mind. I did this in the respect of oxidation. So there were kids who were being shown that things can explode if they're a fine powder. It's very exciting for primary school kids. You can blow the lid off a tin. He had a big kids explosion. Yeah. "Wow. how's that possible? cause it's a powder, so how's that possible? It's just custard powder." And indeed the demonstration was done and there was a bit of discussion and then I talked to the kids. As you say Robin, they were excited by it, but they hadn't, the questioning hadn't revealed what was actually happening, which was combustion. And for combustion there has to be a fuel, there has to be heat, there has to be oxygen. You know, you can immediately start to ask some questions which build up complex understandings of the process of oxidation. "Why doesn't just a pile of custard powder combust as opposed to when it's finally distributed in the air?" Because you've got the same, you've got the fuel and you've got the heat, but there isn't enough oxygen available to it until you distribute it. Now what I've found is that if you push the questions with very young children, they can develop very complex ideas. But you have to know what question it is, which practical example you're going to use, a real world example. And it has to relentlessly, I mean a bit relentlessly drive towards the mathematical structure. And I think the question, the kind of question that you've just talked about, the kind of example, you could easily get sort of distracted, couldn't you, by just surface features of the issue of pizza dough. You have to make sure it constantly drives at the underlying mathematical structure. You highlighted here was the verbal problems. What role do verbal problems play in seeing the maths in things? Because I think a lot of people think mathematics, when I say a lot of people, I'm not talking about educators, I think mostly educators understand, but when you talk to parents or people who have not come through any kind of formal training on how to teach or why you teach or anything like that, they would think of mathematics as, call it "sums", which I know is not the right word, but that's what they think. Why are verbal problems important? Yeah, I mean verbal problems are important because they present the complexity of the context in which we find the mathematical structures. Most obvious example of a parents would be family debt, for example. That presents as a very complex problem to families and often families make poor decisions about incurring debt. And we've had big shocks recently about, in terms of what's been happening in the global economy, which has had impacts on interest rates which have retaken families by surprise. Well maths is at the heart of that, issues of growth in the economy, wage, wage-price relationships. It's only when we think about these more complex settings that adults experience that I think we can really hook onto is a discourse whereby they can understand the importance of mathematical education in schools. But it's seeing the maths in things and knowing that if you approach certain things using mathematical techniques and mathematical understanding, you can take actions which are good for you, good for society, and good for the economy. That's where I would immediately start bridging between the immediate experience of parents and the realities of what their children need to learn at a pretty early age. I think it strikes me too that something that is schools, it's really important that we foster that attitude because I think there is still an attitude that mathematics is just, or at least it's not responding to the world mathematically, it's just about finding correct answers in some people's view, like parents or possibly people outside of school. And it strikes me that the importance of what's being taught about here, there's such rich conversation to be had in terms of responding mathematically to the world and the creativity and the art, the artistic nature of it, and that it's a language in itself. And I think that for me, what comes across loud and clear is the importance that the subject itself, we need to make sure that it's seen in that way too to everyone. Yeah, I think that's absolutely right. And let's just look at a very simple thing, a seemingly simple thing. I think Andy, in the past, you and I have talked about the teaching of negative numbers and the fact that an awful lot of materials and textbooks teach negative numbers through the temperature scale. That is not negative numbers and it gives rise to a misconception amongst children. The late Richard Dunn said that "You need to teach negative numbers not as the movement, that way and that way, that way is positive, that way is negative with zero being the line in the middle, you need to teach it as the absence of something." Now, just going back to family finances, we talk about a hole of debt, don't we? A big hole of debt opening up. Well, that's exactly how Richard Dunn would teach negative numbers. He would say "Imagine a house in which in the front you've got a pile, a pile which is one, a pile which is twice as big, which is two a pile, which is three times as big of earth, which is three. Well, in the back you've got a hole, which is one, a hole twice as big, which is two and a hole three times as big, which is three." Now, that's an appropriate way to teach or give bridge between a mathematical problem, a real world instance rather and a mathematical problem and the concept. And it's conceptually correct. Minus one is the absence of something. It's not just to the left of something. And don't get me started on the number line, but you're absolutely right. I mean I think that it's understanding that numbers have different types of representations and that just, there's this idea of cardinal numbers, there's this idea of nominal numbers, there's this idea of other manifestations through measurement and ordinal numbers and all these kinds of things. And sometimes we just take the easy way out and we just think what's the easiest real world type example I can come up with a negative number, let me use that. But not really challenging "what is the concept that I'm trying to teach here?" That's why it's dangerous to let teachers write all their own content. I m.ean this is not me taking a stab at teachers, but people who make it their life's business to do this sort of thing, spend an awful lot of time thinking and researching and looking at the impact of doing it one way versus another. It takes a lot of time and it takes a lot of expertise and you can't leave some of those things to chance because you can get misconceptions in really early on that are hard to unpick later on if you do it wrong. That's why this topic is so important, Andy. I couldn't summarise the way in which we should approach this better than you have just done. It is very easy to choose a concept which is of immediate appeal, but can actually give rise to a misconception in the child's mind or in a group of children's mind. We know from Singapore maths, we know from the instances of verbal problems contexts in the textbooks that there is very thorough selection of particular instances to encourage application and to reveal the mathematical structure, and careful variation in those against a particular concept. It is something that has to be done systematically and in the light of good evidence. And a lot of the evidence is from good practise. And it's from good practise, but it's also from looking at what the long-term effect is of doing something and saying, "okay, well we did it this way, what happened, how do you judge whether or not your year three lesson is effective?" Maybe it's by what happens when those children go into key stage three? I had a great example of this. I was talking with some teachers where we were doing some international comparisons and it was some Japanese teachers and some Chinese teachers, and one particular question came up, which was "there is a pile of sand, where do you have to cut it to get one third of the pile of sand?" And I said, "The way that's expressed is very interesting and very challenging." And I asked the teacher and I said, "well, when did you first use that as an example?" He said, "No, well I've always used it. We've been using it for 500 years. It goes back to the first mathematical textbooks in China. Why wouldn't you use it?" You see, this is very different from the kind of approach that we teach, which is, "Oh, you've got to be innovative, you've always got to use new stuff, don't use the same stuff over and over." It's a completely different approach. Why do you use it? Cause it works because it's been carefully chosen. Cause it reveals a mathematical structure. Because you can elaborate it to make it more complicated if the kids have already grasped it. You can ask the additional question, "ok that's when it hit the ground, at the bottom of the well but how do you know that it actually, how can you actually know that it hit the bottom" and then incorporate the speed of sound into it? You can elaborate these things. That's right. And otherwise you end up with answers like "how many children travelled in that car? It'd be three and one third children". No, there's no such thing as a third of a child, but that's what happens. And I'm sure people who write exam papers find those types of answers all the time because it's an example of a well-structured question. You can't put a third of a child in a car, so that's not the answer. So "how many children actually went in a car? Was it four or three?" Yeah. I came across this teacher who was using a particular set of examples in a primary school in terms of mathematics and science, and she said, "Oh, these are so boring I've used these for years and I think I'm going to use something different." And I said, "Well, why do you want to use something different? Do they work?" she said, "Oh yeah, they work brilliantly". "So just carry on using them. The only person who's bored is you, just don't let your boredom show." That's right. That's fair. Carry on. They're brilliant questions. I'm still working on the one third of a child in the car. That's very disturbing. But I think what we just described is something about pedagogy. It's something about how you create your own materials as a teacher. It's how you judge between materials. It's what people who produce materials should be concerned with. And we see this in the best and highest quality textbooks from Hong Kong, from Shanghai, from Singapore. Tim, one final question before we let you go. It seems to me, I might be wrong. It seems to me that there's a large, if you look at the mathematics journey that we push most people down, there's a more of an emphasis on calculus versus statistics and probability. Do you think we should have more statistics and probability in mathematics? If you look at where the world is going in the future, I think the push towards calculus has always been an engineering view of the world, but is there an argument for putting more statistics and probability back into mathematics? That's a very general question, I know, but I'd just be interested in knowing what you think. Oh, you might think it's a general question but there's quite a specific thing going on in England about this. Okay. I'm going to reveal quite something that may sound quite traditional, but we introduced different roots in our advanced level qualifications. In other words, 16 to 18 prior to university, roundabout the 1970s, we began to introduce a statistics route and before it, there'd been a strong mechanics route. Boy, you should have seen the kids migrate to statistics. Actually, that created quite a problem in our system. If you set them up in opposition, one with another, then you can get ready migration. I don't think it's about setting them up in opposition. I think you can have a kind of different flavour, but if you completely remove mechanics from 16 to 18, and you introduce it as well, you can do statistics or mechanics, then I think actually you've got a problem. If you look at say, the German system, then you can major a bit more in statistics, but hell, you can't escape the mechanics. Likewise, if you are really committed to mechanics, you still need to do some statistics. I think that's about the right sort of balance. I do like an upper advanced system where people can follow their preferences and their aspirations. I think that's a time in education when we can begin to really follow people's aspirations and preferences because that means they're enjoying themselves and they really probably learn more. That's a good thing for them. But I think it sounds terrible. I think it's a question of balance. We don't want one to the exclusion of the other. Tim, always a pleasure speaking with you. Thank you so much for joining us today. Well, thank you very much, Andy. It's been a real pleasure today. Thank you very much. Thank you. Joining us on the School of School podcast.

You know the saying “a picture is worth a thousand words” the same applies to graphs: they're a very effective means of conveying information visually—without a thousand words in addition to being a “picture,” a graph is also a math-based model math is one way of working with (or manipulating) economic models. Ap microeconomics: exam study guide circular flow diagram: this is so crucial to understand for both micro and macro in other words, one monopolistic provider can meet market demand at a lower atc than if multiple firms were to split the market among themselves study the graph below and understand how the. The following is an adapted excerpt from my book microeconomics made simple: basic microeconomic principles explained in 100 pages or less “consumer surplus” refers to the value that we can use a chart of supply and demand to show consumer surplus in a market example: the following chart. In this scenario, more corn will be demanded even if the price remains the same, meaning that the curve itself shifts to the right (d2) in the graph below in other words, demand will increase other factors can shift the demand curve as well, such as a change in consumers' preferences if cultural shifts cause the market to. Consumer surplus is a term used to describe the difference between the price of a good and how much the consumer is willing to pay demand curves are graphed with the same axis as supply curves in order to allow the two curves to be combined into a single graph: the y-axis (vertical line) of the graph is price and the. Video created by university of california, irvine for the course the power of microeconomics: economic principles in the real world 2000+ courses from schools like stanford and yale - no application required build career skills in data. The core ideas in microeconomics supply, demand and equilibrium. Reffonomics videos microeconomics terms in 3 minutes or less afc, avc, and atc showing fc, vc, and tc long-run average total cost fc, vc, and tc graph, part ii production costs: utility maximization afc, avc, and atc graph perfect competition graph, part iii perfectly competitive characteristics part i. Principles of microeconomics, v 10 by libby understand how graphs show the relationship between two or more variables and explain how a graph elucidates the nature of the relationship define the the key to understanding graphs is knowing the rules that apply to their construction and interpretation this section. Even if a graph is not required, it may be to your advantage to draw one anyway a correct graph can indicate that you understand what is happening even if you use the wrong economic terminology on the other hand, graphs are not magical tools that ensure high scores they are useful in making arguments, but they don't. A summary of seven important details that cut across ap microeconomics graphs enabling students to easily grasp micro econ graphs and important exam tips. B the graphical analysis that will be used in the class will rely upon the cartesian coordinate system this system is shown in the graph below there exist two variables, x and y, which may both take either positive or negative values any specific pairs of values for x and y can be represented on the graph by a single point. Please read/background info i this resource is not meant to teach you economics rather it is meant to serve as a concise guide for you to review economic knowledge you have already learned (translation: you still need to pay attention in class) ii very few parts of this study guide are bolded so pay special attention to. Learning to think like an economist can be a daunting task for beginners introductory economics courses often begin with a jargon-loaded discussion of opportunity costs and marginal benefits versus marginal costs—in other words, what is the benefit of continuing to read the rest of this post, and what else. Key terms economics resource maintenance production distribution consumption positive questions normative questions intermediate goal final goal wealth in the graph shown above, at point b, society is producing the maximum possible microeconomics is the study of national and international economic trends 12. Reading: creating and interpreting graphs it's important to know the terminology of graphs in order to understand and manipulate them let's begin with a visual representation of the terms (shown in figure 1), and then we can discuss each one in greater detail a standard graph with an x- and y-axis there is a positive. Graphs and microeconomics you will see a remarkable number axes - in economics, a graph is not complete without a label on each of its axes telling you what it measures production possibility is just about the hardest bit of algebra we will encounter all term, and it is not really very hard suppose the supply curve is. The term game here implies the study of any strategic interaction between people applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers & acquisitions pricing, fair division, duopolies, oligopolies, social network formation, agent-based computational economics,. Take a look at this graph to help you understand the when and where shutdown point while we're on the topic, what is the supply curve for each firm looking at the graph you'll note the mc curve the supply curve for each firm is simply its marginal cost (mc) curve above the minimum point on the average variable cost. Microeconomics examines smaller units of the overall economy it is different than macroeconomics, which focuses primarily on the effects of interest rates, employment, output and exchange rates on governments and economies as a whole both microeconomics and macroeconomics examine the effects of actions in terms. Microeconomics macroeconomics deals with aggregate economic quantities, such as national output and national income macroeconomics has its roots in it is the inverse of that coefficient solution to 4: in the demand function, change the value of i to 3,000 from 2,300 and collect constant terms: q p p eb d eb eb =. 2 draw the diagram (0 words) the diagram (and it's titles, etc) do not count in your word count you need to diagram the problem explained in the article and also diagram your solution sometimes both the problem and the solution can be shown on one diagram sometimes not of course don't include a diagram (or any. The normal way of expressing a relative price is in terms of a “basket” of demanded of that good demand is illustrated by the demand curve and the demand schedule the term quantity demanded refers to a point on a demand curve—the quantity we graph the demand schedule as a demand curve with the quantity. 21 the cicular-flow-diagram 22 the production possibilities frontier 23 microeconomics and macroeconomics 24 positive versus normative analysis 25 graphs of a single variable 26 graphs of two economists use the term marginal chanes to describe small incremental adjustments to an exiting plan of action.

Received 31 May 2016; accepted 8 August 2016; published 11 August 2016 The Burgers equation was first presented by Bateman and treated later by J. M. Burgers (1895-1981) then it is widely named as Burgers’ equation . Burgers’ equation is nonlinear partial differential equation of second order which is used in various fields of physical phenomena such as boundary layer behaviour, shock weave formation, turbulence, the weather problem, mass transport, traffic flow and acoustic transmission . In addition, the two dimentional Burgers’ equations have played an important role in many physical applications such as investigating the shallow water waves and modeling of gas dynamics . In order to a great applications for burgers’ equations many researchers have been interested in solving it by various techniques. Analytic solution of one dimensional Burgers’ equation are get by many standard methods such as Backland transformation method, differential transformation method and tanh-coth method , while an analytical solution of two dimensional Burgers’ equations was first presented by Fletcher using the Hopf-Cole transfor- mation . The finite difference, finite element, spectral methods, Adomian decomposition method, the varia- tional iteration method, homotopy perturbation method HPM and Eulerian-Lagrangian method gave an nu- merical solution of Burgers’ equations - . Recently, the OHAM was proposed by Marinca and Herisaun - . OHAM is independent of the existence of a embedding parameter in the problem then overcome the limitations of perturbation technique. However, OHAM is the most generalized form of HPM as it uses a general auxiliary function H(p). This method has been studied by a number of researchers for solving linear and nonlinear partial differential equations - . In - proved OHAM is more efficient to solve Burgers’ equations. In 2006, a new method by Daftardar-Gejji and Jafari for solving nonlinear functional appeared . Convergence of it has been proved in . This method is named later as Daftardar-Jafari method DJM in . J. Ali et al. used DJM in the OHAM for solving non-linear differential equations and they named this method as OHAM with DJ polynomials OHAM-DJ . In 2016, OHAM-DJ has been used to solve linear and nonlinear Klein-Gordon equations . The motive of this paper is to show the efficiency of OHAM-DJ for solving the system of Burger’s equations. We consider the system of Burger’s equations as the following : with the initial conditions: and the boundary conditions: where and is its boundary, and are the velocity components to be determined, and are known functions and R is the Reynolds number. This paper is organized into three sections. In Section 2 methodology of OHAM-DJ is presented. In Section 3 application of this method is solved and absolute error of approximate solutions of proposed method is com- pared with exact solutions. In all cases the proposed method yields better results. 2. Methodology of OHAM-DJ Consider (1.1) and let where are boundary operators. According to the basic idea of OHAM , we can construct the optimal homotopy: where is an embedding parameter while and is initial approximation of Equation (1.1) which satisfies the boundary condition, and are nonzero auxiliary functions for, and. Clearly, When and, it holds that, and, respectively. Therefore, as p change from 0 to 1, the solution and varies from to and to respectively, where the initial approximations and are obtained from (2.3) and (2.4). Now, choosing The auxiliary functions and as the form where, are constants to be determined later. Assume that the solutions of (1.1) has the form: The nonlinear terms are decomposed as where are (DJ) polynomials, . For simplicity these polynomials are expressed as: Substiting, (2.5),(2.6), (2.7) and (2.9) into (2.3), and comparing the coefficients of like powers of p, we get The convergence of (2.6) depend upon the auxiliary constants and, which known convergence control parameters or optimal convergence control parameters , if it is convergent at we have Substituting (2.11) into (1.1) we get the residuals and, these parameters can be optimal identified by various methods . Optimization method is one of theses methods to find out the optimal convergence control parameters by means of the minimum of the squared residuals. 3. Numerical Examples In this section, two numerical examples are used to prove the efficiency and the accuracy of the method which we proposed for the system of Burgers’ equations. 3.1. Example 1 Consider the system of two dimensional of Burgers’ equations with the initial conditions as following with the initial conditions: The exact solutions are Accordance to the methodology of OHAM-DJ, Their solutions are By substituting (3.6) into (3.1) we get the residuals and using the optimization method we have computed that and. Finally, putting the values of and into (3.6), to get the approximate solutions (Tables 1-3, Figure 1 and Figure 2). Figure 1. Approximation solutions by OHAM-DJ of example 1, t = 0.01,. Figure 2. Exact solutions of example 1, t = 0.01,. Table 1. Comparison of OHAM-DJ solutions with exact solutions at mesh point x = 2, y = 1 (example 1). 3.2. Example 2 We consider the following two-dimensional Burgers’ equations On square domain, with the initial condition: for which the exact solution is. Where the and in (1.1) are symmetry in this example, and the initial condition are symmetry also. Table 2. Comparison of OHAM-DJ solutions with exact solutions at mesh point x = 1, y = 2 (example 1). Table 3. Comparison of OHAM-DJ solutions with exact solutions at mesh point x = 1.5, y = 2 (example 1). Their solutions are Substituting (3.11) into (3.7) we get the residuals and using the optimization method we have computed that. Finally, putting the values of into (3.11) to get the approximate solutions (Table 4 and Table 5, Figure 3 and Figure 4). In this work, the OHAM-DJ is applied to obtain numerical solutions of the system of Burgers’ equations. The method is efficient and easy to implement where the first or second order solutions rapidly converges to the exact solutions. Furthermore, OHAM-DJ does not need any discretization in time or in space. Thus the solutions of system of Burgers’ equations are not influenced by computer round off errors. The method can be easily Table 4. Comparison of OHAM-DJ solutions with exact solutions at mesh point x = 1, y = 1, (example 2). Table 5. Comparison of OHAM-DJ solutions with exact solutions at mesh point x = 1, y = 1.5, (example 2). Figure 3. Approximation solutions by OHAM-DJ of example 2, t = 0.01,. Figure 4. Exact solutions of example 2, t = 0.01,. extended to other nonlinear equations. Nutshell, OHAM-DJ is a better numerical method for solving nonlinear equations. This paper was funded by King Abdulaziz City for Science and Technology (KACST) in Saudi Arabia. The authors therefore, thank them for their full collaboration. Srivastava, V.K., Ashutosh and Tamsir, M. (2013) Generating Exact Solution of Three Dimensional Coupled Unsteady Nonlinear Generalized Viscous Burgers’ Equations. International Journal of Mathematical Sciences, 5, 1-13. Fletcher, C.A. (1983) Generating Exact Solutions of the Two-Dimensional Burgers’ Equations. International Journal for Numerical Methods in Fluids, 3, 213-216. Fletcher, C.A.J. (1983) A Comparison of Finite Element and Finite Difference Solutions of the One- and Two-Dimensional Burgers’ Equations. Journal of Computational Physics, 51, 159-188. Kutluay, S., Bahadir, A.R. and Ozdes, A. (1999) Numerical Solution of One-Dimensional Burgers Equation: Explicit and Exact-Explicit Finite Difference Methods. Journal of Computational and Applied Mathematics, 103, 251-261. M.A. Abdou and A.A. Soliman. Variational iteration method for solving burger's and coupled burger's equations. Journal of Computational and Applied Mathematics, 181(2):245-251, 2005. Desai, K.R. and Pradhan, V.H. (2012) Solution of Burger’s Equation and Coupled Burger’s Equations by Homotopy Perturbation Method. International Journal of Engineering Research and Applications (IJERA), 2, 2033-2040. Young, D.L., Fan, C.M., Hu, S.P. and Atluri, S.N. (2008) The Eulerian-Lagrangian Method of Fundamental Solutions for Two-Dimensional Unsteady Burgers’ Equations. Engineering Analysis with Boundary Elements, 32, 395-412. Marinca, V. and Herisanu, N. (2015) The Optimal Homotopy Asymptotic Method: Engineering Applications. Springer International Publishing, Gewerbestrasse. Marinca, V., Herisanu, N., Bota, C. and Marinca, B. (2009) An Optimal Homotopy Asymptotic Method Applied to the Steady Flow of a Fourth-Grade Fluid Past a Porous Plate. Applied Mathematics Letters, 22, 245-251. Marinca, V. and Herisanu, N. (2008) Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer. International Communications in Heat and Mass Transfer, 35, 710-715. Gupta, A.K. and Ray, S.S. (2014) Comparison between Homotopy Perturbation Method and Optimal Homotopy Asymptotic Method for the Soliton Solutions of Boussinesq-Burger Equations. Computers & Fluids, 103, 34-41. Ullah, H., Nawaz, R., Islam, S., Idrees, M. and Fiza, M. (2015) The Optimal Homotopy Asymptotic Method with Application to Modified Kawahara Equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 18, 82-88. Iqbal, S., Idrees, M., Siddiqui, A.M. and Ansari, A.R. (2010) Some Solutions of the Linear and Nonlinear Klein-Gordon Equations Using the Optimal Homotopy Asymptotic Method. Applied Mathematics and Computation, 216, 2898-2909. Nawaz, R., Ullah, H., Islam, S. and Idrees, M. (2013) Application of Optimal Homotopy Asymptotic Method to Burger Equations. Journal of Applied Mathematics, 2013, Article ID: 387478. Daftardar-Gejji, V. and Jafari, H. (2006) An Iterative Method for Solving Nonlinear Functional Equations. Journal of Mathematical Analysis and Applications, 316, 753-763. Bhalekar, S. and Daftardar-Gejji, V. (2011) Convergence of the New Iterative Method. International Journal of Differential Equations, 2011, Article ID: 989065. Ali, J., Shah, S., Islam, S. and Khan, H. (2013) Application of Optimal Homotopy Asymtotic Method with Daftardar-Jaffari Polynomials to Non-Linear Differential Equations. World Applied Sciences Journal, 28, 1456-1462. Shah, Z., Nawaz, R., Shah, S., Shah, S.I.A. and Shah, M. (2016) Use of the Daftardar-Jafari Poly-Nomials in Optimal Homotopy Asymptotic Method for the Solution of Linear and Nonlinear Klein-Gordon Equations. Journal of Applied Environmental and Biological Sciences, 6, 71-81. Yu, X., Zhao, G. and Zhang, R. (2011) The New Numerical Method for Solving the System of Two-Dimensional Burgers’ Equations. Computers & Mathematics with Applications, 62, 3279-3291.

Which Math or Stats Course Should I Take ? The Department of Mathematics and Statistics offers several courses (STA 108, MAT 112, MAT 118, MAT 120, MAT 190, MAT 196, MAT 253) that satisfy the Mathematics (GMT or quantitative reasoning) requirement of the General Education Program. This document and conversations with your advisor will help you decide which one(s) to take. STA 108 Elementary Introduction to Probability and Statistics helps you understand you understand what is going on in today’s data-driven world around you. STA 108 will expose the students to the basic statistical rudiments necessary to be an informed member of society. STA 108 is a course that teaches how to collect, organize, analyze, and make sense out of collected data. Take STA 108 if: - You want to learn how to read numbers/data correctly, make predictions, and draw your own conclusions from it. - You want to learn how the result of a survey, poll, or study makes or doesn’t make methodological sense. - You want to learn how the mean (or average) is not the most trusted measure that we should use in our daily lives. All students planning to take Elementary Statistics should take the Statistics Readiness and Diagnostic Test to determine which path into Statistics is right for them. (Ancient and) Contemporary Topics in Mathematics MAT 112 Contemporary Topics in Mathematics covers basic mathematics starting with discoveries by ancient greeks to present day applications of their ideas. God made the integers, all else is the work of humans, Leopold Kronecker, 1886 Following Kronecker’s premise, we assume knowledge of the integers along with the operations addition (plus), subtraction (minus), and multiplication (times). We give examples of the work of humans that is called mathematics that is build on the integers and present practical applications of this work. The material covered includes applications of mathematics that are relevant for the digital age and to other liberal arts disciplines. The course culminates with the topic of public key cryptography. The presentation is rigorous but basic enough for its intended audience to follow. Take MAT 112 if: - You are looking for a course that only requires minimal prior knowledge, that is, addition, subtraction, and multiplication of integers. - You are not comfortable with algebra. - You want to see several ‘unexpected’ applications of mathematics and learn a little about several deep areas of mathematics. - You want to understand what your computer does, when it establishes a secure connection to webpage. - You want to see that public key cryptography is not magic. To get a better idea of what this course is about check out the interactive MAT 112 notes. Mathematics with Business Applications MAT 118 Algebra with Business Applications is an introductory survey of algebra with emphasis on techniques and applications related to business. It also serves as a one-semester preparation for MAT 120 Calculus with Business Applications. It is not intended for students that plan to take MAT 196 Calculus A. Take MAT 118 if: - You want to learn real-world applications of algebra to various subjects that you might use in your personal life or job. - You are thinking of becoming a business major: MAT 118 is a prerequisite to MAT 120 Business Calculus which is a requirement of most degrees in the business school. MAT 118 teaches you the algebra techniques you need to master in order to succeed in MAT 120 In MAT 118 Students will complete projects in case studies to apply these techniques to real-world situations. Students will formulate decisions based on quantitative arguments and communicate these decisions in laymen terms. MAT 120 Calculus with Business Applications provides students an opportunity to appreciate certain concepts in fundamental mathematics, especially functions, limits, derivatives, and applications of the derivative with emphasis on applications in business and social sciences. MAT 120 is a terminal course and not adequate preparation for MAT 296 Calculus B. Take MAT 120 if: - Your program requires you to take MAT 120 Calculus with Business Applications. - You liked algebra and are interested in a brief overview of differential calculus. - You want to learn how math can be useful for business & economics majors. All students who need to take MAT 120 with sufficient background should take the MAT 120 Placement Test to see whether you need to take MAT 118 before MAT 120 or can directly enroll in MAT 120. MAT 196 Calculus A, MAT 296 Calculus B and MAT 396 Calculus C are the main courses of our calculus sequence. The course MAT 190 Precalculus gets you ready for MAT 196 and there is also a support course MAT 181 Foundations of Calculus which can be taken concurrently with MAT 196. Take Calculus if: - Your program requires you to take Calculus. - You will need Calculus for graduate or professional school. - You want to strengthen your degree with a Mathematics or Statistics Minor. - You like mathematics. All students planning to take Calculus should take the Calculus Readiness and Diagnostic Test to determine which path into Calculus is right for them.

But then, that’s the key to calculus: recognizing that 99.999999 effectively approaches 100. In this case, n = 1 and l = 0. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. The proton has one indivisible positive charge while the electron has one indivisible negative charge. Note that the creators of this cartoon didn’t have the wherewithall to make a ‘right’ atom, giving the nucleus four plus charges and the shell three minus… this would be a positively charged ion of Beryllium. The process of normalization is just to make certain that the value ‘under the curve’ contained by the square of the wave function, counted up across all of space in the integral, is 1. I then divide out the exponential so that I don’t have it cluttering things up. The hydrogen atom problem is a classic problem mainly because it’s one of the last exactly solvable quantum mechanics problems you ever encounter. If the savvy reader so desires, the prescriptions given here can generate any hydrogenic wave function you like… just refer back to my Ylm post where I talk some about the spherical harmonics, or by referring directly to the Ylm tables in wikipedia, which is a good, complete online source of them anyway. Hydrogen atom is simplest atomic system where Schrödinger equation can be solved analytically and compared to experimental measurements. The divergence operation uses Green’s formulas to say that a volume integral of divergence relates to a surface integral of flux wrapping across the surface of that same volume… and then you simply chase the constants. For the power series to be a solution to the given differential equation, each coefficient is related to the one previous by a consistent expression. With the new version of U, the differential equation rearranges to give a refined set of differentials. That’s part of why solving the radial equation is challenging. The modification I made allows me to write U as a portion that’s an unknown function of radius and a second portion that fits as a negative exponent. Why not, I figured; the radial solution is actually a bit more mind boggling to me than the angular parts because it requires some substitutions that are not very intuitive. Operationally, this is just another choice for spherically symmetric potential (i.e. Change ), You are commenting using your Google account. The Bohr radius ao is a relic of the old Bohr atom model that I started off talking about and it’s used as the scale length for the modern version of the atom. A small exercise I sometimes put myself through is defining the structure of del. The simplest case to consider is the hydrogen atom, with one positively charged proton in the nucleus and just one negatively charged electron orbiting around the nucleus. The value produced by divergence is a scalar quantity with no direction which could be said to reflect the ‘poofiness’ of a vector field at any given point in the space where you’re working. Here are the first few generalized Laguerre polynomials: Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). Hard to sweat the small stuff. The ‘8’ wedged in here is crazily counter intuitive at this point, but makes the quantization work in the method I’ve chosen! If you were to consider an infinitesimal volume of these perpendicular dimensions, at this locally cartesian point, it would be a volume that ‘approaches’ cubic. It’s basically just saying “What if my solution is some polynomial expression Ar^2 -Br +C,” where I can include as many ‘r’s as I want. Coulomb). The recurrence relation also gives a second very important outcome: The energy quantum number must be bigger than the angular momentum quantum number. Some of the higher energy, larger angular momentum hydrogenic wave functions start looking somewhat crazy and more beautiful, but I really just had it in mind to show the math which produces them. Calculate the Wave Function of a Hydrogen Atom Using the…, Find the Eigenfunctions of Lz in Spherical Coordinates, Find the Eigenvalues of the Raising and Lowering Angular Momentum…, How Spin Operators Resemble Angular Momentum Operators, If your quantum physics instructor asks you to find the wave function of a hydrogen atom, you can start with the radial Schrödinger equation, Rnl(r), which tells you that. In between the Sakurai problems, I decided to tackle a small problem I set for myself. ‘Quantized’ is a word invoked to mean ‘discrete quantities’ and comes back to that pesky little feature Deepak Chopra always ignores: the first thing we ever knew about quantum mechanics was Planck’s constant –and freaking hell is Planck’s constant small! The Laplace operator combines gradient with divergence as literally the divergence of a gradient, denoted as ‘double del,’ the upside-down triangle squared. Calculate the Wave Function of a Hydrogen Atom Using the Schrödinger Equation. Further, the electrons are not stacked into a decent representation for the actual structure: cyclic orbitals would be P-orbitals or above, where Beryllium has only S-orbitals for its ground state, which possess either no orbital angular momentum, or angular momentum without any defined direction. Click hereto get an answer to your question ️ The radial wave equation for hydrogen atom is Ψ = 116√(pi) (1a0)^3/2 [ ( x - 1 ) ( x^2 - 8x + 12 ) ] e^-x/2 where, x = 2r/a0 ; a0 = radius of first Bohr orbit.The minimum and maximum position of radial nodes from nucleus are: Also, in that last line, there’s an “= R” which fell off the side of the picture –I assure you it’s there, it just didn’t get photographed. I then perform the typical Quantum Mechanics trick of making it a probability distribution by normalizing it. The Hydrogen Atom Lecture 24 Physics 342 Quantum Mechanics I Monday, March 29th, 2010 We now begin our discussion of the Hydrogen atom. All that I do to find the divergence differential expression is to take the full integral and remove the infinite sum so that I’m basically doing algebra on the infinitesmal pieces, then literally divide across by the volume element and cancel the appropriate differentials. A general form for the radial wave equations appears at the lower right, fabricated from the back-substitutions. is a generalized Laguerre polynomial. You have to solve this by separation of variables. Further, you may not realize it yet, but something rather amazing happened with that number Q. Divergence creates a scalar from a vector which represents the intensity of ‘divergence’ at some point in a smooth function defined across all of space. The first image, where the box size was a little small, was perhaps the most striking of what I’ve seen thus far…, I knew basically that I was going to find a donut, but it’s oddly beautiful seen with the outsides peeled off. Since I was just sitting on all the necessary mathematical structures for hydrogen wave function 21-1 (no work needed, it was all in my notebook already), I simply plugged it into mathematica to see what the density plot would produce. This little bit of math is defining the geometry of the coordinate variables in spherical polar coordinates. I keep finding interesting structures here. I spent some significant effort thinking about this point as I worked the radial problem this time; for whatever reason, it has always been hazy in my head which powers of the sum are allowed and how the energy and angular momentum quantum numbers constrained them. Radius would be some complicated combination of x, y and z. So then, this framework allows you to define the calculus occurring in spherical polar space. A scalar function defines the topography of the hill… it says simply that at some pair of coordinates in a plane, the geography has an altitude. Here is how you construct a specific hydrogen atom orbital from all the gobbledigook written above. After all those turns and twists, this is a solution to the radial differential equation, but not in closed form. It isn’t exactly crippling to the field because the solutions to all the other atoms are basically variations of the hydrogen atom and all, with some adjustment, have hydrogenic geometry or are superpositions of hydrogen-like functions that are only modified to the extent necessary to make the energy levels match. There are three possible area integrals because the normal vector is in three possible directions, one each for Rho, Theta and Phi. After the hydrogen atom, the water gets deeper and the field starts to focus on tools that give insight without actually giving exact answers. Psi basically just becomes R. The first thing to do is take out the units. 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NCERT Exemplar Solutions Class 8 Science Chapter 11 – Free PDF Download The NCERT Exemplar for Class 8 Science Chapter 11 Force and Pressure explains the topics mentioned in CBSE Class 8 Chapter 11. By studying this Exemplar thoroughly, students are able to strengthen their exam preparation. This will help them determine their strengths and weaknesses. NCERT Exemplar Problems provided here are not meant to provide you only with a question bank for examinations, but are primarily meant to enhance your learning process. Practising Exemplar will help you solve different kinds of questions and numericals, which are important from the examination point of view. Force and Pressure is one of the most important chapters of CBSE Class 8 Science. In this chapter, students will learn about the basic concepts of force and pressure. To understand the concepts of force and pressure and to score good marks in the exam, students must understand the basics of the chapter and solve the important questions. To help students, we have provided the NCERT Exemplar for Class 8 Science Chapter 11 Force and Pressure. Download the PDF of the NCERT Exemplar Class 8 Science Solutions for Chapter 11 – Force and Pressure Access Answers to the NCERT Exemplar Class 8 Science Chapter 11 – Force and Pressure Multiple Choice Questions 1. In Fig 11.1, two boys A and B are shown applying force on a block. If the block moves towards the right, which one of the following statements is correct? (a) The magnitude of force applied by A is greater than that of B. (b) The magnitude of force applied by A is smaller than that of B. (c) The net force on the block is towards A. (d) The magnitude of force applied by A is equal to that of B. Answer is (a) Magnitude of force applied by A is greater than that of B. The magnitude of force applied by A is bigger than that of B as a result of the block moves towards right i.e. towards B. 2. In the circuit shown in Fig.11.2, when the circuit is completed, the hammer strikes the gong. Which of the following force is responsible for the movement of the hammer? (a) gravitational force alone (b) electrostatic force alone (c) magnetic force alone (d) frictional force alone Answer is (c) magnetic force alone As electric current flows through the coil it behaves like an electromagnet which creates magnetic force. Hence the answer is magnetic force alone. 3. During dry weather, while combing hair, sometimes we experience hair flying apart. The force responsible for this is (a) force of gravity. (b) electrostatic force. (c) the force of friction. (d) magnetic force. The answer is (b) electrostatic force. On combing the hair, comb and hair get oppositely charged due to electrostatic force. 4. Fig.11.3 shows a container filled with water. Which of the following statements is correct about the pressure of water? (a) The pressure at A > Pressure at B > Pressure at C. (b) The pressure at A = Pressure at B = Pressure at C. (c) The pressure at A < Pressure at B > Pressure at C. (d) The pressure at A < Pressure at B. The answer is (d) Pressure at A < Pressure at B Increase in water leads to an increase in depth. 5. Two objects repel each other. This repulsion could be due to (a) frictional force only (b) electrostatic force only (c) magnetic force only (d) either a magnetic or an electrostatic force The answer is (d) either a magnetic or an electrostatic force Explanation: when two objects are experiencing repulsive force because there may be an electrostatic force or a magnetic force. 6. Which one of the following forces is a contact force? (a) force of gravity (b) force of friction (c) magnetic force (d) electrostatic force The answer is (b) force of friction Force of attraction acts only when the bodies are in contact. 7. A water tank has four taps fixed at points A, B, C, D as shown in Fig. 11.4. The water will flow out at the same pressure from taps at (a) B and C (b) A and B (c) C and D (d) A and C Answer is (a) B and C B and C are at the same level, hence pressure will be the same at B and C. 8. A brick is kept in three different ways on a table as shown in Fig. 11.5. The pressure exerted by the brick on the table will be (a) maximum in position A-C (b) maximum in position B (c) maximum in position (d) equal in all cases. Answer is (a) maximum in position A-C Since the area of contact is minimum pressure will be maximum in A. Very Short Answer Questions 9. A ball of dough is rolled into a flat chapatti. Name the force exerted to change the shape of the dough. 10. Where do we apply a force while walking? While walking we apply force on the ground. 11. A girl is pushing a box towards the east direction. In which direction should her friend push the box so that it moves faster in the same direction? Towards the east. 12., In the circuit shown in Fig.11.6, when the key is closed, the compass needle placed in the matchbox deflects. Name the force which causes this deflection. Answer is Magnetic force. 13. During dry weather, clothes made of synthetic fibre often stick to the skin. Which type of force is responsible for this phenomenon? Answer is Electrostatic force 14. While sieving grains, small pieces fall down. Which force pulls them down? Force of gravity. 15. Does the force of gravity act on dust particles? Yes, the force of gravity act on dust particles. 16. A gas-filled balloon moves up. Is the upward force acting on it larger or smaller than the force of gravity? The upward force is larger than the force of gravity. 17. Does the force of gravitation exist between two astronauts in space? Yes, the force of gravitation exists between two astronauts in space. Short Answer Questions 18. A chapati maker is a machine which converts balls of dough into chapati. What effect of force comes into play in this process? Force works on the dough to convert it to chapati. 19. Fig.11.7 shows a man with a parachute. Name the force which is responsible for his downward motion. Will he come down with the same speed without the parachute? Force of gravity is responsible for his downward motion. If he comes down without parachute his speed will be higher. 20. Two persons are applying forces on two opposite sides of a moving cart. The cart still moves with the same speed in the same direction. What do you infer about the magnitudes and direction of the forces applied? Force applied is of equal magnitude in the opposite direction hence the cart moves with the same speed in the same direction. 21. Two thermocol balls held close to each other move away from each other. When they are released, name the force which might be responsible for this phenomenon. Explain. Two Thermocol balls held close to each other move away from each other, which is because of electrostatic force. The balls having similar charges move away due to repulsion between similar charges. 22. Fruits detached from a tree fall down due to force of gravity. We know that a force arises due to the interaction between two objects. Name the objects interacting in this case. Earth and fruits. 23. A man is pushing a cart down a slope. Suddenly the cart starts moving faster and he wants to slow it down. What should he do? He should apply a force to pull the cart up the slope. 24. Fig. 11.8 shows a car sticking to an electromagnet. Name the forces acting on the car? Which one of them is larger? Magnetic force (in the upward direction) force of gravity or the weight of the car (downward) act on car. Magnetic force is larger than the force of gravity. Long Answer Questions 25. An archer shoots an arrow in the air horizontally. However, after moving some distance, the arrow falls to the ground. Name the initial force that sets the arrow in motion. Explain why the arrow ultimately falls down. Archer puts muscular force to stretch the string. This will change the shape of the arrow. When the string is released arrow regains its original position which gives it the initial force to set the motion. Because of gravitational from it comes down towards after some time. 26. It is difficult to cut cloth using a pair of scissors with blunt blades. Explain. The blunt blade has a larger area than shard edged blades. Because of this, blunt blade produces a low pressure which makes it difficult to cut the cloth. Whereas in sharp blade surface area is much is very less which increase the pressure produced. This makes the cutting of cloth easier with sharp blades. 27. Two rods of the same weight and equal length have different thickness. They are held vertically on the surface of sand as shown in Fig.11.9. Which one of them will sink move? Why? In Rod B area of contact is smaller. Hence the weight of the rod (Force) produces more pressure. In Case of the rod, the same force produces less pressure. 28. Two women are of the same weight. One wears sandals with pointed heels while the other wears sandals with flat soles. Which one would feel more comfortable while walking on a sandy beach? Give reasons for your answer. Women’s height are same and they apply the same weight when they walk. But women wearing sandal with a flat heel will be more comfortable while walking on a sandy beach. This is because flat soles have larger area compared to the sandals with pointed heels. Also, the pressure exerted by the pointed heels will be more compared to that with sandals having flat soles. This pressure will make the sandals with pointed soled sink in the sand which will make difficult to walk on sand. 29. It is much easier to burst an inflated balloon with a needle than by a finger. Explain. The pressure exerted on an inflated balloon by the needle will be more as it has a smaller area of contact compared to the finger. This larger pressure pierces the surface of the balloon easily which will make the balloon burst. 30. Observe the vessels A, B, C and D shown in Fig.11.10 carefully. The volume of water taken in each vessel is as shown. Arrange them in the order of decreasing pressure at the base of each vessel. Explain. B, D, A, C. Because the pressure of a liquid column depends upon the height of the liquid column and not on volume of the liquid. Sub-topics of the CBSE syllabus Class 8 Science Chapter 11 Force and Pressure - Force – A Push or Pull - Forces Are Due to an Interaction - Exploring Forces - A Force Can Change the State of Motion - Force Can Change the Shape of an Object - Contact Forces - Non-contact Forces - Pressure Exerted by Liquids and Gases - Atmospheric Pressure BYJU’S presents outstanding NCERT Solutions, study materials, sample papers, previous years’ question papers and video and animation lessons for a thorough understanding and to memorise topics for a longer period. To get access to all study materials we provide, log on to BYJU’S website or download BYJU’S – The Learning App. |NCERT Solutions for Class 8 Science Chapter 11| |CBSE Notes for Class 8 Science Chapter 11| Frequently Asked Questions on NCERT Exemplar Solutions for Class 8 Science Chapter 11 What are the topics covered under Chapter 11 of NCERT Exemplar Solutions for Class 8 Science? 1. Force – A Push or Pull 2. Forces Are Due to an Interaction 3. Exploring Forces 4. A Force Can Change the State of Motion 5. Force Can Change the Shape of an Object 6. Contact Forces 7. Non-contact Forces 9. Pressure Exerted by Liquids and Gases 10. Atmospheric Pressure What are non-contact forces in Chapter 11 of NCERT Exemplar Solutions for Class 8 Science? Can students rely on NCERT Exemplar Solutions for Class 8 Science Chapter 11 from BYJU’S? |NCERT Exemplar Class 8 Science Chapter 12 Friction| |NCERT Exemplar Class 8 Science Chapter 13 Sound| |NCERT Exemplar Class 8 Science Chapter 14 Chemicals Effects of Electric Current| |NCERT Exemplar Class 8 Science Chapter 15 Some Natural Phenomena|

BiographySijue Wu's school and undergraduate education were in China. She studied at Beijing University, being awarded her first degree in 1983 and a Master's Degree in 1986. Even before the award of the Master's Degree, she had a paper published, namely Hilbert transforms for convex curves in Rn. She then went to the United States to undertake research. Her doctoral studies were undertaken at Yale University with Ronald Raphael Coifman as her thesis advisor. She submitted her thesis, Nonlinear Singular Integrals and Analytic Dependence, in 1990 and was awarded a Ph.D. She begins her introduction to her thesis as follows:- This thesis is composed of three interrelated parts: w-Calderón-Zygmund operators, a wavelet characterization for weighted Hardy spaces, and the analytic dependence of minimal surfaces on their boundaries.After the award of her doctorate, Wu was appointed as Courant Instructor at the Courant Institute, New York University. She was a member at the Institute for Advanced Study at Princeton in the autumn of 1992 and was then she was appointed Assistant Professor at Northwestern University, holding this position for four years until 1996. Her publications during this period included: A wavelet characterization for weighted Hardy spaces (1992); (with Italo Vecchi) On L1 -vorticity for 2-D incompressible flow (1993); Analytic dependence of Riemann mappings for bounded domains and minimal surfaces (1993) and w-Calderón-Zygmund operators (1995). After spending the year 1996-97 as a member of the Institute for Advanced Study at Princeton, she was appointed as Assistant Professor at the University of Iowa. In 1997 she published the important paper Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Shu Ming Sun begins a very informative review as follows:- Everyone is familiar with the motion of water waves in everyday experience, and there has been an extremely rich variety of phenomena observed in the motion of such waves. However, the full equations governing the motion of the waves are notoriously difficult to work with because of the free boundary and the inherent nonlinearity, which are non-standard and non-local. Although many approximate treatments, such as linear theory and shallow-water theory as well as numerical computations, have been used to explain many important phenomena, it is certainly of importance to study the solutions of the equations which include the effects neglected by approximate models. The well-posedness of the fully nonlinear problem is one of the main mathematical problems in fluid dynamics. Here, the motion of two-dimensional irrotational, incompressible, inviscid water waves under the influence of gravity is considered.Promoted to Associate Professor at Iowa in 1998, Wu was appointed as an Associate Professor at the University of Maryland, College Park, in 1998. The university announced her appointment as follows:- Sijue Wu comes to us from the University of Iowa. Her research interests centre on harmonic analysis and partial differential equations, in particular nonlinear equations from fluid mechanics. Her recent work concerns the full nonlinear water wave problem and the motion of general two-fluid flows.At the 107th Annual Meeting of the American Mathematical Society in January 2001 in New Orleans, Wu was awarded the 2001 Satter Prize. The citation reads :- The Ruth Lyttle Satter Prize in Mathematics is awarded to Sijue Wu for her work on a long-standing problem in the water wave equation, in particular for the results in her papers (1) "Well-posedness in Sovolev spaces of the full water wave problem in 2-D" (1997); and (2) "Well-posedness in Sobolev spaces of the full water wave problem in 3-D" (1999). By applying tools from harmonic analysis (singular integrals and Clifford algebra), she proves that the Taylor sign condition always holds and that there exists a unique solution to the water wave equations for a finite time interval when the initial wave profile is a Jordan surface.Of the paper (2) Emmanuel Grenier writes:- In this very important paper the author investigates the motion of the interface of a 3D inviscid, incompressible, irrotational water wave, with an air region above a water region and surface tension zero.In her response Wu thanked her teachers, friends, and colleagues, making special mention of her thesis advisor Ronald Coifman for the constant support he had given her and Lihe Wang for his friendship and his help. Also in 2001 Wu received a Silver Morningside Medal at the International Congress of Chinese Mathematicians held in Taiwan in December:- ... for her establishment of local well-posedness of the water wave problems in a Sobolev class in arbitrary space dimensions.In August 2002 Wu was an invited speaker at the International Congress of Mathematicians held in Beijing where she delivered the lecture Recent progress in mathematical analysis of vortex sheets. She gave the following summary of her lecture:- We consider the motion of the interface separating two domains of the same fluid that move with different velocities along the tangential direction of the interface. We assume that the fluids occupying the two domains are of constant densities that are equal, are inviscid, incompressible and irrotational, and that the surface tension is zero. We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of the solutions after the singularity formation time.Wu was awarded a Radcliffe Institute Advanced Study Fellowship for the academic year 2002-2003. Her project Mathematical Analysis of Vortex Dynamics was described in an announcement of the award :- Recently, Wu's research has focused on nonlinear equations from fluid dynamics. Using harmonic analysis technique, she has established the local well-posedness of the full two- and three-dimensional waterwave problem. This settled a longstanding problem. As a Radcliffe fellow, Wu will continue her study of vortex sheet dynamics, a phenomenon that arises from the mixing of fluids, such as occurs during aircraft takeoffs. A vortex sheet is the interface separating two domains of the same fluid across which the tangential component of the velocity field is discontinuous. Achieving a better understanding of the motion of a vortex sheet requires proper mathematical modelling; Wu's long-term goal is to establish a successful model. She will also work on the boundary layer problem, another problem arising from fluid dynamics.One outcome of this project, and of an NFS grant she was awarded for 2004-2009, was the paper Mathematical analysis of vortex sheets (2006). Helena Nussenzveig Lopes begins a review of this paper by explaining what vortex sheets are:- Vortex sheets are an idealized model of flows undergoing intense shear. In planar flows they are mathematically described as curves along which the velocity is tangentially discontinuous. Vortex sheets arise in a wide range of physical problems, and hence it is of fundamental importance to understand their evolution. The Birkhoff-Rott equations provide a mathematical description of the evolution of a vortex sheet. However, they have been shown to be ill-posed in several function spaces. It is a longstanding open problem to determine a function space in which these equations are well-posed, or, alternatively, to describe the evolution past singularity formation; this is the problem addressed in the present paper.Wu was named Robert W and Lynne H Browne Professor of Mathematics at the University of Michigan and delivered her inaugural lecture Mathematical Analysis of Water Waves on 29 October 2008. The Browne Professorship recognizes the Wu's outstanding contributions to science and teaching. Finally, let us mention her recent important paper Almost global wellposedness of the 2-D full water wave problem (2009). - 2001 Satter Prize, Notices Amer. Math. Soc. 48 (4) (2001), 411-412. - 2002-2003 Radcliffe Institute Fellows, Sijue Wu, Mathematics, Mathematical Analysis of Vortex Dynamics. http://www.radcliffe.edu/fellowships/fellows_2003swu.aspx Additional Resources (show) Other websites about Sijue Wu: Honours awarded to Sijue Wu Written by J J O'Connor and E F Robertson Last Update February 2010 Last Update February 2010

Virtual corrections to the inclusive decay 111Work supported in part by Schweizerischer Nationalfonds and the Department of Energy, contract DE-AC03-76SF00515 Stanford Linear Accelerator Center Stanford University, Stanford, California 94309, USA Tobias Hurth 222 address after March 1996: ITP, SUNY at Stony Brook, Stony Brook NY 11794-3840, USA and Daniel Wyler Institute for Theoretical Physics, University of Zürich Winterthurerstr. 190, CH-8057 Zürich, Switzerland We present in detail the calculation of the virtual corrections to the matrix element for . Besides the one-loop virtual corrections of the electromagnetic and color dipole operators and , we include the important two-loop contribution of the four-Fermi operator . By applying the Mellin-Barnes representation to certain internal propagators, the result of the two-loop diagrams is obtained analytically as an expansion in . These results are then combined with existing Bremsstrahlung corrections in order to obtain the inclusive rate for . The new contributions drastically reduce the large renormalization scale dependence of the leading logarithmic result. Thus a very precise Standard Model prediction for this inclusive process will become possible once also the corrections to the Wilson coefficients are available. Submitted to Physical Review D In the Standard model (SM), flavor-changing neutral currents only arise at the one-loop level. This is why the corresponding rare B meson decays are particularly sensitive to “new physics”. However, even within the Standard model framework, one can use them to constrain the Cabibbo-Kobayashi-Maskawa matrix elements which involve the top-quark. For both these reasons, precise experimental and theoretical work on these decays is required. In 1993, was the first rare B decay mode measured by the CLEO collaboration . Recently, also the first measurement of the inclusive photon energy spectrum and the branching ratio in the decay was reported . In contrast to the exclusive channels, the inclusive mode allows a less model-dependent comparison with theory, because no specific bound state model is needed for the final state. This opens the road to a rigorous comparison with theory. The data agrees with the SM-based theoretical computations presented in [3, 4, 5], given that there are large uncertainties in both the experimental and the theoretical results. In particular, the measured branching ratio overlaps with the SM-based estimates in [3, 4] and in [6, 7]. In view of the expected increase in the experimental precision, the calculations must be refined correspondingly in order to allow quantitative statements about new physics or standard model parameters. So far, only the leading logarithmic corrections have been worked out systematically. In this paper we evaluate an important class of next order corrections, which we will describe in detail below 333Some of the diagrams were calculated by Soares . We start within the usual framework of an effective theory with five quarks, obtained by integrating out the heavier degrees of freedom which in the standard model are the top quark and the -boson. The effective Hamiltonian includes a complete set of dimension-6 operators relevant for the process (and ) with being the Fermi coupling constant and being the Wilson coefficients evaluated at the scale , and with being the CKM matrix elements. The operators are as follows: In the dipole type operators and , and ( and ) denote the electromagnetic (strong) coupling constant and field strength tensor, respectively. and stand for the left and right-handed projection operators. It should be stressed in this context that the explicit mass factors in and are the running quark masses. QCD corrections to the decay rate for bring in large logarithms of the form , where or and (with ). One can systematically resum these large terms by renormalization group techniques. Usually, one matches the full standard model theory with the effective theory at the scale . At this scale, the large logarithms generated by matrix elements in the effective theory are the same ones as in the full theory. Consequently, the Wilson coefficients only contain small QCD corrections. Using the renormalization group equation, the Wilson coefficients are then calculated at the scale , the relevant scale for a meson decay. At this scale the large logarithms are contained in the Wilson coefficients while the matrix elements of the operators are free of them. As noted, so far the decay rate for has been systematically calculated only to leading logarithmic accuracy i.e., . To this precision it is consistent to perform the ’matching’ of the effective and full theory without taking into account QCD-corrections and to calculate the anomalous dimension matrix to order . The corresponding leading logarithmic Wilson coefficients are given explicitly in [6, 12]. Their numerical values in the naive dimensional scheme (NDR) are listed in table 1 for different values of the renormalization scale . The leading logarithmic contribution to the decay matrix element is then obtained by calculating the tree-level matrix element of the operator and the one-loop matrix elements of the four-Fermi operators (). In the NDR scheme the latter can be absorbed into a redefinition of 444For the analogous transition, the effects of the four-Fermi operators can be absorbed by the shift . In the ‘t Hooft-Veltman scheme (HV) , the contribution of the four-Fermi operators vanishes. The Wilson coefficients and in the HV scheme are identical to and in the NDR scheme. Consequently, the complete leading logarithmic result for the decay amplitude is indeed scheme independent. Since the first order calculations have large scale uncertainties, it is important to take into account the next-to-leading order corrections. They are most prominent in the photon energy spectrum. While it is a -function (which is smeared out by the Fermi motion of the -quark inside the meson) in the leading order, Bremsstrahlung corrections, i.e. the process , broaden the shape of the spectrum substantially. Therefore, these important corrections have been taken into account for the contributions of the operators and some time ago and recently also of the full operator basis [4, 14, 15]. As expected, the contributions of and are by far the most important ones, especially in the experimentally accessible part of the spectrum. Also those (next-to-leading) corrections, which are necessary to cancel the infrared (and collinear) singularities of the Bremsstrahlung diagrams were included. These are the virtual gluon corrections to the contribution of the operator for and the virtual photon corrections to for . A complete next-to-leading calculation implies two classes of improvements: First, the Wilson coefficients to next-leading order at the scale are required. To this end the matching with the full theory (at ) must be done at the level and the renormalization group equation has to be solved using the anomalous dimension matrix calculated up to order . Second, the virtual corrections for the matrix element (at scale ) must be evaluated and combined with the Bremsstrahlung corrections. The higher order matching has been calculated in ref. and work on the Wilson coefficients is in progress. In this paper we will evaluate all the virtual correction beyond those evaluated already in connection with the Bremsstrahlung process. We expect them to reduce substantially the strong scale dependence of the leading order calculation. Among the four-Fermi operators only contributes sizeably and we calculate only its virtual corrections to the matrix element for . The matrix element vanishes because of color, and the penguin induced four-Fermi operators can be neglected 555 This omission will be a source of a slight scheme and scale dependence of the next-to-leading order result. because their Wilson coefficients 666It is consistent to calculate the corrections using the leading logarithmic Wilson coefficients. are much smaller than , as illustrated in table 1. However, we do take into account the virtual corrections to associated with the magnetic operators (which has already been calculated in the literature) and (which is new). Since the corrections to and are one-loop diagrams, they are relatively easy to work out. In contrast, the corrections to , involve two-loop diagrams, since this operator itself only contributes at the one-loop level. Since the virtual and Bremsstrahlung corrections to the matrix elements are only one (well-defined) part of the whole next-to-leading program, we expect that this contribution alone will depend on the renormalization scheme used. Even within the modified minimal subtraction scheme used here, we expect that two different “prescriptions” how to treat , will lead to different answers. Since previous calculations of the Bremsstrahlung diagrams have been done in the NDR scheme and also the leading logarithmic Wilson coefficients are available in this scheme, we also use it here. For future checks, however, we also consider in Appendix A the corresponding calculation in the HV scheme. The remainder of this paper is organized as follows. In section 2 we give the two-loop corrections for based on the operator together with the counterterm contributions. In section 3 the virtual corrections for based on are reviewed including some of the Bremsstrahlung corrections. Then, in section 4 we calculate the one-loop corrections to associated with . Section 5 contains the results for the branching ratio for and especially the drastic reduction of the renormalization scale dependence due to the new contributions. Appendix A contains the result of the two-loop calculation in the HV scheme and, finally, to make the paper self-contained, we include in Appendix B the Bremsstrahlung corrections to the operators , and . 2 Virtual corrections to in the NDR scheme In this section we present the calculation of the matrix element of the operator for up to order in the NDR scheme. The one-loop matrix element vanishes and we must consider several two-loop contributions. Since they involve ultraviolet singularities also counterterm contributions are needed. These are easy to obtain, because the operator renormalization constants are known with enough accuracy from the order anomalous dimension matrix . Explicitly, we need the contributions of the operators to the matrix element for , where denote the order contribution of the operator renormalization constants. In the NDR scheme, the non-vanishing counterterms come from the one-loop matrix element of and as well as from the tree level matrix element of the operator . We also note that there are no contributions to from counterterms proportional to evanescent operators multiplying the Wilson coefficient . 2.1 Regularized two-loop contribution of The dimensionally regularized matrix element of the operator for can be divided into 4 classes of non-vanishing two-loop diagrams, as shown in Figs. 1–4. The sum of the diagrams in each class (=figure) is gauge invariant. The contributions to the matrix element of the individual classes 1–4 are denoted by and , where e.g. is The main steps of the calculation are the following: We first calculate the Fermion loops in the individual diagrams, i.e., the ’building blocks’ shown in Fig. 5 and in Fig. 6, combining together the two diagrams in Fig. 6. As usual, we work in dimensions; the results are presented as integrals over Feynman parameters after integrating over the (shifted) loop-momentum. Then we insert these building blocks into the full two-loop diagrams. Using the Feynman parametrization again, we calculate the integral over the second loop-momentum. As the remaining Feynman parameter integrals contain rather complicated denominators, we do not evaluate them directly. At this level we also do not expand in the regulator . The heart of our procedure which will be explained more explicitly below, is to represent these denominators as complex Mellin-Barnes integrals . After inserting this representation and interchanging the order of integration, the Feynman parameter integrals are reduced to well-known Euler Beta-functions. Finally, the residue theorem allows to write the result of the remaining complex integal as the sum over the residues taken at the pole positions of certain Beta- and Gamma-functions; this naturally leads to an expansion in the ratio , which numerically is about . We express the diagram in Fig. 5 (denoted by ) in a way convenient for inserting into the two-loop diagrams. As we will use subtraction later on, we introduce the renormalization scale in the form , where is the Euler constant. Then, corresponds to subtracting the poles in . In the NDR scheme, is given by 777The fermion/gluon and the fermion/photon couplings are defined according to the covariant derivative where is the four-momentum of the (off-shell) gluon, is the mass of the charm quark propagating in the loop and the term is the ”-prescription”. The free index will be contracted with the gluon propagator when inserting the building block into the two-loop diagrams in Figs. 1 and 2. Note that is gauge invariant in the sense that . where is the four-momentum of the photon. The index in eq. (2.4) is understood to be contracted with the polarization vector of the photon, while the index is contracted with the gluon propagator in the two-loop diagrams in Figs. 3 and 4. The matrix in eq. (2.4) is defined as In a four-dimensional context these quantities can be reduced to expressions involving the Levi-Cività tensor, i.e., (in the Bjorken-Drell convention). The dimensionally regularized expressions for the read where and are given by The range of integration in is restricted to the simplex , i.e., and . Due to Ward identities, not all the are independent. The identities given in ref. in the context of the full theory simplify in our case as follows: They allow to express and in terms of the other which have a more compact form. These relations read Of course, eq. (2.1) can be checked explicitly for all values of , using partial integration and certain symmetry properties of the integrand. We are now ready to evaluate the two-loop diagrams. As both and are transverse with respect to the gluon, the gauge of the gluon propagator is irrelevant. Also, due to the absence of extra singularities in the limit of vanishing strange quark mass, we set from the very beginning (the question of charm quark mass “singularities” will be discussed later). As an example, we present the calculation of the two-loop diagram in Fig. 1c in some detail. Using in eq. (2.3), the matrix element reads In eq. (2.13), and are the Dirac spinors for the and the quarks, respectively, and . In the next step, the four propagator factors in the denominator are Feynman parametrized as where , , and . Then the integral over the loop momentum is performed. Making use of the function in eq. (2.14), the integral over is easy. The remaining variables , and are transformed into new variables , and , all of them varying in the interval [0,1]. The substitution reads Taking into account the corresponding jacobian and omitting the primes() of the integration variables this leads to where , and are matrices in Dirac space depending on the Feynman parameters , , , in a polynomial way. is given by In what follows, the ultraviolet regulator remains a fixed, small positive number. where and denotes the integration path which is parallel to the imaginary axis (in the complex -plane) hitting the real axis somewhere between and . In this formula, the ”momentum squared” is understood to have a small positive imaginary part. In ref. [19, 21] exact solutions to Feynman integrals containing massive propagators are obtained by representing their denominators according to the formula (2.18) with subsequent calculation of the corresponding massless integrals. In our approach, we use formula (2.18) in order to simplify the remaining Feynman parameter integrals in eq. (2.16). We represent the factors and in eq. (2.16) as Mellin-Barnes integrals using the identifications By interchanging the order of integration, we first carry out the integrals over the Feynman parameters for any given fixed value of on the integration path . These integrals are basically the same as for the massless case (in eqs. (2.16) and (2.17)) up to a factor (in the integrand) of Note that the polynomials , and have such a form that the Feynman parameter integrals exist in the limit . If the integration path is chosen close enough to the imaginary axis, the factor in eq. (2.20) does not change the convergence properties of the integrals, i.e., the Feynman parameter integrals exist for all values of lying on . It is easy to see that the only integrals involved are of the type For the integration we use the residue theorem after closing the integration path in the right -halfplane. One has to show that the integral over the half-circle vanishes if its radius goes to . As we explicitly checked, this is indeed the case for , which is certainly satisfied in our application. The poles which lie inside the integration contour are located at The other two-loop diagrams are evaluated similarly. The non-trivial Feynman integrals can always be reduced to those given in eq. (2.21) after some suitable substitutions. The only change is that there are poles in addition to those given in eq. (2.1) in those diagrams where the gluon hits the -quark line; they are located at The sum over the residues naturally leads to an expansion in through the factor in eq. (2.20). This expansion, however, is not a Taylor series because it also involves logarithms of , which are generated by the expansion in . A generic diagram which we denote by has then the form where the coefficients and are independent of . The power in eq. (2.24) is in general a natural multiple of and is a natural number including 0. In the explicit calculation, the lowest turns out to be . This implies the important fact that the limit exists; thus, there cannot be large logarithms (from a small up-quark mass) in these diagrams. From the structure of the poles one can see that the power of the logarithm is bounded by independent of the value of . To illustrate this, take as an example. There are 3 poles located near n=100, viz., at , respectively (see eq. (2.1)). Taking the residue at one of them, yields a term proportional to coming from the remaining two poles. In addition there can be an explicit term from the integration of the two loop momenta. Therefore the most singular term can be . Multiplying this with in eq. (2.20) leads to where can be 4 at most. We have retained all terms up to . Comparing the numerical result with the one obtained by truncating at leads to a difference of about only. We have made further checks of our procedure. For example we have calculated diagram 1b directly. Expanding the result, we reproduce the expressions obtained by applying the Mellin-Barnes integral at the Feynman parameter level as described above. A similar exercise for the imaginary part of diagram 1c shows that the exact and the expanded result (up to terms) in these examples agree at the level. In addition, we checked that the imaginary part of the sum of all diagrams coincides numerically with the results of Soares [23, 8] (note, however that in the physical region only the diagrams in Fig. 1 and Fig. 3 contribute to the imaginary part). In ref. , Soares applied dispersion techniques to calculate the real part. However, using the imaginary part in the physical region, only the real part of the diagrams in Fig. 1 and Fig. 3 is obtained. We have checked that our numbers for these two sets of diagrams indeed coincide with the results of Soares. However, the contribution of the diagrams in Fig. 2 and Fig. 4 is missing in because the additional unphysical cuts were not taken into account. We also note that a separate consideration of the subtraction terms would be required to obtain the correct dependence. We mention that the Dirac algebra has been done with the algebraic program REDUCE 888Some checks have been done with TRACER . The Feynman parameter integrals and the determination of the residues have been done with the symbolic program MAPLE . We now give the results for the diagrams shown in Figs. 1 to 4. As mentioned already, the individual diagrams in each figure are not gauge invariant but only their sum. Note, that the leading ultraviolet singularity in the individual two-loop diagrams is in general of order . In the gauge invariant sums the cancel and we are left with -poles only. The results read (using and ): In these expressions, the symbol denotes the Riemann Zeta function, with ; and are the charge factors for up- and down-type quarks, respectively. The matrix element is the tree level matrix element of the operator ; its explicit form is In formula (2.29) should be identified with the running mass in principle (see eq. (1)). However, as the corrections to are explicitly proportional to , can be identified with the pole mass as well (apart from corrections which we systematically neglect). The operators mix under renormalization and thus the counterterm contributions must be taken into account. As we are interested in this section in contributions to which are proportional to , we have to include, in addition to the two-loop matrix elements of , also the one-loop matrix elements of the four Fermi operators () and the tree level contribution of the magnetic operator . In the NDR scheme the only non-vanishing contributions to come from . (For the contribution comes from the diagram in which the internal -quark emits the photon). The operator renormalization constants are listed in the literature in the context of the leading order anomalous dimension matrix. The entries needed in our calculation are we find the following contributions to the matrix elements We note that there is no one-loop contribution to the matrix element for from the counterterm proportional to , where the evanescent operator (see e.g. the last ref. in ) reads 2.3 Renormalized contribution proportional to Of course, the ultraviolet singularities cancel in . Inserting , and , we get the main result of this paper, which in the NDR scheme reads: Here, and denote the real and the imaginary part of , respectively. The quantity is defined as and . In Fig. 7 we show the real and the imaginary part of . For the imaginary part must vanish exactly; indeed we see from Fig. 7 that the imaginary part based on the expansion retaining terms up to indeed vanishes at to high accuracy. 3 corrections to The virtual corrections associated with the operator as shown in Fig. 8b (together with the selfenergy diagrams and the counterterms) have been taken into account in the work of Ali and Greub, see e.g. [3, 4, 14], where was retained. Since we neglect in this work, we are interested only in the limit . Because of the mass singularities in the virtual corrections (which will be cancelled when also taking into account Bremsstrahlung corrections), we only keep as a regulator. Including the lowest order contribution, the result then becomes (using ) in the NDR scheme Note, that eq. (3.1) contains all the counterterm contributions. The -poles in this equation are therefore of infrared origin as indicated by the notation. The last term in the curly bracket in eq. (3.2) represents a dependence of ultraviolet origin. The additional dependence, which is generated when expanding in is cancelled at the level of the decay width together with the -poles when adding the Bremsstrahlung correction due to the square of the diagrams associated with the operator . As all intermediate formulae are given in the literature, we only give the final result for the corrections (virtual+Bremsstrahlung) to the decay width. Denoting this contribution by we get in the limit where the lowest order contribution reads

Calculating Margin Of Error Formula MSNBC, October 2, 2004. A larger sample size produces a smaller margin of error, all else remaining equal. Copyright © 2016 Statistics How To Theme by: Theme Horse Powered by: WordPress Back to Top Stat Trek Teach yourself statistics Skip to main content Home Tutorials AP Statistics Stat Tables The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample: Margin of error = Critical value x http://bestwwws.com/margin-of/calculating-a-margin-of-error-formula.php As a rough guide, many statisticians say that a sample size of 30 is large enough when the population distribution is bell-shaped. For some margin of error formulas, you do not need to know the value of N. 95% Confidence Interval Margin of Error If you have a sample that is drawn from Typically, you want to be about 95% confident, so the basic rule is to add or subtract about 2 standard errors (1.96, to be exact) to get the MOE (you get This section describes how to find the critical value, when the sampling distribution of the statistic is normal or nearly normal. see this here Meaning Of Margin Of Error The condition you need to meet in order to use a z*-value in the margin of error formula for a sample mean is either: 1) The original population has a normal What is a Survey?. Suppose the population standard deviation is 0.6 ounces. Enter the population size N, or leave blank if the total population is large. The critical value for a 90% level of confidence, with corresponding α value of 0.10, is 1.64. The forumula is FPCF = sqrt[(N-n)/(N-1)]. You need to make sure that is at least 10. Margin Of Error Calculation In Excel The number of Americans in the sample who said they approve of the president was found to be 520. The standard error calculation can be done by the mathematical formula SE = (√((p(1-p)/n) )). Margin Of Error Example Problems View Mobile Version Sign In Help SurveyMonkey ÷ Home How It Works Examples Survey Templates Survey Tips Survey Types Academic Research Customer Satisfaction Education Employee Healthcare Market Research Non Profit Events Using the maximum margin of error formula above, we calculate MOE = (0.98)sqrt[1/865] = (0.98)(0.034001) = 0.033321 or 3.3321%. http://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-proportion/ Click here for a minute video that shows you how to find a critical value. When working with and reporting results about data, always remember what the units are. Margin Of Error Calculation Confidence Interval Typically, you want to be about 95% confident, so the basic rule is to add or subtract about 2 standard errors (1.96, to be exact) to get the MOE (you get p.64. Andale Post authorMarch 7, 2016 at 4:06 pm Thanks for catching that, Mike. Margin Of Error Example Problems For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. http://stattrek.com/estimation/margin-of-error.aspx A random sample of size 1600 will give a margin of error of 0.98/40, or 0.0245—just under 2.5%. Meaning Of Margin Of Error However, when the total population for a survey is much smaller, or the sample size is more than 5% of the total population, you should multiply the margin of error by Calculating Margin Of Error Using Confidence Interval To be 99% confident, you add and subtract 2.58 standard errors. (This assumes a normal distribution on large n; standard deviation known.) However, if you use a larger confidence percentage, then These are essentially the same thing, only you must know your population parameters in order to calculate standard deviation. click site Sampling theory provides methods for calculating the probability that the poll results differ from reality by more than a certain amount, simply due to chance; for instance, that the poll reports The sample proportion is the number in the sample with the characteristic of interest, divided by n. The general formula for the margin of error for a sample proportion (if certain conditions are met) is where is the sample proportion, n is the sample size, and z* is Calculating Margin Of Error In A Survey Your email Submit RELATED ARTICLES How to Calculate the Margin of Error for a Sample… Statistics Essentials For Dummies Statistics For Dummies, 2nd Edition SPSS Statistics for Dummies, 3rd Edition Statistics Margin Of Error Calculation Statistics Here are the steps for calculating the margin of error for a sample proportion: Find the sample size, n, and the sample proportion. If p moves away from 50%, the confidence interval for p will be shorter. In cases where the sampling fraction exceeds 5%, analysts can adjust the margin of error using a finite population correction (FPC) to account for the added precision gained by sampling close One example is the percent of people who prefer product A versus product B. In other words, if you have a sample percentage of 5%, you must use 0.05 in the formula, not 5. In other words, 95 percent of the time they would expect the results to be between: 51 - 4 = 47 percent and 51 + 4 = 55 percent. Margin Of Error Equation Stats The chart shows only the confidence percentages most commonly used. However, the margin of error only accounts for random sampling error, so it is blind to systematic errors that may be introduced by non-response or by interactions between the survey and For the eponymous movie, see Margin for error (film). The critical value for a 99% level of confidence, with corresponding α value of 0.01, is 2.54.Sample SizeThe only other number that we need to use in the formula to calculate http://bestwwws.com/margin-of/confidence-interval-margin-of-error-formula.php Easy! Questions on how to calculate margin of error? All Rights Reserved. Take the square root of the calculated value. Check out our Youtube channel for video tips on statistics! So in this case, the absolute margin of error is 5 people, but the "percent relative" margin of error is 10% (because 5 people are ten percent of 50 people). Retrieved February 15, 2007. ^ Braiker, Brian. "The Race is On: With voters widely viewing Kerry as the debate’s winner, Bush’s lead in the NEWSWEEK poll has evaporated". Since we don't know the population standard deviation, we'll express the critical value as a t statistic. Check out our Statistics Scholarship Page to apply! Often, however, the distinction is not explicitly made, yet usually is apparent from context.

+-- William Hinshaw | 1789-1854 +-- Zimri Hinshaw --------+ | 1822-1900 | | +-- Ruth Hinshaw | 1791-1836 William Martin Hinshaw ---+ B: 1852 | +-- William Clark D: 1928 | | +-- Martha McMath Clark --+ 1817-1888 | +-- Courtney (Clark) M: Anna Marie White +-- Fred Bahnson Hinshaw, 1876-1948 +-- Bertha Eugenia Hinshaw, 1878-1953 +-- Guy Francis Hinshaw, 1880-1977 +-- Hugh Martin Hinshaw, 1883-1921 +-- Clinton White Hinshaw, 1886-1956 +-- Mabel Lee Hinshaw, 1888-1973 +-- Laura Estelle Hinshaw, 1891-1972 |William Martin Hinshaw     [ID 00534]||Click here to switch to Ahnentafel view:| Born Mar 25 1852, Chatham County, North Carolina.1,43,a,b,141,c,d,e,f In 1870, when William was 18, he moved from Chatham County, N.C. to Winston, Forsyth County, N.C.    He attended Boys School in Winston for two years.    When he finished school he sold goods for his brother George, who was head of the Hinshaw Company.    The Hinshaw Company eventually consolidated with "Hodgin, Sullivan & Co.", but William remained as salesman for ten years.g He married Anna Marie White, Jun 1 1875.g    (Ann Marie White)a    (Annie Marie White)h    (Jan 2 1873).i    Anna, daughter of Joshua Simeon White & Charlotte Elizabeth Rights, was born Mar 7 1855, Winston, Forsyth County, North Carolina.g,141,j,c,d,e,f,i Ann's sister, Carolin Elizabeth White, had been the first child born (Sep 21 1852) in Winston, North Carolina (founded 1849).    Her parents home was the 5th house built in Winston.g After they were married, William and Ann lived for awhile with Ann's parents.    William built his first house on the northeast corner of Fourth and Cherry Streets, across from his brother George's home on Cherry Street.    On Jan 23 1880 he bought a lot and built a home on the south side of Fifth Street, where today is located the Forsyth County Main Library.g On Feb 21 1877, William was deeded a lot in Winston where he and his brother George built the Hinshaw Brothers Store.    On Dec 31 1881, William sold his interest to George and Wade H. Bynum, General Partners, and James W. Allison and E.B. Addison, Special Partners, operating as "Hinshaw and Bynum".    William then built a store on the west side of Trade Street, between Fourth and Fifth Streets, and William's family lived above the store.    William owned this store for ten years.g William and family were shown in the 1880 census (Jun 16 1880), Winston, Forsyth County, North Carolina:c William's family fell on hard times so son Fred travelled seeking work, which he found in Charleston, West Virginia.    Fred wrote home telling his parents of his good fortune (no need to build a fire in the morning - just turn the gas valve!).    William and sons Hugh and Clinton went to Charleston to find work.    Eventually William returned to Winston, Fred moved north, and Clinton settled in Huntington, West Virginia.175 William and family were shown in the 1900 census (Jun 19 1900), 84 Brooktown Avenue, Winston, Forsyth County, North Carolina:d William and family were shown in the 1920 census (Jan 6 1920), 232 North Greene, Winston-Salem, Forsyth County, North Carolina:f Anna's obituary was published in the "Laurensville Herald" (Laurens, Laurens County, South Carolina) on Friday, December 24, 1926:i MOTHER OF MRS. WOLFF DIED SATURDAY, 18TH MRS. ANNA M. HINSHAW, mother of MRS. B.M. WOLFF, of this city, died at her home in Winston-Salem, N.C., Saturday afternoon at 1:20 o'clock after a lingering illness.    She had been critically ill for several weeks.    MRS. WOLFF was at her bedside when the end came.    MR. WOLFF went up upon receipt of the news and attended the funeral Sunday. The following account of her appeared in the Winston-Salem Journal Sunday morning: MRS. ANNA M. HINSHAW, 71, wife of W.M. HINSHAW, 232 North Green St., passed away at a local hospital yesterday. MRS. HINSHAW was born in this city March 7, 1855, the daughter of the late J.S. and CHARLOTTE RIGHTS WHITE.    She spent her entire life in this city and was one of the best known and best loved women in the community. Early in life she became a member of the Centenary Meth. Church and devoted most of her life to church work. She married W.M. HINSHAW Jan. 2, 1873 and the two have lived on N. Green St. for many years. Surviving are her husband, 3 daughters, MRS. B.M. WOLFF, of Laurens, MISS MABEL HINSHAW of this city, and MRS. JOHN CALDWELL of Newton; 3 sons, GUY F. HINSHAW, of this city, CLINTON W. HINSHAW, Huntington, W.Va. and FRED B. HINSHAW of N.Y., one sister, MRS. J.R. WALKER of this city; one brother, FRANK WHITE of Exmore, Va., and 17 grandchildren. William Martin Hinshaw died Jul 8 1928, buried Salem Cemetery, Winston-Salem, Forsyth County, North Carolina.b,141 1. "The Hinshaw and Henshaw Families", by William Hinshaw; edited by Milo Custer; private printing, Bloomington, Illinois, 1911; Frank I. Miller Co., printers. LDS microfilm number 1402822. 43. A handwritten document titled "Hinshaws - Early Chatham Co. & Cane Creek Quaker Meeting", by William C. & Nolan Moran. 141. Family group sheets contributed by Thornton Neal Hinshaw 175. Family group sheets contributed by Hugh Malcolm Hinshaw (a) Contribution from Hugh Malcolm Hinshaw. (b) Contribution from Lee Hinshaw. (c) 1880 census, Winston, Forsyth County, North Carolina; roll T9-0963, ED 86, page 473A, line #19, dwelling #198, family #216. (d) 1900 census, 1st Ward, Winston, Forsyth County, North Carolina; roll T623-1195, ED 37, page 5A, line #33, dwelling #73, family #84. (e) 1910 census, 1st Ward, Winston, Forsyth County, North Carolina; roll T624-1111, ED 70, page 100A, line #21, dwelling #249, family #249. (f) 1920 census, Winston-Salem, Forsyth County, North Carolina; roll T625-1298, ED 86, page 4B, line #58, dwelling #59, family #69. (g) Biography of William Martin Hinshaw written in 1964 by Mabel Lee Hinshaw for her brother Clinton and sister Laura, with additional information from Lee Morrow Hinshaw. (h) Contribution from Pat Winfree () citing: obituary of Joshua Simeon White (probated May 23 1911). (i) Obituary of Anna M. Hinshaw; "Laurensville Herald", Dec 24 1926; posted Sep 23 2006 by "Nancie O." () to SCSPARTA-L. (j) Contribution from Pat Winfree (). (k) Contribution from Lee Hinshaw citing: Family record book written by Aileen Hinshaw. If you have additional information on this person, please share! If you would like to be automatically notified by email whenever an update is made affecting this page then enter your email address and click the "Enter" button below: How is this person related to other ancestors? To find out, enter the database ID number of another ancestor, and then click the "Enter" button below: Return to HFA Home Page This site uses spambot thwarting technology to hide email addresses from all known email harvesting programs used by spammers. [This page was computer generated]

Hugo Dyonizy Steinhaus Jasło, Galicia, Austrian Empire (now Poland) BiographyHugo Steinhaus was born in Galicia into a family of Jewish intellectuals. The town of his birth, Jasło, was in Galicia, about half way between Kraków and Lemberg (now called Lviv) (although a bit nearer Kraków than Lemberg). Galicia was attached to Austria in the 1772 partition of Poland. However, by the time Steinhaus was born in Jasło, Austria had named the region the Kingdom of Galicia and Lodomeria and given it a large degree of administrative autonomy. Steinhaus's uncle was an important person being a politician in the Austrian parliament. Steinhaus studied for one year in Lemberg, spent one term in Munich but then spent five years studying mathematics at the University of Göttingen. There he was influenced by an amazingly strong group of mathematicians including Felix Bernstein, Carathéodory, Courant, Herglotz, Hilbert, Klein, Koebe, The main influence on the direction that Steinhaus's research would take was none of the major mathematical figures at Göttingen but rather the influence came from Lebesgue. Steinhaus studied Lebesgue's two major books Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ (1904) and Leçons sur les séries trigonmétriques Ⓣ (1906) around 1912 after completing his doctorate. After military service in the Polish Legion at the beginning of World War I, Steinhaus lived in Kraków. He relates in how, despite the war in 1916, it was safe to walk in Kraków:- During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praised. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society.The mathematical society which Steinhaus proposed was started as the Mathematical Society of Kraków and, shortly after the war ended, it became the Polish Mathematical Society. Steinhaus described the beginnings of the new mathematical society in in a passage which tells us quite a lot about his life in Kraków at the time:- As initiator of the idea, I made my room available for meetings and, as the first step in preparations, nailed an oilcloth blackboard to the wall. When the French manager of the boarding house saw what I had done, she was terrified - what was the proprietor going to say? I calmed her down reminding her that the owner of the building was my uncle's brother-in-law, and she forgave my transgression. However, I had made a mistake. Mr L took the position of a traditional, hard-nosed landlord and was unmoved by the lofty goal the blackboard was supposed to serve. The society expanded - it was the first ray of light of this kind in Poland.Also at this time Steinhaus started a collaboration with Banach and their first joint work was completed in 1916. Steinhaus took up an appointment as an assistant at the Jan Kazimierz University in Lwów (as Lemberg University had become) and, around 1920, he was promoted to Extraordinary Professor. Banach was by this time on the staff at Lwów and the school rapidly grew in importance. Kac, who was a student of Steinhaus in Lwów during the 1930s, described the influence of Lebesgue's work on the Lwów school:- The influence of Lebesgue on the Lwów school was very direct. The school, founded ... by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory. There is no doubt that none of these theories would have achieved today's level of prominence without an essential understanding of the Lebesgue measure and integral. On the other hand, the ideas of Lebesgue measure and integral found their most striking and fruitful applications there in Lwów.Steinhaus was the main figure in the Lwów School up till 1941. In 1923 he published in Fundamenta Mathematicae the first rigorous account of the theory of tossing coins based on measure theory. In 1925 he was the first to define and discuss the concept of strategy in game theory. Steinhaus published his second joint paper with Banach in 1927 Sur le principe de la condensation des singularités Ⓣ. In 1929, together with Banach, he started a new journal Studia Mathematica and Steinhaus and Banach became the first editors. The editorial policy was:- ... to focus on research in functional analysis and related topics.Another important publishing venture in which Steinhaus was involved, begun in 1931, was a new series of Mathematical Monographs. The series was set up under the editorship of Steinhaus and Banach from Lwów and Knaster, Kuratowski, Mazurkiewicz, and Sierpiński from Warsaw. An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series. Steinhaus is best known for his book Mathematical Snapshots written in 1937. Kac, writing in says:- ... to understand and appreciate Steinhaus's mathematical style, one must read (or rather look at) snapshots. ... designed to appeal to "the scientist in the child and the child in the scientist" ... it expresses, not always explicitly and at times even unconsciously, what Steinhaus thought mathematics is and should be. To Steinhaus mathematics was a mirror of reality and life much in the same way as poetry is a mirror, and he liked to "play" with numbers, sets, and curves, the way a poet plays with words, phrases, and sounds.Stark describes Steinhaus lectures in Lwów:- My class was guided by Professor Steinhaus. It was a very big class, and the analysis lecture was attended by over 220 students squeezed into a smallish and poorly ventilated lecture room, standing in the aisles, and sitting on the window sills. ... His figure, perched high on the podium by a small five by five foot blackboard dominated the crowded room. ... despite Steinhaus's attention to preparation, the lectures were too difficult for the average student.The mathematicians of the Lwów school did a great deal of mathematical research in the cafés of Lwów. The Scottish Café was the most popular with the mathematicians in general but not with Steinhaus who (according to Ulam):- ... usually frequented a more genteel tea shop that boasted the best pastry in Poland. This was Ludwik Zalewski's Confectionery at 22 Akademicka Street. It was in the Scottish Café, however, that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Steinhaus, who sometimes joined his colleagues in the Scottish Café, contributed ten problems to the book, including the final one written on 31 May 1941 only days before the Nazi troops entered the town. You can see more about the Scottish Café at THIS LINK. When the prospect of war was looming in 1938, Steinhaus proposed Lebesgue for an honorary degree from Lwów. Steinhaus joked to Kac that :- It will not be a bad record to leave behind, to have had Banach as the first and Lebesgue as the last doctoral candidate.The reception for Lebesgue, after the award of his degree, was held in the Scottish Café but only fifteen mathematicians attended, showing that the school of mathematics in Lwów had shrunk considerably due to the political situation. Steinhaus spent the war years from June 1941 hiding from the Nazis, suffering great hardships, going hungry most of the time but always thinking about mathematics :- ... even then his sharp restless mind was at work on a multitude of ideas and projects.In 1945 Steinhaus moved to the University of Wrocław but made many visits to universities in the United States including Notre Dame. Kac in writes:- ... it was he who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the Second World War to the position of new strength and respect which it now occupies.After the end of World War II the Scottish Book, which seems to have been preserved through the war by Steinhaus, was sent by him to Ulam in the United States. The book was translated into English by Ulam and published. Steinhaus, now in the University of Wrocław, decided that the tradition of the Scottish Book was too good to end. In 1946 he extended the tradition to Wrocław starting the New Scottish Book. Let us finally examine some of Steinhaus's mathematical contributions which we have not mentioned above. In 1944 Steinhaus proposed the problem of dividing a cake into pieces so that it is proportional (each person is satisfied with their share) and envy free (each person is satisfied nobody is receiving more than a fair share). For the problem is trivial, one person cuts the cake, the other chooses their piece. Steinhaus found a proportional but not envy free solution for . An envy free solution to Steinhaus's problem for was found in 1962 by John H Conway and, independently, by John Selfridge. For general the problem was solved by Steven Brams and Alan Taylor in 1995. Steinhaus's bibliography, see , contains 170 articles. He did important work on functional analysis, but he himself described his greatest discovery in this area as Stefan Banach. Some of Steinhaus's early work was on trigonometric series. He was the first to give some examples which would lead to marked progress in the subject. He gave an example of a trigonometric series which diverged at every point, yet its coefficients tended to zero. He also gave an example of a trigonometric series which converged in one interval but diverged in a second interval. As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications. In particular he is associated with the theory of independent functions, arising from his work in probability theory, and he was the first to make precise the concepts of "independent" and "uniformly distributed". In addition to his famous book Mathematical Snapshots he also wrote the highly acclaimed One Hundred Problems .... - R Kaluza, The life of Stefan Banach (Boston, 1996). - H Steinhaus, Reminiscences (Polish) (Cracow, 1970). - Brief scientific biography of Hugo Steinhaus (1887-1972), Zastos. Mat. 13 (1972/73), III-IV. - A Dawidowicz, Reminiscences of Leon Chwistek, Hugo Steinhaus and Wlodzimierz Stozek (Polish), Wiadomosci matematyczne 23 (2) (1981), 232-240. - A Garlicki, Hugo Steinhaus-an episode in Berdechow (Polish), Wiadomosci matematyczne 30 (1) (1993), 121-124. - Hugo Steinhaus (14. I. 1887-25. II. 1972), Colloq. Math. 25 (1) (1972), i-ii. - M Kac, Hugo Steinhaus-a reminiscence and a tribute, Amer. Math. Monthly 81 (1974), 572-581. - B Koszela, The contribution of Józef Marcinkiewicz, Stefan Mazurkiewicz and Hugo Steinhaus in developing Polish mathematics: A biographical sketch (Polish), Mathematics at the turn of the twentieth century (Katowice, 1992), 104-110. - H Kowarczyk, The cooperation of Hugo Steinhaus with the medical sciences and physicians (Polish), Wiadomosci matematyczne (2) 17 (1973), 65-69. - List of the scientific works of Hugo Steinhaus, Wiadomosci matematyczne (2) 17 (1973), 12-28. - J Lukaszewicz, The role of Hugo Steinhaus in the development of the applications of mathematics (Polish), Wiadomosci matematyczne (2) 17 (1973), 51-63. - E Marczewski, The Wroclaw years of Hugo Steinhaus (Polish), Wiadomosci matematyczne (2) 17 (1973), 91-100. - C Ryll-Nardzewski, The works of Hugo Steinhaus on conflict situations (Polish), Wiadomosci matematyczne (2) 17 (1973), 29-38. - A Sarski, Hugo Steinhaus (1887-1972) (Bulgarian), Fiz.-Mat. Spis. Bulgar. Akad. Nauk. 15 (48) (4) (1972), 328-329. - M Stark, Hugo Steinhaus as a teacher during his Lwów period (Polish), Wiadomosci matematyczne (2) 17 (1973), 77-84. - K Urbanik, The ideas of Hugo Steinhaus on probability theory (Polish), Wiadomosci matematyczne (2) 17 (1973), 39-50. Additional Resources (show) Other pages about Hugo Steinhaus: Written by J J O'Connor and E F Robertson Last Update February 2000 Last Update February 2000

Complex Systems Modelling, Analysis, and ControlView this Special Issue Locally and Globally Exponential Synchronization of Moving Agent Networks by Adaptive Control The exponential synchronization problem is investigated for a class of moving agent networks in a two-dimensional space and exhibits time-varying topology structure. Based on the Lyapunov stability theory, adaptive feedback controllers are developed to guarantee the exponential synchronization between each agent node. New criteria are proposed for verifying the locally and globally exponential synchronization of moving agent networks under the constraint of fast switching. In addition, a numerical example, including typical moving agent network with the Rössler system at each agent node, is provided to demonstrate the effectiveness and applicability of the proposed design approach. Over the past decade years, the analysis of complex systems from the viewpoint of networks has become an important interdisciplinary issue . Complex networks have been intensively studied in many fields, such as social, biological, mathematical, and engineering sciences. Generally, a complex network is made up of interconnected nodes in which a node is a basic unit with detailed contents. These interactions between nodes determine many basic properties of a network. To better understand the complex dynamical behaviors of many natural systems, we need to study their operating mechanism, dynamic behavior, synchronization, antijamming ability, and so on. Recently, synchronization of complex dynamical networks has received a great deal of attentions from various fields of science and engineering [1–3]. Particularly, synchronization of large-scale complex networks of coupled chaotic oscillators has been extensively investigated in the fields of science and engineering. The synchronization properties of a complex network are mainly determined by its topological structures connections between nodes. In the current study of complex networks, most of the existing works on synchronization consider static networks, whose topological structures do not change as time evolves [4–10]. The Master-stability function (MSF) approach allows us to determine the stability of a linearly coupled dynamical network with a constant coupling (or Laplacian) matrix. However, numerous real-world networks such as biological, communication, social, and epidemiological networks generally evolve with time-varying topological structures. Henceforth, researchers have devoted more and more efforts to complex networks with time-varying topologies. Stilwell et al. prove that if the network of oscillators synchronizes for the static time-average of the topology, then the network will synchronize with the time-varying topology if the time-average is achieved sufficiently fast. At the same conditions, Lu et al. [13, 14] found that the directed network with switching topology can reach global synchronization for sufficiently large coupling strength if there exists a spanning-directed tree in the network. Inspired by the above discussions, in this paper we investigate the adaptive exponential synchronization problem for a specific time-varying network model. The model arises from the interaction of mobile agents proposed by Frasca et al. and can be widely used to explore various practical problems, for example, clock synchronization in mobile robots , synchronized bulk oscillations , and task coordination of swarming animals . How one controls the appearance of synchronized states of the dynamical network is of great significance in theory and potential applications. For the mobile agent network model, we introduce adaptive control to regulate its synchronization, as an attempt to explain the control of complex time-varying systems. Although synchronization control of moving agent network has great application potential in a variety of areas, there has been very little existing literature on the exponential synchronization problem. Therefore, we adopt the constraint of fast switching to derive exponent synchronization conditions. By using Lyapunov stability theory, adaptive controllers are designed for synchronization of moving agent network with time-varying topological structures. The adaptive controllers can ensure the states of moving agent network fast synchronization. The current paper is organized as follows. A general moving agent network model and several mathematical preliminaries are introduced in Section 2. In Section 3, several locally and globally adaptive synchronization criteria for the moving agent networks are deduced. A representative example is given to show the effectiveness of the proposed network synchronization criteria in Section 4. Conclusions are finally drawn in Section 5. 2. Moving Agent Network Model and Preliminaries We consider as moving agents distributed in a two-dimensional planar space of size , with periodic boundary conditions. Each agent moves with velocity and direction of motion . The velocity is the same for all individuals (denoted by ) and is updated in direction through the angle for each time unit. The agents are considered as random walkers. Hence, the motion law of the th agent is given as follows: where , is the position of agent in the plane at time , are independent random variables chosen at each time unit with uniform probability in the interval , and is the motion integration step size. Each agent interacts at a given time with only those agents located within a neighborhood of an interaction radius, defined as [15, 19, 20]. When two agents interact, the state equations of each agent are changed to include diffusive coupling with the neighboring agent. Under these hypotheses, the state dynamics of agent node can be formulated as where , are the state variables of the node ; is a smooth nonlinear vector-valued function which governs the local dynamics of oscillator; is the coupling strength; are the elements of a time-varying Laplacian matrix: which defines the neighborhood of agents at a given time and depends on the trajectory of each agent. In detail, for arbitrary two agents , with distance at time , , if ; , if ; , where is the number of neighbors of the th agent at time ; are the control inputs. In this paper, the control objective is to make the states of network (2) exponentially synchronize with a manifold defined in (3) by introducing a simple adaptive controller into each individual node where is a solution of an isolated node, We assume that is an arbitrary desired state which can be an equilibrium point, a periodic orbit, an aperiodic orbit, or even a chaotic orbit in the phase space. Next, the rigorous mathematical definition of exponential synchronization for dynamical network (2) is introduced. Definition 1. Let be a solution of dynamical network (2), where are initial conditions of node and is continuously differentiable on . If there are control inputs , and further there exist constants , and a nonempty subset with , , such that for all , and (Hereafter, denote as the Euclidean norm.) where , , is a solution of the system (4) with the initial condition , then the dynamical network (2) is said to realize exponential synchronization such that is the exponential rate. And is called the region of synchrony for the dynamical network (2) (see [21, 22]). 3. Synchronization of Moving Agent Networks In this section, we discuss the exponential synchronization of moving agent network (2) by designing adaptive controllers for each agent node. Several network synchronization criteria are given. 3.1. Local Synchronization Linearizing error system (7) about the synchronized states , we can get where and is the Jacobian matrix of evaluated at . In the following, we give several useful hypotheses. Assumption 2. Suppose that there exists a nonnegative constant satisfying Generally, Assumption 2 is likely to be satisfied. For example, in many chaotic systems such as Lorenz system, Rössler system, and Chen system, there exists a constant satisfying . Assumption 3. Suppose there exists a constant such that coupling matrix satisfying where is the time-average of the coupling matrix . Lemma 4. Suppose that a coupled network with fixed topology defined by admits a stable synchronization manifold and Assumption 3 holds. Then the network with a time-variant topological structure defined by (2) admits a stable synchronization manifold. According to analysis of , under the constraint of fast switching, , where is the probability that a link is activated and thus , and is the all-to-all coupling matrix with zero-row sum. That is Theorem 5. Suppose that Assumptions 2 and 3 hold. Then the dynamical moving agent network (2) is locally exponential synchronization under the following sets of adaptive controllers: and updating laws where is the exponential rate available to be designed, and , where is defined in the Assumption 2. Proof. Select a Lyapunov function as follows: where constant is to be given below. Then the time derivative of along the solution of the error system (8) is given as follows: According to Assumption 2, , so Here, according to Lemma 4, under the constraint of fast switching, one substitute for Since , Therefore, Since is a nonnegative constant, one can select suitable constants to make . Therefore, By calculating integration on both sides of the above inequality, we get . According to (15), one can get , so Let , so . That is, So, one gets where . Therefore, in closed-loop under the controllers (13) and updating laws (14), it follows that the error system (7) is locally exponentially stable at the equilibrium set , , with the exponential rate . Consequently, the synchronous solution of the dynamical network (2) is locally exponentially stable. Then the dynamical network (2) is said to realize locally exponential synchronization under the controllers (13) and updating laws (14). The proof is thus completed. 3.2. Global Synchronization Rewrite node dynamics as , where is a constant matrix and is a smooth nonlinear function. Thus, network (2) is described by where . Similarly, one can get the error system where and . Assumption 7. Suppose that there exists a nonnegative constant , satisfying Theorem 8. Suppose that Assumptions 3 and 7 hold. Then the dynamical moving agent network (2) is globally synchronized under the following adaptive controllers: and updating laws where , is a nonnegative constant satisfying , is defined in the Assumption 7, and is the exponential rate available to be designed. Proof. since is a given constant matrix, there exists a nonnegative constant such that . Similarly, construct Lyapunov function (15); then one has And according to Assumption 7, there exists a nonnegative constant , satisfying ; therefore, According to Lemma 4, one substitute for Since , are nonnegative constants, one can select suitable constants to make . Therefore, By similar calculation, one gets . Then, one has Similar, one can get where . Therefore, in closed-loop under the controllers (28) and updating laws (29), it follows that the error system (7) is globally exponentially stable at the equilibrium set , , with the globally exponential rate . Consequently, the synchronous solution of the dynamical network (2) is globally exponential stable. Then the dynamical network (2) is said to realize globally exponential synchronization under the controllers (28) and updating laws (29). The proof is thus completed. Remark 9. In this paper, the coupling scheme of dynamical network (2) is a linear relationship. If the network coupling scheme is general nonlinear relationship, the network (2) is rewritten as follows: Suppose that ( is a nonnegative constant) hold. Then, the synchronous solution of moving agent network (36) is globally exponentially stable under the adaptive controllers (28) and updated laws (29) by the similar proof. In this section, one example is given for illustrating the proposed synchronization criteria. Consider a dynamical network consisting of 5 identical Rössler oscillators, where state dynamics of each agent is described by where , . The following parameters have been used: , , . Each agent node interacts at a given time with only those agents located within a neighborhood of an interaction radius. Here, we let periodic boundary conditions size . The initial position of agent in the plane is chosen at random. The initial orientation , other time units being chosen at each time unit with uniform probability in the interval . Each agent is moving 40 time unit (). Then the position of each agent during the movement is shown in Figure 1. When two agents interact (let interaction radius ), the state equations of each agent are changed to include diffusive coupling with the neighboring agent, acting on the state variable . Based on these assumptions, the state dynamics of each agent can be described in terms of the following equations: where , is given by the Rössler dynamics, , are the elements of a time-varying matrix , and , . Obviously, one gets Similar to [23–25], since Rössler chaotic system has a chaotic attractor which is confined to abounded region , there exists a constant satisfying for and . Therefore, can be got from the method similar to . Thus, Assumption 7 holds. Assume that , to guarantee the fast-switching condition Assumption 3. Thus, Assumptions 3 and 7 hold. According to Theorem 8, the synchronous solution of dynamical moving agent network (38) is globally exponentially stable. The other parameters are assigned as follows: , , , and . The synchronous error is shown in Figures 2, 3, and 4. We learn from these figures that the synchronization errors can be globally exponentially stable for dynamical network (38). Therefore, we conclude that the states of the networks (38) can be globally exponentially synchronized with the state of each isolated Rössler system. Locally and globally adaptive exponential synchronization of moving agent network has been investigated in this paper. The network with decentralized controllers is considered as a large-scale nonlinear system with time-varying topological structure. An adequate Lyapunov function is constructed to deal with the problem of controlled synchronization so as to ensure the closed-loop system stability. Several network synchronization criteria for such network with time-varying topological have been obtained. And a numerical simulation of coupled Rössler system network is given, which demonstrates the effectiveness of the proposed synchronization scheme. This research was supported in part by the Natural Science Foundation of Hebei under Grant no. F2012501030, Fundamental Research Funds for the Central Universities under Grant no. N100323012 from the Ministry of Education, and the National Natural Science Foundation of China under Grant no. 51105068. S. A. Pandit and R. E. Amritkar, “Characterization and control of small-world networks,” Physical Review E, vol. 60, no. 2B, pp. R1119–R1122, 1999.View at: Google Scholar M. Barahona and L. M. Pecora, “Synchronization in small-world systems,” Physical Review Letters, vol. 89, no. 5, pp. 054101/1–054101/4, 2002.View at: Google Scholar S. Zheng, S. Wang, G. Dong, and Q. Bi, “Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 284–291, 2012.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Physical Review Letters, vol. 80, no. 10, pp. 2109–2112, 1998.View at: Google Scholar D. Li, J.-a. Lu, X. Wu, and G. Chen, “Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 844–853, 2006.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Participants will not answer certain items on a questionnaire, perhaps because they were tired, perhaps they did not want to answer, or maybe they did not see the item. Note on the Excel image above that at least 6 participants did not indicate their gender. For some binary variables, one category more naturally should be coded as one. If you are a new researcher, it might be helpful to get a blank questionnaire and write the code for each response so you do not forget what code you assigned to each response. Because these codes are not written on the questionnaire itself, it is best to make a sheet that reminds you of the codes given to each response for future reference. Bureau of Labor Statistics. The easiest way to handle items that were skipped on the questionnaire is to simply discard the item for that particular participant. To help double-check your work, after you have entered the data in the computer, ask a peer read out the responses to each questionnaire as you check the entry in the computer Tips for Coding Data As previously stated, this step is called coding the data because each response is given a number or letter code and then entered into the computer. I recommend starting with a letter that somehow represents the questionnaire or the sample, perhaps "L" for Literacy, and then 3 numbers, starting with, Flipping the categories e. Below are some tips that can help you determine which code to give for which response. For other items, they might have circled two responses such as Strongly Agree and Disagree. For example, below is the first completed questionnaire. The research question was as follows: Because you will not be entering in the entire response of every participant, this process is called Coding the Data because each response is given a code. After conducting an interview with a manager, she asks for the names of other managers that use virtual teams. In most cases, I recommend the second option where the participant is included in the analyses for English, Maths, and Social Studies. This would be impossible unless the questionnaires were numbered. Can we conclude that the time people live in Sycamore is significantly more than the national average? When it comes to responses to Likert-Scale items, enter the number of their response: The researcher now has three options: Click here to see ALL problems on test Question For example, if a participant did not indicate their gender, I will type Z. If a participant does not complete a phase of the study, they generally are discarded from the entire study. These are items that actually say the opposite of what was intended. Economic status was categorized as follows: I typically enter Z for missing items because z is rarely coded for any other response. Two of the options might be Primary and Post-Graduate. For example, you might not be able to read the numbers they wrote for their age. There were four packaging sizes: When doing data entry, take each questionnaire and enter the correct code for the response given to each item. Notice the top row has bolded identifiers for each item in the questionnaire: See below for an example of the correct codes when reversing items. Therefore, you can easily enter a or b instead of typing out "male" and "female. If this is the case, since it is not clear what the correct answer is, it is typically best to enter that item as missing, Z. For example, consider an item that asks participants to indicate their level of education completed. The mean of a zero-one variable represents the proportion in the category represented by the value one e. Determine which level of measurement— nominal, ordinal, interval, or ratio—is used in the followi A sample of families in the Sycamore, OK area shows the mean time living in a single family residence is With this same data set, coding gender as (1=m,2=f) means that females make $70, while males make $60, meaning that females make only about % more than males. 2. gender (edit: male is 1, female is coded as 2) 3. field of study (edit: 11 total fields, I simply used existing dept designations, for example 11. That is a good point and it works perfectly if gender is coded as 1 for males and 2 for females (as I state in my example above), but it's not clear that is the case for the original question asker, Southern Soul. Here is the coded data for the questionnaire given above. Notice the top row has bolded identifiers for each item in the questionnaire: S/No is the Serial Number written on the questionnaire itself, Gend is gender (item 1), Age (item 2), SchTeach is the type of school taught at (item 3), etc. • On a questionnaire which asks for gender, males are coded as 1 and females are coded as 2. INTERVAL • Respondents are asked to rate a list of high-tech companies as excellent, good, fair, or poor in terms of their service delivery. A. to determine the number of males and females buying Nike or Reebok shoes. B. to determine the number of males and females voting democrat, republican, or independent. C. to determine the number of males and females included in the sample.Download

Lewis structures show each atom and its position in the structure of the molecule using its chemical symbol. Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, Lewis dot diagrams for 44 chemical elements, Optional atomic numbers, element names, & symbols. Lewis dot structures are a key concept in understanding how atoms bond. Two Lewis structures must be drawn: Each structure has one of the two oxygen atoms double-bonded to the nitrogen atom. A simple Lewis model also does not account for the phenomenon of aromaticity. Lewis dot structures reflect the electronic structures of the elements, including how the electrons are paired. In 1902, while Lewis was trying to explain valence to his students, he depicted atoms as constructed of a concentric series of cubes with electrons at each corner. Atomic Structure Links. Answer . Answer . (ii) 1 mole of carbon is burnt in 16 g of dioxygen. For example, all elements which fall within the first column, or Group I, has one (1) valence electron. a) Draw the lewis structure by repeating steps 1-4. is 3.0 × 10 –25 J, calculate its … To draw an electron dot diagram, place one dot around the symbol for every valence electron. Lewis structure of the given molecule and ions are, 283 Views. Selection for center atom. The number of dots equals the number of valence electrons in the atom. Deprecated: Function create_function() is deprecated in /home/clients/ce8dc658147c71f5a2e0706832294e69/web/index.php on line 7 Optional facts for each element include the electron dot diagrams, atomic numbers, element symbols, and element names. carbon, because it has two electrons in its outermost p sublevel. Explain the formation of a chemical bond. Developing Effective Teams Let's Ride *No strings attached. (refer to the first picture) b) find the element(s) that has more dots than the rest. 6. Lewis electron dot diagrams may be drawn to help account for the electrons participating in a chemical bond between elements. Compare … We can draw it if we know the molecular formula of the compound. H 3 PO 4 contains three elements phosphorous, oxygen and hydrogen. Due to the greater variety of bonding schemes encountered in inorganic and organometallic chemistry, many of the molecules encountered require the use of fully delocalized molecular orbitals to adequately describe their bonding, making Lewis structures comparatively less important (although they are still common). The Lewis symbol for helium: Helium is one of the noble gases and contains a. Lewis Dot Diagrams of Selected Elements The first shell (n=1) can have only 2 electrons, so that shell … The answer is the number of electrons that make up the bonds. represents one valence electron. This page was last edited on 13 January 2021, at 12:40. The lewis structure is also called an electron dot structure which determines the number of valence electrons present in an atom. The second oxygen atom in each structure will be single-bonded to the nitrogen atom. Each oxygen may take a maximum of 3 lone pairs, giving each oxygen 8 electrons including the bonding pair. The overall charge on the compound must equal zero, that is, the number of electrons lost by one atom must equal the number of electrons gained by the other atom. This video shows how to use the periodic table to draw Lewis structures and figure out how many … Hydrogen atoms bonded to carbon are not shown—they can be inferred by counting the number of bonds to a particular carbon atom—each carbon is assumed to have four bonds in total, so any bonds not shown are, by implication, to hydrogen atoms. In condensed structural formulas, many or even all of the covalent bonds may be left out, with subscripts indicating the number of identical groups attached to a particular atom. Notably, the naive drawing of Lewis structures for molecules known experimentally to contain unpaired electrons (e.g., O2, NO, and ClO2) leads to incorrect inferences of bond orders, bond lengths, and/or magnetic properties. calcium, because it is an alkaline earth metal with two inner shell electrons. Although main group elements of the second period and beyond usually react by gaining, losing, or sharing electrons until they have achieved a valence shell electron configuration with a full octet of (8) electrons, hydrogen (H) can only form bonds which share just two electrons. A Lewis diagram counts the valence electrons. Each Cl atom interacts with eight valence electrons total: the six in the lone … B. In this case, the atoms must form a double bond; a lone pair of electrons is moved to form a second bond between the two atoms. 1. (The exception is helium, He, which only has one energy level or orbital. Deprecated: Function create_function() is deprecated in /home/clients/ce8dc658147c71f5a2e0706832294e69/web/index.php on line 7 how the molecule might react with other molecules. Each Cl atom interacts with eight valence electrons total: the six in the lone … Comprehensive data on the chemical element Samarium is provided on this page; including scores of properties, element names in many languages, most known nuclides of Samarium. The Lewis structure indicates that each Cl atom has three pairs of electrons that are not used in bonding (called lone pairs) and one shared pair of electrons (written between the atoms). If you use up all of your dots before each element has 8 dots around it, then you must share some of the dots. We can draw it if we know the molecular formula of the compound. The central atom of a molecule is usually the least electronegative atom or … The heavier elements will follow the same trends depending on their group. Lewis Electron Dot Diagrams . ClO3-Which of the following is/are possible … How to Draw Lewis Structures Lewis Structures 1) Find the element on the periodic table. The left shows an iodine atom with one lone pair. In such cases it is usual to write all of them with two-way arrows in between (see Example below). The number of dots equals the number of valence electrons in the atom. All elements in the vertical column (or group) 1 have 1 valence electron. How Can I Spread The Dots Around Iodine Atome How To Draw The Dot Structure For I2 The Basics Of Chemical Bonding How To Draw The Lewis Dot Structure For I- (Iodide Ion. Once all lone pairs are placed, atoms (especially the central atoms) may not have an octet of electrons. A Lewis dot structure is like a simplified Bohr-Rutherford model. Satisfy the octet rule. And from the given option only carbon of group 14 have 4 valence electrons in its valence shell which means that given Lewis dot structure with four dots represents carbon atom. The nitrogen atom has only 6 electrons assigned to it. In terms of Lewis structures, formal charge is used in the description, comparison, and assessment of likely topological and resonance structures by determining the apparent electronic charge of each atom within, based upon its electron dot structure, assuming exclusive covalency or non-polar bonding. lithium, because it is a group 1 element with two bonding electrons. The rest of the electrons just go to fill all the other atoms' octets. When we write the . It is sometimes useful to calculate the formal charge on each atom in a Lewis structure. The number of dots equals the number of valence electrons in the atom. Moreover, they also describe how these valence electrons are participating in the bond formation to form a molecule. Nitrogen has 5 valence electrons; each oxygen has 6, for a total of (6 × 2) + 5 = 17. Valence electrons are primarily responsible for the chemical properties of elements. 2) Determine the number of Lewis structures are a useful way to summarize certain information about bonding and may be thought of as “electron bookkeeping”. Place the first dots alone on each side and then pair up any remaining dots. All the. A simpler method has been proposed for constructing Lewis structures, eliminating the need for electron counting: the atoms are drawn showing the valence electrons; bonds are then formed by pairing up valence electrons of the atoms involved in the bond-making process, and anions and cations are formed by adding or removing electrons to/from the appropriate atoms.. On this Lewis Dot Diagrams of the Elements printable you can choose the color of the table and facts to include for each elements. A Lewis structure can be drawn for any covalently bonded molecule, as well as coordination compounds. Each of the different possibilities is superimposed on the others, and the molecule is considered to have a Lewis structure equivalent to some combination of these states. Draw a double-headed arrow between the two resonance forms. Lewis Structures for N2. This “cubic atom” explained the eight groups in the periodic table and represented his idea that chemical bonds are formed by electron transference to give each atom a complete set of eight outer electrons (an “octet”). 37 Sophia partners guarantee credit transfer. Steps for drawing Lewis Dot Structure. The ion has a charge of −1, which indicates an extra electron, so the total number of electrons is 18. Electron Dot Diagrams Recall that the valence electrons of an atom are the electrons located in the highest occupied principal energy level. 1s2 2s2 2p6 3s2 Lewis Dot Diagram of Tellurium (Te). The maximum number of … The total of the formal charges on an ion should be equal to the charge on the ion, and the total of the formal charges on a neutral molecule should be equal to zero. 8. Write Lewis dot symbols for atoms of the elements Mg, Na, B, O, N, Br. When counting electrons, negative ions should have extra electrons placed in their Lewis structures; positive ions should have fewer electrons than an uncharged molecule. This is especially true in the field of organic chemistry, where the traditional valence-bond model of bonding still dominates, and mechanisms are often understood in terms of curve-arrow notation superimposed upon skeletal formulae, which are shorthand versions of Lewis structures. The atoms are first connected by single bonds. Step-by-step tutorial for drawing the Lewis Structure for N2. This atom will be 2sp hybridized with remaining 2p x and 2p y atomic orbitals. Lewis dot structures help predict molecular geometry. Electrons shared in a covalent bond are counted twice. 299 Institutions have … After you have completed the customization of the table you can download it as a PDF. This … Therefore, there is a resonance structure. Draw a Lewis electron dot diagram for an atom or a monatomic ion. Electrons in covalent bonds are split equally between the atoms involved in the bond. This atom will be 2sp hybridized with remaining 2p x and 2p y atomic orbitals. The periodic table has all of the information needed to draw a Lewis dot structure. In a skeletal formula, carbon atoms are not signified by the symbol C but by the vertices of the lines. A Lewis structure can be drawn for any covalently bonded molecule, as well as coordination compounds. See More. the physical properties of the molecule (like boiling point, surface tension, etc. For the main group elements 1, the valence electrons are the electrons in the highest energy level (valence shell). The number of electrons in a given shell can be predicted from the quantum numbers associated with that shell along with the … Would you expect the group 18 elements to have the same electron dot diagram as neon? You've seen what the Bohr diagrams for the first 20 elements. It will hold more than 8 electrons. This college course is 100% free and is worth 1 semester credit. In accordance with what we discussed above, here are the Lewis symbols for the first twenty elements in the periodic table. We also know the Electron dot structures as Lewis dot formula. Optional facts for each element include the electron dot diagrams, atomic numbers, element symbols, and element names.

Select Board & Class give 7 examples of inertia of rest with explanation How to find the least count of a spring balance? give examples of inertia of direction with explanation how many types of inertia are there name them define one newton force what is the formula of recoil velocity of a gun????????? 2) What can you say about the speed of a moving object if no force is acting on to it.? 3) Two forces of 5N &22N are acting in a body in the same direction what will be the resultant force& in which direction will it act? If the two forces in the above example would have been acting in the opposite direction What would be the resultant force& in which direction will it act? 4) If we push the box with a small force, the box does not move, why? 5) What should be the force acting on an object moving with uniform velocity? explain why a cricketer moves his hands backwards while catching a fast moving cricket ball. please explain newton's third law of motion! 1 .A car of mass 1000kg and a bus of mass 8000kg are moving with the same velocity of 36 if a man jumps out from a boat , the boat moves backwards . why? Derive the equation for the conservation of momentum 1) The minute hand of a clock is 7 cm long. Calculate the distance covered and the displacement of minute hand of the clock from 9.00 AM to 9.30 AM. 2) An athlete completes a round of a circular track of diameter 200 m in 20s. Calculate the distance travelled by the athlete. 3)A boy is running on a straight road. He runs 500 m towards north in 2 minutes 10 seconds and then turns back and runs 200 m in 1 minute. Calculate his average speed and magnitude of average velocity during the whole journey. 4) Akshil drove his car with speed of 20 km/h while going to his college. When he returned to his home along the same route , the speed of the car is 30 km/h. Calculate the average speed of the car during the entire journey. a horizontal force equal to the weight of a bodymoves it from rest ,wat is the acceleration produced in it? name and define three different types of inertia and give an example each why a moving ship takes longer time as compared to a car when equal breaks are appiled Define force and write its si. Unit From a rifle of mass 4kg, a bullet of mass 50g is fired with an initial velocity of 35 ms-1. Calculate the initial recoil velocity of the rifle. The bullet of mass 10g moving with a velocity of 400m/s gets embeded in a freely suspended wooden block of mass700g .what is the velocity aquired by a block a boy weighing 30 kg is riding a bicycle weighing 50 kg . if the bicycle is moving at a speed of 9 km/h towards the west , find the linear momentum of the bicycle-boy system in SI units " examples of newton's second law of motion why are road accidents at high speed are very much worse than accidents at low speed??? Why does the speed of an object change with time? A body of mass 5 Kg is moving with a velocity of 10 m/s. A force is applied on it so that in 25 s , it attains a velocity of 35 m/s. calculate the value of the force applied. ALL THE FORMULAS IN THIS CHAPTER FORCE AND LAWS OF MOTION CAN ANY ONE SAY ME ASAP on what factors does inertia of a body depend? a rubber ball of mass 50 g falls from a height of 10 cm and rebounds to a height of 50 cm. determine the change in linear momentum and average force between the ball and the ground , taking time of contact as 0.1 s What is the real definition of Second Law Of Motion how does a karate player breaks a slab of ice with a single blow? An object of mass 2 kg is sliding with a constant velocity of 4 m s–1 on a frictionless horizontal table. The force required to keep the object moving with the same velocity is (a) 32 N (b) 0 N (c) 2 (d) 8 N a 40 kg skater moving at 4ms eastwards collides head on with a 60 kg skater travelling at 3ms westwards . if the two skates remain in contact, what is their final velocity? a truck starts from rest and rolls down a hill with a constant acceleration .it travels a distance of 400m in 20s . find the acceleration . find the force acting on it if its mass is 7 metric tonne (1 metric tonne = 1000kg). Explain recoiling of gun with third law of motion? A Bead Of Mass m is attached to one end of a spring of natural length R And spring constant K = (√3+1)mg/R . other end of the spring is attached toa point A ON the vertical ring of radius R MAKING AN ANGLE OF 30 DEGREE WITH ITS CIRCUMFERENCE THE NORMAL REACTION AT THE BEAD WHEN THE SPRING IS ALLOWED TO MOVE FROM POINT A a stone of 1kg is thrown with a velocity of 20m/s. across a frozen surface of a lake and comes to rest after travelling a distance of 50m. what is the force of friction between the stone and the ice. a hunter has a machine gun that can fire 50 g bullets with a velocity of 150 m/s . a 60 kg tiger springs at him with a velocity of 10 m/s . how many bullets must the hunter fire per second into the tiger in order to stop him in his track why does one get hurt seriously while jumping on a hard floor ? plz help Why shockers are provided in automobile vehicles? a bullet of mass 20g is horizontally fired with a velocity of 150m/s frm a pistol of mass 2kg. What is the recoil velocity of the pistol? Find the initial velocity of the car which is stopped in 10 sec by applying brakes .The retardation due to brakes is 2.5m/sec sq..... a train is moving with acceleration along a straight line with respect to ground aperson in train finds that- A.newtond law2 is false and 3law is true B.newtons 3 law is false but newtons 2law is true C.all laws r false D.all laws are true A bullet fired from a gun makes a clean whole in a window whereas a stone pebble thrown brakes the whole window.why? Do all motions requires a cause? How much momentum will a dumbbell of mass 10 Kg transfer to the floor if it falls from a height of 80 cm? Take its downwards acceleration to be 10 m/s2? A gardener waters the plants by a pipe of diameter 1 mm. The water comes out from the pipe atthe rate of 10 cm3/sec. The reactionary force exerted on the hand of the gardener is? can someone plz tell me the derivations of equations of motions without graph .. what is the difference between magnitude and direction why chinaware is wrapped in straw while transportation? a bullet of mass 10 g is fired from a rifle with a velocity of 800 m/s . after passing through a mud wall 180cm thick, the velocity drops to 100m/s. calculate the average resistance of the wall. A gunman gets a jerk on firing a bullet.why?? action and reaction forces do not balance each other. why? Place a water - filled tumbler on tray. Hold the tray and turn around as fast as u can. Why does water not spill? Few examples of balanced and unbalanced force? Q- two carts A and B of mass 10 kg each are placed on a horizontal track. They are joined tightly by a light but strong rope C . a man holds the cart A and pulls it towards the right with a force of 70N. The total force of friction by the track and the air on each cart is 15N,acting toward the left. Find:- (a) the acceleration of the cart (b) the force exerted by the rope on the car B why is it difficult to balance our body when we accidentally step on a peel of banana give reasons object of mass 1 kg travelling in a straight line with a velocity of 10 m s−1 collides with, and sticks to, a stationary wooden block of mass 5 kg. Then they both move off together in the same straight line. Calculate the total momentum just before the impact and just after the impact. Also, calculate the velocity of the combined object. object of mass 100 kg is accelerated uniformly from a velocity of 5 m to 8 m s−1 in 6 s. Calculate the initial and final momentum of the object. Also, find the magnitude of the force exerted on the object. how a karate player can break a pile of tiles with a single blow of his hand if you shake the branches of tree,the fruits fall down. why? Define inertia. On what factor does it depend? what are the different kinds of inertia? give one example of each. A ball of mass 400g dropped from a height of 5m. A boy on the ground hits the ball vertically upwards with a bat with an force of 100N so that it attains a vertically height of 20m. The time for which the ball remains in contact with the bat? (g = 10m/s2) Show that Newton's first law of motion is contained in second law ? When a carpet is beaten with a stick dust comes out. why? Copyright © 2022 Aakash EduTech Pvt. Ltd. All rights reserved. E.g: 9876543210, 01112345678 We will give you a call shortly, Thank You Office hours: 9:00 am to 9:00 pm IST (7 days a week)

Cambridge tracts in theoretical computer science series by a. His natural deduction calculus also supports a notion of analytic proof, as was shown by dag prawitz. Then there exists a function fsuch that fa 2afor each a2f. An introduction to mathematical thought processes, 6th edition. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Vlll contents 7c the converse pt proof 96 7d condensed detachment 102 8 counting a types inhabitants 108 8a inhabitants 108 8b examples of the search strategy 114 8c the search algorithm 118 8d the counting algorithm 124 8e the structure of a nfscheme 127 8f stretching, shrinking and completeness 2 9 technical details 140 9a the structure of a term 140 9b. There are a number of very good introductions to proof theory. Majorization and the realizability interpretation 99 3. This book is both a concise introduction to the central results and methods of structural proof theory, and a work. Metody dowodzenia twierdzen i automatyzacja rozumowan na. Twierdzenie o eliminacji reguly ciecia i dowody w postaciach normalnych. Volume 1 is a selfcontained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Reguly operowania spojnikami a reguly strukturalne. Proof theory is concerned almost exclusively with the study of formal proofs. Basic proof theory 2ed cambridge tracts in theoretical. That is one of the merits of categorical proof theory. Better to call a mathematician a pluralist than a formalist. G s means that there is a proof tree for s using the. In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. Metody dowodzenia twierdzen i automatyzacja rozumowan. Schwichtenberg, jul 27, 2000, computers, 417 pages. I start by discussing aspects of the dialectica interpretation from the point of view of categorical proof theory. In mathematics, the notion of a set is a primitive notion. Troelstra encyclopedia of life support systems eolss 7. How to read and do proofs actually gives you many different methods forward backwards method should be introduced in all proof books in your tool box to prepare you for tackling any proof. Examples are given of several areas of application, namely. Ii focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and heyting algebras the publication first takes a look at the topology of metric spaces, algebra, and finitetype arithmetic and theories of operators. The axiom of pair, the axiom of union, and the axiom of. String theory is a quantum theory of 1d objects called strings. It satisfies all of your conditions, but it is not an elementary book. Basic proof theory cambridge tracts in theoretical computer science. Cambridge core programming languages and applied logic basic proof theory by a. A copy of the license is included in the section entitled gnu free documentation license. Ii proof theory and constructive mathematics anne s. Basic proof methods david marker math 215, introduction to advanced mathematics, fall 2006. Studies in logic and the foundations of mathematics, volume 123. How to selfexplain to improve your understanding of a proof, there is a. Basic problems on knot theory are also explained there. Constructivism in mathematics, vol 2, volume 123 1st edition. Basic proof theory 2ed cambridge tracts in theoretical computer. There is no largest ordinal, and there is no set of all ordinals. Buy basic proof theory 2ed cambridge tracts in theoretical computer. This introduction to the basic ideas of structural proof theory contain. This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of firstorder logic. I start by discussing aspects of the dialectica interpretation. The author does an excellent job explaining things and even does proof analysis that breaks down the methods that are used. Constructivism in mathematics, vol 1, volume 121 1st edition. This proof will be omitted, though the theorem is equivalent to the axiom of choice. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. Realizabilities are powerful tools for establishing consistency and independence results for theories based on intuitionistic logic. The first is known as the mortality salience ms hypothesis. These strings come in open free endpoints and closed connected endpoints varieties. Basic proof theory download ebook pdf, epub, tuebl, mobi. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. Since the programme called for a complete formalization of the relevant parts of mathematics, including the logical steps in mathematical arguments, interest in proofs as. After reading this proof, one reader made the following selfexplanations. That is, a proof is a logical argument, not an empir. The notion of analytic proof was introduced into proof theory by gerhard gentzen for the sequent calculus. Troelstra discovered principles ect 0 and gc 1 which precisely characterize formal number and function realizability for intuitionistic arithmetic and analysis, respectively. Subsystems of set theory and second order number theory. Building on troelstras results and using his methods, we introduce the notions of church domain and domain of continuity in order to demonstrate the optimality of almost negativity in ect 0 and gc 1. Proof techniques 1 introduction to mathematical arguments. Download it once and read it on your kindle device, pc, phones or tablets. This type theory is the basis of the proof assistant nuprl10. Decidability problems for the prenex fragment of intuitionistic logic. Analyzing realizability by troelstras methods sciencedirect. Coherent sequences sen in some canonical way, beyond the natural requirement that c. This paper contains a number of loosely linked sections. Use features like bookmarks, note taking and highlighting while reading how to read and do proofs. This alone assures the subject of a place prominent in human culture. But even more, set theory is the milieu in which mathematics takes place today. Type theory talks about how things can be constructed syntax, expressions. Basic proof theory free ebook pdf file anne s troelstra. Schwichtenberg harold schellinx 1 journal of logic, language and information volume 7, pages 221 223 1998 cite this article. Basic proof theory cambridge university press introduction to proof theory lix basic proof theory, a. This book is both a concise introduction to the central results and methods of structural proof theory and a work of research that will be of interest to specialists. Ieee symposium on logic in computer science lics96, pp. Purchase constructivism in mathematics, vol 2, volume 123 1st edition. Volume 2 contains mainly advanced topics of a proof theoretical and semantical nature. Proof theory began in the 1920s as a part of hilberts program. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide. Proof techniques 1 introduction to mathematical arguments by michael hutchings. Building on troelstras results and using his methods, we introduce the. Schwichtenberg of the book basic proof theory which is published in 2001 wrote in their introduction that their intention was to fill the gap between this and all other introductionary books in proof theory. This puts type theory somewhere in between the research elds of software technology and proof theory, but there is more. Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence. Structural proof theory structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. Troelstra and schwichtenberg did not think interesting proof theory stops at cutelimination, or at gentzens elaborate proof of the consistency of arithmetic using transfinite induction tarski claimed this latter item advanced his understanding of the issue not one epsilon. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Basic proof theory propositional logic see the book by troelstra and schwichtenberg 1. Troelstra, choice sequences, a chapter of intuitionistic mathematics article pdf available in bulletin of the american mathematical society. Theory and problems of set theory and related topics schaums outline. An introduction to mathematical thought processes, 6th edition kindle edition by solow, daniel. Show that there exists a unique set c such that x2cif and only if either aand bor x2band. Proof theory is not an esoteric technical subject that was invented to support a formalist doctrine in the philosophy of mathematics. Metody dowodzenia twierdzen i automatyzacja rozumowan na poczatek. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Many tmt studies converge on three primary points that support the theory. Troelstra, choice sequences, a chapter of intuitionistic mathematics article pdf available in bulletin of the american mathematical society november 1979. For example, a deck of cards, every student enrolled in math 103, the collection of all even integers, these are all examples of sets of things.738 443 803 1346 1234 1250 199 1356 1300 821 1342 999 155 872 282 535 652 477 563 28 1256 1250 1463 1283 596 1272 890 681 1352 997 516 870 965 1281 1232 40 165 1014 347 813

Study of the Penetration and Diffusion Characteristics of Inorganic Solidified Foam in Rock Fractures. Continuous oxygen is one of the essential conditions for spontaneous combustion of a coal seam [1, 2]. Mine fire or coal fires are induced by the air leakage passages that provide oxygen for loose coal . However, the fracture channels of the fire area are often complex and crisscross each other, even extending to the high places. In recent years, increasing numbers of scholars began to study foam sealing materials, both at home and abroad, because such materials can diffuse and accumulate in the high fractures and provide stereoscopic covering. The materials of inert gas bubble , inhibitor foam [5, 6], gel foam [7-10], three-phase foam [11-13], and foamed grout [14-16] are the primary foam materials used for a coal fire. Each of the above-mentioned materials has its own advantages and disadvantages. To address the disadvantages of the above-mentioned materials, we performed research on ISF fire-fighting technology . ISF can solve the problems of cooling, plugging, insulation, and compressive strength. The fresh state of ISF (foam fluid) can accumulate to a high position as well as cover and cool high temperature coal. Simultaneously, ISF has good insulating ability, thermal stability, and adjustable coagulation time . When it is solidified, ISF has high porosity and compressive strength. The injection of ISF is a very complicated process. To ensure that the foam fluid effectively reaches the coal fire area and seals the fractures, it is necessary to conduct research on the penetration and diffusion laws of the injection process. However, because the injection of ISF is a process with strong concealment, it is difficult to obtain fluid mechanics parameters and collect internal migration images of the foam fluid itself inside the loose coal and rock fractures. In previous studies, a similar, complex model worked out by Gustafson and Stille for predicting grout penetration in real rock was examined by Fujita et al. considering 1D condition [19, 20]. The latter also reviewed various other grout penetration models developed and used in Japan and elsewhere. Yang et al. investigated the rheological properties of cement grouts with different water cement ratios and their flow in fractures . The present study deals with some of these issues and describes and discusses theoretical grout flow models for predicting flow into plane-parallel fractures of candidate cement grout materials that behave as Bingham fluids under static pressure and as Newton liquids under oscillatory pressure. However, little attention has been paid to the penetration characteristics of foam fluid in the rock fractures . Therefore, its flowing diffusion range, stacking height, and influence factors have not been well understood which causes its blind application in fire controlling. To deal with this problem and provide basis for the design parameters of ISF, this paper puts forward a perfusion experiment of ISF in a fractured rock model that was built according to the site application conditions in Luwa coal mine through actually detecting some of the key parameters of the hydrodynamics of a foam fluid in the fractures, identifying the relationship of various factors in the process of injection, and then summarizing the penetration and diffusion laws of a foam fluid inside the fractures. Combined with the diffusion morphological image of the foam fluid in different directions, the diffusion regularity of the foam fluid is deduced. 2. Experimental Procedures 2.1. The Test System. The whole test system (Figure 1) is divided into three parts: elevating perfusion apparatus, fractured rock model, and monitoring system. The fractured rock model consists of a loose coal pillar with the porosity of 0.15 and the surrounding gob with the porosity of 0.35. The ISF was prepared by the system composed of a foam generator and a self-made mixer. The prepared foam fluid was lifted in an elevating perfusion apparatus, which can cause the foam to flow with a certain pressure head, thereby enabling the fracture rock model to be infused. The main steps of model test are as follows. First, build the similarity model according to the design requirements of filling mediums, and then bury the tiny earth pressure gauges in the process of layering. Second, connect the monitoring equipment; debug the high-speed motion analyser and the static strain gauge. Third, pour the prepared ISF fluid into the fractured rock model using a designed outlet pressure based on the regulating lifting height of the elevating perfusion apparatus. Four, record the real-time diffusion image and the pressure data from the detector, and then analyse and determine the penetration and diffusion laws combined with the viscosity parameters of the ISF fluid. The interface microstructure was investigated using scanning electron microscopy (SEM) (FEI Quanta TM 250 SEM system) with the size of the test specimen being a 10 mm x 10 mm x 10 mm prism. 2.2. Arrangement of the Testing Points for Penetration Pressure Monitoring. There are ten tiny earth pressure gauges buried in the fractured rock model. The coordinates of the testing points are listed in Table 1. In the injection process of ISF, the penetration pressure of the foam fluid in the fracture channels can be monitored by the tiny earth pressure gauges embedded in the fractured rock model. The data collection interval is two seconds. The penetration test ends after all the penetration pressure monitoring data have been collected; the whole experiment takes approximately 15 minutes to complete. The size of the tiny earth pressure gauge is 15 mm (diameter) x 6.4 mm (thickness), maximum measurement value is 0.1 Mpa, and the accuracy is [+ or -] 1.0% F.S. The computation formula between fluid penetration pressure and dependent variable is as follows: P = a + b[epsilon]. (1) In formula (1), P is the value of penetration pressure, kPa; [epsilon] is the dependent variable, x[10.sup.-6][epsilon]; a and b are the offered parameters. After calibrating the foam fluid, the determined offered parameters are listed in Table 2. 3. The Test Result Analysis 3.1. The Penetration Pressure Analysis. According to the offered parameters of the 0.1 MPa tiny earth pressure gauges shown in Table 2, the penetration pressure of foam fluid can be calculated. The pressure values over time are shown in Figures 2 and 3. From Figures 2 and 3, overall, the 1#~10# monitoring point data fluctuates up and down within a certain range. In addition to the 1# monitoring point (the injection inlet), overall, the monitoring points are divided into three groups: the first group contains the 2#, 3#, and 4# monitoring points, the second group consists of the 5#, 6#, and 7# monitoring points, and the third group is composed of the 8#, 9#, and 10# monitoring points. With the increase of diffusion distance, the penetration pressure decreases, in agreement with the traditional grouting penetration diffusion law. The penetration pressure of the 1# monitoring point fluctuates with the value of 16.37 kPa, which was obtained from the beginning. The first group of 2#, 3#, and 4# monitoring points was buried separately within the loose coal pillar, the centre line, and the surrounding gob. The penetration diffusion distance of the 2#, 3#, and 4# monitoring points is 255 mm, 250 mm, and 255 mm, respectively. The 3# monitoring point first collects the penetration pressure at the moment of 120 s, and the 2# monitoring point and 4# monitoring point collect the penetration pressure at the moment of 160 s and 144 s, respectively. This phenomenon occurs primarily because the porosity of loose coal pillar is smaller than the surrounding gob, and the frictional resistance loss in the fracture channel is high. As a result, under the condition of the same 255 mm diffusion distance, it takes a longer time for the foam fluid to diffuse to the 2# monitoring point. In addition, the average penetration pressure of the 2# monitoring point is 13.13 kPa, which is smaller than the average penetration pressure of the 3# monitoring point (14.97 kPa) and the average penetration pressure of the 4# monitoring point (14.36 kPa). The set of 5#, 6#, and 7# monitoring points is buried in the same manner as the first group of 2#, 3#, and 4# monitoring points. The 5#, 6#, and 7# monitoring points are located along the centre line, the loose coal pillar, and the surrounding gob, and the corresponding diffusion distance is 510 mm, 500 mm, and 510 mm, respectively. As the first group of data, the 6# monitoring point in the centre line first collected the penetration pressure at the moment of 324 s, with the average penetration pressure value of 11.94 kPa. Next is the 7# monitoring point arranged in the goaf at 364 s, with the average penetration pressure value of 10.72 kPa. Finally, the 5# monitoring point in the loose coal pillar collects the penetration pressure at the time of 382 s: the average value of penetration pressure is 10.05 kPa. The penetration diffusion time and the penetration pressure relationship of the third group of monitoring points 8#, 9#, and 10# are similar to those of the first two groups. The 8#, 9#, and 10# monitoring points' penetration diffusion distances are 776 mm, 750 mm, and 776 mm, respectively. The 9# monitoring point in the centre line first detected the penetration pressure with the value of 7.63 kPa at the moment of 644 s. Subsequently, the 10# monitoring point arranged in the goaf detected the penetration pressure at 712 s: the average penetration pressure value is 5.98 kPa. Finally, the 5# monitoring points in the loose coal pillar detected the penetration pressure at 734 s: the average value is 5.52 kPa. Although the penetration diffusion distance difference of the three groups is approximately 250 mm, the penetration diffusion time and the penetration pressure average value difference of the three groups are increasing because the foam fluid diffuses in loose coal and rock fracture channel with energy loss, thereby causing the speed to be increasingly low. Therefore, under the same penetration diffusion distance, the time required for the latter half of the distance becomes increasingly long, and the pressure difference becomes increasingly high. 3.2. Penetration Diffusion Image Analysis. Here, the data from the monitoring points buried within the model is analysed. To more vividly illustrate the diffusion form of the foam fluid, a high-speed motion analyser was adopted in the whole experiment; the X-Y plane of the foam fluid penetration diffusion pattern is shown in Figure 4. From Figure 4, with the increase of diffusion time, the diffusion form of the foam fluid in the X-Y plane becomes ellipsoidal. The diffusion velocity in the loose coal pillar and the surrounding gob is not the same, as the diffusion area of the foam fluid in the surrounding gob is larger than that of the loose coal pillar. This difference is primarily because the fracture porosity of the loose coal pillar is small, causing the foam fluid to experience greater free diffusion resistance. From the diffusion area size at different moments, the diffusion area is increasing with time, but the diffusion area of growth becomes smaller. This behaviour occurs primarily because the foam fluid is a time-varying Bingham fluid, resulting in the viscosity increasing from 4360 MPa-s at 60 s to 4451 MPa-s at 360 s, causing the foam fluid flows to slow. Simultaneously, with the increase of the diffusion range, the penetration pressure drop of the foam fluid in the complex fracture network is gradually reduced and the front diffusion of foam fluid slows. In addition, the base material in the pore wall of the foam fluid undergoes hydration and condensation reactions [23, 24], thus causing the stability of the bubble to enhance that in turn makes it difficult to change the shape of the structure. In the SEM image obtained from the bubble wall (Figure 5), we can observe that the rod-shaped ettringite crystals fill the capillary pores. Surface products such as C-S-H gel can be observed as the major ISF microstructure component. CH as a pore product with a polycrystalline shape is another dominant cement hydration product. The phenomenon results in a large quantity of bubbles uniting into a group. The bubbles in the group flow as a whole and block each other. The fluidity of the foam fluid reduces gradually. Figure 6 shows the condensation effect of foam fluid in a coal and rock fracture surface after 3 hours. The figure reveals that the foam fluid can commendably diffuse and cover the fracture surface, and the foam can form a layer of a certain thickness, effectively isolating the coal and oxygen compounds. To perform in-depth analysis of the internal penetration diffusion of the ISF in the fractured rock model, the XZ and YZ sections were recorded by the CCD camera, as shown in Figures 7 and 8, respectively. From Figure 7, in the XZ penetration section, with the increase of the penetration diffusion distance, the foam fluid penetration in the fracture channel changes from dense to loose in the X direction, mainly because the penetration pressure and the velocity of the foam fluid are high near the perfusion inlet, thereby enabling the foam to diffuse fully. However, the penetration pressure loss of the foam fluid is higher when the position is further away from the perfusion inlet. From the Z direction, the fracture penetration is fuller at the same Z height of the perfusion inlet; the next place is above the plane of the perfusion inlet, and the penetration diffusion effect is the worst in the sections under the height of the perfusion inlet. It indicates that the penetration diffusion ability of the foam fluid is stronger in its initial direction of velocity vector than the direction of the vector upward or downward. Upward penetration is fuller than the downward penetration because the peripheral interface cracks in the process of planar penetration channels are open and the crack channel resistance is low, indicating that the foam fluid does not act in the same manner as normal cement slurry. This difference in downward penetration mainly occurs because the density of foam fluid is smaller than that of ordinary cement, with the value being approximately 1/5 the density of a common slurry. As a result, the gravitational effect on the foam fluid is not obvious. Another primary factor is that the foam fluid is composed of bubbles; as a result, adhesion of cement and fly ash particles occurs in the bubble hole wall or the particles exist in a foam liquid membrane to form a skeleton, thereby preventing gravity settling of the particles. The size of a single bubble is approximately 400 microns, and each bubble links together, thereby causing the bubble size to be greater than a single general cement or fly ash particle in the slurry. Form the penetration and diffusion in the YZ section shown in Figure 8, the left is loose and the right is dense in the Y direction, largely because the left part is the loose coal pillar model with the porosity of 0.15 and the right part is the surrounding gob model with the porosity of 0.35. As the perfusion inlet is in the cross section, the foam fluid will be more easily diffuse into the right side with large porosity. In the Z direction, the distribution of penetration and diffusion is uniform, except the top with the surface penetration, mainly because perfusion inlet of foam fluid is in the X direction, which is vertical to the YZ section. The penetration and diffusion of foam fluid in this YZ section occurred when the fracture has been blocked by the X direction penetration. The up and down diffusion process was mainly affected by the width of fracture in the Z direction. Thus, in the YZ plane, the penetration and diffusion results show that the left part is loose and right part is dense. 3.3. Penetration and Diffusion Law. In the grouting engineering field, there is a large difference for different flow patterns of slurry diffuse in the geotechnical engineering [25-27]. To meet the requirements of engineering, reasonable grouting parameters for different slurry must be determined. However, the determination of the grouting parameters definitely requires guidance from grouting theory. The relatively mature diffusion grouting theory is based on the spherical, cylindrical, and sleeve valve pipe models [28-31]. The effective diffusion radius formula is only applicable to a Newtonian fluid. However, the inorganic solidified foam is a Bingham fluid, which is a non-Newtonian fluid; as a result, such foam cannot be modelled by the above formulas. Therefore, in this paper, the diffusion theory of the ISF in porous media with fracture channels is derived. As the ISF fluid diffuses completely into the fractures channel, we assume that it is a whole unit mass of foam fluid that reaches a certain position along a fracture channel. Therefore, we can draw lessons from the model of a Bingham fluid in pipe penetration , as shown in Figure 9. Assuming that the fracture radius is [r.sub.0], a foam fluid unit is taken from the centre line position of the fracture channel; its length is dZ and its radius is r. The pressure of the left and right ends of the foam fluid unit dZ is p + dp and p, respectively. The differential pressure is dp. The shear stress around the foam fluid unit surface by the left direction (in the opposite direction of flow velocity) is [tau]. Thus, we obtain the balance equation for flow stress of a foam fluid unit as follows: [pi][r.sup.2]dp = -2[pi]r[tau]dl. (2) In the region of 0 [less than or equal to] r [less than or equal to] [r.sub.p], the foam fluid column is stationary relative to the adjacent layer fluid. The foam fluid moves in a piston as a whole, and the movement velocity is v = [v.sub.p]. In the region of [r.sub.p] [less than or equal to] r [less than or equal to] [r.sub.0], the foam fluid column is moving relative to the adjacent layer fluid. So the following equation can be obtained: [r.sub.p] = -2[[tau].sub.s] x dl/dp. (3) The basic rheological equation of a Bingham fluid slurry can be expressed as [tau] = [[tau].sub.s] + [[eta].sub.p][gamma]. (4) Based on (2) and (4), the following can be obtained: [gamma] = -dv/dr = ([tau] - [[tau].sub.s])/[[eta].sub.p] = -[1/[[eta].sub.p]] x (r/2 x [d.sub.p]/dl + [[tau].sub.s]). (5) Consider boundary conditions r = [r.sub.0], v = 0 and then calculate integral for both sides of (5). [mathematical expression not reproducible]. (6) For 0 [less than or equal to] r [less than or equal to] [r.sub.p], [v.sub.p] = -1/[[eta].sub.p] [1/4 dp/dl ([r.sup.2.sub.0] - [r.sup.2.sub.p]) + [[tau].sub.s] ([r.sub.0] - [r.sub.p])]. (7) Therefore, the flow of fracture channel per unit time is the sum of the shear zone and the piston area, namely, [mathematical expression not reproducible]. (8) The average flow velocity of the section of fracture channel can be expressed as V = K/[beta] (-dp/dl) [1 - 4/3 ([lambda]/-dp/dl) + 1/3[([lambda]/- dp/dl).sup.4]]. (9) We make three assumptions for the penetration and diffusion of foam fluid in the fracture channel: (1) the coal pillar and circumferential crack area are all homogeneous and isotropic; (2) the foam fluid is a Bingham fluid; and (3) the spreading form of the foam fluid is spherical diffusion. In the injection process of a foam fluid, the injection content, Q, satisfies Q = VAt. (10) In (10), A is a sphere when the foam fluid diffuses in the fracture channel, A = 4[pi][l.sup.2]; t is the grouting time; -dp/dl is much larger than [lambda] in the injection process. Thus, (9) can be simplified as V = K/[beta] (-dp/dl) [1 - 4/3 ([lambda]/-dp/dl)]. (11) Combining (2) and (4) and then calculating the integral for both sides of the equation, the following is obtained: p = Q[beta]/4[pi]tKl - 4/3 [lambda]l + c. (12) Considering the boundary conditions of injecting foam fluid (i.e., for p = p0, I = l0, and for p = p1, I = l1), the following is obtained: [DELTA]p = [empty set][beta]/3tK[l.sub.0] [l.sup.3.sub.1] - [empty set] [beta]/3tK [l.sup.2.sub.1] + 4/3 [lambda][l.sub.1] - 4/3[lambda][l.sub.0]. (13) Equation (13) is the calculation formula of the effective diffusion radius for foam fluid diffusing in the fracture channel, where [DELTA]p is the penetration pressure difference between two monitoring points in the fracture channel; [empty set] is the porosity: the porosity of coal pillar in the test model is 0.15 and the porosity of the goaf is 0.35; [beta] is the viscosity ratio of foam fluid and water; t is the grouting time; is the permeability coefficient; [lambda] is the ratio of two times the static shear force and the fracture channel radius; [l.sub.0] is the radius of injection inlet with the value of 15 mm; [l.sub.1] is the penetration and diffusion distance. Penetration time is a major issue in the grouting of rock fractures. The foam fluid state will be calculated as a time function, making a distinction between flow rate control and pressure control. The indoor environment temperature is 10[degrees]C, and the viscosity of the water is 1.3077 Mpa x s, based on the viscosity of the water table. The viscosity of the ISF fluid was measured using a NDJ-5s rotary viscometer, setting the rotor number to 2 and the rotor speed to 6. The viscosity varies with time, as shown in Figure 10. The change of viscosity over time was fitted in Figure 10; the fitting function obtained is given by u = a[e.sup.bt] + c, (14) where a = 0.033, b = 0.166, and c = 4.335; the correlation coefficient is [R.sup.2] = 0.98. As a result, we can obtain the viscosity ratio of foam fluid and water using [beta] = [u.sub.ISF]/[u.sub.W] = [a[e.sup.bt] + c]/[u.sub.W] = [[0.033e.sup.0166t] + 4.335]/[1.3077 x [10.sup.-3]]. (15) From Figures 2 and 3, the average penetration pressure and the diffuse time of monitoring points 1#-10# are listed in Table 3. Based on (13), the relationship among the differential pressure of the 1# monitoring point, the diffusion time, and the diffusion distance can be expressed as [mathematical expression not reproducible]. (16) We substituted the related parameters in the Table 3 into (16) to obtain [mathematical expression not reproducible]. (17) To further amend the formula of effective diffusion distance of foam fluid, the diffusion time and diffusion distance of the remaining monitoring points 2#, 4#, 5#, 7#, 8#, 9#, and 10# were substituted into the formula, and then the predictive values were calculated. The comparison of the predicted values and the test values is provided in Table 4. From Table 4, the predicted values were overall in accord with the experimental results. The predicted values were all less than the experimental results; this difference occurs mainly because the foam fluid impacts the tiny earth pressure gauges and causes it to be slightly mobile. As a result, the follow-up monitoring data may contain some errors. However, because all the relative error of monitoring is within 10%, the prediction formula is reasonable. Thus, we can obtain the relations of the diffusion distance, penetration pressure difference, and diffusion time given by (18). Based on it, it can be obtained that the penetration pressure difference between two monitoring points ([DELTA]p) was affected by the porosity ([empty set]) and the penetration and diffusion distance ([l.sub.1]). In Figure 7, the porosity of the model is the same. With the increase of the penetration and diffusion distance, the penetration pressure near the perfusion inlet is higher than the position which is further away from the perfusion inlet. For the same penetration and diffusion distance, the greater the porosity is, the smaller the seepage pressure difference is, and the greater the seepage pressure is, which can prove the diffusion morphology in Figure 8. [DELTA]p = [empty set]((a[e.sup.bt] + c)/[u.sub.W])/3tK[l.sub.0] [l.sup.3] - [empty set]((a[e.sup.bt] + c)/[u.sub.W])/3tK [l.sup.2] + 4/3[lambda]l - 4/3[lambda][l.sub.0]. (18) A similar fractured rock model was built according to the coal pillar and the surrounding gob. The penetration pressure, image, and law in the complex fracture were investigated. Some conclusions can be drawn as follows. (1) The pressure fluctuated up and down within a certain range overall. With the increase of diffusion distance, the penetration pressure decreased, consistent with the traditional grouting penetration diffusion law. Affected by the porosity of filling medium, a longer time is required to monitor the penetration pressure for the loose coal pillar than the gob, and the value of penetration pressure in loose coal pillar was lower than that of the gob at the same diffusion distance. (2) The diffusion form of the foam fluid in the X-Y plane is ellipsoidal, with the diffusion area increasing with time, although the diffusion area of growth becomes smaller with time; in the XZ section, the foam fluid diffusion in the fracture channel is from dense to loose in the direction, and, from the direction, the fracture penetration is fuller at the same Z height of the injection inlet; in the YZ section, the left part is loose and the right part is dense in the direction, and the distribution of penetration and diffusion is uniform, except the top, with surface penetration. (3) The viscosity of the ISF fluid was measured by a NDJ5s rotary viscometer. 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Yi Lu, (1, 2, 3) Tao Wang, (3) and Qing Ye (3) (1) Work Safety Key Lab on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, Hunan University of Science and Technology, Xiangtan, Hunan (411201), China (2) Hunan Province Key Laboratory of Safe Mining Techniques of Coal Mines, Hunan University of Science and Technology, Xiangtan, Hunan (411201), China (3) School of Resource, Environment and Safety Engineering, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China Correspondence should be addressed to Yi Lu; firstname.lastname@example.org Received 5 July 2017; Revised 12 August 2017; Accepted 22 August 2017; Published 28 September 2017 Academic Editor: Marco Rossi Caption: Figure 1: The test system. Caption: Figure 2: The change of penetration pressure over time from monitoring points 1#-5#. Caption: Figure 3: The change of penetration pressure over time from monitoring points 6#-10#. Caption: Figure 4: The penetration and diffusion form of the foam fluid in the X-Y plane. Caption: Figure 5: SEM image obtained from the bubble wall. Caption: Figure 6: Condensation effect of a fluid penetration diffusion front in the coal and rock surface after 3 hours. Caption: Figure 7: Penetration and diffusion form of the foam fluid in the X-Z section. Caption: Figure 8: Penetration and diffusion form of the foam fluid in the Y-Z section. Caption: Figure 9: Schematic of the column flow of a unit of foam fluid in a fracture channel. Caption: Figure 10: Foam fluid viscosity varies over time. Table 1: Coordinates of the monitoring points. Number of the tiny X coordinate Y coordinate Z coordinate earth pressure gauges (mm) (mm) (mm) 1# 0 0 300 2# 232 93 250 3# 250 0 250 4# 232 -93 250 5# 464 186 200 6# 500 0 200 7# 464 -186 200 8# 696 279 100 9# 750 0 100 10# 696 -279 100 Table 2: Offered parameters of the 0.1 MPa tiny earth pressure gauges. Number of the tiny Offered parameters earth pressure gauges a h 1# -0.80875 0.06873 2# -1.58700 0.05042 3# -1.44946 0.05691 4# -1.22361 0.05934 5# -0.92195 0.06540 6# -1.43612 0.06260 7# -1.02616 0.06676 8# -0.34060 0.08350 9# -1.11560 0.06587 10# -1.02340 0.06231 Table 3: Penetration pressure and diffusion time of the monitoring points. Monitoring points 1# 2# 3# 4# 5# 6# Average penetration 16.47 13.13 14.97 14.36 10.05 11.94 pressure (kPa) Diffusion time (s) 0 160 120 144 382 324 Differential pressure 0 3.24 1.5 2.01 6.32 4.53 with 1# (kPa) Viscosity of ISF 4.360 4.370 4.360 4.365 4.451 4.448 fluid (Pa x s) Diffusion distance 0 255 250 255 510 500 (mm) Porosity 0.35 0.15 0.35 0.35 0.15 0.35 Monitoring points 7# 8# 9# 10# Average penetration 10.72 5.52 7.63 5.98 pressure (kPa) Diffusion time (s) 364 734 644 712 Differential pressure 5.65 10.85 8.74 10.39 with 1# (kPa) Viscosity of ISF 4.454 4.557 4.523 4.548 fluid (Pa x s) Diffusion distance 510 776 750 776 (mm) Porosity 0.35 0.15 0.35 0.35 Table 4: Comparison of the predicted values and the test values. Monitoring point 2# 4# 5# 7# 8# 9# 10# Predicted value 2.94 1.92 6.05 5.29 9.93 7.96 9.51 (kPa) Test value (kPa) 3.24 2.01 6.32 5.65 10.85 8.74 10.39 Relative error (%) 9.25 4.48 4.27 6.37 8.47 8.92 8.46 |Printer friendly Cite/link Email Feedback| |Title Annotation:||Research Article| |Author:||Lu, Yi; Wang, Tao; Ye, Qing| |Publication:||Advances in Materials Science and Engineering| |Date:||Jan 1, 2017| |Previous Article:||Semianalytical Solution and Parameters Sensitivity Analysis of Shallow Shield Tunneling-Induced Ground Settlement.| |Next Article:||Mode II Fracture of GFRP Laminates Bonded Interfaces under 4-ENF Test.|

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played? Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34? A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . . A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter. The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . . There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island... How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . . In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct. Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . . Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why! Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to. . . . A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they? This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . . ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm. Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . . Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation? Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat. . . . Show that among the interior angles of a convex polygon there cannot be more than three acute angles. You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . . Can you find all the 4-ball shuffles? The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . . If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . . Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. A huge wheel is rolling past your window. What do you see? Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees? This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . . Here are some examples of 'cons', and see if you can figure out where the trick is. Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total. Replace each letter with a digit to make this addition correct. I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades? Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches. Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice? I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour? Can you fit Ls together to make larger versions of themselves? Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? What are the missing numbers in the pyramids? Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . . A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . . You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance? Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?

Advances in Antenna Array Processing for RadarView this Special Issue Array Processing for Radar: Achievements and Challenges Array processing for radar is well established in the literature, but only few of these algorithms have been implemented in real systems. The reason may be that the impact of these algorithms on the overall system must be well understood. For a successful implementation of array processing methods exploiting the full potential, the desired radar task has to be considered and all processing necessary for this task has to be eventually adapted. In this tutorial paper, we point out several viewpoints which are relevant in this context: the restrictions and the potential provided by different array configurations, the predictability of the transmission function of the array, the constraints for adaptive beamforming, the inclusion of monopulse, detection and tracking into the adaptive beamforming concept, and the assessment of superresolution methods with respect to their application in a radar system. The problems and achieved results are illustrated by examples from previous publications. Array processing is well established for radar. Publications of this topic have appeared for decades, and one might question what kind of advances we may still expect now. On the other hand, if we look at existing radar systems we will find very few methods implemented from the many ideas discussed in the literature. The reason may be that all processing elements of a radar system are linked, and it is not very useful to simply implement an isolated algorithm. The performance and the property of any algorithm will have an influence on the subsequent processing steps and on the radar operational modes. Predictability of the system performance with the new algorithms is a key issue for the radar designer. Advanced array processing for radar will therefore require to take these interrelationships into account and to adapt the related processing in order to achieve the maximum possible improvement. The standard handbooks on radar [1, 2] do not mention this problem. The book of Wirth is an exception and mentions a number of the array processing techniques described below. In this tutorial paper, viewpoints are presented which are relevant for the implementation of array processing methods. We do not present any new sophisticated algorithms, but for the established algorithms we give examples of the relations between array processing and preceding and subsequent radar processing. We point out the problems that have to be encountered and the solutions that need to be developed. We start with spatial sampling, that is, the antenna array that has to be designed to fulfill all requirements of the radar system. A modern radar is typically a multitasking system. So, the design of the array antenna has to fulfill multiple purposes in a compromise. In Section 3, we briefly review the approaches for deterministic pattern shaping which is the standard approach of antenna-based interference mitigation. It has the advantage of requiring little knowledge about the interference scenario, but very precise knowledge about the array transfer function (“the array manifold”). Adaptive beamforming (ABF) is presented in Section 4. This approach requires little knowledge about the array manifold but needs to estimate the interference scenario from some training data. Superresolution for best resolution of multiple targets is sometimes also subsumed under adaptive beamforming as it resolves everything, interference and targets. These methods are considered in Section 5. We consider superresolution methods here solely for the purpose of improved parameter estimation. In Section 6, we briefly mention the canonical extension of ABF and superresolution to space-time array processing. Section 7 is the final and most important contribution. Here we point out how direction estimation must be modified if adaptive beams are used, and how the radar detector, the tracking algorithm, and the track management should be adapted for ABF. 2. Design Factors for Arrays Array processing starts with the array antenna. This is hardware and is a selected construction that cannot be altered. It must therefore be carefully designed to fulfill all requirements. Digital array processing requires digital array outputs. The number and quality of these receivers (e.g., linearity and number of ADC bits) determine the quality and the cost of the whole system. It may be desirable to design a fully digital array with AD-converters at each antenna element. However, weight and cost will often lead to a system with reduced number of digital receivers. On the other hand, because of the -decay of the received power, radar needs antennas with high gain and high directional discrimination, which means arrays with many elements. There are different solutions to solve this contradiction.(i)Thinned arrays: the angular discrimination of an array with a number of elements can be improved by increasing the separation of the elements and thus increasing the aperture of the antenna. Note that the thinned array has the same gain as the corresponding fully filled array.(ii)Subarrays: element outputs are summed up in an analog manner into subarrays which are then AD-converted and processed digitally. The size and the shape of the subarrays are an important design criterion. The notion of an array with subarrays is very general and includes the case of steerable directional array elements. 2.1. Impact of the Dimensionality of the Array Antenna elements may be arranged on a line (1-dimensional array), on a plane (a ring or a 2-dimensional planar array), on a curved surface (conformal array), or within a volume (3D-array, also called Crow’s Nest antenna, [3, Section ]). A 1-dimensional array can only measure one independent angle; 2D and 3D arrays can measure the full polar coordinates in . Antenna element design and the need for fixing elements mechanically lead to element patterns which are never omnidirectional. The elements have to be designed with patterns that allow a unique identification of the direction. Typically, a planar array can only observe a hemispherical half space. To achieve full spherical coverage, several planar arrays can be combined (multifacetted array), or a conformal or volume array may be used. 2.1.1. Arrays with Equal Patterns For linear, planar, and volume arrays, elements with nearly equal patterns can be realized. These have the advantage that the knowledge of the element pattern is for many array processing methods not necessary. An equal complex value can be interpreted as a modified target complex amplitude, which is often a nuisance parameter. More important is that, if the element patterns are really absolutely equal, any cross-polar components of the signal are in all channels equal and fulfill the array model in the same way as the copolar components; that is, they produce no error effect. 2.1.2. Arrays with Unequal Array Patterns This occurs typically by tilting the antenna elements as is done for conformal arrays. For a planar array, this tilt may be used to realize an array with polarization diversity. Single polarized elements are then mounted with orthogonal alignment at different positions. Such an array can provide some degree of dual polarization reception with single channel receivers (contrary to more costly fully polarimetric arrays with receivers for both polarizations for each channel). Common to arrays with unequal patterns is that we have to know the element patterns for applying array processing methods. The full element pattern function is also called the array manifold. In particular, the cross-polar (or short -pol) component has a different influence for each element. This means that if this component is not known and if the -pol component is not sufficiently attenuated, it can be a significant source of error. 2.2. Thinned Arrays To save the cost of receiving modules, sparse arrays are considered, that is, with fewer elements than the full populated grid. Because such a “thinned array” spans the same aperture, it has the same beamwidth. Hence, the angular accuracy and resolution are the same as the fully filled array. Due to the gaps, ambiguities or at least high sidelobes may arise. In early publications like , it was advocated to simply take out elements of the fully filled array. It was early recognized that this kind of thinning does not imply “sufficiently random” positions. Random positions on a grid as used in [3, Chapter 17] can provide quite acceptable patterns. Note that the array gain of a thinned array with elements is always , and the average sidelobe level is . Today, we know from the theory of compressed sensing that a selection of sufficiently random positions can produce a unique reconstruction of a not too large number of impinging wave fields with high probability . 2.3. Arrays with Subarrays If a high antenna gain with low sidelobes is desired one has to go back to the fully filled array. For large arrays with thousands of elements, the large number of digital channel constitutes a significant cost factor and a challenge for the resulting data rate. Therefore, often subarrays are formed, and all digital (adaptive) beamforming and sophisticated array processing methods are applied to the subarray outputs. Subarraying is a very general concept. At the elements, we may have phase shifters such that all subarrays are steered into a given direction and we may apply some attenuation (tapering) to influence the sidelobe level. The sum of the subarrays then gives the sum beam output. The subarrays can be viewed as a superarray with elements having different patterns steered into the selected direction. The subarrays should have unequal size and shape to avoid grating effects for subsequent array processing, because the subarray centers constitute a sparse array (for details, see ). The principle is indicated in Figure 1, and properties and options are described in [6, 7]. In particular, one can also combine new subarrays at the digital level or distribute the desired tapering over the analog level and various digital levels . Beamforming using subarrays can be mathematically described by a simple matrix operation. Let the complex array element outputs be denoted by . The subarray forming operation is described by a subarray forming matrix by which the element outputs are summed up as . For subarrays and antenna elements, is of size . Vectors and matrices at the subarray outputs are denoted by the tilde. Suppose we steer the array into a look direction by applying phase shifts and apply additional amplitude weighting at the elements (real vector of length ) for a sum beam with low sidelobes, then we have a complex weighting which can be included in the elements of the matrix . Here, denotes the centre frequency, , denote the coordinates of the th array element, denotes the velocity of light, and , denote the components of the unit direction vector in the planar antenna -coordinate system. The beams are formed digitally with the subarray outputs by applying a final weighting as In the simplest case, consists of only ones. The antenna pattern of such a sum beam can then be written as where and denotes the plane wave response at the subarray outputs. All kinds of beams (sum, azimuth and elevation difference, guard channel, etc.) can be formed from these subarray outputs. We can also scan the beam digitally at subarray level into another direction, . Figure 2 shows a typical planar array with 902 elements on a triangular grid with 32 subarrays. The shape of the subarrays was optimized by the technique of such that the difference beams have low sidelobes when a −40 dB Taylor weighting is applied at the elements. We will use this array in the sequel for presenting examples. An important feature of digital beamforming with subarrays is that the weighting for beamforming can be distributed between the element level (the weighting incorporated in the matrix ) and the digital subarray level (the weighting ). This yields some freedom in designing the dynamic range of amplifiers at the elements and the level of the AD-converter input. This freedom also allows to normalize the power of the subarray outputs such that . As will be shown in Section 4, this is also a reasonable requirement for adaptive interference suppression to avoid pattern distortions. 2.4. Space-Time Arrays Coherent processing of a time series can be written as a beamforming procedure as in (1). For a time series of array snapshots , we have therefore a double beamforming procedure of the space-time data matrix of the form where , denote the weight vectors for spatial and temporal beamforming, respectively. Using the rule of Kronecker products, (3) can be written as a single beamforming operation where is a vector obtained by stacking all columns of the matrix on top. This shows that mathematically it does not matter whether the data come from spatial or temporal sampling. Coherent processing is in both cases a beamforming-type operation with the correspondingly modified beamforming vector. Relation (4) is often exploited when spatial and temporal parameters are dependent (e.g., direction and Doppler frequency as in airborne radar; see Section 6). 3. Antenna Pattern Shaping Conventional beamforming is the same as coherent integration of the spatially sampled data; that is, the phase differences of a plane wave signal at the array elements are compensated, and all elements are coherently summed up. This results in a pronounced main beam when the phase differences match with the direction of the plane wave and result in sidelobes otherwise. The beam shape and the sidelobes can be influenced by additional amplitude weighting. Let us consider the complex beamforming weights , . The simplest way of pattern shaping is to impose some bell-shaped amplitude weighting over the aperture like (for suitable constants , ), or . The foundation of these weightings is quite heuristical. The Taylor weighting is optimized in the sense that it leaves the conventional (uniformly weighted) pattern undistorted except for a reduction of the first sidelobes below a prescribed level. The Dolph-Chebyshev weighting creates a pattern with all sidelobes equal to a prescribed level. Figure 3 shows examples of such patterns for a uniform linear array with 40 elements. The taper functions for low sidelobes were selected such that the 3 dB beamwidth of all patterns is equal. The conventional pattern is plotted for reference showing how tapering increases the beamwidth. Which of these taperings may be preferred depends on the emphasis on close in and far off sidelobes. Another point of interest is the dynamic range of the weights and the SNR loss, because at the array elements only attenuations can be applied. One can see that the Taylor tapering has the smallest dynamic range. For planar arrays the efficiency of the taperings is slightly different. (a) Low sidelobe patterns of equal beamwidth (b) Taper functions for low sidelobes The rationale for low sidelobes is that we want to minimize some unknown interference power coming over the sidelobes. This can be achieved by solving the following optimization problem, : denotes the angular sector where we want to influence the pattern, for example, the whole visible region , and is a weighting function which allows to put different emphasis on the criterion in different angular regions. The solution of this optimization is For the choice of the function , we remark that for a global reduction of the sidelobes when , one should exclude the main beam from the minimization by setting on this set of directions (in fact, a slightly larger region is recommended, e.g., the null-to-null width) to allow a certain mainbeam broadening. One may also form discrete nulls in directions by setting . The solution of (5) then can be shown to be This is just the weight for deterministic nulling. To avoid insufficient suppression due to channel inaccuracies, one may also create small extended nulls using the matrix . The form of these weights shows the close relationship to the adaptive beamforming weights in (11) and (17). An example for reducing the sidelobes in selected areas where interference is expected is shown in Figure 4. This is an application from an airborne radar where the sidelobes in the negative elevation space have been lowered to reduce ground clutter. 4. Adaptive Interference Suppression Deterministic pattern shaping is applied if we have rough knowledge about the interference angular distribution. In the sidelobe region, this method can be inefficient because the antenna response to a plane wave (the vector ) must be exactly known which is in reality seldom the case. Typically, much more suppression is applied than necessary with the price paid by the related beam broadening and SNR loss. Adaptive interference suppression needs no knowledge of the directional behavior and suppresses the interference only as much as necessary. The proposition for this approach is that we are able to measure or learn in some way the adaptive beamforming (ABF) weights. In the sequel, we formulate the ABF algorithms for subarray outputs as described in (1). This includes element space ABF for subarrays containing only one element. 4.1. Adaptive Beamforming Algorithms Let us first suppose that we know the interference situation; that is, we know the interference covariance matrix . What is the optimum beamforming vector ? From the Likelihood Ratio test criterion, we know that the probability of detection is maximized if we choose the weight vector that maximizes the signal-to-noise-plus-interference ratio (SNIR) for a given (expected) signal , The solution of this optimization is is a free normalization constant and denotes interference and receiver noise. This weighting has a very intuitive interpretation. If we decompose and apply this weight to the data, we have . This reveals that ABF does nothing else but a pre-whiten and match operation: if contains only interference, that is, , then , the prewhitening operation; the operation of on the (matched) signal vector restores just the matching necessary with the distortion from the prewhitening operation. This formulation for weight vectors applied at the array elements can be easily extended to subarrays with digital outputs. As mentioned in Section 2.3, a subarrayed array can be viewed as a superarray with directive elements positioned at the centers of the subarrays. This means that we have only to replace the quantities , by , . However, there is a difference with respect to receiver noise. If the noise at the elements is white with covariance matrix it will be at subarray outputs with covariance matrix . Adaptive processing will turn this into white noise. Furthermore, if we apply at the elements some weighting for low sidelobes, which are contained in the matrix , ABF will reverse this operation by the pre-whiten and match principle and will distort the low sidelobe pattern. This can be avoided by normalizing the matrix such that . This can be achieved by normalizing the element weight as mentioned in Section 2.3 (for nonoverlapping subarrays). Sometimes interference suppression is realized by minimizing only the jamming power subject to additional constraints, for example, , for suitable vectors and numbers , . Although this is an intuitively reasonable criterion, it does not necessarily give the maximum SNIR. For certain constraints however both solutions are equivalent. The constrained optimization problem can be written in general terms as and it has the solution Examples of special cases are as follows:(i)Single unit gain directional constraint: . This is obviously equivalent to the SNIR-optimum solution (9) with a specific normalization.(ii)Gain and derivative constraint: , with suitable values of the Lagrange parameters , . A derivative constraint is added to make the weight less sensitive against mismatch of the steering direction.(iii)Gain and norm constraint: , . The norm constraint is added to make the weight numerically stable. This is equivalent to the famous diagonal loading technique which we will consider later.(iv)Norm constraint only: . Without a directional constraint the weight vector produces a nearly omnidirectional pattern, but with nulls in the interference directions. This is also called the power inversion weight, because the pattern displays the inverted interference power. As we mentioned before, fulfilling the constraints may imply a loss in SNIR. Therefore, several techniques have been proposed to mitigate the loss. The first idea is to allow a compromise between power minimization and constraints by introducing coupling factors and solve a soft constraint optimization with . The solution of the soft-constraint optimization is One may extend the constrained optimization by adding inequality constraints. This leads to additional and improved robustness properties. A number of methods of this kind have been proposed, for example, in [9–12]. As we are only presenting the principles here we do not go into further details. The performance of ABF is often displayed by the adapted antenna pattern. A typical adapted antenna pattern with 3 jammers of 20 dB SNR is shown in Figure 5(a) for generic array of Figure 2. This pattern does not show how the actual jamming power and the null depth play together. (a) Adapted antenna pattern Plots of the SNIR are better suited for displaying this effect. The SNIR is typically plotted for varying target direction while the interference scenario is held fixed, as seen in Figure 5(b). The SNIR is normalized to the SNR in the clear absence of any jamming and without ABF. In other words, this pattern shows the insertion loss arising from the jamming scenario with applied ABF. The effect of target and steering direction mismatch is not accounted for in the SNIR plot. This effect is displayed by the scan pattern, that is, the pattern that arises if the adapted beam scans over a fixed target and interference scenario. Such a plot is rarely shown because of the many parameters to be varied. In this context, we note that for the case that the training data contains the interference and noise alone the main beam of the adapted pattern is fairly broad similar to the unadapted sum beam and is therefore fairly insensitive to pointing mismatch. How to obtain an interference-alone covariance matrix is a matter of proper selection of the training data as mentioned in the following section. Figure 5 shows the case of an untapered planar antenna. The first sidelobes of the unadapted antenna pattern are at −17 dB and are nearly unaffected by the adaptation process. If we have an antenna with low sidelobes, the peak sidelobe level is much more affected; see Figure 6. Due to the tapering we have a loss in SNIR of 1.7 dB compared to the reference antenna (untapered without ABF and jamming). (a) Adapted antenna pattern 4.2. Estimation of Adaptive Weights In reality, the interference covariance matrix is not known and must be estimated from some training data . To avoid signal cancellation, the training data should only contain the interference alone. If we have a continuously emitting interference source (noise jammer) one may sample immediately after or before the transmit pulse (leading or rear dead zone). On the other hand, if we sample the training data before pulse compression the desired signal is typically much below the interference level, and signal cancellation is negligible. Other techniques are described in . The maximum likelihood estimate of the covariance matrix is then This is called the Sample Matrix Inversion algorithm (SMI). The SMI method is only asymptotically a good estimate. For small sample size, it is known to be not very stable. For matrix invertibility, we need at least samples. According to Brennan’s Rule, for example, , one needs samples to obtain an average loss in SNIR below 3 dB. For smaller sample size, the performance can be considerably worse. However, by simply adding a multiple of the identity matrix to the SMI estimate, a close to optimum performance can be achieved. This is called the loaded sample matrix estimate (LSMI) The drastic difference between SMI and LSMI is shown in Figure 7 for the planar array of Figure 2 for three jammers of 20 dB input JNR with 32 subarrays and only 32 data snapshots. For a “reasonable” choice of the loading factor (a rule of thumb is for an untapered antenna) we need only snapshots to obtain a 3 dB SNIR loss, if denotes the number of jammers (dominant eigenvalues) present, . So the sample size can be considerably lower than the dimension of the matrix. The effect of the loading factor is that the dynamic range of the small eigenvalues is compressed. The small eigenvalues possess the largest statistical fluctuation but have the greatest influence on the weight fluctuation due to the matrix inversion. One may go even further and ignore the small eigenvalue estimates completely; that is, one tries to find an estimate of the inverse covariance matrix based on the dominant eigenvectors and eigenvalues. For high SNR, we can replace the inverse covariance matrix by a projection matrix. Suppose we have jammers with amplitudes in directions . If the received data has the form , or short , then Here, we have normalized the noise power to 1 and . Using the matrix inversion lemma, we have is a projection on the space orthogonal to the columns of . For strong jammers, the space spanned by the columns of will be the same as the space spanned by the dominant eigenvectors. We may therefore replace the estimated inverse covariance matrix by a projection on the complement of the dominant eigenvectors. This is called the EVP method. As the eigenvectors are orthonormalized, the projection can be written as . Figure 7 shows the performance of the EVP method in comparison with SMI, LSMI. Note the little difference between LSMI and EVP. The results with the three methods are based on the same realization of the covariance estimate. For EVP, we have to know the dimension of the jammer subspace (dimJSS). In complicated scenarios and with channel errors present, this value can be difficult to determine. If dimJSS is grossly overestimated, a loss in SNIR occurs. If dimJSS is underestimated the jammers are not fully suppressed. One is therefore interested in subspace methods with low sensitivity against the choice of the subspace dimension. This property is achieved by a “weighted projection,” that is, by replacing the projection by where is a diagonal weighting matrix and is a set orthonormal vectors spanning the interference subspace. does not have the mathematical properties of a projection. Methods of this type of are called lean matrix inversion (LMI). A number of methods have been proposed that can be interpreted as an LMI method with different weighting matrices . The LMI matrix can also be economically calculated by an eigenvector-free QR-decomposition method, . One of the most efficient methods for pattern stabilization while maintaining a low desired sidelobe level is the constrained adaptive pattern synthesis (CAPS) algorithm, , which is also a subspace method. Let be the vector for beamforming with low sidelobes in a certain direction. In full generality, the CAPS weight can be written as where the columns of the matrix span the space orthogonal to and is again a unitary matrix with columns spanning the interference subspace which is assumed to be of dimension . is a directional weighting matrix, , denotes the set of directions of interest, and is a directional weighting function. If we use no directional weighting, , the CAPS weight vector simplifies to where denotes the projection onto the space spanned by the columns of and . 4.3. Determination of the Dimension of Jammer Subspace (dimJSS) Subspace methods require an estimate of the dimension of the interference subspace. Usually this is derived from the sample eigenvalues. For complicated scenarios and small sample size, a clear decision of what constitutes a dominant eigenvalue may be difficult. There are two principle approaches to determine the number of dominant eigenvalues, information theoretic criteria and noise power tests. The information theoretic criteria are often based on the sphericity test criterion; see, for example, , where denote the eigenvalues of the estimated covariance matrix ordered in decreasing magnitude. The ratio of the arithmetic to geometric mean of the eigenvalues is a measure of the equality of the eigenvalues. The information theoretic criteria minimize this ratio with a penalty function added; for example, the Akaike Information Criterion (AIC) and Minimum Description Length (MDL) choose dimJSS as the minimum of the following functions: The noise power threshold tests (WNT) assume that the noise power is known and just check the estimated noise power against this value, . This leads to the statistic and the decision is found if the test statistic is for the first time below the threshold: The symbol denotes the -percentage point of the -distribution with degrees of freedom. The probability to overestimate dimJSS is then asymptotically bounded by . More modern versions of this test have been derived, for example, . For small sample size, AIC and MDL are known for grossly overestimating the number of sources. In addition, bandwidth and array channels errors lead to a leakage of the dominant eigenvalues into the small eigenvalues, . Improved eigenvalue estimates for small sample size can mitigate this effect. The simplest way could be to use the asymptotic approximation using the well-known linkage factors, , More refined methods are also possible; see . However, as explained in , simple diagonal loading can improve AIC and MDL for small sample size and make these criteria robust against errors. For the WNT this loading is contained in the setting of the assumed noise level . Figure 8 shows an example of a comparison of MDL and AIC without any corrections, MDL and WNT with asymptotic correction (25), and MDL and WNT with diagonal loading of . The threshold for WNT was set for a probability to overestimate the target number of = 10%. The scenario consists of four sources at = −0.7, −0.55, −0.31, −0.24 with SNR of 18, 6, 20, 20.4 dB and a uniform linear antenna with 14 elements and 10% relative bandwidth leading to some eigenvalue leakage. Empirical probabilities were determined from 100 Monte Carlo trials. Note that the asymptotic correction seems to work better for WNT than for MDL. With diagonal loading, all decisions with both MDL and WNT were correct (equal to 4). A more thorough study of the small sample size dimJSS estimation problem considering the “effective number of identifiable signals” has been performed in , and a new modified information theoretic criterion has been derived. 5. Parameter Estimation and Superresolution The objective of radar processing is not to maximize the SNR but to detect targets and determine their parameters. For detection, the SNR is a sufficient statistic (for the likelihood ratio test); that is, if we maximize the SNR we maximize also the probability of detection. Only for these detected targets we have then a subsequent procedure to estimate the target parameters: direction, range, and possibly Doppler. Standard radar processing can be traced back to maximum likelihood estimation of a single target which leads to the matched filter, . The properties of the matched filter can be judged by the beam shape (for angle estimation) and by the ambiguity function (for range and Doppler estimation). If the ambiguity function has a narrow beam and sufficiently low sidelobes, the model of a single target is a good approximation as other targets are attenuated by the sidelobes. However, if we have closely spaced targets or high sidelobes, multiple target models have to be used for parameter estimation. A variety of such estimation methods have been introduced which we term here “superresolution methods.” Historically, these methods have often been introduced to improve the limited resolution of the matched filter. The resolution limit for classical beamforming is the 3 dB beamwidth. An antenna array provides spatial samples of the impinging wavefronts, and one may define a multitarget model for this case. This opens the possibility for enhanced resolution. These methods have been discussed since decades, and textbooks on this topic are available, for example, . We formulate here the angle parameter estimation problem (spatial domain), but corresponding versions can be applied in the time domain as well. In the spatial domain, we are faced with the typical problems of irregular sampling and subarray processing. From the many proposed methods, we mention here only some classical methods to show the connections and relationships. We have spectral methods which generate a spiky estimate of the angular spectral density like. Capon’s method: and MUSIC method (Multiple Signal Classification): with , and spanning the dominant subspace. An LMI-version instead of MUSIC would also be possible. The target directions are then found by the highest maxima of these spectra ( 1- or 2-dimensional maximizations). An alternative group of methods are parametric methods, which deliver only a set of “optimal” parameter estimates which explain in a sense the data for the inserted model by or dimensional optimization . Deterministic ML method (complex amplitudes are assumed deterministic): Stochastic ML method (complex amplitudes are complex Gaussian): where denotes the completely parameterized covariance matrix. A formulation with the unknown directions as the only parameters can be given as The deterministic ML method has some intuitive interpretations: (1), which means that the mean squared residual error after signal extraction is minimized.(2), which can be interpreted as maximizing a set of decoupled sum beams .(3) with , where we have partitioned the matrix of steering vectors into . This property is valid due to the projection decomposition lemma which says that for any partitioning we can write . If we keep the directions in fixed, this relation says that we have to maximize the scan pattern over while the sources in the directions of are deterministically nulled (see (7)). One can now perform the multidimensional maximization by alternating 1-dimensional maximizations and keeping the remaining directions fixed. This is the basis of the alternating projection (AP) method or IMP (Incremental MultiParameter) method, [22, page 105]. A typical feature of the MUSIC method is illustrated in Figure 9. This figure shows the excellent resolution in simulation while for real data the pattern looks almost the same as with Capon’s method. (a) Simulated data (b) Measured real data A result with real data with the deterministic ML method is shown in Figure 10. Minimization was performed here with a stochastic approximation method. This example shows in particular that the deterministic ML-method is able to resolve highly correlated targets which arise due to the reflection on the sea surface for low angle tracking. The behavior of the monopulse estimates reflect the variation of the phase differences of direct and reflected path between 0° and 180°. For 0° phase difference the monopulse points into the centre, for 180° it points outside the 2-target configuration. The problems of superresolution methods are described in [21, 23]. A main problem is the numerical effort of finding the maxima (one -dimensional optimization or 1-dimensional optimizations for a linear antenna). To mitigate this problem a stochastic approximation algorithm or the IMP method has been proposed for the deterministic ML method. The IMP method is an iteration of maximizations of an adaptively formed beam pattern. Therefore, the generalized monopulse method can be used for this purpose, see Section 7.1 and . Another problem is the exact knowledge of the signal model for all possible directions (the vector function ). The codomain of this function is sometimes called the array manifold. This is mainly a problem of antenna accuracy or calibration. While the transmission of a plane wave in the main beam direction can be quite accurately modeled (using calibration) this can be difficult in the sidelobe region. For an array with digital subarrays, superresolution has to be performed only with these subarray outputs. The array manifold has then to be taken at the subarray outputs as in (2). This manifold (the subarray patterns) is well modeled in the main beam region but often too imprecise in the sidelobe region to obtain a resolution better than the conventional. In that case it is advantageous to use a simplified array manifold model based only on the subarray gains and centers, called the Direct Uniform Manifold model (DUM). This simplified model has been successfully applied to MUSIC (called Spotlight MUSIC, ) and to the deterministic ML method. Using the DUM model requires little calibration effort and gives improved performance, . More refined parametric methods with higher asymptotic resolution property have been suggested (e.g., COMET, Covariance Matching Estimation Technique, ). However, application of such methods to real data often revealed no improvement (as is the case with MUSIC in Figure 9). The reason is that these methods are much more sensitive to the signal model than the accuracy of the system provides. A sensitivity with an very sharp ideal minimum of the objective function may lead to a measured data objective function where the minimum has completely disappeared. 5.2. Target Number Determination Superresolution is a combined target number and target parameter estimation problem. As a starting point all the methods of Section 4.3 can be used. If we use the detML method we can exploit that the objective function can be interpreted as the residual error between model (interpretation 2) and data. The WNT test statistic (23) is just an estimate of this quantity. The detML residual can therefore be used for this test instead of the sum of the eigenvalues. These methods may lead to a possibly overestimated target number. To determine the power allocated to each target a refined ML power estimate using the estimated directions can be used with as in (30). This estimate can even reveal correlations between the targets. This has been successfully demonstrated with the low angle tracking data of Figure 10. In case that some target power is too low, the target number can be reduced and the angle estimates can be updated. This is an iterative procedure of target number estimation and confirmation or reduction. This way, all target modeling can be accurately matched to the data. The deterministic ML method (28) together with the white noise test (24) is particularly suited for this kind of iterative model fitting. It has been implemented in an experimental system with a 7-element planar array at Fraunhofer FHR and was first reported in [21, page 81]. An example of the resulting output plot is shown in Figure 11. The estimated directions in the , -plane are shown by small dishes having a color according to the estimated target SNR corresponding to the color bar. The circle indicates the 3 dB contour of the sum beam. One can see that the two targets are at about 0.5 beamwidth separation. The directions were estimated by the stochastic approximation algorithm used in Figure 10. The test statistic for increasing the target number is shown by the right most bar. The thresholds for increasing the number are indicated by lines. The dashed line is the actually valid threshold (shown is the threshold for 2 targets). The target number can be reduced if the power falls below a threshold shown in two yellow bars in the middle. The whole estimation and testing procedure can also be performed adaptively with changing target situations. We applied it to two blinking targets alternating between the states “target 1 on”, “both targets on”, “target 2 on”, “both targets on”, and so forth. Clearly, these test works only if the estimation procedure has converged. This is indicated by the traffic light in the right up corner. We used a fixed empirically determined iteration number to switch the test procedure on (=green traffic light). All thresholds and iteration numbers have to be selected carefully. Otherwise, situations may arise where this adaptive procedure switches between two target models, for example, between 2 and 3 targets. The problem of resolution of two closely spaced targets becomes a particular problem in the so called threshold region, which denotes configurations where the SNR or the separation of the targets lead to an angular variance departing significantly from the Cramer-Rao bound (CRB). The design of the tests and this threshold region must be compatible to give consistent joint estimation-detection resolution result. These problems have been studied in [27, 28]. One way to achieve consistency and improving resolution proposed in is to detect and remove outliers in the data, which are basically responsible for the threshold effect. A general discussion about the achievable resolution and the best realistic representation of a target cluster can be found in . 6. Extension to Space-Time Arrays As mentioned in Section 2.4, there is mathematically no difference between spatial and temporal samples as long as the distributional assumptions are the same. The adaptive methods and superresolution methods presented in the previous sections can therefore be applied analogously in the time or space-time domain. In particular, subarraying in time domain is an important tool to reduce the numerical complexity for space-time adaptive processing (STAP) which is the general approach for adaptive clutter suppression for airborne radar, . With the formalism of transforming space-time 2D-beamforming of a data matrix into a usual beamforming operation of vectors introduced in (4), the presented adaptive beamforming and superresolution methods can be easily transformed into corresponding subarrayed space-time methods. Figure 12 shows an example of an efficient space-time subarraying scheme used for STAP clutter cancellation for airborne radar. 7. Embedding of Array Processing into Full Radar Data Processing A key problem that has to be recognized is that the task of a radar is not to maximize the SNR, but to give the best relevant information about the targets after all processing. This means that for implementing refined methods of interference suppression or superresolution we have also to consider the effect on the subsequent processing. To get optimum performance all subsequent processing should exploit the properties of the refined array signal processing methods applied before. In particular it has been shown that for the tasks of detection, angle estimation and tracking significant improvements can be achieved by considering special features. 7.1. Adaptive Monopulse Monopulse is an established technique for rapid and precise angle estimation with array antennas. It is based on two beams formed in parallel, a sum beam and a difference beam. The difference beam is zero at the position of the maximum of the sum beam. The ratio of both beams gives an error value that indicates the offset of a target from the sum beam pointing direction. In fact, it can be shown that this monopulse estimator is an approximation of the Maximum-Likelihood angle estimator, . The monopulse estimator has been generalized in to arrays with arbitrary subarrays and arbitrary sum and difference beams. When adaptive beams are used the shape of the sum beam will be distorted due to the interference that is to be suppressed. The difference beam must adaptively suppress the interference as well, which leads to another distortion. Then the ratio of both beams will no more indicate the target direction. The generalized monopulse procedure of provides correction values to compensate these distortions. The generalized monopulse formula for estimating angles with a planar array and sum and difference beams formed into direction is where is a slope correction matrix and is a bias correction. is the monopulse ratio formed with the measured difference and sum beam outputs and , respectively, with difference and sum beam weight vectors , (analogous for elevation estimation with ). The monopulse ratio is a function of the unknown target directions . Let the vector of monopulse ratios be denoted by . The correction quantities are determined such that the expectation of the error is unbiased and a linear function with slope 1 is approximated. More precisely, for the following function of the unknown target direction: we require These conditions can only approximately be fulfilled for sufficiently high SNR. Then, one obtains for the bias correction for a pointing direction , For the elements of the inverse slope correction matrix , one obtains with or and or , and denotes the derivative . In general, these are fixed antenna determined quantities. For example, for omnidirectional antenna elements, and phase steering at the elements we have , where is the antenna element gain, and . It is important to note that this formula is independent of any scaling of the difference and sum weights. Constant factors in the difference and sum weight will be cancelled by the corresponding slope correction. Figure 13 shows theoretically calculated bias and variances for this corrected generalized monopulse using the formulas of for the array of Figure 2. The biases are shown by arrows for different possible single target positions with the standard deviation ellipses at the tip. A jammer is located in the asterisk symbol direction with JNR = 27 dB. The hypothetical target has a SNR of 6 dB. The 3 dB contour of the unadapted sum beam is shown by a dashed circle. The 3 dB contour of the adapted beam will be of course different. One can see that in the beam pointing direction the bias is zero and the variance is small. The errors increase for target directions on the skirt of the main beam and close to the jammer. The large bias may not be satisfying. However, one may repeat the monopulse procedure by repeating the monopulse estimate with a look direction steered at subarray level into the new estimated direction. This is an all-offline procedure with the given subarray data. No new transmit pulse is needed. We have called this the multistep monopulse procedure . Multistep monopulse reduces the bias considerably with only one additional iteration as shown in Figure 14. The variances appearing in Figure 13 are virtually not changed with the multistep monopulse procedure and are omitted for better visibility. 7.2. Adaptive Detection For detection with adaptive beams, the normal test procedure is not adequate because we have a test statistic depending on two different kinds of random data: the training data for the adaptive weight and the data under test. Various kinds of tests have been developed accounting for this fact. The first and basic test statistics were the GLRT, , the AMF detector, , and the ACE detector, . These have the form The quantities , , are here all generated at the subarray outputs, denotes the plane wave model for a direction . Basic properties of these tests are The AMF detector represents an estimate of the signal-to-noise ratio because it can be written as This provides a meaningful physical interpretation. A complete statistical description of these tests has been given in very compact form in [32, 33]. These results are valid as well for planar arrays with irregular subarrays and also mismatched weighting vector. Actually, all these detectors use the adaptive weight of the SMI algorithm which has unsatisfactory performance as mentioned in Section 4.2. The unsatisfactory finite sample performance is just the motivation for introducing weight estimators like LSMI, LMI, or CAPS. Clutter, insufficient adaptive suppression and surprise interference are the motivation for requiring low sidelobes. Recently several more complicated adaptive detectors have been introduced with the aim of achieving additional robustness properties, [34–38]. However, another and quite simple way would be to generalize the tests of (36), (37), (38) to arbitrary weight vectors with the aim of inserting well known robust weights as derived in Section 4.1. This has been done in . First, we observe that the formulation of (40) can be used for any weight vector. Second, one can observe that the ACE and GLRT have the form of a sidelobe blanking device. In particular it has already been shown in that diagonal loading provides significant better detection performance. A guard channel is implemented in radar systems to eliminate impulsive interference (hostile or from other neighboring radars) using the sidelobe blanking (SLB) device. The guard channel receives data from an omnidirectional antenna element which is amplified such that its power level is above the sidelobe level of the highly directional radar antenna, but below the power of the radar main beam, [1, page 9.9]. If the received signal power in the guard channel is above the power of the main channel, this must be a signal coming via the sidelobes. Such signals will be blanked. If the guard channel power is below the main channel power it is considered as a detection. With phased arrays it is not necessary to provide an external omnidirectional guard channel. Such a channel can be generated from the antenna itself; all the required information is in the antenna. We may use the noncoherent sum of the subarrays as guard channel. This is the same as the average omnidirectional power. Some shaping of the guard pattern can be achieved by using a weighting for the noncoherent sum: If all subarrays are equal, a uniform weighting may be suitable; for unequal irregular subarrays as in Figure 2 the different contributions of the subarrays can be weighted. The directivity pattern of such guard channel is given by . More generally, we may use a combination of noncoherent and coherent sums of the subarrays with weights contained in the matrices , , respectively, Examples of such kind of guard channels are shown in Figure 15 for the generic array of Figure 2 with −35 dB Taylor weighing for low sidelobes. The nice feature of these guard channels is (i) that they automatically scan together with the antenna look direction, and (ii) that they can easily be made adaptive. This is required if we want to use the SLB device in the presence of CW plus impulsive interference. A CW jammer would make the SLB blank all range cells, that is, would just switch off the radar. To generate an adaptive guard channel we only have to replace in (42) the data vector of the cell under test (CUT) by the pre-whitened data . Then, the test statistic can be written as , where for ACE and for GLRT. Hence is just the incoherent sum of the pre-whitened subarray outputs; in other words, can be interpreted as an AMF detector with an adaptive guard channel and the same with guard channel on a pedestal. Figure 16 shows examples of some adapted guard channels generated with the generic array of Figure 2 and −35 dB Taylor weighting. The unadapted patterns are shown by dashed lines. (a) Uniform subarray weighting (b) Power equalized weighting (c) Power equalized + difference weighting (a) Weighting for equal subarray power (b) Difference type guard with weighting for equal subarray power This is the adaptive generalization of the usual sidelobe blanking device (SLB) and the AMF, ACE and GLRT tests can be used as extension of the SLB detector to the adaptive case, , called the 2D adaptive sidelobe blanking (ASB) detector. The AMF is then the test for the presence of a potential target and the generalized ACE or GRLT are used confirming this target or adaptive sidelobe blanking. A problem with these modified tests is to define a suitable threshold for detection. For arbitrary weight vector it is nearly impossible to determine this analytically. In the detection margin has been introduced as an empirical tool for judging a good balance between the AMF and ASB threshold for given jammer scenarios. The detection margin is defined as the difference between the expectation of the AMF statistic and the guard channel, where the expectation is taken only over the interference complex amplitudes for a known interference scenario. In addition one can also calculate the standard deviation of these patterns. The performance against jammers close to the main lobe is the critical feature. The detection margin provides the mean levels together with standard deviations of the patterns. An example of the detection margin is shown in Figure 17 (same antenna and weighting as in Figures 15 and 16). Comparing the variances of the ACE and GLRT guard channels in revealed that the GLRT guard performs significantly better in terms of fluctuations. The GLRT guard channel may therefore be preferred for its better sidelobe performance and higher statistical stability. 7.3. Adaptive Tracking A key feature of ABF is that overall performance is dramatically influenced by the proximity of the main beam to an interference source. The task of target tracking in the proximity of a jammer is of high operational relevance. In fact, the information on the jammer direction can be made available by a jammer mapping mode, which determines the direction of the interferences by a background procedure using already available data. Jammers are typically strong emitters and thus easy to detect. In particular, the Spotlight MUSIC method working with subarray outputs is suited for jammer mapping with a multifunction radar. Let us assume here for simplicity that the jammer direction is known. This is highly important information for the tracking algorithm of a multifunction radar where the tracker determines the pointing direction of the beam. We will use for angle estimation the adaptive monopulse procedure of Section 7.1. ABF will form beams with a notch in the jammer direction. Therefore one cannot expect target echoes from directions close to the jammer and therefore it does not make sense to steer the beam into the jammer notch. Furthermore, in the case of a missing measurement of a tracked target inside the jammer notch, the lack of a successful detection supports the conclusion that this negative contact is a direct result of jammer nulling by ABF. This is so-called negative information . In this situation we can use the direction of the jammer as a pseudomeasurement to update and maintain the track file. The width of the jammer notch defines the uncertainty of this pseudo measurement. Moreover, if one knows the jammer direction one can use the theoretically calculated variances for the adaptive monopulse estimate of as a priori information in the tracking filter. The adaptive monopulse can have very eccentric uncertainty ellipses as shown in Figure 13 which is highly relevant for the tracker. The large bias appearing in Figure 13, which is not known by the tracker, can be reduced by applying the multistep monopulse procedure, . All these techniques have been implemented in a tracking algorithm and refined by a number of stabilization measures in . The following special measures for ABF tracking have been implemented and are graphically visualized in Figure 18.(i)Look direction stabilization: the monopulse estimate may deliver measurements outside of the 3 dB contour of the sum beam. Such estimates are also heavily biased, especially for look directions close to the jammer, despite the use of the multistep monopulse procedure. Estimates of that kind are therefore corrected by projecting them onto the boundary circle of sum beam contour.(ii)Detection threshold: only those measurements are considered in the update step of the tracking algorithm whose sum beam power is above a certain detection threshold (typically 13 dB). This guarantees useful and valuable monopulse estimates. It is well known that the variance of the monopulse estimate decreases monotonically with this threshold increasing. (iii)Adjustment of antenna look direction: look directions in the jammer notch should generally be avoided due to the expected lack of good measurements. In case that the proposed look direction lies in the jammer notch, we select an adjusted direction on the skirt of the jammer notch.(iv)Variable measurement covariance: a variable covariance matrix of the adaptive monopulse estimation according to is considered only for a mainlobe jammer situation. For jammers in the sidelobes, there is little effect on the angle estimates, and we can use the fixed covariance matrix of the nonjammed case. (v)QuadSearch and Pseudomeasurements: if the predicted target direction lies inside the jammer notch and if, despite all adjustments of the antenna look direction, the target is not detected, a specific search pattern is initiated (named QuadSearch) which uses look directions on the skirt of the jammer notch to obtain acceptable monopulse estimates. If this procedure does not lead to a detection, we know that the target is hidden in the jammer notch and we cannot see it. We use then the direction of the jammer as a pseudobearing measurement to maintain the track file. The pseudomeasurement noise is determined by the width of the jammer notch.(vi)LocSearch: in case of a permanent lack of detections (e.g., for three consecutive scans) while the track position lies outside the jammer notch, a specific search pattern is initiated (named LocSearch) that is similar to the QuadSearch. The new look directions lie on the circle of certain radius around the predicted target direction.(vii)Modeling of target dynamics: the selection of a suitable dynamics model plays a major role for the quality of tracking results. In this context, the so-called interacting multiple model (IMM) is a well-known method to reliably track even those objects whose dynamic behavior remains constant only during certain periods.(viii)Gating: in the vicinity of the jammer, the predicted target direction (as an approximation of the true value) is used to compute the variable angle measurement covariance. Strictly speaking, this is only valid exactly in the particular look direction. Moreover, the tracking algorithm regards all incoming sensor data as unbiased measurements. To avoid track instabilities, an acceptance region is defined for each measurement depending on the predicted target state and the assumed measurement accuracy. Sensor reports lying outside this gate are considered as invalid. In order to evaluate our stabilization measures we considered a realistic air-to-air target tracking scenario . Figure 19 provides an overview of the different platform trajectories. In this scenario, the sensor (on a forward looking radar platform flying with a constant speed of 265 m/s) employs the antenna array of Figure 2 (sum beamwidth BW = 3.4°, field of view 120°, scan interval 1 s) and approaches the target (at velocity 300 m/s), which thereupon veers away after a short time. During this time, the target is hidden twice in the jammer notch of the standoff jammer (SOJ)—first for 3 s and then again for 4 s. The SOJ is on patrol (at 235 m/s) and follows a predefined race track at constant altitude. Figure 20 shows exemplary the evaluation of the azimuth measurements and estimates over time in a window where the target first passes through the jammer notch. The different error bars of a single measurement illustrate the approximation error of the variable measurement covariance: denotes the true azimuth standard deviation (std) which is generated in the antenna simulation; corresponds to the std which is used in the tracking algorithm. More precisely, the tracking program computes the adaptive angle measurement covariance only in the vicinity of the jammer with a diameter of this zone of 8.5°. Outside of this region, the tracking algorithm uses a constant std of 0.004 for both components of the angle measurement. The constant std for the other parameters are 75 m and 7.5 m/s for range and range-rate measurements. The signal-to-noise and jammer-to-noise ratios were set to 26 dB and 27 dB at a reference range of 70 km. From Figure 20 the benefits of using pseudobearing measurements become apparent. From these investigations, it turned out that tracking only with adaptive beamforming and adaptive monopulse nearly always leads to track loss in the vicinity of the jammer. With additional stabilization measures that did not require the knowledge of the jammer direction (projection of monopulse estimate, detection threshold, LocSearch, gating) still track instabilities occurred culminating finally in track loss. An advanced tracking version which used pseudomeasurements mitigated this problem to some degree. Finally, the additional consideration of the variable measurement covariance with a better estimate of the highly variable shape of the angle uncertainty ellipse resulted in significantly fewer measurements that were excluded due to gating. In this case all the stabilization measures could not only improve track continuity, but also track accuracy and thus track stability, . This tells us that it is absolutely necessary to use all information of the adaptive process for the tracker to achieve the goal of detection and tracking in the vicinity of the interference. 8. Conclusions and Final Remarks In this paper, we have pointed out the links between array signal processing and antenna design, hardware constraints and target detection, and parameter estimation and tracking. More specifically, we have discussed the following features.(i)Interference suppression by deterministic and adaptive pattern shaping: both approaches can be reasonably combined. Applying ABF after deterministic sidelobe reduction allows reducing the requirements on the low sidelobe level. Special techniques are available to make ABF preserve the low sidelobe level.(ii)General principles and relationships between ABF algorithms and superresolution methods have been discussed, like dependency on the sample number, robustness, the benefits of subspace methods, problems of determining the signal/interference subspace, and interference suppression/resolution limit. (iii)Array signal processing methods like adaptive beamforming and superresolution methods can be applied to subarrays generated from a large fully filled array. This means applying these methods to the sparse superarray formed by the subarray centers. We have pointed out problems and solutions for this special array problem.(iv)ABF can be combined with superresolution in a canonical way by applying the pre-whiten and match principle to the data and the signal model vector. (v)All array signal processing methods can be extended to space-time processing (arrays) by defining a corresponding space-time plane wave model. (vi)Superresolution is a joint detection-estimation problem. One has to determine a multitarget model which contains the number, directions and powers of the targets. These parameters are strongly coupled. A practical joint estimation and detection procedure has been presented.(vii)The problems for implementation in real system have been discussed, in particular the effects of limited knowledge of the array manifold, effect of channel errors, eigenvalue leakage, unequal noise power in array channels, and dynamic range of AD-converters.(viii)For achieving best performance an adaptation of the processing subsequent to ABF is necessary. Direction estimation can be accommodated by using ABF-monopulse; the detector can be accommodated by adaptive detection with ASLB, and the tracking algorithms can be extended to adaptive tracking and track management with jammer mapping. With a single array signal processing method alone no significant improvement will be obtained. The methods have to be reasonably embedded in the whole system, and all functionalities have to be mutually tuned and balanced. This is a task for future research. The presented approaches constitute only a first ad hoc step, and more thorough studies are required. Note that in most cases tuning the functionalities is mainly a software problem. So, there is the possibility to upgrade existing systems softly and step-wise. The main part of this work was performed while the author was with the Fraunhofer Institute for High Frequency Physics and Radar Techniques (FHR) in Wachtberg, Germany. M. I. Skolnik, Radar Handbook, McGraw Hill, 2nd edition, 1990. M. A. Richards, J. A. Scheer, and W. A. Holden, Principles of Modern Radar, SciTech Publishing, 2010. W. D. Wirth, Radar Techniques Using Array Antennas, IEE Publishers, 2001. R. Klemm, Principles of Space-Time Adaptive Processing, IET Publishers, London, UK, 3rd edition, 2006. U. Nickel, “Subarray configurations for digital beamforming with low sidelobes and adaptive interference suppression,” in Proceedings of the IEEE International Radar Conference, pp. 714–719, Alexandria, Egypt, May 1995.View at: Google Scholar U. Nickel, “Properties of digital beamforming with subarrays,” in Proceedings of the International Conference on Radar (CIE '06), pp. 6–19, Shanghai, China, October 2006.View at: Google Scholar W. Bürger, “Sidelobe forming for ground clutter and jammer suppression for airborne active array radar,” in Proceedings of the IEEE International Symposium on Phased Array Systems and Technology, Boston, Mass, USA, 2003.View at: Google Scholar Y. Hua, A. B. Gershman, and Q. 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« PreviousContinue » By considering the arc AM, and its supplement AM', and recollecting what has been said, we readily see that, sin (an arc)=sin (its supplement) cos (an arc)=-cos (its supplement) It is no less evident, that if one or several circumferences were added to any arc AM, it would still terminate exactly at the point M, and the arc thus increased would have the same sine as the arc AM; hence if C represent a whole circumference or 360°, we shall have sin x=sin (C+x)=sin x=sin (2C+x), &c. The same observation is applicable to the cosine, tangent, &c. Hence it appears, that whatever be the magnitude of x the proposed arc, its sine may always be expressed, with a proper sign, by the sine of an arc less than 180°. For, in the first place, we may subtract 360° from the arc x as often as they are contained in it; and y being the remainder, we shall have sin x=sin y. Then if y is greater than 180°, make y=180° +z, and we have sin y=—sin z. Thus all the cases are reduced to that in which the proposed arc is less than 180°; and since we farther have sin (90°+x)=sin (90°-x), they are likewise ultimately reducible to the case, in which the proposed arc is between zero and 90°. XIV. The cosines are always reducible to sines, by means of the formula cos A=sin (90°-A); or if we require it, by means of the formula cos A=sin (90°+A): and thus, if we can find the value of the sines in all possible cases, we can also find that of the cosines. Besides, as has already been shown, that the negative cosines are separated from the positive cosines by the diameter DE; all the arcs whose extremities fall on the right side of DE, having a positive cosine, while those whose extremities fall on the left have a negative cosine. Thus from 0° to 90° the cosines are positive; from 90° to 270° they are negative; from 270° to 360° they again become positive; and after a whole revolution they assume the same values as in the preceding revolution, for cos (360°+x)=cos x. From these explanations, it will evidently appear, that the sines and cosines of the various arcs which are multiples of the quadrant have the following values: sin 0°=0 sin 90°-R sin 180°=0 sin 270°-R sin 360°=0 sin 450°=R. sin 540°-0 sin 630°-R sin 720°=0 sin 810°-R cos 90° 0 COS 0° R And generally, k designating any whole number we shall sin 2k. 90° 0, cos (2k+1). 90°=0, cos 4k. 90°=R, sin (4k+1). 90°=R, sin (4k-1). 90° ——R, What we have just said concerning the sines and cosines renders it unnecessary for us to enter into any particular detail respecting the tangents, cotangents, &c. of arcs greater than 180°; the value of these quantities are always easily deduced from those of the sines and cosines of the same arcs: as we shall see by the formulas, which we now proceed to explain. THEOREMS AND FORMULAS RELATING TO SINES, COSINES, TANGENTS, &c. XV. The sine of an arc is half the chord which subtends a double arc. in other words, the sine of a third part of the right angle is equal to the half of the radius XVI. The square of the sine of an arc, together with the square of the cosine, is equal to the square of the radius; so that in general terms we have sin A+cos A=R2. This property results immediately from the right-angled triangle CMP, in which MP2+CP2-CM2. It follows that when the sine of an arc is given, its cosine may be found, and reciprocally, by means of the formulas cos A=√(R2-sin A), and sin A=±√(R2-cos2A). The sign of these formulas is +, or —, because the same sine MP answers to the two arcs AM, AM', whose cosines CP, CP', are equal and have contrary signs; and the same cosine CP answers to the two arcs AM, AN, whose signs MP, PN, are also equal, and have contrary signs. Thus, for example, having found sin 30°=1R, we may deduce from it cos 30°, or sin 60°= √(R2—R2)=√ R2=†Ř√3. XVII. The sine and cosine of an arc A being given, it is required to find the tangent, secant, cotangent, and cosecant of the The triangles CPM, CAT, CDS, being similar, we have the proportions: CP: PM :: CA: AT; or cos A: sin A:: R: tang A: CP: CM:: CA: CT; or cos A: R:: R: sec A= PM: CP:: CD: DS; or sin A: cos A:: R: cot A= PM: CM:: CD: CS; or sin A: R:: R: cosec A R sin A cos A R cos A which are the four formulas required. It may also be observed, that the two last formulas might be deduced from the first two, by simply putting 90°-A instead of A. From these formulas, may be deduced the values, with their proper signs, of the tangents, secants, &c. belonging to any arc whose sine and cosine are known; and since the progressive law of the sines and cosines, according to the different arcs to which they relate, has been developed already, it is unnecessary to say more of the law which regulates the tangents and secants. By means of these formulas, several results, which have already been obtained concerning the trigonometrical lines, may be confirmed. If, for example, we make A=90°, we shall have sin A=R, cos A=0; and consequently tang 90°= R2 an expression which designates an infinite quantity; for, the quotient of radius divided by a very small quantity, is very great, and increases as the divisor diminishes; hence, the quotient of the radius divided by zero is greater than any finite quantity, The tangent being equal to R.- S and cotangent to R. Cos it follows that tangent and cotangent will both be positive when the sine and cosine have like algebraic signs, and both negative, when the sine and cosine have contrary algebraic signs. Hence, the tangent and cotangent have the same sign in the diagonal quadrants: that is, positive in the 1st and 3d, and negative in the 2d and 4th; results agreeing with those of Art. XII. It is also apparent, from the above formulas, that the secant has always the same algebraic sign as the cosine, and the cosecant the same as the sine. Hence, the secant is positive on the right of the vertical diameter DE, and negative on the left of it; the cosecant is positive above the diameter BA, and negative below it: that is, the secant is positive in the 1st and 4th quadrants, and negative in the 2d and 3d: the cosecant is positive in the 1st and 2d, and negative in the 3d and 4th. XVIII. The formulas of the preceding Article, combined with each other and with the equation sin 2A+cos 2A=R2, furnish some others worthy of attention. First we have R2 + tang2 AR2 + R2 (sin2 A+ cos2 A). cos 2A R2 sin A hence R2+tang2 A÷sec2 A, a formula which might be immediately deduced from the rightangled triangle CAT. By these formulas, or by the right-angled triangle CDS, we have also R2+cot2 Acosec2 A. Lastly, by taking the product of the two formulas tang A= Resin A and cot A= R cos A formula which gives. cot A= we have tang Ax cot A=R2, a Hence cot A cot B: tang B: tang A; that is, the colangents of two arcs are reciprocally proportional to their tangents. The formula cot Ax tang A=R might be deduced immediately, by comparing the similar triangles CAT, CDS, which give AT CA: CD DS, or tang A: R:: R: cot A. XIX. The sines and cosines of two arcs, a and b, being given, it is required to find the sine and cosine of the sum or difference of these arcs. Let the radius AC=R, the árc AB-a, the arc BD=b, and consequently ABD=¿ + b. the points B and D, let fall the C FK KEP The similar triangles BCE, ICK, give the proportions, CB: CI: BE: IK, or R: cos b : sin a: IK=" CB: CI: CE: CK, or R: cos b:: cos a: CK=· sin a cos b. cos a cos b. The triangles DIL, CBE, having their sides perpendicular, each to each, are similar, and give the proportions, CB: DI :: CE : DL, or R : sin b :: cos a: DL= cos a sin b.. R CB: DI: BE: IL, or R : sin b:: sin a: But we have IK+DL=DF=sin (a+b), and CK-IL=CF÷cos (a+b). The values of sin (a-b) and of cos (a-b) might be easily deduced from these two formulas; but they may be found directly by the same figure. For, produce the sine DI till it meets the circumference at M; then we have BM=BD=b, and MI=ID=sin b. Through the point M, draw MP perpendicular, and MN parallel to, AC: since MIDI, we have MN =IL, and IN-DL. 'But we have IK-IN-MP-sin (a-b), and CK+MN=CP=cos (a-b); hence

By Jan Cnops Dirac operators play an immense position in numerous domain names of arithmetic and physics, for instance: index concept, elliptic pseudodifferential operators, electromagnetism, particle physics, and the illustration conception of Lie teams. during this basically self-contained paintings, the fundamental rules underlying the concept that of Dirac operators are explored. beginning with Clifford algebras and the basics of differential geometry, the textual content specializes in major homes, specifically, conformal invariance, which determines the neighborhood habit of the operator, and the original continuation estate dominating its worldwide habit. Spin teams and spinor bundles are lined, in addition to the kinfolk with their classical opposite numbers, orthogonal teams and Clifford bundles. The chapters on Clifford algebras and the basics of differential geometry can be utilized as an advent to the above themes, and are compatible for senior undergraduate and graduate scholars. the opposite chapters also are obtainable at this point in order that this article calls for little or no prior wisdom of the domain names lined. The reader will gain, despite the fact that, from a few wisdom of complicated research, which provides the best instance of a Dirac operator. extra complex readers---mathematical physicists, physicists and mathematicians from assorted areas---will get pleasure from the clean method of the speculation in addition to the hot effects on boundary worth theory. Read or Download An Introduction to Dirac Operators on Manifolds PDF Similar differential geometry books For the reason that 1994, after the 1st assembly on "Quaternionic buildings in arithmetic and Physics", curiosity in quaternionic geometry and its functions has persevered to extend. development has been made in developing new periods of manifolds with quaternionic buildings (quaternionic Kaehler, hyper Kaehler, hyper-complex, etc), learning the differential geometry of unique periods of such manifolds and their submanifolds, knowing kinfolk among the quaternionic constitution and different differential-geometric constructions, and likewise in actual purposes of quaternionic geometry. Singular areas with higher curvature bounds and, particularly, areas of nonpositive curvature, were of curiosity in lots of fields, together with geometric (and combinatorial) staff thought, topology, dynamical structures and likelihood conception. within the first chapters of the booklet, a concise creation into those areas is given, culminating within the Hadamard-Cartan theorem and the dialogue of the best boundary at infinity for easily attached whole areas of nonpositive curvature. The quantity develops the principles of differential geometry in an effort to comprise finite-dimensional areas with singularities and nilpotent features, on the comparable point as is general within the basic concept of schemes and analytic areas. the speculation of differentiable areas is constructed to the purpose of delivering a handy gizmo together with arbitrary base alterations (hence fibred items, intersections and fibres of morphisms), infinitesimal neighbourhoods, sheaves of relative differentials, quotients by means of activities of compact Lie teams and a conception of sheaves of Fr? This memoir is either a contribution to the speculation of Borel equivalence relatives, thought of as much as Borel reducibility, and degree conserving staff activities thought of as much as orbit equivalence. right here $E$ is related to be Borel reducible to $F$ if there's a Borel functionality $f$ with $x E y$ if and provided that $f(x) F f(y)$. - Initiation to Global Finslerian Geometry - Real and complex singularities : Sao Carlos Workshop 2004 - Comprehensive Introduction To Differential Geometry, 2nd Edition, Volume 4 - Differential Forms in Geometric calculus - Elegant Chaos: Algebraically Simple Chaotic Flows - Fat manifolds and linear connections Extra info for An Introduction to Dirac Operators on Manifolds A)]k = e;/Ca)(eM(a) . (a)]k) everywhere, for all k > O. In general, the mapping b ~ e:\:/(a)(eM(a) . b) defines the orthogonal projection of a Clifford number b E a p,q having zero scalar part onto the Clifford algebra generated by TaM. It is assumed here that the function! is defined on the whole of the manifold; if this is not the case, ! is silently extended to M with zero. A vector-valued Clifford field is also called a tangent vector field. 2. Derivatives and differentials 31 In the language of bundles on abstract manifolds a Clifford field is also called a section of the Clifford bundle (see the appendix), or a Clifford section for short. Finally we construct the spinor connection, which again is a slightly modified derivative. 57) Embedded spin structure. An isometric embedding in some (pseudo-)Euclidean space was sufficient to define Clifford fields in terms of functions with values in the embedding Clifford algebra. If we want to introduce spin structures we shall need a much stronger condition. This will mean a loss of generality. However, the results and properties stated here will be valid in the general case, unless stated otherwise, and can be obtained by methods quite similar to the ones used here to derive them. There also is a certain amount of arbitrariness, as is usual with square roots. We have to fix the spinor space in a certain reference point before we can define the spinor sections. But we do not want the choice to be too big, and for that we need to introduce a spin structure. This is the description of all Chapter 2. Manifolds 50 possible isomorphisms (as spinor spaces) from the spinor space in an arbitrary point of the manifold to a canonical spinor space, for which we choose the spinor space of the reference point. An Introduction to Dirac Operators on Manifolds by Jan Cnops

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Photo Of Ball Mill Motor @ 1,191 rpm Parallel Gearbox with 1,191 rpm input 226 rpm output. Helical gears. Ball Mill @ 226 rpm pinion gear 18 rpm bull gear. Helical gears. Clutch between gearbox ball mill Bull gear is 2-piece split design. Critical Speed Ball Mill Calculation Calculation Of Ball Mill Critical Speed. Sep 08, 2020 Ball mill critical speed calculation ball mill rpm calculation how to calculate critical speed of ball mill ore crusher quarry crusher rock crusher rollcrusher lecture 2 slideshare 23 jan 2011 the grinding in ball mill is therefore caused due to speed at which a mill charge will centrifuge is known as the. How to calculate charge in ball mill Description Calculations for mill motor power, mill speed and media charge L = Internal length of the mill in cms. after lining. how to calculate sag mill ball charge BINQ Mining. Jan 01, 2013 TechnoMine Services, LLC. mill charge and speed. Rpm Optimal Ball Mill Ball Mill Power Calculation Stone Crusher Machine. Ball mill power calculation stone crusher machine.100 tph ball mill need kw powerball mill power calculation example 1 a wet grinding ball mill in closed circuit is to be fed 100 tph of a material with a work index of 15 and a sie distribution of 80 passing inch 6350 microns the required produ. In order to design a ball mill and to calculate the specific energy of grinding, it is necessary to have equation (s) formula for critical speed of ball mill If the actual speed of a 6 ft diameter ball mill is 25 rpm, calculate lifter face inclination angle φ, steel ball distribution curve with a lower inclination. Ball Mill Critical Speed Calculator In a ball mill of diameter 2000 mm, The critical speed of ball mill is given by, where R = radius of ball mill r = radius of ball. . what is the optimum rotation speed for a . how i calculate the optimum speed of a ball mill. what is the optimum rotation speed for a ball mill. Ball mill critical speed calculation - seshadrivaradhan.in. The calculator takes the input shaft speed and the torque required for a given load What is the ball mill critical speed and how to improve ball mill efficiency . Rod and ball mills circuit sizing method - The Cement Grinding Office. Show Critical Speed Of Ball Mill Calculation Jun 26, 2017 Ball Nose Milling Without a Tilt Angle. Ball nose end mills are ideal for machining 3-dimensional contour shapes typically found in the mold and die industry, the manufacturing of turbine blades, and fulfilling general part radius requirements.To properly employ a ball nose end mill (with no tilt angle) and gain the optimal tool life and part finish, follow the 2-step process below (see Figure 1). May 28, 2012 8.1 Calculation of Cement Mill Power Consumption 8.2 Calculation of . 3.2 Calculation of the Critical Mill Speed G weight of a grinding ball in View Open - University of the Witwatersrand. Calculation For Inclination Of Ball Mill Jul 18, 2021 Now, Click on Ball Mill Sizing under Materials and Metallurgical Now, Click on Critical Speed of Mill under Ball Mill Sizing. The screenshot below displays the page or activity to enter your values, to get the answer for the critical speed of mill according to the respective parameters which is the Mill Diameter (D) and Diameter of Balls (d) Now, enter the value appropriately and accordingly. Use the SFM and the diameter of the mill to calculate the RPM of your machine. Use the RPM, IPT, CLF and the number of flutes to calculate the feed rate or IPM. Ball End Mills. Bull Nose End Mill. Flat End Mills. Metric End Mills. Milling Bits. Miniature End Mills. Calculate Critical Speed Of Ball Mill 220.127.116.11 Ball mills. The ball mill is a tumbling mill that uses steel balls as the grinding media. The length of the cylindrical shell is usually 1–1.5 times the shell diameter ( Figure 8.11). The feed can be dry, with less than 3 moisture to minimize ball coating, or slurry containing 20–40 water by weight. All calculations are based on industry formulas and are intended to provide theoretical values. End Mill Diameter Revolutions per Minute 318.057 RPM Millimeters per Minute Solution Millimeters per Tooth (Chipload) Revolutions per Minute. Ball Mill Torque Calculation Krosline How To Calculate Rpm Of A Motor To Ball Mill. Selection Sheet - Eaton Ball mills - These mills use forged steel balls up to 5 inches Use of a clutch permits the mill motor to be The required clutch torque is calculated from the power rating of the mill motor clutch shaft rpm and an appropriate service factor. Ball mill calculation kw and rpm. BALL MILL DRIVE MOTOR CHOICES - Artec Machine Systems. ball mills, the starting torque restrictions of some of the newer mill drive configurations, and the softness. How To Calculate Ball Mill Rotational Speed Calculations The critical speed of ball mill is given by, where R = radius of ball mill r = radius of ball. For R = 1000 mm and r = 50 mm, n c = 30.7 rpm. But the mill is operated at a speed of 15 rpm. Therefore, the mill is operated at 100 x 15 30.7 = 48.86 of critical speed. If 100 mm dia balls are replaced by 50 mm dia balls, and the. Milling Speed and Feed Calculator. Determine the spindle speed (RPM) and feed rate (IPM) for a milling operation, as well as the cut time for a given cut length. Milling operations remove material by feeding a workpiece into a rotating cutting tool with sharp teeth, such as an end mill or face mill. Ball Mill Parameter Selection & Calculation Ball Mill Parameter Selection Calculation Power . 30 08 2019 Critical Speed_ When the ball mill cylinder is rotated, there is no relative slip between the grinding medium and the cylinder wall, and it just starts to run in a state of rotation with the cylinder of the mill This instantaneous speed of the mill is as follows N0 — mill working speed, r min K’b — speed ratio, There are. Jul 30, 2010 The RPM for a 0.500 would be 2400 RPM. 4 300 0.500 =2400 For drilling I use an exception of 50 , so 1200 RPM. But plunging any endmill directly into the material is hard. You basicly have to slow it down till it stops chattering or the chatter is exceptable. Drilling a hole first would help a lot.

A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2017; you can also visit the original URL. The file type is Levin's method produces a parameterization of the intersection curve of two quadrics in the form where a(u) and d(u) are vector valued polynomials, and s(u) is a quartic polynomial. ... enhanced version of Levin's method is proposed that, besides classifying the morphology of the intersection curve of two quadrics, produces a rational parameterization of the curve if the curve is singular ... We thank Professor Helmut Pottmann and the referee for their helpful comments. We also thank Barry Joe for commenting on an earlier version of the paper. ...doi:10.1016/s0167-8396(03)00081-5 fatcat:dr3pbke2lvbybjpw4vwxbx4t5i Curves and Surfaces in Computer Vision and Graphics Algorithms are presented for constructing G t continuous meshes of degree two (quadric) and degree three (cubic) implicitly defined, piecewise algebraic surfaces, which exactly fit any given collection ... of points and algebraic space curves, of arbitrary degree. ... It follows then from Bezout's theorem 3.1 for surface intersection, that the two quadrics 51 and 52 must meet in a plane curve (either an irreducible conlc or straight lines). ...doi:10.1117/12.19736 fatcat:64kdqd4m65gfjorjr7m7ltv3ye This method can handle objects with interacting quadric surfaces and avoids the combinatorial search for tracing all the quadric surfaces in an intermediate wire-frame by the existing methods. ... A B-rep oriented method for reconstructing curved objects from three orthographic views is presented by employing a hybrid wire-frame in place of an intermediate wire-frame. ... This work was supported by the 973 Program of China (Grant No. 2004CB719404) and the Program for New Century Excellent Talents in University (Grant No. NCET-04-0088). ...doi:10.1016/j.cad.2006.04.009 fatcat:h4gws3ywkbh5vgmwwweq4x5ggq Algebraic Geometry and its Applications In this short article we summarize a number of recent applications of constructive real algebraic geometry to geometric modelling and robotics, that we have been involved with under the tutelage of Abhyankar ... The three surface intersection points are shown as the common intersections of the space curves for each pair of surfaces. ... Here we consider constructive methods for both local and global real parameterizations of curves and surfaces. ...doi:10.1007/978-1-4612-2628-4_25 fatcat:wuqw3apqqnfdnmvt5xe5s7fdxa (F-ANGR) 87c:14063 Théoréme de Torelli affine pour les intersections de deux quadriques. [The affine Torelli theorem for intersections of two quadrics] Invent. Math. 80 (1985), no. 3, 375-416. ... The main result of this paper deals with affine complete intersections U of two quadrics in affine space of even dimension. ... The number of hyperpolygons use d are optimal, in that they are the order of the minimu m number required for a smooth Gouraud like shading o f the hypersurfaces . ... Abstrac t Algorithms are presented for polygonalizing implicitl y defined, quadric and cubic hypersurfaces in n > 3 dimensional space and furthermore displaying their projections in 3D . ... Acknowledgements : I sincerely thank Insung Ihm an d Andrew Royappa for the many hours they spent in fron t of the graphics workstation, helping me visualize in fou r and higher dimensional space . ...doi:10.1145/91385.91428 dblp:conf/si3d/Bajaj90 fatcat:fpdkcsjxmndt5lol3knwo4q4a4 Constraint evaluation results in the activation of methods which compute rigid motions from surface information. ... The order in which constraints are evaluated may also be used as a language for specifying the sequence of assembly and set-up operations. ... Closed form solution are available for parametric representation of curves of intersection between any two natural quadrics [ 11,101. Vertices may be computed by efficient numeric methods . ...doi:10.1145/319120.319129 dblp:conf/si3d/Rossignac86 fatcat:ehudqr6oknan5mfxj4avz52rbu treatment of the intersection of two quadrics is too brief and too difficult to be of greatest service. ... It seems odd that it was thought necessary to speak in such detail of quadric surfaces after developing so large a number of formulas for surfaces defined by any analytic function, but the outline of the ... Woon suggests two methods for calculating points on a quadric surface intersection curve ( QSIC). ... For non-planar intersections , the curve lies in a quadric surface which has a "base curv e" which is either a line , a parabola , a hyperbola , or an box " , which is a cube or r ectangula r par all epiped ...doi:10.1145/563274.563320 dblp:conf/siggraph/Levin76 fatcat:wmskkxqcojbuzfkaub6sx7o4v4 Woon suggests two methods for calculating points on a quadric surface intersection curve ( QSIC). ... For non-planar intersections , the curve lies in a quadric surface which has a "base curv e" which is either a line , a parabola , a hyperbola , or an box " , which is a cube or r ectangula r par all epiped ...doi:10.1145/965143.563320 fatcat:dgaptn4rwvcg7nenfxzniwpaoi Topics in Surface Modeling C k Least-Squares Approximate Surface Fit: Coustruct a real algebraic surface S, which C k -1 interpolates a collection of points Pi in m..1 and given space curves Cj in m.3 as before, with associated ... Fil wilh SlIrface Patches: Construct a mesh of real algebraic surface patches S;, which C k inlNpolaLcs a collection of points Pi in m. 3 and given space curves C j in m. 3 , with associated "IlOl'lllal ... It follows then frolll Bezout's theorem 4.1 for surface intersection, that the two quadrics 51 and S'l must meet in ,L plane curve (either an irreducible conic or straight lines). ...doi:10.1137/1.9781611971644.ch2 fatcat:pyxpssexaff6pczvuz7cy7vhxm We present the first complete, exact and efficient C++ implementation of a method for parameterizing the intersection of two implicit quadrics with integer coefficients of arbitrary size. ... Unlike existing implementations, it correctly identifies and parameterizes all the connected components of the intersection in all cases, returning parameterizations with rational functions whenever such ... Until recently, the only known general method for computing a parametric representation of the intersection between two arbitrary quadrics was that of J. Levin . ...doi:10.1145/997817.997880 dblp:conf/compgeom/LazardPP04 fatcat:6mekaucuu5ekhod7n2ponfawme treatment of the intersection of two quadrics is too brief and too difficult to be of greatest service. ... [May, intersections. The method of Monge is given the preference, but other methods are given in outline. ...doi:10.1090/s0002-9904-1914-02513-3 fatcat:h6wigtbqyneghcp2cbvknn4t3i Individual faces of a polyhedron are replaced by low degree implicit algebraic surface patches with local support. ... This paper presents efficient algorithms for generating families of curved solid objects with boundaty topology related to an input polyhedron. ... Acknowledgements : We are grateful to Vinod Anupam, Andrew Royappa and Dan Schikore for their assistance in the implementation of the smoothing algorithms. ...doi:10.1145/142920.134014 fatcat:7whadzf2z5hnfcbmaz5cmpiu5m The conchoid surface F d of a surface F with respect to a fixed reference point O is a surface obtained by increasing the distance function with respect to O by a constant d. ... This contribution studies conchoid surfaces of quadrics in Euclidean R 3 and shows that these surfaces admit real rational parameterizations. ... Acknowledgments This work has been partially supported by the 'Ministerio de Economia y Competitividad' under the project MTM2011-25816-C02-01. ...doi:10.1016/j.jsc.2013.07.003 fatcat:fltp23pn2rcjhg3ihphfj6m2g4 « Previous Showing results 1 — 15 out of 1,456 results

Last Updated on September 16, 2022 In geometry, the basic shape of a triangle has two sides that are 8 inches long and 10 inches long. This property of triangles can also be applied to other shapes, including cubes, hexagons, and trapezoids. If there are more than two sides, a triangle is called a psilocylindrical polygon. It is also referred to as a right-angled triangle. Triangles have two sides of lengths 8 and 10 In geometrical terms, triangles with two sides of lengths eight and ten are called right triangles. Their sum is greater than one. They also have two sides of length six and eight. However, triangles with three sides are called degenerate triangles. In this case, the two smaller sides are equal in length. Thus, these triangles are also right triangles. However, it is important to note that the sides of degenerate triangles have different lengths and corresponding angles. To know which sides are equal in length, we first need to know which side of a triangle is longer. The length of the longest side is eight and the shortest side is ten. Therefore, the sides of triangles with two sides of lengths eight and ten are equal. However, a triangle with two sides eight and ten would have three sides of length five and four, and a triangle with three sides of length five and six would have two sides that are equal in length. In addition, a triangle with two sides of length eight and ten cannot be formed from a random set of side lengths. It must have a third side whose sum is greater than eight and ten. For example, a triangle with two sides of length eight and ten is not a right triangle. The missing side can have any length between eight and ten, as long as it is shorter than eight. Angles measure 90 degrees, 30 degrees, and 100 degrees The angles are measured in degree units. An angle has several different types. A right angle, as its name suggests, is 90 degrees. This angle is often represented by a square box in between the angles’ arms. There are also other types of angles, known as obtuse angles or reflex angles. Obtuse angles measure less than 180 degrees while reflex angles exceed 360 degrees. Common examples of reflex angles are 180deg, 220deg, and 350deg. Besides being important to science, angles are also used in our everyday lives. Athletes use angles to improve their performance, whether it’s passing a soccer ball or spinning a disc to make it fly far. We use angles to measure our body rotation in sports. In discus throwing, we need to rotate at a specific angle to hit the disc far. In soccer, we need to use a certain angle to pass the ball. Although degrees are the most common unit of measurement, you can also measure angles in seconds and minutes. Usually, a degree equals 60 seconds or 60 minutes. To measure an angle, position the protractor at its vertex, where both sides of the angle meet. If you are unsure of where to place the vertex of an angle, measure it again until you find the correct number. Another way to calculate the angle measurement is to draw a triangle. A triangle has three angles, one is equal and the third is equal. The sum of all the angles in a triangle should equal 180 degrees. The same principle applies to angles in a right triangle. If the angle measurement is unknown, use an equals sign to find the value. Using a triangle’s properties, you can solve angles in three different ways. First, you must draw a triangle. Next, label the angles so that you can easily find the angles by degrees. They have two sides of lengths 5 and 8 A triangle has two sides of lengths 5 and 8, and a side of length 7. Likewise, a triangle has two unequal sides. If one side is seven inches long and the other is eight inches long, the side with the largest measurement must be longer than the other. If neither of the two sides is long enough to reach the missing side, the triangle is not a triangle. It is possible, however, for a triangle to have three sides of equal lengths. The minimum value of the third side of a triangle must be greater than the minimum value of the other two sides. However, a triangle does not contain a third side if it contains two sides of lengths of five and eight. That’s why a triangle with three sides of five and eight sides is not a triangle. The minimum and maximum value of this third side must be greater than the sum of the two sides. They have two sides of lengths 5 and 8 inches A triangle has two sides of lengths 5 and 8. The sides must be equal in length to make a right-angled triangle. This is not always the case, though. If the two shorter sides are equal in length, the triangle will have one side of length 8 inches longer than the other side. Therefore, the area of a triangle with two sides of length 5 and 8 inches is six square units. About The Author Pat Rowse is a thinker. He loves delving into Twitter to find the latest scholarly debates and then analyzing them from every possible perspective. He's an introvert who really enjoys spending time alone reading about history and influential people. Pat also has a deep love of the internet and all things digital; she considers himself an amateur internet maven. When he's not buried in a book or online, he can be found hardcore analyzing anything and everything that comes his way.

Given a prime number p and a natural number m not divisible by p, we propose the problem of finding the smallest number such that for , every group G of order has a non-trivial normal p-subgroup. We prove that we can explicitly calculate the number in the case where every group of order is solvable for all r, and we obtain the value of for a case where m is a product of two primes. Throughout this note, p will be a fixed prime number. We use to denote the p-core of G, that is, its largest normal p-subgroup. We propose the following optimization problem: Given a number m not divisible by p, find the smallest r0 such that every group having order n =prm, with , has a nontrivial p-core . Denote such number r0 by . In Theorem 2.1, we will prove that is well-defined for any prime p and number m (with ). In Theorem 2.3 we explicitly determine the value of in the case that all groups whose order have the form prm are solvable (for example, if m is prime or if both p and m are odd). Finally, in Section 3, we calculate Λ(2,15), a case that is not covered by the previous theorem. We remark that the motivation for this research came from the search for examples of finite groups G such that the Brown complex of nontrivial p-subgroups of G (see for example for the definition and properties) is connected but not contractible. It is known that is contractible when G has a nontrivial normal p-subgroup, and Quillen conjectured in that the converse is also true. Theorem 2.1. For any prime number p and natural number m such that , there is a number such that if , any group of order has a non-trivial p-core . Proof. Let G be a group of order with . Let P be a Sylow p-subgroup of G. Since the kernel of the action of G on the set of cosets of P is precisely , we obtain that G embeds in Sm, and so pr divides (m-1)!. Hence, if is the largest power of p dividing ((m-1)!), we obtain that . For t,q natural numbers, let be the product (note that can also be defined as , where is the q-factorial of t), and if is a prime factorization of m, with the qi pairwise distinct and for each i, we let . We prove that if is the largest power of p dividing , then . Theorem 2.2. Let where and . If , then there is a group of order n with . Proof. Let K be the group , that is, a product of elementary abelian groups, where and are distinct primes and Cq denotes the cyclic group of order q. Then Γ(m) divides the order of , and hence so does ps. Let H be a subgroup of of order ps. For every S∈ H and k∈ K define the map by . Then is a subgroup of . If we identify H with the subgroup of maps of the form TS,0 and K with the subgroup of maps of the form , then G is just the semidirect product of K by H. Hence |G|=n. We have that G acts transitively on K in a natural fashion, and the stabilizer of 0∈ K is H, a p-Sylow subgroup of G. Hence the stabilizers of points in K are precisely the Sylow subgroups of G, so their intersection contains only the identity , as we wanted to prove. The next theorem will show that the lower bound given by Theorem 2.2 is tight in some cases. Theorem 2.3. Let , where . If G is a group of order n and ps does not divide Γ(m) then either: 1. , or 2. G is not solvable. Proof. Let G be solvable with order and . Let F(G) be the Fitting subgroup of G. Consider the map , sending g to given by conjugation by g. The restriction of c to P, a p-Sylow subgroup of G, has kernel . Since (Theorem 7.67 from ), and F(G) does not contain elements of order p by our assumption on , we have P ∩ CG(F(G))=1 and so P acts faithfully on F(G). If is the prime factorization of m, we have that F(G) is the direct product of the for . Hence . Let such that the action induced by cg on , is the identity. Since cg acts on each factor as the identity, then by Theorem 5.1.4 from , we have that it acts as the identity on each . By the faithful action of P on F(G), we have that g=1. This implies that P acts faithfully on . But then |P| divides the order of the automorphism group of , which is a product of elementary abelian groups of respective orders with for all i. Hence divides Corollary 2.4. Let ps be the largest power of p that divides . If m is prime, or if both p,m are odd, then . Proof. By Burnside's p,q-theorem, and the Odd Order Theorem, we have that all groups that have order of the form for some r are solvable. Therefore, for all , by Theorem 2.3 we have that all groups of order have non-trivial p-core. At this moment, we can prove that in some cases, the group constructed in 2.2 is unique. Theorem 2.5. Let where and s>0. If , but for all proper divisors m' of m, then up to isomorphism, the group constructed in the proof of Theorem 2.2 is the only solvable group of order n with . Proof. With the notation of the argument of the proof of 2.3, if G is a solvable group of order n with , we must have that and for all i in order to satisfy the divisibility conditions. Hence is elementary abelian and a qi-Sylow subgroup for all i, and so G is the semidirect product of a p-Sylow subgroup P of with F(G), where the action of P on F(G) by conjugation is faithful. Hence G is isomorphic to the group constructed in the proof of Theorem 2.3. One case in that we may apply Theorem 2.5 is when n=864. There are 4725 groups of order , but only one of them has the property of having a trivial 2-core. An example that cannot be tackled with the previous results is the case p=2, . In this case, . Not all groups with order of the form are solvable, however, we will prove that is actually 4. (The group S5 attests that .) Theorem 3.1. Every group G of order for r ≥ 4 is such that . Proof. Let G be a group of order for . Suppose that . From Theorem 2.3, we obtain that G is not solvable. We will prove then that . Suppose otherwise, and let T=O3(G). Then , and so G/T is solvable. Since , from Theorem 2.3, we have that . Let such that . Suppose . Since , and G/L is solvable, we have that divides , that is, . Now, L is also solvable and , hence if we had we would have , and G would have a non-trivial subnormal 2-subgroup, which contradicts our assumption that . Hence j=1. But then , which contradicts that . Hence . By a similar argument, we get that . From we obtain that G is not simple. Hence G has a proper minimal normal subgroup M. From the previous paragraph, we obtain that M is not abelian, since in that case we would have that . The only possibility is that . We have then a morphism sending g to cg, the conjugation by g. Since , and , in any case the kernel of c is a nontrivial normal 2-subgroup.

I was born on December 1, 1943, at the Brooklyn Jewish Hospital. I had a normal childhood, engaging in the usual games and sports. My father introduced me to mathematics at the level of elementary algebra when I was seven. Intermittently, he would teach me more. Soon I was hooked. In the seventh grade, I made a great new friend, Mel Hochster, a fellow math enthusiast, later my college roommate, now, an eminent mathematician. I attended Erasmus Hall, a large public high school with many famous alumni. There were some very bright students and the honours classes were at a good level. Eventually, I became captain of the math team. At Harvard, my teachers Shlomo Sternberg and Raul Bott were charismatic and encouraging. As a junior, with no practice, I tied for 21st in the country on the Putnam exam. This relatively modest accomplishment meant a lot to me. As a senior, I took a graduate course in PDE from a young Assistant Professor named Jim Simons. In graduate school at Princeton, after deciding to study differential geometry, I consulted Jim, who was a specialist in that area. Coincidentally, he had just moved to Princeton and was working as a code breaker at the Institute for Defense Analyses. My advisor was the legendary Salamon Bochner, but my teacher was Jim. For a year, he told me what to read and patiently answered all my questions. Then he suggested a thesis problem. After I solved it, it morphed into something very different, a finiteness theorem for manifolds of a given dimension admitting a Riemannian metric with bounds on curvature and diameter and a lower bound on volume. This needed a corresponding lower bound for the injectivity radius, which I think of as my first real theorem. The finiteness theorem brought a certain change in perspective to Riemannian geometry, now subsumed under Cheeger–Gromov compactness. The major part of my career has been spent at Stony Brook (1969–1989) and the Courant Institute (1989–). First, I spent an exciting year at Berkeley and another at Michigan. Significant stays in Brazil, Finland, IHES in France and IAS in Princeton were enormously fruitful. I have had exceptionally brilliant collaborators and some great students. Several collaborators are mentioned below. Unfortunately, space constraints forced the omission of many others. When I started doing research, my viewpoint was geometric and topological. As I learned more analysis, my work evolved into a mixture of all three fields. Several times, I noticed things which were hiding in plain sight, but which proved to have far reaching consequences. In retrospect, a significant part of my work involved finding structure in contexts which might initially have seemed too naive or too rough. Occasionally, a specific problem led to new developments that went far beyond what was needed for the original application. With strong mutual connections and the mention of a few highlights, my work could be summarized as follows. (1) Curvature and geometric analysis; see below. (2) A lower bound for the first nonzero eigenvalue of the Laplacian, which has had a vast, varied and seemingly endless number of descendants. (3) Analysis on singular spaces: The precursor was my proof of the Ray–Singer conjecture on the equality of Ray–Singer torsion, an analytic invariant and Reidemeister torsion, a topological invariant. Simultaneously, Werner Muller gave a different proof. Independently, I discovered Poincaré duality for singular spaces, in the guise of L2-cohomology. Later, I showed it was equivalent to the contemporaneously defined intersection homology theory of Goresky–MacPherson. I pioneered index theory and spectral theory on piecewise constant curvature pseudomanifolds. Applications included a local combinatorial formula for the signature. Adiabatic limits of η-invariants and local families index for manifolds with boundary were joint with Jean–Michel Bismut. (4) Metric measure spaces: I showed that properly formulated, all of first order differential calculus is valid for metric measure spaces whenever the measure is doubling and a Poincaré inequality holds in Heinonen-Koskela’s sense. Examples include non-selfsimilar fractals with dimension any real number. Related work with Bruce Kleiner and Assaf Naor had applications to theoretical computer science. Curvature. My thesis (1967) and my first paper with Detlef Gromoll (1969) on the soul theorem for complete manifolds of nonnegative curvature were purely geometric. In 1971, we proved the fundamental splitting theorem for complete manifolds of nonnegative Ricci curvature. The statement was geometric but the proof involved partial differential equations (PDE). Both works with Detlef were early examples of rigidity theorems. Here is the principle. When geometric hypotheses are sufficiently in tension, they can mutually coexist only in highly non-generic situations where specific special structure is present. Similarly, Misha Gromov and I characterized collapse with bounded curvature in terms of generalized circular symmetry (1980–1992) in the end joining forces with Kenji Fukaya (1992). Work with Toby Colding (1995–2000) on Ricci curvature was a mixture of geometry and PDE. We proved quantitative versions of rigidity theorems which, together with scaling, vastly increased their range of applicability. Specifically, if the hypotheses of rigidity theorems fail to hold by only a sufficiently small amount, then the conclusions hold up to an arbitrarily small error. Quantitative rigidity theorems were the basis of our structure theory for weak geometric limits (Gromov–Hausdorff limits) of sequences of smooth Riemannian manifolds with Ricci bounded below. These geometric objects play the role that distributions play in analysis. In particular, limit spaces can have singularities living on lower dimensional subsets and we proved a sharp bound on their dimension. Aaron Naber and I gave the first quantitative theory of such singular sets (2011–2021). Beyond bounding their dimension, we bounded their size. Our flexible techniques were rapidly applied to numerous nonlinear elliptic and parabolic geometric PDE’s. In 2015, we proved a longstanding conjecture on noncollapsed Gromov–Hausdorff limits of sequences of n-dimensional Einstein manifolds: Singular sets have dimension at most n – 4 . 28 October 2021 Hong Kong

We study the energy conditions in the framework of the modified gravity with higher-derivative torsional terms in the action. We discuss the viability of the model by studying the energy conditions in terms of the cosmographical parameters like Hubble, deceleration, jerk, snap and lerk parameters. In particular, We consider two specific models that are proposed in literature and examine the viability bounds imposed by the weak energy condition. Energy conditions in gravity with higher-derivative torsion terms Tahereh Azizi 111and Miysam Gorjizadeh 222 Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, IRAN Key Words: Teleparallel gravity, higher-derivative torsion terms, Energy conditions Recent astronomical observations [1, 2, 3] have shown that the current universe experiences an accelerated expansion. To explain this unexpected phenomenon, two remarkable approaches have been suggested. In the first approach, an exotic matter component with negative pressure is considered in the right hand side of the Einstien equations dubbed as, dark energy in the literature. There are several candidates for the dark energy proposal such as the cosmological constant , canonical scalar field (quintessence) , phantom field , chaplygin gas and so on (for a review see ). The second approach is based on the modifying the left hand side of the field equations dubbed as, dark gravity. Some examples of the modified gravity models are theories, string inspired gravity, braneworld gravity, etc ( for review see [9, 10, 11] and references therein). One of the interesting modified gravity model is the gravity which is the torsion scalar. This scenario is based on the “teleparallel” equivalent of General Relativity (TEGR) [12, 13] which uses the Weitzenböck connection that has no curvature but only torsion. Note that the Lagrangian density of the Einstien gravity is constructed from the curvature defined via the Levi-Civita connection. In the context of the TEGR, the dynamical object is a vierbein field , , which form the orthogonal bases for the tangent space at each point of spacetime. The metric tensor is obtained from the dual vierbein as where is the Minkowski metric and is the component of the vector in a coordinate basis. Note that the Greek indices are refer to the coordinates on the manifold while Latin indices label the tangent space. The Lagrangian density of the teleparallel gravity is constructed from the torsion tensor which is defined as One can write down the torsion scalar where and is the contorsion tensor defined as Using the torsion scalar as the teleparallel Lagrangian leads to the same gravitational equations of the general relativity. Similar to the modified gravity, one can modify teleparallel gravity by considering an arbitrary function of the torsion scalar in the action of the theory, which leads to the theories of gravity [14, 15, 16]. It is worth noticing that the field equations of the gravity are second order differential equations and so it is more manageable compared to the theories whose the field equations are nd order equations. Consequently, the modified TEGR models have attracted a lot of interest in literature (see and references therein ). Recently, a further modification of the teleparallel gravity has been proposed with constructing a torsional gravitational modifications using higher-derivative terms such as and terms in the Lagrangian of the theory . In this regard, the dynamical system of this model has been studied via performing the phase-space analysis of the cosmological scenario and consequently, an effective dark energy sector that comprises of the novel torsional contributions is obtained. The aim of this paper is to explore the energy conditions of this modified teleparallel gravity by taking into account the further degrees of freedom related to the higher derivative terms. Indeed, one procedure for analyzing the viability of the modified gravity models is studying the energy conditions to constrain the free parameters of them. In this respect, one can impose the null, weak, dominant and strong energy conditions ,which arise from the Raychaudhuri equation for the expansion [19, 20], to the modified gravity model. In the literature, this approach has been extensively studied to evaluate the possible ranges of the free parameter of the generalized gravity models. For instants, the energy bound have been explored to constrain theories of gravity [21, 22, 23, 24] and some extensions of gravity [25, 26, 27, 28, 29, 30, 31, 32, 33], modified Gauss-Bonnet gravity [34, 35, 36, 37] and scalar-tensor gravity [38, 39]. The energy condition have also been analysed in gravity [40, 41, 42] and generalized models of gravity [43, 44, 45, 46]. To study the energy conditions in this modified teleparallel gravity, we consider the flat FRW universe model with perfect fluid matter and define an effective energy density and pressure originates from the higher-derivative torsion terms. Then we discuss the energy conditions in term of the cosmographical parameters such as the Hubble, deceleration, jerk, snap and lerk parameters. Particularly, we consider two specific models that are proposed in literature and using the present-day values of the cosmographical parameters, we analyze the weak energy condition to determine the possible constraints on the free parameter of the presented models. The paper is organized as follows: In section 2 we review the modified gravity with higher-derivative torsion terms, the equations of motion and the resulted modified Friedmann equations related to the model. Section 3 is devoted to the energy conditions in this modified teleparallel gravity. In section 4, we explore the weak energy condition in two specific models of the scenario by using present-day values of the cosmographic quantities. Finally, our conclusion will be appeared in section 4. 2 The field equations The action of the modified teleparallel gravity with higher-derivative torsion terms is defined as where , and , and for simplicity we have set . The is the matter action includes the general matter field which can ,in general, have an arbitrary coupling to the vierbein field. If the matter couples to the metric in the standard form, then varying the action (1) with respect to the vierbein yields the generalized field equations as follows where for simplicity, we have used the notation and . Note that and denote derivative with respect to the torsion scalar and , with , respectively and is the matter energy momentum tensor which is defined as . In order to study the cosmological implication of the model, we consider a spacially flat Friedmann-Robertson-Walker (FRW) universe with metric . This metric arises from the following diagonal vierbein where is the scale factor. Now we assume that the matter content of the universe is given by the perfect fluid with the energy density and pressure . Thus using the field equations (2), we obtain the generalized Friedmann equations as follows where dot denotes the derivative with respect to cosmic time and is the Hubble parameter. With the definition of the veirbien (3), the torsion scalar and the functions and respectively are given by where the energy density and pressure of the effective dark energy sector are respectively defined as Since we have assumed a minimally coupled matter to the veirbien field, the standard matter is satisfied in continuity equation , i.e. . So from the friedmann equations (9) and (10), one can easily verified that the dark energy density and pressure satisfy the conservation equation In the rest of this paper we analyze the viability of this modified teleparallel gravity scenario by studying the energy conditions to constrain the free parameters of the model. 3 Energy conditions The energy conditions are originated from the Raychaudhuri equation together with the requirement that gravity is attractive for a space-time manifold which is endowed by a metric . In the case of a congruence of timelike and null geodesics with tangent vector field and respectively, the Raychaudhuri equation is given by the temporal variation of expansion for the respective curve as follows [19, 20, 24] Here is the Ricci tensor and , and are, respectively, the expansion scalar, shear tensor and rotation tensor associated with the congruence of timelike or null geodesics. Note that the Raychaudhuri equation is a purely geometric equation hence, it makes no reference to a specific theory of gravitation. Since the shear is a purely spatial tensor (), for any hypersurface of orthogonal congruence (), the conditions for attractive gravity reduces to where the first condition is refer to the strong energy condition (SEC) and the second condition is named null energy condition (NEC). From the field equations in general relativity and its modifications, the Ricci tensor is related to the energy-momentum tensor of the matter contents. Thus inequalities (16) give rise to the respective physical conditions on the energy-momentum tensor as where is the trace of the energy-momentum tensor. For perfect fluid with energy density and pressure , the SEC and NEC are defined by and respectively, while the dominant energy condition (DEC) and weak energy condition (WEC) are defined respectively, by and . Note that the violation of the NEC leads to the violation of all other conditions. Since the Raychaudhuri equation is a purely geometric equation, the concept of energy conditions can be extended to the case of modified theories of gravity with the assumption that the total matter contents of the universe act like a perfect fluid. Hence, the respective conditions can be defined by replacing the energy density and pressure with an effective energy density and effective pressure, respectively as follows To get some insight on the meaning of the above energy conditions, in the next section, we consider two specific functions for the Lagrangian (1), to obtain the constraints on the parametric space of the model. 4 Constraints on specific Models In order to analyze the torsional modified gravity model with higher-derivative terms from the point of view of energy conditions, we use the standard terminology in studying energy conditions for modified gravity theories. To this end, we investigate such energy bounds in terms of the cosmographic parameters , i.e. the Hubble, deceleration, jerk, snap and lerk parameters, defined respectively as where the superscripts represent the derivative with respect to time. In terms of these parameters, the Hubble parameter as well as its higher time derivatives are given by 4.1 Model I: where , and are constants. It has been shown that for a wide range of the model parameters the universe can result in a dark-energy dominated, accelerating universe and the model can describe the thermal history of the universe, i.e. the successive sequence of radiation, matter and dark energy epochs, which is a necessary requirement for any realistic scenario. Since for a theoretical model to be cosmologically viable, it should satisfy at least the weak energy condition, we examine specially the weak energy condition in our analysis. More ever, for simplicity we consider vacuum, i.e. . Inserting the cosmographical parameters (LABEL:hubble-dot) in Equ. (20), the bounds on the model parameters imposed by the weak energy condition are given by respectively. The subscript stands for the present value of the cosmographic quantities. Now, we take the following observed values for the , , , and . The numerical results for satisfying the weak energy condition are given in figure 1. For the parametric spaces of the model, we have fixed the value of to and plot the and versus and . As the figure shows, the WEC is satisfied in the specific form of Equ. (25) for a suitable choice of the subspaces of the model parametric space. 4.2 Model II: In second case, we consider a class of models in which the action does not depend on but only on given by the following functional form where , , and are constants. It has been found that in this model the universe will be led to a dark energy dominated, accelerating phase for a wide region of the parameter space. The scale factor behaves asymptotically either as a power law or as an exponential law, while for large parameter regions the exact value of the dark-energy equation-of-state parameter can be in great agreement with observations . To examine the model via energy conditions, in a similar procedure to the previous subsection, we consider the vacuum. Using the cosmographical parameters as before, the condition to justification of the WEC are obtained as

Exploratory factor analysis |This page summarises key points about the use of exploratory factor analysis particularly for the purposes of psychometric instrument development. For a hands-on tutorial about the steps involved, see EFA tutorial.| Assumed knowledge[edit | edit source] Purposes of factor analysis[edit | edit source] There are two main purposes or applications of factor analysis: - 1. Data reduction Reduce data to a smaller set of underlying summary variables. For example, psychological questionnaires often aim to measure several psychological constructs, with each construct being measured by responses to several items. Responses to several related items are combined to create a single score for the construct. A measure which involves several related items is generally considered to be more reliable and valid than relying on responses to a single item. - 2. Exploring theoretical structure Theoretical questions about the underlying structure of psychological phenomena can be explored and empirically tested using factor analysis. For example, is intelligence better understood as a single, general factor, or as consisting of multiple, independent dimensions? Or, how many personality factors are there and what are they? History[edit | edit source] There are several requirements for a dataset to be suitable for factor analysis: - Normality: Statistical inference is improved if the variables are multivariate normal - Linear relations between variables - Test by visually examining all or at least some of the bivariate scatterplots: - Is the relationship linear? - Are there bivariate outliers? - Is the spread about the line of best fit homoscedastic (even (or cigar-shaped) as opposed to fanning in or out))? - If there are a large number of variables (and bivariate scatterplots), then consider using Matrix Scatterplots to efficiently visualise relations amongst the sets of variables within each factor (e.g., a Matrix Scatterplot for the variables which belong to Factor 1, and another Matrix Scatterplot for the variables which belong to Factor 2 etc.) - Factorability is the assumption that there are at least some correlations amongst the variables so that coherent factors can be identified. Basically, there should be some degree of collinearity among the variables but not an extreme degree or singularity among the variables. Factorability can be examined via any of the following: - Inter-item correlations (correlation matrix) - are there at least several small-moderate sized correlations e.g., > .3? - Anti-image correlation matrix diagonals - they should be > ~.5. - Measures of sampling adequacy (MSAs): - Kaiser-Meyer-Olkin (KMO) (should be > ~.5 or .6) and - Bartlett's test of sphericity (should be significant) - Sample size: The sample size should be large enough to yield reliable estimates of correlations among the variables: - Ideally, there should be a large ratio of N / k (Cases / Items) e.g., > ~20:1 - e.g., if there are 20 items in the survey, ideally there would be at least 400 cases) - EFA can still be reasonably done with > ~5:1 - Bare min. for pilot study purposes, as low as 3:1. - Ideally, there should be a large ratio of N / k (Cases / Items) e.g., > ~20:1 For more information, see these lecture notes. Types (methods of extraction)[edit | edit source] The researcher will need to choose between two main types of extraction: - Principal components (PC): Analyses all variance in the items. This method is usually preferred when the goal is data reduction (i.e., to reduce a set of variables down to a smaller number of factors and to create composite scores for these factors for use in subsequent analysis). - Principal axis factoring (PAF): Analyses shared variance amongst the items. This method is usually preferred when the goal is to undertake theoretical exploration of the underlying factor structure. Rotation[edit | edit source] The researcher will need to choose between two main types of factor matrix rotation: - Orthogonal (Varimax - in SPSS): Factors are independent (i.e., correlations between factors are less than ~.3) - Oblique (Oblimin - in SPSS): Factors are related (i.e., at least some correlations between factors are greater than ~.3). The extent of correlation between factors can be controlled using delta. - Negative values "decrease" factor correlations (towards full orthogonality) - "0" is the default - Positive values (don't go over .8) "permit" higher factor correlations. If the researcher hypothesises uncorrelated factors, then use orthogonal rotation. If the researchers hypothesises correlated factors, then use oblique rotation. In practice, researchers will usually try different types of rotation, then decide on the best form of rotation based on the rotation which produces the "cleanest" model (i.e., with lowest cross-loadings). Determining the number of factors[edit | edit source] There is no definitive, simple way to determine the number of factors. The number of factors is a subjective decision made by the researcher. The researcher should be guided by several considerations, including: - Theory: e.g., How many factors were expected? Do the extracted factors make theoretical sense? - Eigen values: - Kaiser's criterion: How many factors have eigen-values over 1? Note, however, that this cut-off is arbitrary, so is only a general guide and other considerations are also important. - Scree-plot: Plots eigen-values. Look for the 'elbow' minus 1 (i.e., where there is a notable drop); the rest is 'scree'. Extract the number of factors that make up the 'cliff' (i.e., which explain most of the variance). - Total variance explained: Ideally, try to explain approximately 50 to 75% of the variance using the least number of factors - Interpretability: Are all factors interpretable? (especially the last one?) In other words, can you reasonably name and describe each set of items as being indicative of an underlying factor? - Alternative models: Try several different models with different numbers of factors before deciding on a final model and number of factors. Depending on the Eigen Values and the screen plot, examine, say, 2, 3, 4, 5, 6 and 7 factor models before deciding. - Remove items that don't belong: Having decided on the number of factors, items which don't seem to belong should be removed because this can potentially change and clarify the structure/number of factors. Remove items one at a time and then re-run. After removing all items which don't seem to belong, re-check whether you still have a clear factor structure for the targetted number of factors. It may be that a different number of factors (probably one or two fewer) is now more appropriate. For more information, see criteria for selecting items. - Number of items per factor: The more items per factor, the greater the reliability of the factor, but the law of diminishing returns would apply. Nevertheless, a factor could, in theory, be indicated by as little as a single item. - Factor correlations - What are the correlations between the factors? If they are too high (e.g., over ~.7), then some of the factors may be too similar (and therefore redundant). Consider merging the two related factors (i.e., run an EFA with one less factor). - Check the factor structure across sub-samples - For example, is the factor structure consistent for males and females? (e.g., in SPSS this can be done via Data - Split file - Compare Groups or Organise Output by Groups - Select a categorical variable to split the analyses by (e.g., Gender) - Paste/Run or OK - Then re-run the EFA syntax) Mistakes in factor extraction may consist in extracting too few or too many factors. A comprehensive review of the state-of-the-art and a proposal of criteria for choosing the number of factors is presented in . Criteria for selecting items[edit | edit source] In general, aim for a simple factor structure (unless there is a particular reason why a complex structure would be preferable). In a simple factor structure each item has a relatively strong loading on one factor (target loading; e.g., > |.5|) and relatively small loadings on other factors (cross-loadings; e.g., < |.3|). Consider the following criteria to help decide whether to include or remove each item. Remember that these are rules of thumb only – avoid over-reliance on any single indicator. The overarching goal is to include items which contribute to a meaningful measure of an underlying factor and to remove items that weaken measurement of the underlying factor(s). In making these decisions, consider: - Communality - indicates the variance in each item explained by the extracted factors; ideally, above .5 for each item. - Primary (target) factor loading - indicates how strongly each item loads on each factor; should generally be above |.5| for each item; preferably above |.6|. - Cross-loadings - indicate how strongly each item loads on the other (non-target) factors. There should be a gap of at least ~.2 between the primary target loadings and each of the cross-loadings. Cross-loadings above .3 are worrisome. - Meaningful and useful contribution to a factor - read the wording of each item and consider the extent to which each item appears to make a meaningful and useful (non-redundant) contribution to the underlying target factor (i.e., assess its face validity) - Reliability - check the internal consistency of the items included for each factor using Cronbach's alpha and check the "Alpha if item removed" option to determine whether removal of any additional items would improve reliability) - See also: How do I eliminate items? (lecture notes) Name and describe the factors[edit | edit source] Once the number of factors has been decided and any items which don't belong have been removed, then - Give each extracted factor a name - Be guided by the items with the highest primary loadings on the factor – what underlying factor do they represent? - If unsure, emphasise the top loading items in naming the factor - Describe each factor - Develop a one sentence definition or description of each factor Data analysis exercises[edit | edit source] Pros and cons[edit | edit source] - Basic terms - Anti-image correlation matrix: Contains the negative partial covariances and correlations. Diagonals are used as a measure of sampling adequacy (MSA). Note: Be careful not to confuse this with the anti-image covariance matrix. - Bartlett's test of sphericity: Statistical test for the overall significance of all correlations within a correlation matrix. Used as a measure of sampling adequacy (MSA). - Common variance: Variance in a variable that is shared with other variables. - Communality: The proportion of a variable's variance explained by the extracted factor structure. Final communality estimates are the sum of squared loadings for a variable in an orthogonal factor matrix. - Complex variable: A variable which has notable loadings (e.g., > .4) on two or more factors. - Correlation: The Pearson or product-moment correlation coefficient. - Composite score: A variable which represents combined responses to multiple other variables. A composite score can be created as unit-weighted or regression-weighted. A composite score is created for each case for each factor. - Correlation matrix: A table showing the linear correlations between all pairs of variables. - Data reduction: Reducing the number of variables (e.g., by using factor analysis to determine a smaller number of factors to represent a larger set of factors). - Eigen Value: Column sum of squared loadings for a factor. Represents the variance in the variables which is accounted for by a specific factor. - Exploratory factor analysis: A factor analysis technique used to explore the underlying structure of a collection of observed variables. - Extraction: The process for determining the number of factors to retain. - Factor: Linear combination of the original variables. Factors represent the underlying dimensions (constructs) that summarise or account for the original set of observed variables. - Factor analysis: A statistical technique used to estimate factors and/or reduce the dimensionality of a large number of variables to a fewer number of factors. - Factor loading: Correlation between a variable and a factor, and the key to understanding the nature of a particular factor. Squared factor loadings indicate what percentage of the variance in an original variable is explained by a factor. - Factor matrix: Table displaying the factor loadings of all variables on each factor. Factors are presented as columns and the variables are presented as rows. - Factor rotation: A process of adjusting the factor axes to achieve a simpler and pragmatically more meaningful factor solution - the goal is a usually a simple factor structure. - Factor score: Composite score created for each observation (case) for each factor which uses factor weights in conjunction with the original variable values to calculate each observation's score. Factor scores are standardised to according to a z-score. - Measure of sampling adequacy (MSA): Measures which indicate the appropriateness of applying factor analysis. - Oblique factor rotation: Factor rotation such that the extracted factors are correlated. Rather than arbitrarily constraining the factor rotation to an orthogonal (90 degree angle), the oblique solution allows the factors to be correlated. In SPSS, this is called Oblimin rotation. - Orthogonal factor rotation: Factor rotation such that their axes are maintained at 90 degrees. Each factor is independent of, or orthogonal to, all other factors. In SPSS, this is called Varimax rotation. - Parsimony principle: When two or more theories explain the data equally well, select the simplest theory e.g., if a 2-factor and a 3-factor model explain about the same amount of variance, interpret the 2-factor model. - Principal axis factoring (PAF): A method of factor analysis in which the factors are based on a reduced correlation matrix using a priori communality estimates. That is, communalities are inserted in the diagonal of the correlation matrix, and the extracted factors are based only on the common variance, with unique variance excluded. - Principal component analysis (PC or PCA): The factors are based on the total variance of all items. - Scree plot: A line graph of Eigen Values which is helpful for determining the number of factors. The Eigen Values are plotted in descending order. The number of factors is chosen where the plot levels off (or drops) from cliff to scree. - Simple structure: A pattern of factor loading results such that each variable loads highly onto one and only one factor. - Unique variance: The proportion of a variable's variance that is not shared with a factor structure. Unique variance is composed of specific and error variance. - Common factor: A factor on which two or more variables load. - Common factor analysis: A statistical technique which uses the correlations between observed variables to estimate common factors and the structural relationships linking factors to observed variables. - Error variance: Unreliable and inexplicable variation in a variable. Error variance is assumed to be independent of common variance, and a component of the unique variance of a variable. - Image of a variable: The component of a variable which is predicted from other variables. Antonym: anti-image of a variable. - Indeterminacy: If it is impossible to estimate population factor structures exactly because an infinite number of factor structures can produce the same correlation matrix, then there are more unknowns than equations in the common factor model, and we say that the factor structure is indeterminate. - Latent factor: A theoretical underlying factor hypothesised to influence a number of observed variables. Common factor analysis assumes latent variables are linearly related to observed variables. - Specific variance: (1) Variance of each variable unique to that variable and not explained or associated with other variables in the factor analysis. (2) The component of unique variance which is reliable but not explained by common factors. References[edit | edit source] - Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4, 272–299. - Tabachnick, B. G. & Fidell, L. S. (2001). Principal components and factor analysis. In Using multivariate statistics (4th ed., pp. 582–633). Needham Heights, MA: Allyn & Bacon. - Iantovics, L.B., Rotar, C., Morar, F.: Survey on establishing the optimal number of factors in exploratory factor analysis applied to data mining, Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 9(2), 2019, e1294. See also[edit | edit source] - Lecture notes - Data analysis tutorial - Internal consistency - Composite scores - Practice quiz - Psychometric instrument development - Sample write-ups - Survey research and design in psychology - Wikipedia & Wikibooks - Exploratory factor analysis (Wikipedia) - Factor analysis in psychometrics (Wikipedia) - Principal component analysis (Wikipedia) - Principal component analysis (Wikibooks) [edit | edit source] - Darlington, R. B., Factor analysis. - Exploratory factor analysis (Lecture slides on slideshare.net) - Exploratory factor analysis (Lecture on ucspace.canberra.edu.au) - Factor analysis links (del.icio.us) - Factor analysis resources: Understanding & using factor analysis in psychology & the social sciences (Wilderdom) - Open and free online course on exploratory data analysis (Carnegie Mellon University) - Principal components and factor analysis (statsoft.com) - Factor analysis: Principal components factor analysis: Use of extracted factors in multivariate dependency models (bama.ua.edu)

Chemistry 2 Lecture 3 Particle on a ring approximation Learning outcomes from Lecture 2 •Be able to explain why confining a particle to a box leads to quantization of its energy levels • Be able to explain why the lowest energy of the particle in a box is not zero • Be able to apply the particle in a box approximation as a model for the electronic structure of a conjugated molecule (given equation for En). Assumed knowledge for today Be able to predict the number of π electrons and the presence of conjugation in a ring containing carbon and/or heteroatoms such as nitrogen and oxygen. The de Broglie Approach • The wavelength of the wave associated with a particle is related to its momentum: p = mv = h / λ • For a particle with only kinetic energy: E = ½ mv2 = p2 / 2m = h2 / 2mλ2 Particle-on-a-ring • Particle can be anywhere on ring • Ground state is motionless Particle-on-a-ring • Ground state is motionless • In higher levels, we must fit an integer number of waves around the ring 1 wave λ = 2πr 2 waves λ = 2πr/2 3 waves λ = 2πr/3 The Schrödinger equation • The total energy is extracted by the Hamiltonian operator. • These are the “observable” energy levels of a quantum particle Energy eigenfunction Hamiltonian operator Energy eigenvalue The Schrödinger equation • The Hamiltonian has parts corresponding to Kinetic Energy and Potential Energy. In terms of the angle θ: 2 2 ˆ H V ( 2mr 2 2 ) Potential Energy Hamiltonian operator Kinetic Energy “The particle on a ring” • The ring is a cyclic 1d potential E must fit an integer number of wavelengths 0 0 2p “The particle on a ring” p-system of benzene is like a bunch of electrons on a ring “The particle on a ring” • On the ring, V = 0. Off the ring V = ∞. sin j 2 2 ˆ H sin j 2 2 2mr 2 2 j 2mr 2 sin j j j = 1, 2, 3…. “The particle on a ring” • On the ring, V = 0. Off the ring V = ∞. cos j 2 2 ˆ H cos j 2 2 2mr 2 2 j 2mr 2 cos j j j = 0, 1, 2, 3…. Particle-on-a-ring • Ground state is motionless = constant “The particle on a ring” • The ring is a cyclic 1d potential E must fit an integer number of wavelengths 0 0 2p “The particle on a ring” j 2p j j 2 2 2m r mL 2 2 2 2 2 j = 0, 1, 2, 3…. length of circumference radius of ring j=3 j=2 j=1 j=0 “The particle on a ring” j=3 All singly degenerate Doubly degenerate above j=0 n=4 j=2 n=3 j=1 n=2 n=1 j=0 box ring Application: benzene Question: how many p-electrons in benzene? Answer: Looking at the structure, there are 6 carbon atoms which each contribute one electron each. Therefore, there are 6 electrons. benzene Question: what is the length over which the pelectrons are delocalized, if the average bond length is 1.40 Å? Answer: There are six bonds, which equates to 6 × 1.40 Å = 8.40 Å benzene Question: if the energy levels of the electrons are given by n = 2ℏ2j2p2/mL2, what is the energy of the HOMO in eV? Answer: since there are 6 p-electrons, and therefore the HOMO must have j=1. We know that L = 6 × 1.40 Å = 8.4 0Å. From these numbers, we get j = 3.41×10-19 j2 in Joules. The energy of the HOMO is thus 1 = 3.41×10-19J = 2.13 eV. j=3 j=2 j=1 j=0 benzene Question: what is the energy of the LUMO, and thus the HOMO-LUMO transition? j=3 Answer: j = 3.41×10-19 j2 in Joules. The energy of the LUMO is thus 2 = 1.365×10-18J = 8.52 eV. The energy of the HOMO-LUMO transition is thus 6.39 eV. j=2 j=1 j=0 benzene Question: how does the calculated value of the HOMO-LUMO transition compare to experiment? Answer: The calculated energy of the HOMO-LUMO transition is 6.39 eV. This corresponds to photons of wavelength l = hc/(6.39× 1.602×10-19) ~ 194 nm, which is not so far from the experimental value (around 200 nm). j=3 Hiraya and Shobatake, J. Chem. Phys. 94, 7700 (1991) j=2 j=1 j=0 Learning Outcomes • Be able to explain why confining a particle on a ring leads to quantization of its energy levels • Be able to explain why the lowest energy of the particle on a ring is zero • Be able to apply the particle on a ring approximation as a model for the electronic structure of a cyclic conjugated molecule (given equation for En). Next lecture • Quantitative molecular orbital theory for beginners Week 10 tutorials • Schrödinger equation and molecular orbitals for diatomic molecules Practice Questions 1. The particle on a ring has an infinite number of energy levels (since j = 0, 1,2, 3, 4, 5 …) whereas for a ring CnHn has only n p-orbitals and so n energy levels. C6H6, for example, only has levels with j = 3 (one level), j = 1 (two levels), j = 2 (two levels) and j = 3 (one level) (a) Using the analogy between the particle on a ring waves and the πorbitals on slide 17, draw the four π molecular orbitals for C4H4 and the six π molecular orbitals for C6H6 (b) Using qualitative arguments (based on the number of nodes and/or the number of in-phase or out-of-phase interactions between neighbours) construct energy level diagrams and label the orbitals as bonding, non-bonding or antibonding (c) Based on your answer to (b), why is C6H6 aromatic and C4H4 antiaromatic?

I read somewhere that LEDs could self limit current up to a certain point and had a few lying around in my box so I decided to test this statement. Its true for roughly the voltage drop across the LED. But the question comes is what happened when I applied 20V to the LED? It made a loud popping sound. What happened to the internals of the LED? 8\$\begingroup\$ Um - They popped - blown, gone, killed.... \$\endgroup\$– Linker3000May 24, 2011 at 21:50 2\$\begingroup\$ I also noticed that my white LEDs went from white to blue and then to purple before they popped. @Linker3000 Define "popped", "blown", "gone", "killed". They are all different words with different meanings and each of the words has large number of meanings which are not related to process that occurred here. To me it looks like the question is about the physical process of LED destruction, which is very important to understand so that its early stages could be detected in a circuit that appears to be running fine. \$\endgroup\$– AndrejaKoMay 24, 2011 at 21:55 6\$\begingroup\$ You turned it into a DED, a Dark Emitting Diode. Although you might have gotten lucky and turned it into a SED, or Smoke Emitting Diode. \$\endgroup\$– user3624May 25, 2011 at 2:49 2\$\begingroup\$ You released the Magic Smoke! \$\endgroup\$– Connor WolfMay 26, 2011 at 23:32 1\$\begingroup\$ What happened was approximately the same thing that happened to a neon bulb I put accross 230 Vac as a kid at age 8, without the > 100k series resistor. It made a bang. Also, your parents might have been running towards your work bench, checking if you were still o.k. \$\endgroup\$– zebonautJun 3, 2011 at 22:40 Increasing heat from power dissipation causes a failure of the LED die. The change in colour, e.g. red and green LEDs going yellow at high currents, is probably because the die is actually glowing hot, i.e. near failure. Note that the red LED has a fall in wavelength, but the green one has an increase in wavelength. White LEDs going blue could be explained by the yellow-emitting phosphor in the LED being less effective at high currents. White LEDs are often constructed from a blue LED coated with a special phosphor which emits yellow light when blue light hits it creating a fairly even white light. So perhaps you are seeing more blue than the yellow phosphor can convert. 1\$\begingroup\$ Maybe the phosphor already died at that point. The orange/red glow is probably from heat alone. Once upon a time I had an EPROM glow red when I accidentally reversed the polarity of the power supply. \$\endgroup\$– starblueMay 25, 2011 at 19:28 An LED is a Light Emitting Diode. The key part of that name is "Diode." An LED is a diode. Diodes do not limit forward current very well. The extremely steep current/voltage curve (an exponential curve) is probably their second most important functional characteristic which results in sales of diodes. When you put a forward voltage on a diode (an LED, BE junction of a BJT, or whatever), it's current DOUBLES with each incremental 26mV of voltage. So if you applied 0.7V (700mV) and got 50mA, then you should get about 100mA at 726mV, 200mA at 752mV, 400mA at 778mV, and so on. So what would you expect at 20000mV? The theoretical answer is about 7 times 10 to the 222nd power amps. That is a '7' with about 222 zeros after it. But your home's circuit breaker (thankfully) limits current draw to something less than Quadra-Bazillions of times the total of all power plants on planet Earth. Your LED draws maybe 20 amps for a few microseconds, turning a very small volume inside it about as hot as the Sun, and then it is all over. If you do this with larger parts, the shrapnel can kill you. An electrian, at a factory I worked at, had the misfortune of crossing two phases of industrial strength AC with his screwdriver. As far as could be figured out, he was not electricuted: the plastic screwdriver handle protected him from that. However, the short instantly vaporized the metal in the screwdriver. The resulting explosion killed him. So provide external current limiting in line with your LED, or... wear safety goggles. EDIT: I got carried away with the point I was trying to make and erroneously said current doubles with each 25 mV across the diode junction. The actual factor is 'e', where 'e' = the base of the natural logarithm = about 2.7. So the current would increase by a factor of 2.7, not 2, for each 25mV. The damage to the device looks pretty much the same for large voltages, though.... \$\begingroup\$ A friend of a friend was working in a busbar cabinet, bolting down something with a spanner when he was startled by a spider - so he threw the spanner at it! As you said, the tool vaporized, but the resulting induction field twisted half the box off the wall. The engineer survived. No word on the spider. \$\endgroup\$ May 27, 2011 at 13:48 \$\begingroup\$ You are describing the Wikipedia topic Shockley diode equation. \$\endgroup\$ Jan 7, 2018 at 3:42 The weakest electrical portion of the LED fused. That part would vary based on LED construction. I've had this happen and blow off the tip of a 3mm LED with surprising energy, when I was young and messing with such questions. It hit the ceiling pretty hard.

Chapter 9: Description of Siemens LMS Vibration Control Routines in Simulink environment 9.2 S IMULINK MODEL DESCRIPTION The Simulink model represents the real “heart” of the control system. It is a very complex system in term of logic and implemented functions. It is composed by different blockset drowned at each level as Mux, Demux, MatLab embedded functions, switch and transfer functions and it is written in order to be an HIL closed loop system, capable to reproduce the same characteristics and behaviours of the test facility control system. The main control environment is endowed of appropriate “scopes” in order to see the output in real time. In Figure 51 and Figure 52 the Simulink environment at different levels is shown. How is possible to see, the control system shows different masks. They shadow control logic, but, going deeper, there are some function impossible to analyse in order to better understand the logic. In these terms, the Simulink model will be considered as a “black box” when the simulation run. pag. 88 Figure 51: Main control environment pag. 89 Figure 52: Detailed sine control environment pag. 90 However, from a system point of view, is possible to describe the qualitative way in which the control system operates. The code is able to perform the vibration control of a loaded system or device using up to 40 notchers (or DoFs) in order to monitoring their response. It is written in order to guarantee always 4 pilot curves. In fact, the original version of the model considered provide just one output coming from DEMUX. It is repeated considering 3 gain factors that amplifies the input signal. Subsequently, it becomes composed by 4 output. Then, is provided to the following MUX that recollect them with the other 40 signals coming from notchers in which some of them comes from the “C” matrix of the state space system and the remaining part from the “ground”. Figure 53 (a) Figure 53: Modification to the Simulink model: before (a), after (b) This implemented modelling strategy were tested with the reduced models mass- spring- damper described in “Chapter 8: The Virtual Shaker Testing approach (VST)” but is not appropriate in order to perform the VST using a condensed model in which the pilots are extracted with the Craig- Bampton theory, according to the location of the physical accelerometers attached to the vibration platform. For this reason, the Simulink model were modified in order to extract exactly 4 signals from DEMUX. Particularly, they refer to the pilots of the Craig- Bamption theory located into the first four position of the “C” matrix. Figure 53 (b). After that, the signals come into the real “sine controller” which, for each time step, perform actions as notching level calculation, control amplitude, time delay or sinusoidal input means cola. In Figure 54 is shown as is appear the mask of the sine control in which the input variables are recalled. One of the key points of the sine control are the control and estimation strategy. The first one represents the way in which the control takes place. It concerns the trend, and the values, of the pilots. Particularly, three possibilities are available: • Maximum: the control signal generates a control profile characterized by the maximum value of the pilots • Average: the control signal generates a control profile characterized by the sum of the pilot signals divided by the number of control channel, chosen in “Flow.m” • Minimum: the control signal generates a control profile characterized by the minimum value of the pilots The second one gives the way in which all the signals are evaluated. Basically, there are four possibilities: • Peak: it takes the greatest amplitude of the sample signal. If the system is characterized by noisy, this kind of evaluation could introduce some instabilities • Average: if the signal is characterized by “N” sample time, it calculates the average of the absolute value of them. It takes the complete signal. • RMS: it calculates the average of the squared values of “N” sample time during one period. As the average method, it evaluates the complete signal. It is able to produce a low drive signal • Harmonic: it is considered as the appropriate valuer for fundamental frequency research. It provides magnitude and phase response Figure 54: Sine control block parameters When the simulation starts, it requires the initial condition in order to join into the loop. They usually are the homogeneous conditions in which, using the formulation of a dynamic second order system, are displacement and speed equal to zeros. However, the block scheme implemented in Simulink is the “Discrete State Space”, shown in Figure 55. It come from the “Continuous State Space” system. In fact, how is described in “ pag. 92 Chapter 2: The State space systems”, this kind of mathematical formulation represent the best way in which to perform the control of a dynamic or electro-dynamic system. Particularly, the discrete formulation comes from the continuous quantizing the matrix operator with the "𝑡𝑠" sampling time consistent of the data acquisition system of the vibration facility. Nevertheless, from the algorithm point of view, it is able to solve the differential equation using the same procedure of the finite difference. Figure 55: Simulink block for Discrete State Space On the other hand, the code is not able to undergoes to modification in terms of solver. In fact, it is written usign a “discrete logic”, so is not possible to replace the “Conotinuous state space block” using the Runge-Kutta method. For sake of clarity, if Runge- Kutta is the exact approximation of a differential equation in time domain, the discrete state space solver is the approximation of this. In these term, the extracted curves using the discrete method will match in some points with them extrated using the continuous method. When the simulation advance up to the maximum value of the frequency (typically 100 Hz for sine sweep) the Simulink model send out two matrix: “spectra.mat” and “spectra_notch.mat” in which are recollect the requested output and classified by columns in term of time, frequency related notcher values and control profiles. The output that is possible to extract to the sine control system are: • 4 pilot curves: they describe the variation in time domain of the considered pilot. How we said before, talking in terms of signals, is necessary to guarantee 4 signals to the control system. They could be overlapped or not • Up to 40 notched curves in frequency and time domain • The control signal profile compared to the abort and alarm limits • Two “on/off diagram” in which are shown if one, or more Dofs are notched and which of them • The drive amplitude of the input • The cola: it shows the sinusoidal input imposed to the coil • The enhancement of the frequency during the simulation Particularly, the control curve (i.e. the acceleration of the pilots handles by one of the controls and strategy criteria) is compared to the upper and lower abort and alarm limit. pag. 93 Usually, during a vibration test if the control curve exceeds one of the two abort limit (±6 dB) it is interrupted. However, into the LMS vibration control environment we are to see the control curve during all the simulation span, in frequency or time domain, even if it exceeds the upper or the lower value. Probably, in fact, the aim of do not interrupt the running simulation is to permit to the engineer to observe where and when the control “burst” or “disappears”. This, in order to apply the required modification to the control environment in terms of compression factor, notch profile, control strategy or damping model if the FEM model is unprovided. In this way, the sine control lives in a wide set of simulation distinguished by the possible variation of each control parameter and under a precise strategy to distinguish the type of control. Obviously, the final result of the VST activity is to suggest the more appropriate value at each control parameter, before the real vibration test into the facility centre, in order to observe some specified behaviours during the base excitation and guarantee that them do not provoke any rupture. For these reason does not exist a unique and unequivocal result, but it depends on what the test facility engineers want to observe. In these terms, the VST shall emanate a variety of output, based on what they want.

Emergent Universe Scenario, Bouncing and Cyclic Universes in Degenerate Massive Gravity Shou-Long Li, H. Lü, Hao Wei, Puxun Wu and Hongwei Yu Department of Physics and Synergetic Innovation Center for Quantum Effect and Applications, Hunan Normal University, Changsha 410081, China School of Physics, Beijing Institute of Technology, Beijing 100081, China Center for Joint Quantum Studies, School of Science, Tianjin University, Tianjin 300350, China We consider alternative inflationary cosmologies in massive gravity with degenerate reference metrics and study the feasibilities of the emergent universe scenario, bouncing and cyclic universes. We focus on the construction of the Einstein static universe, classes of exact solutions of bouncing and cyclic universes in degenerate massive gravity. We further study the stabilities of the Einstein static universe against both homogeneous and inhomogeneous scalar perturbations and give the parameters region for a stable Einstein static universe. firstname.lastname@example.org email@example.com firstname.lastname@example.org email@example.com firstname.lastname@example.org General relativity (GR), as a classical theory describing the non-linear gravitational interaction of massless spin-2 fields, is widely accepted at low energy limit. Nevertheless, there are still several motivations to modify GR, based on both theoretical considerations (e.g. [1, 2]) and observations (e.g.[3, 4].) One proposal, initiated by Fierz and Pauli , is to assume that the mass of graviton is nonzero. Unfortunately, the interactions for massive spin-2 fields in Fierz-Pauli massive gravity have long been thought to give rise to ghost instabilities . Recently, the problem has been resolved by de Rham, Gabadadze and Tolley (dRGT) , and dRGT massive gravity has attracted great attention and is studied in various areas such as cosmology [7, 8, 9, 10] and black holes [11, 12]. We refer to e.g. [13, 14, 15] and reference therein for a comprehensive introduction of massive gravity. There are several extensions of dRGT massive gravity for different physical motivations, such as bi-gravity , multi-gravity , minimal massive gravity , mass-varying massive gravity , degenerate massive gravity and so on . Thereinto, the degenerate massive gravity was initially proposed by Vegh to study holographically a class of strongly interacting quantum field theories with broken translational symmetry. Later this theory has been studied widely in the holographic framework [22, 23, 24, 25] and black hole physics [26, 27, 28, 29, 30]. However, the cosmological application of this theory is few. Recently, together with suitable cubic Einstein-Riemann gravities and some other matter fields, degenerate massive gravity was used to construct exact cosmological time crystals with two jumping points, which provides a new mechanism of spontaneous time translational symmetry breaking to realize the bouncing and cyclic universes that avoid the initial spacetime singularity. It is worth noting that the higher derivative gravities are indispensable for the realization of cosmological time crystals. However, if we consider only the infrared modification of GR, we can study instead the feasibilities of bouncing and cyclic models in degenerate massive gravity without the higher-order curvature invariants. Actually, it is valuable to investigate alternative inflationary cosmological models within the standard big bang framework, because traditional inflationary cosmology [32, 33, 34, 35] suffers from both initial singularity problem and trans-Planckian problem . By introducing a mechanism for a bounce in cosmological evolution, both the trans-Planckian problem and initial singularity can be avoided. The bouncing scenario can be constructed via many approaches such as matter bounce scenario , pre-big-bang model , ekpyrotic model , string gas cosmology , cosmological time crystals and so on [42, 43, 44]. The cyclic universe, e.g. , can be viewed as the extension of the bouncing universe since it brings some new insight into the original observable universe . Another direct solution to the initial singularity proposed by Ellis et al. [47, 48], i.e., emergent universe scenario, is assuming that the universe inflates from a static beginning, i.e., the Einstein static universe, and reheats in the usual way. In this scenario, the initial universe has a finite size and some past-eternal inflation, and then evolves to an inflationary era in the standard way. Both horizon problem and the initial singularity are absent due to the initial static state. Actually, these alternative inflationary cosmologies have been studied in different class of massive gravities. The bouncing and cyclic universes have been studied in mass-varying massive gravity . The emergent scenario has been also studied in dRGT massive gravity [50, 51] and bi-gravity [52, 53]. To our knowledge, these alternative inflationary models have not been studied in degenerate massive gravity. For our purpose, we would like to study the feasibilities of emergent universe, bouncing and cyclic universes in massive gravity with degenerate reference metrics. The remaining part of this paper is organized as follows. In Sec. 2, we give a brief review of the massive gravity and its equations of motion. In Sec. 3, we study the emergent universe in degenerate massive gravity with perfect fluid. First we obtain the exact Einstein static universe solutions in sevaral cases. Then we give the linearized equations of motion and discuss the stabilities against both homogeneous and inhomogeneous scalar perturbations. We give the parameters regions of stable Einstein static universes. In Sec. 4, we construct exact solutions of the bouncing and cyclic universes in degenerate massive gravity with a cosmological constant and axions. We conclude our paper in Sec. 5. 2 Massive gravity In this section, following e.g. , we briefly review massive gravity. The four dimensional action of massive gravity is given by where is Plank mass and we assume in the rest discussion, is the action of matters, is the Ricci scalar, represents the determinant of , represents the mass of graviton, are free parameters and are interaction potentials which can be expressed as follows, where the regular brackets denote traces, such as . is given by where is a fixed symmetric tensor and called reference metric, which is given by where is the Minkowski background and are the Stückelberg fields introduced to restore diffeomorphism invariance . In the limit of , massive gravity reduces to GR. The equations of motion are given by Generally, all the Stückelberg fields are nonzero in massive gravity and the rank of the matrix (2.5) is full, i.e. . In Ref. , there are two spatial nonzero Stückelberg fields which break the general covariance in massive gravity. The matrix has rank 2 thus being degenerate. The massive gravity with degenerate reference metrics is called degenerate massive gravity. For our purpose, we set only the temporal Stückelberg field to equal to zero. It follows that massive gravity we consider in this paper has degenerate reference metrics of rank 3. And the unitary gauge of the corresponding Stückelberg fields is defined simply by . So are given by in the basis , where is a positive constant. 3 Emergent universe scenario In this section, we consider the realization of emergent universe scenario in the context of degenerate massive gravity. We consider only spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric because the Stückelberg fields in degenerate massive gravity are chosen in a spatially flat basis. On the other hand, based on the latest astronomical observations [55, 56], the universe is at good consistency with the standard spatially-flat case. In the following discussion, we assume that the matter field is composed of perfect fluids. Firstly we construct the Einstein static universe in several cases. Then we study the stability against both homogeneous and inhomogeneous scalar perturbations. 3.1 Einstein static universe The spatially flat FLRW metric is given by The energy-momentum tensor corresponding to perfect fluids is given by where and represent the energy density and pressure respectively, is the constant equation-of-state (EOS) parameter, and velocity 4-vector is given by where dot denotes the derivative with respect to time. For the sake of obtaining the Einstein static universe, we let scale factor and . We request to avoid the ghost excitation from massive gravity. The energy density can be solved from the Friedmann equation (3.4), The Einstein static universe solution is given by . Because there are several parameters in the Eq. (3.8), we will discuss them in different cases. 3.1.1 Case 1: , , In this case, Eq. (3.8) reduces to a simple linear equation. The Einstein static solution is given by Case (1.1): For the solution is given by Case (1.2): For , the solution is given by 3.1.2 Case 2: , In this case, Eq. (3.8) reduces to a quadratic equation. The Einstein static solutions are given by Case (2.1): For , and the solution is given by Case (2.2): For , and the solution is given by The existence of requires the following two cases: Case (2.3): For , and the solution is given by Case (2.4): For , and the solution is given by 3.1.3 Case 3: In this case, Eq. (3.8) can be rewritten as For and , there are three real solutions which are given by For and , there is one real solution which is given by For , there is one real solution which is given by For , there is one real solution which is given by There are three free parameters and in the solutions. It is hard to analyze the parameters region of existence of all six solutions analytically. Instead we analyze the existence regions numerically and plot the parameters regions of the existence of all solutions in Fig. 1. We find that the solutions and cannot exist. In the previous subsection, we study the existence of the Einstein static universe in massive gravity with degenerate reference metrics. However, the emergent scenario does not thoroughly solve the issue of big bang singularity when perturbations are considered. For example, although the Einstein static universe is stable against small inhomogeneous perturbations in some cases [57, 58, 59, 60], the instability exists in previous parameters range against homogeneous perturbations . So it is valuable to explore the viable Einstein static universe by considering both homogeneous and inhomogeneous scalar perturbations. Actually, the stabilities of the Einstein static universe has been studied in various modified gravities, for examples, loop quantum cosmology , theory [63, 65, 64], theory [66, 67], modified Gauss-Bonnet gravity [68, 69], Brans-Dicke theory [70, 71, 72, 73], Horava-Lifshitz theory [74, 75, 76], brane world scenario [77, 78, 79], Einstein-Cartan theory , gravity , Eddingtong-inspired Born-Infeld theory , Horndeski theory [83, 84], hybrid metric-Palatini gravity and so on [86, 87, 88, 89, 90, 91, 92]. We refer to e.g. and reference therein for more details of stability of the Einstein static universe. In the following discussions, we would consider the stabilities of the Einstein static universe against both homogeneous and inhomogeneous scalar perturbations in degenerate massive gravity. 3.2.1 Linearized Massive Gravity Now we study the linear massive gravity with degenerate reference metrics. We use the symbols bar and tilde representing the background and the perturbation components of the metric respectively. First, we obtain the linearized equations of motion The perturbed metric can be written as where is the background metric which is given by Eq. (3.1) with and is a small perturbation. For our purpose, we consider scalar perturbations in the Newtonian gauge. is given by where and are functions of . For scalar perturbations, it is useful to perform a harmonic decomposition , Now the indexes are lowered and raised by the background metric unless otherwise stated. By using the relation , the inverse metric is perturbed by So the perturbed can also be written as According to Eq. (2.4), we have , i.e., So we have where “ 0 ” and “ ” denote time and space components respectively and the same index does not mean the Einstein rule. For perfect fluids, the perturbations of energy density and pressure are and respectively. The perturbations of velocity are given by where and are also functions of . The perturbed energy momentum tensor is given by It is useful to perform a harmonic decomposition of , In these expressions, summation over co-moving wavenumber are implied. The harmonic function satisfies where is Laplacian operator and is separation constant. For spatially flat universe, we have where the modes are discrete [60, 69]. Substituting Eqs. (3.30) and (3.42) into (3.37), after some algebra, we find where satisfies a second order ordinary differential equation To analyse the stabilities of the Einstein static universe in massive gravity with a degenerate reference metric, we require the condition of existence of the oscillating solution of Eq. (3.45) which is given by In the following discussions, we study the parameters region satisfying reality conditions (3.6) and (3.7), and stability condition (3.47) for the Einstein static flat universes against both homogeneous and inhomogeneous perturbations in different cases. 3.2.2 Case 1: , , The stabilities of the Einstein static universe (3.11) require

- How do you find PID parameters? - How do you tune a PID to a level controller? - What is the main reason to have an integral term in a PID controller? - What is gain in PID control? - What are the advantages of PID controller? - How do I manually tune a PID loop? - What are the disadvantages of PID controller? - How do you reduce PID overshoot? - What is integral gain in PID? - What causes overshoot in PID? - What is the difference between PI and PID controller? - How do you set a PID temp controller? - What is PID and equation of PID? - What is gain in a PID loop? - What are the drawbacks of P controller? - How can I improve my PID control? - How do PID loops work? - What is integral gain? How do you find PID parameters? The PID formula weights the proportional term by a factor of P, the integral term by a factor of P/TI, and the derivative term by a factor of P.TD where P is the controller gain, TI is the integral time, and TD is the derivative time.. How do you tune a PID to a level controller? Tuning PID loops for level controlDo a step test. a) Make sure, as far as possible, that the uncontrolled flow in and out of the vessel is as constant as possible. … Determine process characteristics. Based on the example shown in Figure 3: … Repeat. … Calculate tuning constants. … Enter the values. … Test and tune your work. What is the main reason to have an integral term in a PID controller? The main purpose of the integral term is to eliminate the steady state error. In the normal case there is going to be a small steady state error and the integral is mainly used to eliminate this error. It’s however true that when the error gets to 0 the integral will still be positive and will make you overshoot. What is gain in PID control? The proportional gain (Kc) determines the ratio of output response to the error signal. For instance, if the error term has a magnitude of 10, a proportional gain of 5 would produce a proportional response of 50. In general, increasing the proportional gain will increase the speed of the control system response. What are the advantages of PID controller? The PID controller is used in inertial systems with relatively low noise level of the measuring channel. The advantage of PID is fast warm up time, accurate setpoint temperature control and fast reaction to disturbances. Manual tuning PID is extremely complex, so it is recommended is to use the autotune function. How do I manually tune a PID loop? To tune a PID use the following steps:Set all gains to zero.Increase the P gain until the response to a disturbance is steady oscillation.Increase the D gain until the the oscillations go away (i.e. it’s critically damped).Repeat steps 2 and 3 until increasing the D gain does not stop the oscillations.More items… What are the disadvantages of PID controller? It is well-known that PID controllers show poor control performances for an integrating process and a large time delay process. Moreover, it cannot incorporate ramp-type set-point change or slow disturbance. How do you reduce PID overshoot? General Tips for Designing a PID ControllerObtain an open-loop response and determine what needs to be improved.Add a proportional control to improve the rise time.Add a derivative control to reduce the overshoot.Add an integral control to reduce the steady-state error.Adjust each of the gains , , and. What is integral gain in PID? The integral in a PID controller is the sum of the instantaneous error over time and gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain (Ki) and added to the controller output. The integral term is given by. What causes overshoot in PID? PID Theory While a high proportional gain can cause a circuit to respond swiftly, too high a value can cause oscillations about the SP value. … However, due to the fast response of integral control, high gain values can cause significant overshoot of the SP value and lead to oscillation and instability. What is the difference between PI and PID controller? The PID controller is generally accepted as the standard for process control, but the PI controller is sometimes a suitable alternative. A PI controller is the equivalent of a PID controller with its D (derivative) term set to zero. How do you set a PID temp controller? Tuning a PID Temperature ControllerAdjust the set-point value, Ts, to a typical value for the envisaged use of the system and turn off the derivative and integral actions by setting their levels to zero. … Note the period of oscillation then reduce the gain by 30%.Suddenly decreasing or increasing Ts by about 5% should induce underdamped oscillations.More items… What is PID and equation of PID? PID controller Derivative response. Proportional and Integral controller: This is a combination of P and I controller. Output of the controller is summation of both (proportional and integral) responses. Mathematical equation is as shown in below; y(t) ∝ (e(t) + ∫ e(t) dt) y(t) = kp *e(t) + ki ∫ e(t) dt. What is gain in a PID loop? Gain is the ratio of output to input—a measure of the amplification of the input signal. … The three primary gains used in servo tuning are known as proportional gain, integral gain, and derivative gain, and when they’re combined to minimize errors in the system, the algorithm is known as a PID loop. What are the drawbacks of P controller? The most commonly used controller for the vector control of ac motor is Proportional- Integral (P-I) controller. However, the P-I controller has some disadvantages such as high starting overshoot, sensitivity to controller gains and sluggish response to sudden disturbances. How can I improve my PID control? Increased Loop Rate. One of the first options to improve the performance of your PID controllers is to increase the loop rate at which they perform. … Gain Scheduling. … Adaptive PID. … Analytical PID. … Optimal Controllers. … Model Predictive Control. … Hierarchical Controllers. How do PID loops work? PID controller maintains the output such that there is zero error between the process variable and setpoint/ desired output by closed-loop operations. PID uses three basic control behaviors that are explained below. Proportional or P- controller gives an output that is proportional to current error e (t). What is integral gain? The Integral Gain controls how much of the Control Output is generated due to the accumulated Position Error or Velocity Error while in position control or velocity control, respectively. Position control is defined as when the Current Control Mode is Position PID. … This gain is the most important gain for I-PD control.

Written in EnglishRead online Includes bibliographical references. |Statement||Edited by G. Kallianpur and D. Kolzow.| |Series||Lecture notes in mathematics; 695, Lecture notes in mathematics (Springer-Verlag) -- 695.| |Contributions||Kallianpur, G., Kölzow, D. 1930-| |The Physical Object| |Pagination||xii, 261 p.| |Number of Pages||261| Download Measure theory applications to stochastic analysis Measure Theory Applications to Stochastic Analysis Proceedings, Oberwolfach Conference, Germany, July 3–9, Approximation of Processes and Applications to Control and Communication theory An Analog to the Stochastic integral for A Complex Measure Related to the Schrodinger Equation On the Nearness of Two Solutions in Comparison theorems for One-Dimensional Stochastic. Applications a la representation des martingales -- Nonlinear semigroups in the control of partially-observable stochastic systems -- Optimal control of stochastic systems in a sphere bundle -- Optimal filtering of infinite-dimensional stationary signals -- On the theory of markovian representation -- Likelihood ratios with gauss measure noise. Measure theory applications to stochastic analysis. Berlin ; New York: Springer-Verlag, (OCoLC) Material Type: Conference publication, Internet resource: Document Type: Book, Internet Resource: All Authors / Contributors: G Kallianpur; D Kölzow. This book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with. Book Description Unlike traditional books presenting stochastic processes in an academic way, this book includes concrete applications that students will find interesting such as gambling, finance, physics, signal processing, statistics, fractals, and biology. About this book A breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more. “The book is quite readable and can be used as a textbook for the application of mathematical theory in the area of econometrics. Also, a mathematician might benefit from an intuitive exposition of some different and specific types of integration appearing in the theory of stochastic processes. tation of martingales as stochastic integrals and on the equivalent change of probability measure, as well as elements of stochastic differential equations. These results suffice for a rigorous treatment of important applications, such as filtering theory, stochastic con-trol, and the modern theory of financial economics. Stochastic Analysis Major Applications Conclusion Background and Motivation Re-interpret as an integral equation: X(t) = X(0) + Z t 0 (X(s);s) ds + Z t 0 ˙(X(s);s) dW s: Goals of this talk: Motivate a de nition of the stochastic integral, Explore the properties of Brownian motion, Highlight major applications of stochastic analysis to PDE and. The general theory of static risk measures, basic concepts and results on markets of semimartingale model, and a numeraire-free and original probability based framework for financial markets are also included. The basic theory of probability and Ito's theory of stochastic analysis, as preliminary knowledge, are presented. Topics include a quick survey of measure theoretic probability theory, followed by an introduction to Brownian motion and the Itô stochastic calculus, and finally the theory of stochastic differential equations. The text also includes applications to partial differential equations, optimal stopping problems and options pricing. for stochastic differential equation to [2, 55, 77, 67, 46], for random walks to , for Markov chains to [26, 90], for entropy and Markov operators . For applications in physics and chemistry, see . For the selected topics, we followed in the percolation section. The books [, 30] contain introductions to Vlasov dynamics. Stochastic processes and diffusion theory are the mathematical underpinnings of many scientific disciplines, including statistical physics, physical chemistry, molecular biophysics, communications theory and many more. Many books, reviews and research articles have been published on this topic, from the purely mathematical to the most s: 3. Abstract: This is a textbook for advanced undergraduate students and beginning graduate students in applied mathematics. It presents the basic mathematical foundations of stochastic analysis (probability theory and stochastic processes) as well as some important practical tools and applications (e.g., the connection with differential equations, numerical methods, path integrals, random fields. It is a general study of stochastic processes using ideas from model theory, a key central theme being the question, 'When are two stochastic processes alike?' The authors assume some background in nonstandard analysis, but prior knowledge of model theory and advanced logic is not necessary. minimal prior exposure to stochastic processes (beyond the usual elementary prob-ability class covering only discrete settings and variables with probability density function). While students are assumed to have taken a real analysis class dealing with Riemann integration, no prior knowledge of measure theory. This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Hull—More a book in straight finance, which is what it is intended to be. Not much math. Explains financial aspects very well. Go here for details about financial matters. Duffie— This is a full fledged introduction into continuous time finance for those with a background in measure theoretic probability theory. Too advanced. "Introduction to the theory random processes" is a very good first book in stochastic analysis IMO, while "Introduction to the theory of diffusion processes" is more advanced and dense. He does not really concentrate on Markov semigroups though. $\endgroup$ – m7e May 11 '16 at Completely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance. The objective of this textbook is to provide a very basic and accessible introduction to option pricing, invoking only a minimum of stochastic analysis. Although short, it covers the theory essential to the statistical modeling of stocks, pricing of derivatives (general contingent claims) with martingale theory, and computational finance. Communications on Stochastic Analysis (COSA) is an online journal that aims to present original research papers of high quality in stochastic analysis (both theory and applications) and emphasizes the global development of the scientific community. The journal welcomes articles of interdisciplinary nature. Expository articles of current interest are occasionally also published. In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random ically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such. Subsequent chapters examine several aspects of discrete martingale theory, including applications to ergodic theory, likelihood ratios, and the Gaussian dichotomy theorem. Prerequisites include a standard measure theory course. No prior knowledge of probability is assumed; therefore, most of the results are proved in : M. Rao. The book explores foundations and applications of the two calculi, including stochastic integrals and differential equations, and the distribution theory on Wiener space developed by the Japanese school of probability. Uniquely, the book then delves into the possibilities that arise by using the two flavors of calculus together. Pris: kr. Häftad, Skickas inom vardagar. Köp Measure Theory. Applications to Stochastic Analysis av G Kallianpur, D Kolzow på Relative Strength Index. Jack D. Schwager, the co-founder of Fund Seeder and author of several books on technical analysis, uses the term "normalized" to describe stochastic oscillators that. Measure and Probability Theory with Economic Applications Efe A. Preface (TBW) More on Stochastic Dominance / Economic Applications of Stochastic Dominance Theory. A Selection of Ordering Principles / Applications to Fixed Point Theory / Applications to Variational Analysis / An Application to Convex Analysis. Browse the list of issues and latest articles from Stochastic Analysis and Applications. List of issues Latest articles Volume 38 Volume 37 Volume 36 Volume 35 Volume 34 Volume 33 Volume 32 Books; Keep up to date. Register to receive personalised research and resources by email. Sign me up. Among the list of new applications in mathematics there are new approaches to probability, hydrodynamics, measure theory, nonsmooth and harmonic analysis, etc. There are also applications of nonstandard analysis to the theory of stochastic processes, particularly constructions of Brownian motion as random walks. This book began as the lecture notes fora graduate-level course in stochastic processes. The official textbook for the course was Olav Kallenberg's excellent Foundations of Modern Probability, which explains the references to it for background results on measure theory, functional analysis, the occasional complete punting of a proof, etc. $\begingroup$ I agree with you in that this is not a begginer's book, but I don't think this justifies saying the book is horrible. I mentioned it because Andrew asked for a reference with examples, which can be found, if not in the text, in the exercises. This is probably not the best book to start learning measure theory (more basic references were already cited before) but it is certainly a. Topics include a quick survey of measure theoretic probability theory, followed by an introduction to Brownian motion and the Ito stochastic calculus, and finally the theory of stochastic differential equations. The text also includes applications to partial differential equations, optimal stopping problems and options pricing. A good non-measure theoretic stochastic processes book is Introduction to Stochastic Processes by Hoel et al. (I used it in my undergrad stochastic processes class and had no complaints). I'm gonna be honest though and say those exercises are stuff you should've gone over in an introductory probability class. 1 PROBABILITY SPACES. Underlying the mathematical description of random variables and events is the notion of a probability space (Ω, ℱ, P).The sample space Ω is a nonempty set that represents the collection of all possible outcomes of an experiment. The elements of Ω are called sample sigmafield ℱ is a collection of subsets of Ω that includes the empty set ∅ (the. A proof-based book on Stochastic Integration which 1) stands on Measure Theory but 2) avoids advanced Real Analysis (e.g. Hilbert or Banach spaces, etc.) and Topology or keeps them to a minimum, as I am less familiar with those areas. The book should be rigorous and present proofs to theorems (but avoid getting too technical à la française). Starting with the introduction of the basic Kolmogorov-Bochner existence theorem, this text explores conditional expectations and probabilities as well as projective and direct limits. Topics include several aspects of discrete martingale theory, including applications to ergodic theory, likelihood ratios, and the Gaussian dichotomy theorem. The first part deals with the analysis of stochastic dynamical systems, in terms of Gaussian processes, white noise theory, and diffusion processes. The second part of the book discusses some up-to-date applications of optimization theories, martingale measure theories, reliability theories, stochastic filtering theories and stochastic. “The theory of random measures is an important point of view of modern probability theory. This is an encyclopedic monograph and the first book to give a systematic treatment of the theory. the general theory presented in this book is therefore of great importance, far beyond the applications. measure-theoretic probability theory, Brownian motion, stochas-tic processes including Markov processes and martingale theory, Ito’s stochastic calculus, stochastic di erential equations, and partial di erential equations. Those prerequisites give one entry to the subject, which is why it is best taught to advanced Ph.D. students.This book is devoted to regularity and fractal properties of superprocesses with (1 +β)-branching. Regularity properties of functions is the most classical question in analysis.bility theory, Fizmatgiz, Moscow (), Probability theory, Chelsea (). It contains problems, some suggested by monograph and journal article material, and some adapted from existing problem books and textbooks. The problems are combined in nine chapters which are equipped with short introductions and subdivided in turn into individual.

I guess Chris Huygens has a deserved place as the father of time. We actually have to go back to Descartes and his philosophy to get a sense of Chris impact. Descartes drew philosophically on his jesuit upbringing in formulating through his meditative praxis a consistent philosophy of existence under god. He had noqualms at starting with god in some universal notion derived from centuries of theological philosophy and greek style deduction. Where eastern culture clashed with greek the test was trial by contest. This is the underlying notion of" logical" proof. Whichever individual representing a "conviction of truth"who won the often bloody trial was vindicated as by divine sanction. Today we carry the notion by logically or rather systematically consistent propositions winning the day over competing proportions when empirically tested. So called truth if subject to empirical test is only as good as its last vindication. However logicians, magi and magicians have long been aware of and utilised the tautology. By denying inexperienced "souls" the full knowledge of tautology, and encouraging hypnogogy in the same magicians were able to define the experience of reality pretty much how they wished, and to sparingly utilise naturally occurring consequents of acceptance and attentional focus. We really do "see the world through rose tiinted glasses". So Descartes brilliantly organized on the basis of "pure reason" that is "Logos" as a type of gods wisdom and thinking action,personified in the Christ and the spirit, his take on the structure of the universe under god. Of course it is a fusion of cultural thinking traceable back to the Sumerians, but studiously filtering out any notions that did not support his theology or christology, simply denoting such dissonance as error, evil, sin, pagan etc. Nevertheless despite this bias Descartes philosophy is remarkably similar to philosophies found at this level of erudition in all cultures . Therefore Leibniz, and other philosophers of reason and empiricism including Barrow, Wallis and Newton, were given an established idea to work against. It is to be noted that no one essentially dismissed Descartes, they merel wanted to improve upon his notions.it is thereby to be noted that unless one specifically subscribes to a cultural philosophy other than western the basic presiding philosophy is Cartesian in all of science. Kant later challenges Cartesian philosophy with a variant devised by Newton, and Newton's variant has only just recently been challenged. Leibniz, drawing on empirical evidence from Galileo and Huygens and other mechanics argued that descartes notions of vis were flawed by tautology. That to clear things in a more consistent way one needed to view motion as "independent" of distance, and that the god preserved and conserved "quantity" in bodily interactions was not matter times by the velocity or rather speed of that matter but rather matter times by the speed squared . This rather technical argument takes some reviewing of apprehensions. Until Galileo there was no real distinction beyond matter moving at variable speeds. Speeds if they were measured at all, were taken as axiomatic attributes of moving matter. Some experience was allowed to inform the opinion of the different amounts of matter and the different speeds and the interactive impact of these apprehensions in collisions. It was commonly deduced that heavier items travel slower along the ground than lighter ones, but fall faster! After common thought speed was understood but measured differently in different circumstance: sometimes it would be measured by distance traveled in a day, other times by days taken to travel a distance between known landmarks. Distance was therefore the underlying measure of speed. A greater speed meant a greater distance covered, a variation in the speed meant a variation in the distance covered. So 2 examples of invariance were ver puzzling. The first was Gallileos invariance in speed with regard to mass: heavier objects fall at the same speed as relatively lighter ones. Hang on a minute, the objects do not show a uniform speed so Gallileo concluded that the objects varied their speed in exactly the same way and he wanted to demonstrate this surprising result by proportions. Therefore he measured all sorts of proportions including a notion he called musical time or rhythm. He showed that for constant musical time the proportions of distance achieved were identical. The rhythm of motion , the music of the spheres has its origins ib Galileo's methods. At the same time Huygens and others were investigating the motion of a pendulum. The invariance here is no matter what the initial pressure on the weight the periodic motion seemed to adjust to give the same rhythm. So clearly the farther the pendulum had to fall the faster its speed at the bottom of the swing, but the rhythm of the music of the spheres seemed to remain constant. Chris was able again by taking all sorts of measurements to show a proportion between two pendulums which related the length of the pendulums radius to the rhythm of its period, and the relation was a proportion to the square root of the radius . The common name for this rhythm was "time". Thus "time" if anything is an analogy of musical time that is a constant driving rhythm which of course has a direction associated with it from the spatial representation on music paper. Our confused notion of time has its origin in music, and both invariances are crucial to music making. The gravitational pressure on a mass drives the pendulum of a metronome that beats out time in exactly the proportions Huygens observed. These proportions have their resolution in the Euclidean geometry of the circle. As a consequence of Chris's work Leibniz felt that the conserved quantities would have to depend on mass times by the square of the speeds/velocity, and all by default used definitions of motion dependent on the notion of musical timing. Musical timing only became scientific timing when Huygens introduced the first accurate clocks based on the invariant properties of falling matter and pendulum action. However it should not escape notice that scientific time is based on motion and a comparison of motions is what rhythmical time is. The speed of the contiguous parts of the pendulum vary but the rhythm we apprehend as constant: thus constant speed analogues of Huygens pendulum clocks can and have been made which further highlight the comparisons of speed or motion denoted as time. It is also important to note that periodic motion follows a closed motion trace which can be measured as a distance, and the motion trace in 3d space can be projected onto 2d space to produce a trace. Finally if that 2d space is in relative motion to the anchor point of the pendulum bounded traces can be described which analogise the comparison of a motion against a motion, a distance against a distance, and a rhythm against a rhythm.

By Nobuaki Obata; Taku Matsui; Akihito Hora; Kyōto Daigaku. Sūri Kaiseki Kenkyūjo (ed.) A useful complement to straightforward textbooks on quantum mechanics, this specified advent to the overall theoretical framework of up to date physics makes a speciality of conceptual, epistemological, and ontological concerns. the speculation is built through pursuing the query: what does it take to have fabric items that neither cave in nor explode once they're shaped? the soundness of topic hence emerges because the leader this is why the legislation of physics have the actual shape that they do.The first of the book's 3 components familiarizes the reader with the fundamentals via a quick historic survey and by means of following Feynman's path to the Schrödinger equation. the mandatory arithmetic, together with the targeted thought of relativity, is brought alongside the best way, to the purpose that each one appropriate theoretical suggestions should be safely grasped. half II takes a better glance. because the thought takes form, it's utilized to numerous experimental preparations. a number of of those are critical to the dialogue within the ultimate half, which goals at making epistemological and ontological feel of the speculation. Pivotal to this activity is an realizing of the exact prestige that quantum mechanics attributes to measurements -- with out dragging in "the recognition of the observer." Key to this realizing is a rigorous definition of "macroscopic" which, whereas hardly even tried, is supplied during this ebook Mathematical idea of Quantum debris Interacting with a Quantum box (A Arai); H-P Quantum Stochastic Differential Equations (F Fagnola); Quantum White Noise Calculus (U C Ji & N Obata); Can "Quantumness" Be an foundation of Dissipation? (T Arimitsu); what's Stochastic Independence? (U Franz); Creation-Annihilation techniques on Cellar Complecies (Y Hashimoto); Fock area and illustration of a few Infinite-Dimensional teams (T Matsui & Y Shimada); unfastened Product activities and Their functions (Y Ueda); comments at the s-Free Convolution (H Yoshida); and different papers Read Online or Download Non-commutativity, infinite-dimensionality and probability at the crossroads : proceedings of the RIMS Workshop on Infinite-Dimensional Analysis and Quantum Probability : Kyoto, Japan, 20-22 November, 2001 PDF Best probability books The most aim of credits possibility: Modeling, Valuation and Hedging is to give a accomplished survey of the previous advancements within the sector of credits threat learn, in addition to to place forth the newest developments during this box. an enormous point of this article is that it makes an attempt to bridge the distance among the mathematical concept of credits hazard and the monetary perform, which serves because the motivation for the mathematical modeling studied within the publication. Meta research: A advisor to Calibrating and mixing Statistical proof acts as a resource of uncomplicated tools for scientists eager to mix facts from varied experiments. The authors objective to advertise a deeper realizing of the inspiration of statistical proof. The booklet is created from components - The guide, and the idea. This can be a concise and user-friendly advent to modern degree and integration conception because it is required in lots of elements of research and likelihood conception. Undergraduate calculus and an introductory path on rigorous research in R are the one crucial necessities, making the textual content appropriate for either lecture classes and for self-study. ''This ebook may be an invaluable connection with keep watch over engineers and researchers. The papers contained conceal good the hot advances within the box of contemporary keep watch over thought. ''- IEEE staff Correspondence''This ebook may help all these researchers who valiantly try and hold abreast of what's new within the conception and perform of optimum keep watch over. - The Neoclassical Growth Model - and Ricardian Equivalence - Probabilistic Applications of Tauberian Theorems (Modern Probability and Statistics) - Quantum probability and infinite dimensional analysis : proceedings of the 29th conference, Hammamet, Tunisia, 13-18 October 2008 - The Method Trader - Seminaire de Probabilites XI, 1st Edition - Scientific Reasoning: The Bayesian Approach Additional resources for Non-commutativity, infinite-dimensionality and probability at the crossroads : proceedings of the RIMS Workshop on Infinite-Dimensional Analysis and Quantum Probability : Kyoto, Japan, 20-22 November, 2001 GCrard: O n the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri PoincarC 1 (2000), 443-459. 48. C. GBrard: O n the scattering theory of massless Nelson models, mp-arc 01-103, preprint, 2001. 49. J. Glimm and A. Jaffe: The X((p4)2 quantum field theory without cutoffs. II. The field operators and the approximate vacuum, Ann. of Math. 91 (1970), 362401. 50. J . Glimm and A. Jaffe: “Quantum Physics (Second Edition),” Springer, New York, 1987. 51. M. Griesemer, E. H. Lieb and M. We remark that, in the case of the Nelson model in the three-dimensional space, such a limit exists with only energy renormalization [go]. Recently Hirokawa, Hiroshima and Spohn proved the existence of a ground state of the Nelson model without both infrared and ultraviolet cutoffs. , a model of a quantized Dirac field interacting with the quantum radiation field. This direction of research is taken in [39,27]. Acknowledgments This work is supported by Grant-In-Aid 13440039 for scientific research from the JSPS. A useful example is given in with A(z) = z . A ( 0 ; x ) . In this case we have A'(z; x) := A ( z ;x>_- A(0;x). The Hamiltonians H P F ( A )and &ipole(A) do not have the gauge covariance. 2 The Nelson Type Model This model describes N non-relativistic particles interacting with a scalar Bose field on the d-dimensional Euclidean space Rd [go] (originally d = 3). 3), and g : RdN+ L2(Rd). An example of g is given by N g ( z ) ( l c ) = x x j ( k ) e - i k z j , k E Rd,z= ( z ~ , . In addition, if V = 0 and w ( k ) = d w (rn > 0 is a constant denoting the mass of a boson), then this is the case of the original Nelson model [go].

Unity of pomerons from gauge/string duality We develop a formalism where the hard and soft pomeron contributions to high energy scattering arise as leading Regge poles of a single kernel in holographic QCD. The kernel is obtained using effective field theory inspired by Regge theory of a 5-d string theory. It describes the exchange of higher spin fields in the graviton Regge trajectory that are dual to glueball states of twist two. For a specific holographic QCD model we describe Deep Inelastic Scattering in the Regge limit of low Bjorken , finding good agreement with experimental data from HERA. The observed rise of the effective pomeron intercept, as the size of the probe decreases, is reproduced by considering the first four pomeron trajectories. In the case of soft probes, relevant to total cross sections, the leading hard pomeron trajectory is suppressed, such that in this kinematical region we reproduce an intercept of 1.09 compatible with the QCD soft pomeron data. In the spectral region of positive Maldelstam variable the first two pomeron trajectories are consistent with current expectations for the glueball spectrum from lattice simulations. Regge theory is the study of the analytic structure of the scattering amplitude in the so called complex angular momentum -plane. The assumption that the scattering amplitude in the -plane has a pole, the so called pomeron, such that there are no more singularities at the right of it except at integer values, led to the explanation in the early 60s of the total cross-section behavior with center of mass energy in and experiments, among others. This particular analytic structure suggest that the scattering amplitude in the so called Regge limit of large at fixed , is dominated by the interchange of a infinite set of particles of all spins: the ones belonging to the pomeron Regge trajectory donnachie_pomeron_2002 . Regge theory is particularly appealing since the amplitude obtained in the -plane, , analytically continued to the non-physical scattering region of positive , provides a connection with the exchanged spin bound states of the theory, whose mass is given by . This remarkable fact allowed to explain early scattering data when meson trajectories are exchanged. It also led to the proposal that there exists another set of resonances with the quantum numbers of the vacuum, associated with the pomeron, which in principle will be a family of the so far unobserved glueballs. This idea has been supported through the years by Donnachie and Landshoff, who showed in the early 90s that Regge theory provides an economical description of total elastic cross-sections donnachie_total_1992 . This is known as the soft pomeron trajectory, with an intercept of around , and is well established as a model for total elastic cross-sections of soft particles (for example, and scattering). Deep inelastic scattering (DIS) is another process where Regge theory is important. In this case we consider the imaginary part of the amplitude for , at zero momentum transfer , which gives the total cross section for the scattering of an off-shell photon with a proton. Single Reggeon exchange then predicts a total cross section determined by the intercept, . However this story is bit more evolved. In the system there are two kinematical quantities: the virtuality of the photon and the Bjorken , which in the Regge limit is related to by , with . When HERA data for DIS scattering came out, it was somehow surprising to observe that the rise of the cross section with was actually faster than that predicted by the soft pomeron. The main difference is that, instead of using two soft probes for the scattering process, the off-shell photon virtuality can be well above the QCD confining scale. What is actually observed is a growth of the intercept with from about to . More concretely, if we write the total cross section as then the exponent grows with . Figure 1 shows the latest data points from HERA experiment, restricted to the region of low where Regge kinematics holds. In figure 2 we see the observed behaviour of the exponent . The behaviour of the exponent for low is consistent with the observed intercept of the soft pomeron for soft probes, but for hard probes (larger ) this is no longer the case, suggesting the existence of another trajectory with a bigger intercept, the so called hard pomeron. The nature of both pomerons, and in particular their relation, remains an unsolved problem in QCD. Are the soft and hard pomerons the same or distinct trajectories? Our main motivation in this work is to use holography to shed light into this problem. A very interesting proposal to resolve the above puzzle was again put forward by Donnachie and Landshoff donnachie_small_1998 ; donnachie_new_2001 ; donnachie_perturbative_2002 ; donnachie_elastic_2011 ; donnachie_pp_2013 . They proposed that the hard and soft pomerons are distinct trajectories, with the hard pomeron intercept around . The soft pomeron would be dominant in the soft region, since it is already well established to explain all soft processes, and the hard pomeron with a bigger intercept would dominate in the hard processes. More concretely, the idea is to write the cross section as where the sum runs over distinct trajectories. Then the effect of summing over several trajectories, which compete with each other as one varies the virtuality , has the desired effect of producing a varying effective exponent , as shown in figure 2. We shall follow this perspective and see that it follows naturally using the gauge/string duality as a tool to study QCD strongly coupled phenomena. Since QCD is well established as the theory of strong interactions, Regge theory should be encoded on it. However, it turned out to be a remarkable hard problem to deduce the pomeron -plane analytic structure from QCD. This fact has its roots in that, even nowadays, we mostly know how to compute QCD quantities perturbatively. The most successful approach has been that of the BFKL pomeron Fadin:1975cb ; Kuraev:1977fs ; Balitsky:1978ic , also known as the hard pomeron, and its generalisations. BFKL, in particular, predicts an amplitude for hadron scattering with a branch cut structure in the complex angular momentum plane. Introducing a momentum cut-off, it yields a discrete set of poles that can been confronted with HERA data kowalski_using_2010 . This approach has two undesired features: it does not cover the non-pertubative region of soft probes and it requires a very large number of poles which results in a very large number of fitting parameters. The gauge/string duality gives an alternative approach to look at DIS polchinski_hard_2002 ; polchinski_deep_2003 . Of particular importance to DIS at low is the proposal that the the pomeron trajectory is dual to the graviton Regge trajectory brower_pomeron_2007 . This work sparked several phenomenological studies that considered QCD processes mediated by pomeron exchange, which in general have been very successful in reproducing experimental data hatta_deep_2008 ; cornalba_saturation_2008 ; pire_ads/qcd_2008 ; albacete_dis_2008 ; hatta_relating_2008 ; levin_glauber-gribov_2009 ; brower_saturation_2008 ; brower_elastic_2009 ; gao_polarized_2009 ; hatta_polarized_2009 ; kovchegov_comparing_2009 ; avsar_shockwaves_2009 ; cornalba_deep_2010 ; dominguez_particle_2010 ; cornalba_ads_2010 ; betemps_diffractive_2010 ; gao_polarized_2010 ; kovchegov_$r$_2010 ; levin_inelastic_2010 ; domokos_pomeron_2009 ; domokos_setting_2010 ; brower_string-gauge_2010 ; costa_deeply_2012 ; Brower:2012mk ; costa_vector_2013 ; anderson_central_2014 ; Nally:2017nsp . In this paper we shall explore the above Regge theory ideas for DIS in the new framework of the gauge/string duality. We shall test our predictions using the specific holographic QCD model proposed in gursoy_exploring_2008 ; gursoy_exploring_2008-1 ; gursoy_improved_2011 . This model incorporates features of the strongly coupled QCD regime, like the spectrum of glueballs and mesons, confinement, chiral symmetry breaking among others. It is therefore an ideal ground to study processes dominated by the exchange of glueball trajectories. Our main findings are summarized in figures 2 and 3. We show that low DIS data, and in particular the running of the effective exponent , can be reproduced considering only the first four pomeron trajectories arising from the graviton trajectory in holographic QCD. The glueball trajectories shown in figure 3 are fixed by DIS scattering data, but they are also consistent with results of higher spin glueballs from lattice simulations meyer_glueball_2005 ; meyer_glueball_2005-1 . This paper is organized as follows. In section 2, we redo the computation by Donnachie and Landshoff that tries to reproduce DIS data with a hard and a soft pomeron, determining the functions in (2) from data analysis. Quite remarkably if we translate these functions into the proper gauge/string duality language, they are nothing but wave functions describing the normalizable modes of the graviton Regge trajectory. Section 3 presents the necessary formulae to study DIS using the gauge/string duality. The discussion is standard and already scattered in existing literature. In section 4 we focus on the pomeron trajectory, and in particular in constructing the analytic continuation of the spin equation that describes string fields in the graviton Regge trajectory. This discussion extends that already presented in our previous work Ballon-Bayona:2015wra . In section 5 we do the data analysis, fitting low DIS data in the very large kinematical range of . Our best fit has a per degree of freedom of 1.7, without removing presumed outliers existing in data. This leads us to the pomeron Regge trajectories shown in figure 3. We present our conclusions in section 6. 2 What is DIS data telling us about holographic QCD? The physics of the pomeron in the gauge/string duality was uncovered in brower_pomeron_2007 where pomeron exchange was identified with the exchange of string states in the graviton Regge trajectory. The amplitude for a scattering process in the Regge limit is then of the general form: where the functions represent the external scattering waves functions for a given process and is the so-called kernel of the pomeron which represents the tree level interchange of the aforementioned string states. Leaving aside technicalities which will be discussed in section 3, the pomeron kernel has the following dual representation where the sum runs over the graviton Regge trajectories that arise from quantising string states in the "AdS" box. The quantum number plays a important role in this work, since the contribution of the first few pomeron trajectories will be vital to reproduce the DIS cross section. The prefactor depends on , it factorizes in and , and it has a functional form that depends on the specific QCD holographic dual. We shall see that in general it has the form where is the usual conformal function in the 5D dual metric and the function will be determined by the background fields, for instance by the dilaton field . For the specific holographic model used in this paper we will have . The function in (4) is the -th excited wave function of a Schrödinger problem. We shall see that this fact follows from the spectral representation of the propagator of spin string fields in the graviton Regge trajectory that are exchanged in the dual geometry, analytically continued to . This is a highly non-trivial statement that can be checked by looking at an amplitude of the form (3) and fitting it to data. Once the external state functions and the specific functional form (5) are fixed, we can use data to confirm, or disprove, this fact. More concretely, if we consider a process dominated in the Regge limit by pomeron exchange and choose a specific holographic QCD model, we can test this model since the data should know about the underlying Schrödinger problem formulated in the dual theory. We consider DIS, for which the total cross section can be computed, through the optical theorem, from the imaginary part of the amplitude (3) for at zero momentum transfer. In this case two of the external state functions, say , represent the off-shell photon which couples to the quark bilinear electromagnetic current operator, which is itself dual to a bulk gauge field. The insertion of a current operator in a correlation function is then described by a non-normalizable mode of this bulk gauge field. The other two functions, , describe the target proton in terms of a bulk normalizable mode. We recall that, in QCD language, the functions are known as dipole wave functions of the external states. We wish to find out if the available experimental data is compatible with the holographic recipe, leaving aside technicalities which will be discusses in section 3. As it is well known, the imaginary part of the amplitude (3) at is related to the structure function . Here is the offshellness of the spacelike probe photon, whose dependence enters through the external state wave functions . The Maldelstam variable is related to by the usual expression , so we are in the low regime. As a first approximation, the integration over the variable in the amplitude (3) can be done by considering a Dirac delta function centred at . This is a good approximation only for large , i.e. near the AdS boundary at , but it will be enough for the purpose of this section. In any case it is a quick way to gain some insight about the shape of the kernel and the compatibility of our proposal with the experimental data. The integral can simply be done because the expression factorizes and the external wave functions are normalizable, therefore affecting the contribution of each Regge pole by an overall multiplicative constant. After these steps the expression for , as we will see in the next section, drastically simplifies to where the do not depent neither on nor on , and we denoted by the intercept values of each Reggeon . Here we are keeping the right warp factor and dilaton dependence, but if one takes the conformal limit, and , the qualitative result would be the same. Thus we predict a structure function of the form where is the product of known functions and a Schrödinger wave function with quantum number (the -th excited state). More concretely, a generic confining potential would produce wavefunctions where its number of nodes can be used to label them: the ground state would have one node, the first excited state would have two nodes and so on. Let us now focus on the QCD side of the problem. Using Regge theory arguments Donnachie and Landshoff donnachie_new_2001 proposed that the structure function has precisely the form (7). We can do the same reasoning as them. In order to know more about the functions the simplest thing to do is to first consider some fixed values of the that are physically reasonable, like and . These are reasonable values for the intercepts of the hard and soft pomeron, that are now unified in a single framework, since they appear as distinct Regge trajectories of the dual graviton trajectory in a confining background. Next, for a fixed value of we find the best coefficients and that match the data with the formula , then we can see how these coefficients evolve with . This was already done for a different set of data in donnachie_new_2001 , which served as a starting point for the authors’ proposal for the functional dependence. Of course the shape of the functions depends on the choice of the intercepts but it is well motivated, given the vast experimental evidence to fix the soft pomeron intercept around . Regarding we should be open to different values, but the expectation is that it will be responsible for the faster growth observer in DIS at higher values of . The left panel of figure 4 shows the result of this procedure for the values and , close to what we will show to be the intercepts that give the best fit in our model. The point we want to emphasize is that apparently not much is learned from the shape of these functions. However, if we divide the functions by the appropriate functions, as given by (6), the putative wave functions of the Schrödinger problem emerge. This remarkable fact is shown in the right panel of figure 4, which clearly meets our expectations. We should remark that if we use instead , as first suggested by Donnachie and Landshoff, we do not observe the oscillatory behavior expected for , suggesting perhaps that this value is unphysical. In fact it is known that recent data suggests a smaller donnachie_elastic_2011 . Indeed, as soon as we get below certain threshold value for the oscillatory behaviour becomes evident with the first node of localized very close to the boundary. Moreover, the form of the wavefunctions in our kernel will be very similar to the dashed lines in the figure. We take this as a strong evidence that the DIS data has encoded the dynamics suggested by holographic QCD. In the next sections we will proceed to phenomenologically construct the effective Schrödinger potential that leads to the wavefunctions that fit best the data. The discussion of this section was oversimplified, but it brings out the main idea. In practice, the integral over in the dual representation of the amplitude (3) is not localized, since we also consider lower values of . Also, to get a reasonable fit to the data we need to include the first four pomeron Regge trajectories. This is fine because those trajectories will be dominant with respect to the corrections to the leading hard pomeron trajectory. Eventually one would also need to include the exchange of meson Regge trajectories, but that is for now left out of our work, since those trajectories are still suppressed with respect to the first four Pomerons. 3 Low DIS in holographic QCD In this section we present the essential ingredients of the effective field theory description for low DIS in holographic QCD 111Holographic approaches for DIS in the large regime can be found in polchinski_deep_2003 ; BallonBayona:2007qr ; BallonBayona:2008zi ; BallonBayona:2010ae ; Koile:2013hba ; Koile:2015qsa .. First we briefly describe the kinematics of DIS and its connection to forward Compton scattering amplitude via the optical theorem. Then we present the holographic description of that amplitude, in the Regge limit, via the exchange of higher spin fields. We finish the section deriving a formula similar to (7) which encodes the Regge pole contribution to the DIS structure functions. In DIS a beam of leptons scatters off a hadronic target of momentum . Each lepton interacts with the hadron through the exchange of a virtual photon of momentum . DIS is an inclusive process because the scattering amplitude involves the sum over all possible final states. The relevant quantities are the virtuality and the Bjorken variable . Another quantity is the Mandelstam variable , describing the squared center of mass energy of the virtual photon-hadron scattering process. The DIS cross section is described in terms of the hadronic tensor where is the electromagnetic current operator and denotes a hadronic state of momentum . Current conservation and Lorentz invariance imply that has the decomposition The Lorentz invariant quantities and are the structure functions of DIS. They determine completely the DIS cross section and provide information regarding the partonic distribution in hadrons. The optical theorem relates the hadronic tensor to the imaginary part of the scattering amplitude describing forward Compton scattering. For a photon of incoming momentum and outgoing momenta , and for a hadron of incoming momentum and outgoing momenta , this amplitude admits the following decomposition where we identified and , and defined the transverse projection of the virtual photon polarization as The DIS structure functions are then extracted from the relations In DIS there are two interesting limits that are usually considered. The first is the Bjorken limit, where with fixed. In this limit perturbative QCD provides a good description of the experimental data in terms of partonic distribution functions. The second interesting case is the limit of , the so-called Regge limit of DIS, for which is fixed and is very small. In this limit the hadron becomes a dense gluon medium so that the picture of the hadron made of weakly interacting partons is no longer valid. As explained in section 2, in this paper we investigate DIS in the Regge limit (low ) from the perspective of the pomeron in holographic QCD, which encodes the dynamics of the dense gluon medium. We develop a five dimensional model for the graviton Regge trajectory for a family of backgrounds dual to QCD-like theories in the large- limit. Our formalism leads to the existence of a set of leading Regge poles describing DIS in the Regge limit, the first two interpreted as the hard and soft pomerons. 3.2 Regge theory in holographic QCD Let us now consider the computation of the forward Compton scattering amplitude in holographic QCD. We are interested in elastic scattering between a virtual photon and a scalar particle with incoming momenta and , respectively. In light-cone coordinates , for the external off-shell photon with virtuality we take while for the target hadron of mass we take The Regge limit corresponds to and the case corresponds to forward Compton scattering. The momenta and are, respectively, identified with the and defined in the previous subsection. As explained, we will extract the DIS structure functions from the forward Compton amplitude. First we define with generality the holographic model that may be used. We need to define the external states in DIS and the interaction between them that is dominated by a t-channel exchange of higher spin fields (those in the graviton Regge trajectory). Later on, to compare with the data, we will use a specific holographic QCD model gursoy_exploring_2008 ; gursoy_exploring_2008-1 , but for now we will write general formulae that can be used in other models. The string dual of QCD will have a dilaton field and a five-dimensional metric that are, respectively, dual to the Lagrangian and the energy-momentum tensor. In the vacuum those fields will be of the form for some functions and whose specific form we assume is known. We shall use greek indices in the boundary, with flat metric . We are defining the warp factor with respect to the string frame metric. In DIS the external photon is a source for the conserved current , where the quark field is associated to the open string sector. The five dimensional dual of this current is a massless gauge field . We shall assume that this field is made out of open strings and that is minimally coupled to the metric, so its effective action has the following simple form where and we use the notation for five-dimensional points. We could in principle have higher order terms in and other couplings to the metric in the action, but for the sake of simplicity we shall work with this action. As reviewed in appendix A, after a convenient gauge choice, the gauge field components describing a boundary plane wave solution with polarization take the form where and satisfy the equations Since we are computing an amplitude with a source for the electromagnetic current operator , the boundary conditions for are those of a non-normalizable mode, i.e. and . Note that the field strength also takes a plane wave form . As we shall see, a useful quantity is the stress-like tensor For the target we consider a scalar field that represents an unpolarised proton. This hadronic state is described by a normalizable mode of the form The specific details will not be important. We will simply assume that we can make the integration over the point where this field interacts with the higher spin fields. The effect of such an overall factor can be absorbed in the coupling constant. The next step in our construction is to introduce the higher spin fields that will mediate the interaction terms between the external fields of the scattering process. These fields are dual to the spin twist two operators made of the gluon field that are in the leading Regge trajectory. There are also other twist two operators made out of the quark bilinear. However, as we shall see, the corresponding Regge trajectories are subleading with respect to the first pomeron trajectories here considered. Noting that the higher spin field is in the closed string sector, and that the external fields are in the open sector, we shall consider the minimal coupling for the gauge field and for the scalar field . The higher spin field is totally symmetric, traceless and satisfies the transversality condition . The latter fact implies we do not need to worry in which external fields the derivatives in (21) and (22) act. However, there can be other couplings to the derivatives of the dilaton field and also to the curvature tensor. Here we consider only this leading term in a strong coupling expansion (that is, the first term in the derivative expansion of the effective action). Below we simply assume that the higher spin field has a propagator, without specifying its form. In the next section we focus on the dynamics of this field in detail. In the Regge limit, the amplitude describing the spin J exchange between the incoming gauge field and scalar field can be written as The fields and represent the outgoing gauge and scalar fields. The tensor represents the propagator of the spin J field. After some algebra the amplitude takes the form where and is the determinant of the 3-d transverse metric given by . This is the metric on the transverse space of the dual scattering process. The local energy squared for the dual scattering process is given by . The function is the Fourier transform of the integrated propagator for a field of even spin , and the light-cone coordinates are defined by the relation with . For the case of forward Compton scattering we have that , and . Summing over the contribution of the fields with spin , and using the result in (19), we obtain the amplitude where we have defined and is the eikonal phase defined by In (31) we used a Sommerfeld-Watson transform to convert the sum in into an integral in the complex J-plane. Comparing the expressions (10) and (28) for the forward Compton amplitude, and using (12), we extract the DIS structure functions for holographic QCD: 3.3 Regge poles In the next section we will describe the dynamics of a higher spin field . In particular, we shall see how the propagator admits a spectral representation associated to a Schrödinger problem that describes massive spin glueballs. Assuming that such Schrödinger potential admits an infinite set of bound states for fixed , we will show that The function depends on the particular holographic QCD model and will be obtained below for backgrounds of the form (15). The eigenfunctions and eigenvalues of the Schrödinger equation are and , respectively. Plugging this result in (31) and deforming the contour integral, so that we pick up the contribution from the Regge poles , we find that 222This procedure is standard in Regge Theory (see e.g. donnachie_pomeron_2002 ). In DIS this result implies that the structure functions and take the Regge form where we have defined the functions and the couplings Notice that in (36) we have already used the relation , valid in the Regge limit of DIS. The couplings include our ignorance of the hadron dual wave function, which appears in the integrand of (38), as well as the local couplings in the dual picture between the external fields and the spin field. The formula (36) has the expected form (7) advocated by Donnachie and Landshoff. 4 Pomeron in holographic QCD In the large scattering regime the lowest twist two operators dominate in the OPE of the currents appearing in the computation of the hadronic tensor. Therefore we consider here the interchange of the gluonic twist 2 operators of the form where is the QCD covariant derivative. In the singlet sector there are also twist 2 quark operators of the form , but these are subleading because the corresponding Regge trajectory has lower intercept. From a string theory perspective the equations of motion for the higher spin fields dual to should come by requiring their correspondent vertex operator to have conformal weights in the background dual to the QCD vacuum. We shall follow an effective field theory approach, proposing a general form of the equation in a strong coupling expansion, and then use the experimental data to fix the unknown coefficients. The proposed equation will obey two basic requirements, namely to be compatible with the graviton’s equation for the case and to reduce to the well known case in the conformal limit (pure space with constant dilaton). Let us consider first the conformal case ( and constant dilaton). In AdS space the spin field obeys the equation where is the AdS length scale and is the dimension of . Note that this field is symmetric, traceless () and transverse (). This equation is invariant under the gauge transformation with , but we will modify this in such a way that this gauge symmetry will be broken, as expected for a dual of a QFT with no infinite set of conserved currents. This is trivially achieved by changing the value of in (40) away from the unitarity bound . The transversality condition allows us to consider as independent components only the components , along the boundary direction. These can be further decomposed into irreducible representations of the Lorentz group , so that the traceless and divergenceless sector decouple from the rest and describe the in the dual theory. Finally note that we can analyse the asymptotic form of the spin equation of motion (40) near the boundary, with the result where denotes the source for . Since under the rescaling the AdS field has dimension , we conclude that the operator and its source have, respectively, dimension and , as expected. In the case that concerns us, since QCD is nearly conformal in the UV, we can do a similar analysis near the boundary. Next let us consider the case , where we have some control. This is the case of the energy-momentum tensor dual to the graviton. To describe the TT metric fluctuations we need to assume what is the dynamics of this field. The simplest option is to consider an action for the metric and dilaton field of the form where we work in the string frame. The field is the dilaton without the zero mode, that is absorbed in the gravitational coupling. This class of theories can be used to study four dimensional theories where conformal symmetry is broken in the IR. To make use of the AdS/CFT dictionary one usually impose AdS asymptotics for , which leads to a constraint on the UV form of the potential . This is a good approximation for large- QCD because it is nearly conformal in the UV 333Due to asymptotic freedom conformal symmetry is actually broken mildly in the UV by QCD logarithmic corrections.. The way conformal symmetry is broken in the IR is determined by the potential . As shown in gursoy_exploring_2008-1 , the confinement criteria and the spectrum of glueballs with spin constrain strongly the form of the potential in the IR. The term with the dilaton arises because we work in the string frame; the other term comes from the coupling of metric fluctuations to the background Riemann tensor , with and . In the case of pure AdS space (43) simplifies to with , as expected for the AdS graviton. We shall assume that our equation reduces to the simple form (43) in the case . Of course there could be higher order curvature corrections to this equation. Also, in the QCD vacuum there are scalar operators with a non-zero vev that do not break Lorentz simmetry. The corresponding dual fields will be non-zero and may couple to the metric, just like the above curvature and dilaton terms. Our goal is to write a two derivative equation for the spin fields using effective field theory arguments in an expansion in the derivatives of the background fields. For this it is important to look first at the dimension of the operator , which can be written as , where is the anomalous dimension. In free theory the operator has critical dimension . Knowledge of the curve is important when summing over spin exchanges, since this sum is done by analytic continuation in the -plane, and then by considering the region of real . Figure 5 summarizes a few important facts about the curve . Let us define the variable by , and consider the inverse function . The figure shows the perturbative BFKL result for , which is an even function of and has poles at . This curve is obtained by resuming terms in leading order perturbation theory. Beyond perturbation theory, the curve must pass through the energy-momentum tensor protected point at and . We shall use a quadratic approximation to this curve that passes through this protected point, The use of a quadratic form for the function is known as the diffusion limit and it is used both in BFKL physics and in dual models that consider the AdS graviton Regge trajectory (see for instance costa_deeply_2012 ). Let us now construct the proposal for the symmetric, traceless and transverse spin field in the dilaton-gravity theory (42). After decomposing this field in irreps, the TT part decouples from the other components and describes the propagating degrees of freedom. The proposed equation has the form where and are constants. Several comments are in order: (i) For this equation reduce sto the graviton equation (43); (ii) In the AdS case all terms in the second line vanish and the equation reduces to (40) for the TT components; (iii) The second term comes from the tree level coupling of a closed string, as appropriate for the graviton Regge trajectory in a large approximation; (iv) This action contains all possible terms of dimension inverse squared length compatible with constraints (i) and (ii) above. Notice that the term is absent because it reduces to other two derivative terms of and by the equations of motion. Also, note that the terms with two derivatives are accompanied by a metric factor from covariance of the 5-d theory. The exception is the first term, which itself includes the 5-d metric , and the third that is a mass term related to the dimension of the dual operator, which requires a length scale . It is important to realize that (46) is not supposed to work for any . Instead, we are building the analytic continuation of such an equation, which we want to use around . We expect this to be the case for large coupling, which is the case for the dense gluon medium observed in the low regime. In practice, we will look at the first pomeron poles that appear between (for , as required in the computation of the total cross section). This justifies why we left the coefficients in the second line of (46) constant and consider only the first term in the expansion around 2. Finally let us consider the third term in (46). This mass term is determined by the analytic continuation of the dimension of the exchanged operators . We will write the following formula where is the ’t Hooft coupling, is a constant and is a length scale set by the QCD string, which will be one of our phenomenological parameters. The first term follows directly from the diffusion limit (45), relating the scales and via . The diffusion limit is a strong coupling expansion, so it is natural that the dimension of the operators gets corrected in an expansion in . This is the reason for adding the second term in (47), following exactly what happens in SYM costa_conformal_2012 ; Cornalba:2007fs . This term can be added to correct the IR physics, but it is still subleading in the UV, when compared with the last term. The effect of this correction is to make the scale dependent of the energy scale, while keeping the general shape of curve in figure 5. The last term in (47) was added simply to reproduce the correct free theory result that is necessary to be obeyed near the boundary in the UV. More concretely, in order to obtain a scaling of the form (41), with the free dimension , we need this last term. This follows by considering the asymptotic value of the background fields and then analysing our spin equation near the boundary to obtain . This behaviour is important since it implies Bjorken scaling at the UV. We can regard (47) as an interpolating function between the IR and UV that matches the expected form of the dimension of the spin operator in both regions. This is the same type of approach followed in phenomenological holographic QCD models. To sum up, we shall consider the effective Reggeon equation (46), with (47), to describe the exchange of all the spin fields in the graviton Regge trajectory. This equation contains 5 parameters that will be fixed by the data, namely the constants and . We finish the analysis of the spin equation with a remark. In the same lines of karch_linear_2006 we can try to write a quadratic effective action for the spin symmetric, traceless and transverse field, such that its irrep TT part obeys the proposed free equation. Such an action would have the form where the dots represent terms quadratic in that are higher in the derivatives of either or the background fields. Since in the QCD vacuum only scalars under the irrep decomposition are allowed to adquire a vev, the mass term in (48) includes all such possibilities. We are also treating the dilaton field in a special way, by allowing a very specific coupling in the overall action. In particular, other scalar fields could also have a non-trivial coupling to the kinetic term 444Since we write a 5-d action, one could also have fields with a vev proportional to the 5-d metric . An example is the background Riemann tensor that can couple to the spin field (for instance, the metric fluctuations do). However, for traceless fields only mass terms of the type written in (48) will survive.. It is simple to see that our proposal (46), with (47), corresponds to setting 4.1 Effective Schrödinger problem The amplitude (24) computes the leading term of the Witten diagram describing the exchange of the spin field in the Regge limit, whose propagator obeys the equation for some second order differential operator whose action on the part of the spin field is defined by (46). For Regge kinematics, however, we are only interested in the component of the propagator, in the limit where the exchanged momentum has , as can be seen from the kinematics of the external photons (13). Thus, we can take , which implies that the component of (46) decouples from the other components, taking the following form 555For example, the bulk Laplacian projected in the boundary indices gives , where is the bulk scalar Laplacian. This equation can be re-casted as a 1-d quantum mechanics problem, that is, setting with , and choosing to cancel the term linear in the derivative , equation (52) takes the Schrödinger form where and the potential is given by

- Circuit analysis overview - Kirchhoff's current law - Kirchhoff's voltage law - Kirchhoff's laws - Labeling voltages - Application of the fundamental laws (setup) - Application of the fundamental laws (solve) - Application of the fundamental laws - Node voltage method (steps 1 to 4) - Node voltage method (step 5) - Node voltage method - Mesh current method (steps 1 to 3) - Mesh current method (step 4) - Mesh current method - Loop current method - Number of required equations Labeling voltages on a schematic is not a matter of "right" and "wrong". It simply establishes how the voltage appears in the analysis equations. Created by Willy McAllister. Want to join the conversation? - Why was the current labeled as flowing out of the positive side of the battery? Doesn't positive current always flow into the positive side?(7 votes) - Just think like charges repel. Positive charge is always going to flow away from the positive terminal and into the negative one. Remember the conventional way of labeling current is the path a positive charge would take. Hope that's helpful.(8 votes) - At3:47, If I is going out of the positive terminal why the result is not negative?(5 votes) - It is a positive current of positive charge that goes from the positive terminal through the circuit. David, you confuse electron current with conventional current and what you state above is very true for electron current. This is even explained in an earlier video.(5 votes) - @10:10or so you could have been more clear and specific that the plus side of the V1 is still considered to be 1V less then the minus side of V1. Although this is for many counter intuitive it is still correct yes. I needed to replay this part 3 times to exactly listen what you said and where you were pointing too. Just wanted to add for more clarity.(4 votes) - Welcome to the world of many-teachers vs many-style-of-explanation thing. It is not easy sharing a topic to a 30 people class, let alone to a 8 billion khan class. huhu.. Anyway, I kind of like your question/post. It was challenging to be a good teacher, the kind that all-people-can-understand. I wish we can all be one of them. /(^_^) It was also very challenging to be a good student, the kind that are so fast to reach the gist(point) of a lecture/class. I am a very slow learner, and not as hardworking, let alone doing revision. I envy them very-very much. Nevertheless, let's learn something (if not everything) with all our hearts. And may we become wiser, and our kids (and other people kids) become smarter.(7 votes) - Where does the V1 and V2 come in?(3 votes) - From nowhere. For this video/section, the purpose of the video is to share how to label our required variable/parameters when it was not given/labeled in the problem/real_circuit. It is to share that we CAN work out any circuit and get every parameters value ( V, I, R, power, equipment_rating) but we need to be systematic in approach and consistent in naming/labeling parameters. This is VERY useful when you are troubleshooting an electronics/power system that you don't have much info to start with. :)(5 votes) - Generally, what is the most used assumption in circuits: the electron flow [which is the current comes from the negative terminal] or the conventional flow [in which the current comes from the positive terminal same as the voltage (for simplicity)]?(3 votes) - Hi Voncarlo, The test equipment, right hand rule, and textbooks all assume a current flows from positive to negative. If you read the fine print the charge carriers (primarily electrons) are traveling from negative to positive. This is as good an explanation as any: https://xkcd.com/567/ - lets say i had "assumed" a starting direction for the voltage and when calculating the current, get the current to be negative, does this mean that the direction is opposite to my originally assumed direction? also when i substitute my current solution to get V1 or V2, do i just use V=IR with no signs or do i follow the signs i had indicated previously, like if my assumed current flow at the begining got a negative sign for a certain voltage, will i have V=-IR when proving or just V=IR?(3 votes) - IMHO : does this mean that the direction is opposite to my originally assumed direction? > Yup. will i have V=-IR when proving or just V=IR? > Just V = IR. Since this is a question of current direction, then we technically discussing a vector, in which, I don't have a clear way of typing it. :p I would rather draw. :| Sorry. So, I'll use an analogy. let say, we are an alien.. seeing the blue dot (earth, as human call it). we (alien) wanted to study, how fast does this dot move around the star (human call it the sun). The answer have two part, one scalar, one vector. One is the amount of the blue dot displacement change, and one is the direction of the rotation. We need both. The displacement change amount can be calculated, independent of where we are flying from. But for the rotation around the star, we may get 'clockwise' or 'anticlockwise'. The direction calculated depends on which way are we viewing this dot and star. (human : it's anticlockwise if we view from the earths' north and clockwise if we view from the earths' south) Both are true answer(for the alien), but we(alien) can only calculate/observe one at any one point of reference. Similarly, (back to human) putting V1 and V2 means we assume the direction of the current flow. If we calculated a positive value, means it flow in our assumed direction. if we get a negative value, means it flow in the opposite of our assumed direction. hope that helps. :)(3 votes) - At4:20you use Kirkoff Law, meaning the sum of voltages=0. Is this a proprety we can apply in every and any circuit? or is there some kind of requirement?(2 votes) - I'm looking at this serie on circuit analysis to remember what I learnt before, but you got to wonder, since now all I saw was some really basic notations, and all of a sudden he uses K Law without even an introduction ? Imo spend less than 10 videos about notations and at least one to present this important law ...(2 votes) - why is the second diagram negative one volt? resistors don't have polarity(2 votes) - For future readers with the same question: I believe it is clearer if you imagine someone else handed him the schematic with the voltages already marked—which label the voltages entering and leaving the resistor, not polarities of the the resistor itself. (Nothing changes the fact that the resistor is causing a voltage drop.) And now he is using the KVL to analyze a loop. The signs around the resistor do not alter the effect of the resistor (which is revealed by how many ohms it is). The signs simply affect whether the voltage is a rise or drop in calculations, but note that a negative number for a rise is the same as a drop, and a negative number for a drop is the same as a rise. So when you do the calculation, if you see a situation where you want to reverse the signs around the resistor, you can instead just flip the sign of the value of number of ohms. It doesn't change the role of the resistor in the calculation, it just affects the math we do.(1 vote) - That it works out mathematically makes sense since the arithmetic and algebra are very simple but WHY would you ever depict current flowing into a negative terminal and thereby have to reverse the sign of the current ? It seems like this is just complicating things unnecessarily by deliberately introducing something that is wrong and deciding to mathematically compensate for it.(1 vote) - Hello Galba, In the near future you will encounter systems where the direction of current flow cannot be determined by inspection. To solve using "nodal" and "mesh" analysis you will need to guess. Sometimes we get it correct, sometimes not. It is only after the simultaneous equation are solved that the direction of current is known. Know that current can enter a voltage source from the positive terminal. Think of this as charging a battery. Please leave a comment below if you have any questions. - I still struggle to understand why the sign convention puts differences of potential in the opposite direction of the current for passive components. In my head I think of the current i as a kind of vector, so V would have to be a vector too. Because V=iR, and R>0, V would have to point into the same direction. As an additional problem, this way of thinking still works with Kirchhoff's voltage law (ΣV=0). So why would V have to point into the opposite direction?(1 vote) - I feel your pain. This is one of those humps you have to get over at the beginning. Fear not, it is a small hump. Your reasoning about vector current and voltage is valid, but it happens to not be the one we use. Instead, be sure you've seen this video on the sign convention: https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/modal/v/ee-passive-sign-convention. With circuits we don't bother with a vector notation for current, because there are only two possible directions, this way and that way. This distinction can be taken care of with just the sign of the current, so we don't need vectors. The definition of current we use is "positive current is the direction positive charge moves (or would move if it was present)." That's the first decision. We use this even though we know negative electrons are moving in wires. Next, we want Ohm's Law to give the right answer for the voltage polarity when there is a current flowing in a resistor. Suppose you have a battery connected to a resistor. We define current to flow OUT of the positive battery terminal on its way to the resistor. If you measure the voltage with a voltmeter, the more positive voltage is on the end of the resistor closest to the + battery terminal. So we label the voltage that way, with positive voltage sign next to where the current is coming INTO the resistor. This is the sign convention we use at KA and in almost every EE text I've ever seen. It is totally arbitrary that we do this, but we are super consistent about it. It is possible to use the opposite convention, which means we define current to flow in the direction electrons move. That's what you've described in your question. This is also valid, but it is not commonly used. The only example I've ever seen is in the training material used by the US Navy and other branches of the US military. (search for "NEETS").(3 votes) - [Voiceover] In this video, I want to do a demonstration of the process of labeling voltages on a circuit that we're about to analyze. This is something that sometimes causes stress, or confusion, and I want to just basically try to get out of that stressful situation, so the first thing I want to do, is remind ourselves of the convention, the sign convention for passive components, so we said, If we have a resistor that we draw this way, and we label the voltage plus or minus V on it, then when we label the current arrow, we want to label the current arrow so it goes into the positive terminal of the component, in this case a resistor, so just another quick example, if I draw the resistor sideways like this, if I label the voltages, the minus one on this side, and the plus one on this side, then when I go to apply the current arrow, I put the current arrow into the positive voltage sign, so this here is the sign convention for an individual component. How do we label the voltage, and the current together to be consistent? Now we're going to to over and analyze a circuit. I've drawn a circuit here. It's two identical circuits, and we're going to do it two different ways, with two different voltage labels, so the first thing we do, of course, when we analyze a circuit, is we set up the variables that we want to talk about. We'll do this side first. I'll label this as i, and that's a choice I can make, and then I'm going to label the voltages too, and I'm going to choose to label the voltages like this, plus and minus V, and we'll call this V1, and this will be plus or minus V2, so first, let's carry through and do the analysis of this circuit, and I'll give some component values to this. We'll call this 10 ohms, and this one's 20 ohms. Now I'm going to do an application of Kirchhoff's voltage law, around this loop, and we'll see how it turns out. All right, so Kirchhoff's voltage law will start at this node here, and will go around the circuit this way, and we'll do some equations. We'll say, plus three volts, when we go through this device, we get a voltage rise of three, then we get a voltage drop, because we go from plus to minus, we get a voltage drop of minus V1, and then we get another drop of V2, minus V2, equals zero. That's our KVL equation for this circuit over here, so let's keep going with this analysis. Three volts, minus V1 is i times R1, i times 10, and V2 is i times 20 ohms, and that equals zero, so let's keep going. Three minus i times 10, plus 20, equals zero, and that means that i equals minus three over, minus three goes to this side, 10 plus 20 is 30, and the minus sign goes with the 30, minus 30, so i equals plus amp, so we solve for i, and let's just pick out V1. Let's solve for V1, and we said earlier that V1 was i times R1, so V1 equals i, which is .1 amps, times 10 ohms, equals one volt. If we do it for V2, that equals .1 amps, times 20 ohms, and that equals two volts, and I can do one last little check, I can go back, and I can see KVL, I could do a check to see if this equation came out right, so three volts, minus V1 is one volt, and V2 is two volts, and that equals zero, and I get to put a check mark here, because yes, it's equal to zero. That was a real quick analysis of a simple two resistor circuit, and we got all the voltages and currents. Now I'm going to go do the same thing again, but this time I'm going to do the voltage labels a little bit odd. What I'm going to do here, is I'm going to say two plus. I'm going to define V1 to be in that direction, and we'll keep V2 like it was before, and now I need a current variable, and I'm going to call my current variable i here, just like we did before. Now at this point, you might say, Willie, you did it wrong. You did it wrong. That's not the right voltage label, but I want to show you that I'm going to get the same answer, even though I did it this unusual way, so let's do the same KVL analysis on this circuit, and what I want to show you, is that the arithmetic that we're about to do takes care of the science just fine. All right, so KVL on this circuit says that we'll start at the same place, and go around the same direction, so this says they're three volts, a voltage rise of three volts, we go in the minus sign, and out the positive sign, so that's a voltage rise. Now we get over to R1, and we go in the minus sign of R1, and out the plus sign, so that's plus V1, that's different than we had last time, right? Last time we had minus V1 here, see, and now we have plus V1. This is going to work out okay though. Now we go in the plus sign of V2, and we come out the minus sign, so that's a voltage drop, so we do a subtraction, and that equals zero. All right, we've got different equations, but we've got different definitions of V1, so now I want to write these V terms in terms of the resistance value, and the current value, and this is where we use the sign convention carefully, so now we need to include a term to represent R1, to use Ohm's law here. Now we have to be careful, this is one point we have to be careful, the current is going in the negative terminal of R1, so we're going to say VR1 equals negative i, times R1. Does that make sense? If we define our current variable to be going in the negative sign, then Ohm's law picks up this negative sign, to make it come out right, so down here, we plug in minus iR1, which is minus i times 10, that's a difference, and then we go through V2, and V2 is the same as it was in the other equation, minus V2 is i times R2, which is 20, and that all equals zero. Even right now, if you look at this equation, you see this minus sign, that snuck in here, because of our good use of the sign convention for passive componets. That makes this equation look just like this one here, so let's continue with the analysis. Just need a little bit more room, three volts minus i times 10 plus 20, equals zero. Now we have the same equation as before, so we're going to get the same i, i equals, three goes to the other side, and becomes minus three, 10 plus 20 is 30, with the minus sign. It goes over, same as before, so those came out the same, so now let's go check the voltages, see if we can compute the same voltages, and what he have to notice here, is our reference direction, the original reference direction we have for V generated a minus sign when we used Ohm's law, so we keep doing that, it's okay, so V1, equals minus iR1, equals minus .1 amps times 10, and that equals minus one volt, and V2 equals i times R2, which is equal to .1 times 20 ohms, and that equals plus two volts, and now the difference, we see the difference here. Here is the difference that showed up. V1 in this circuit has the positive voltage at the top, it came out with a value of plus one volt, and when we flipped over V1, negative one volt here, means that this terminal of the resistor, is one volt below this terminal of the resistor, and that means exactly the same thing in this case, as it does in this case, so these two things mean the same thing, and of course, the voltage on number two came out the same, so the purpose of the demonstration was just to show you, that no matter which way you name the voltages, somewhere in here, like right here, Kirchhoff's voltage law will take care of keeping the sign right, and you end up with the same answer at the end, so when you're faced with the problem of labeling a circuit like this, don't stress out about trying to guess ahead, what the sign of the voltage is going to turn out. You just need to pick an orientation, and go with it, and the arithmetic will take care of the positive signs, and the negative signs.