I asked a question regarding negative values of null dyads in dyad census I had previously and thanks to Szhorvat for answering it. It was very helpful. I have a follow-up question I would appreciate any insights if possible. Research in multigraph recognizes social network data often comprise multiple edges and loops in their natural state and simplifying by collapsing edges and removing loops discards information inherent to the network. I’m just wondering if dyad census in igraph would consider offering a way to take into account multiple edges and loops in the analysis? Or is this question irrelevant to dyad census? I hope it’s not weird to think negative values of null dyad is making sense as long as the sum of MAN still comes to 10, the total number of isomorphism classes. And the pattern I had did reflect actors’ behavior well. I cannot find any research about multiple edges in dyad census and would appreciate any reference to any. Thanks so much.
“Dyad census” simply means counting how many pairs of vertices have:
o o, no connection
o --> o, a unidirectional connection
o <-> o, a bidirectional connection
Since dyad census just means counting different types of vertex pairs, negative values obviously make no sense. As I said in the other thread, the negative value is a result of integer overflow, i.e. a computational limitation. This will be partially addressed when R/igraph is updated to use version 0.10 of C/igraph (i.e. likely in igraph 2.0).
Whenever a negative value appears, there should be a warning message, as below:
If you don’t see a warning, please show a complete minimal example that reproduces the issue, so we can track down and correct the problem.
The classic “dyad census” is plain counting of vertex pairs, so self-loops and multi-edges are ignored. There are possible generalizations of the idea, but those would be a different concept.
You may want to look into the literature on weighted motifs (not implemented in igraph at the moment).
Edge multiplicities can be regarded as edge weights. Based on these, one may assign a weight to each subgraph that’s being considered, and instead of counting subgraphs directly, we can add up their weights.


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I found the lab about Randomized Response Surveys, from chapter 6, interesting. This lab has the reader explore how to get accurate data from a survey and how to evaluate that data.
An important mathematical term introduced in this lab would be bias, which is the difference between an estimate’s expected value and the true value to be estimated. Expected value would actually be another term defined in this chapter, there is actually a fair amount of vocabulary.
A good example question from this lab is without doing any simulations, guess the general shape of the functional relation between Pr(Heads) for the penny and the SD of the estimate. How do you think the estimator will behave if Pr(Heads) is near 0? near 1? Record your guess in the form of a sketch of a graph of SD (theta) as a function of theta = Pr(Heads).
I think the general nature of statistics, having to adjust for unexpected circumstances and how you apply logic and patterns to those circumstances is what fascinates me about this lab.


I used to drive a VW bug back then.... It didn't even get to 60mph... it was struggling to reach 55mph even...
Hey, I can hijack my own thread!
Enough OT, let's go back to DoF, or is everyone satisfied so far?
Another way to see it (keeping it simple) is that when comparing two different formats and same AOV (angle of view), then the DOF is approximately the same for a given aperture diameter (not the aperture ratio).
4x5 using a 105mm at f/8 will have approximately the same DOF as 8x10 using a 210mm at f/16, since the aperture diameters for 105mm at f/8 and 210/mm at f/16 are the same
As the focal length of a lens increases the DOF decreases..I believe that stands for the most part with all formats. Switching between formats and lenses one might notice a difference, but if using the appropriate lens for a particular format things should fall into place. You have to test this with the same subject, same distance different formats, same f-stop. As always there is some interesting technical information here but again I believe understanding the basics of this is key.
Just my thought.
Yes, nobody works in aperture dimensions, but you could use an empirical method using the ratio between the focal length of the lenses and adjust the aperture value (f/stops) accordingly.
Let's say that a 160mm lens on 4x5 gives you the same AOV of a 320mm lens on 8x10, then the ratio 320/160=2. You have to multiply by 2 the aperture in the 320mm to get the same DOF as the 160.
In the case of the OP, to get the same DOF in 35mm using a 50mm lens and 6x4.5 using a 80mm lens, you need to multiply by 1.6 the aperture in the 80mm lens. Here the results are approximate since the film proportions and angle of view between the two are not the same.
In the attachment there are two examples with data obtained from the online DOF calculator http://www.dofmaster.com/dofjs.html


Explore Taylor polynomials graphically and analytically, as well as graphically determine the interval where the Taylor polynomial approximates the function it models.
In this problem, students investigate a Taylor polynomial centered at zero.
Students will recall that this Taylor polynomial is also known as a Maclaurin polynomial. Students find the 4th degree Taylor polynomial that approximates f(x) = ln(x + 5) at x = 0. They will notice that the values are closest at the center and become farther apart the further the x-values are from the center.
This problem also gives students the opportunity to explore different degrees of a Taylor polynomial. They use the Taylor command to find the respective polynomials and graph the function with each of the larger power polynomials in turn.
This will demonstrate that a Taylor polynomial will only approximate the function over a given interval, no matter how large the Taylor Polynomial is.
At the end of this activity, students will be able to find the first few terms of a Taylor series approximation to a function for any given value of x. They will also be able to graph a function and its Taylor polynomials of various degrees to show their convergence to a function.
© Copyright 1995-2024 Texas Instruments Incorporated. All rights reserved.


Question: Is it possible for events A and B to be
Is it possible for events A and B to be both mutually exclusive and independent? Justify your answer.
Relevant QuestionsRefer to Exercise 4.23. Define the following events. Event A: A randomly selected government employee having an advanced college degree would accept a promotion Event B: A randomly selected government employee having an ...Of a finance company’s loans, 1% are defaulted (not completely repaid). The company routinely runs credit checks on all loan applicants. It finds that 30% of defaulted loans went to poor risks, 40% to fair risks, and 30% ...Conditional probabilities can be useful in diagnosing disease. Suppose that three different, closely related diseases (A1, A2, and A3) occur in 25%, 15%, and 12% of the population. In addition, suppose that any one of three ...Suppose the random variable y has a Poisson distribution. Compute the following probabilities: a. P(y = 4) given μ = 2 b. P(y = 4) given μ = 3.5 c. P(y > 4) given μ = 2 d. P(1 ≤ y The College Boards, which are administered each year to many thousands of high school students, are scored so as to yield a mean of 513 and a standard deviation of 130. These scores are close to being normally distributed. ...
Post your question


2nd Grade Research and Inquiry Resources
• Children will use the Internet to do research and inquiry on fractions.
• Children will learn more about how to tell if one fraction is larger or smaller than another fraction.
• Children will solve fraction problems on a worksheet by circling the larger of two fractions.
This site is presented by Webmath. The site has math activities and information for grades K• 8. It also provides resources for solving a variety of math problems.
• Have children create fractions using crayons.
• Have children compare their fractions and tell which one is larger.
• Have children connect to http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractions/level1.htm. Then have children:
Play the game. Have children type the number of children sharing the pizza. They then click on Check and see the pizza divided in the right fractions. They click on Next to continue to the next question.
Have children draw a picture of a pizza and divide it into slices according to how many people they will share it with.
Have children write a fraction for one slice of the pizza they drew.


Table of Contents
How do you convert between metric units?
To convert from one unit to another within the metric system usually means moving a decimal point. If you can remember what the prefixes mean, you can convert within the metric system relatively easily by simply multiplying or dividing the number by the value of the prefix.
How do you convert between two units?
In a nutshell, do all conversions of units by multiplying the original measurement by a well-chosen form of the number 1. A bit less briefly: Find the conversion factor for the given and desired units, and write it as a fraction with the given units in the opposite position from the original measurement.
How do you convert between imperial and metric units?
How to Convert Imperial to Metric
- Find the right conversion number (see Metric – Imperial Conversion Charts)
- If the answer is in the thousands or millions (or thousandths or millionths), then drop the zeros and use the correct prefix (see Metric Numbers)
What are examples of metric units?
The metric system has meter, centimeter, millimeter, and kilometer for length; kilograms and gram for weight; liter and milliliter for capacity; hours, minutes, seconds for time.
What is conversion formula?
Conversion Rate = Total number of conversions / Total number of unique visitors * 100. Conversion Rate = Total number of conversions / Total number of leads * 100.
How do you convert standard units?
Standard units are dimensionless. If the original list has units, the original SD has the same units. To transform a measurement to standard units, we divide the measurement (minus the mean) by the SD, which cancels the original units.
What is an example of a famous unit conversion failure?
Some Famous Unit Conversion Errors! Orbiter spacecraft after a 286-day journey to Mars. Thrusters used to help point the spacecraft had, over the course of months, been fired incorrectly because data used to control the wheels were calculated in incorrect units.
What are the 7 basic units of measurement in the metric system?
The seven SI base units, which are comprised of:
- Length – meter (m)
- Time – second (s)
- Amount of substance – mole (mole)
- Electric current – ampere (A)
- Temperature – kelvin (K)
- Luminous intensity – candela (cd)
- Mass – kilogram (kg)
What are the three metric system?
Metric system basics The three most common base units in the metric system are the meter, gram, and liter. The meter is a unit of length equal to 3.28 feet; the gram is a unit of mass equal to approximately 0.0022 pounds (about the mass of a paper clip); and the liter is a unit of volume equal to 1.05 quarts.
What are the 3 formulas for converting temperature units?
Celsius, Kelvin, and Fahrenheit Temperature Conversions
|Celsius to Fahrenheit||° F = 9/5 ( ° C) + 32|
|Fahrenheit to Celsius||° C = 5/9 (° F – 32)|
|Celsius to Kelvin||K = ° C + 273|
|Kelvin to Celsius||° C = K – 273|
|Fahrenheit to Kelvin||K = 5/9 (° F – 32) + 273|
How do you convert units within the metric system?
To convert among units in the metric system, identify the unit that you have, the unit that you want to convert to, and then count the number of units between them. If you are going from a larger unit to a smaller unit, you multiply by 10 successively.
What are the rules for converting measurements?
and write it as a fraction with the given units in the opposite position from the original measurement.
How do you convert Mt to lbs?
METRIC TON TO POUND (mt TO lb) FORMULA. To convert between Metric Ton and Pound you have to do the following: First divide 1000 / 0.45359237 = 2204.62262185 Then multiply the amount of Metric Ton you want to convert to Pound, use the chart below to guide you.
What metric units do you use to measure distance?
Here’s a list giving you the basics: Units of distance: The basic metric unit of distance is the meter (m). Units of fluid volume: The basic metric unit of fluid volume (also called capacity) is the liter (L). Units of mass: Technically speaking, the metric system measures not weight but mass. Units of time: As in the English system, the basic metric unit of time is a second (s).


So it seems the ZL1/CTS-V LSA superchargers are the hot swap these days, they are all over eBay for as little as 500.00 and one of the primary reasons we developed our LSA Supercharger Accessory Drive kit. It's like the 60's again when we had all the 6-71 and 8-71 blowers laying around for your hot rods. Note: GM called their Diesel 2 cycle supercharger for the numbers of cylinders and the cubic inch of each, hence 6-71 and so on. Now we have the ZL1/CTS-V LSA superchargers just begging to be on your LS swap in your muscle car.
I see lots of people talking about ZL1/CTS-V LSA superchargers on the forums, but I see a lot of people are still mystified by how they work. Many think they can put it on their big inch LS build and it will make 9lbs of boost. I will try to let a little bit of the magic out of the bag today, calculating boost from a constant displacement supercharger like the LSA Supercharger on a constant displacement engine.
One of the first things you need to realize is that the LSA supercharger and engine are both constant displacement pumps designed to work together and produce 9lbs of boost, so as you change one constant value greater the other is affected proportionately lower.
ZL1/CTS-V LSA superchargers - Boost Math:
The ZL1/CTS-V LSA superchargers installed on a factory 376 cubic inch LS is rated at 9lbs of boost. The 9lbs of boost is achieved using a 7.8" crank pulley and a 2.95" blower pulley. That gives the LSA Supercharged Engine a 2.6 pulley ratio on the supercharger. In other words the top pulley goes 2.6 revolutions for every revolution on the bottom.
Pulley Ratio Math:
Therefore: 7.8 / 2.95 = 2.644
(crank pulley OD / blower pulley OD = pulley ratio)
Now for this example we are going to calculate how much boost we make if we put this ZL1/CTS-V LSA superchargers on a 427 cubic inch LS engine. It's a 2 step process. The first thing we need to do is figure out how much bigger the 427 cubic inch LS engine is in comparison to the stock 376 cubic inch engine. The second calculation is to adjust the boost down proportionally.
Remember, we have the same amount of air on top but a bigger hole to fill on the bottom lowering the pressure.
427/376=1.135638 Lets round it to 1.136 Save the number for the second calculation.
For Discussion: Percentage is like talking dollar and pennies and where we put the decimal point. 1 dollar is 100 pennies with the pennies being like percentages. Just move the decimal point 2 positions to the right for percentage.
So now 1 or 100%=376 is for the base engine and .136 or13.6%=51.136 for the increase size. So the 427 is 13.56% bigger than the stock 376 cubic inch ZL1.
Boost Ratio Math:
Simply divide the original boost by the overall increase in engine size.
Therefore: 9/1.136 = 7.92 boost.
So, our 427 will make 7.92lbs of boost with the stock pulleys.
To Increasing Boost :
So how do we make more boost? Easy, change the pulley sizes and spin the LSA Supercharger faster. To do this we need to calculate a new pulley ratio with some know pulley sizes. The engine is 13.56% larger therefore a 10% larger crank pulley would get back most of the boost. For example, theATI 10% Overdrive pulley (for the stock ZL1) is 8.58 inches. That should get us back closer to the original 9lbs of boost.
Pulley Ratio Math:
8.58 / 2.95 = 2.908 (crank pulley dia. / blower pulley dia. = pulley ratio)
Now we can calculate boost with the new pulley ratio
Therefore: 7.92/2.6 x 2.908 = 8.85lbs boost
(current boost/current pulley ratio x new pulley ratio = new boost value)


vb learning can converge to bad local minima. better initialization or better learning algorithm may help. for instance, deterministic annealing. also, sometimes changing the model might help. for instance, if one factors q(mu)q(Lambda), then re-formulating so that one gets q(mu,Lambda) might improve the posterior accuracy and the learning
aha, then I will change it to something like "implementing a student t distribution would require extra complexity which is out of the scope of this example"
doesn't the mixture node needs a class as a parameter not a network?
for the case of the constructing the t for the mixture
not sure what you mean, but the mixture in a mixture model is a different mixture than in student t construction. student t construction is based on an infinite mixture. it's basically just a particular gaussian-gamma joint distribution with the gamma distribution marginalized. but in vb approach, one doesn't marginalize the gamma analytically in order to keep the equations in
the exponential family form
i should write an example
what I thought was using two nodes to represent the t, but that won't do as an input to the mixture node... I just glanced at the links and the topic, it is new to me :)
I'm in a hurry for the release, so i'll postpone it
an example would be great :)
thanks a lot, Jaakko!
Hi @jluttine !I want to know is this result can only be shown by graph?Can we show probability density just like a format or some text?Thanks a lot!
Hello, I'm trying to learn a Gaussian mixture from Mnist but I get "ValueError: Must pass 2-d input", can't I use higher dimensional data?


This worksheet is designed for students to practice division by creating equal groups. The worksheet provides rows of various shapes, and students must divide these shapes into groups of a specified size. Each question directs the student to make groups of a different quantity, such as 3, 6, 4, 5, and 2, and then write the corresponding division sentence that represents the division of shapes into these groups. The worksheet visually represents the division problems, allowing students to count and divide the shapes accordingly.
The objective of this worksheet is to teach students the concept of division as the process of making equal groups from a larger set. It helps students understand that division is not only about numerical calculations but also about equal distribution. The worksheet encourages students to count and group objects, reinforcing the idea of division as a form of repeated subtraction. This hands-on approach to division is intended to build foundational skills that students will need for more complex mathematical operations involving division.


is called the weighted least-squares estimate of [beta].
Delete one observation and compute the least-squares estimate of [beta] from the remaining (n - 1) observations.
By minimizing the trace of E'E, Theil shows that the orthogonal least-squares estimate
of the price index is the eigenvector corresponding to [lambda], the largest eigenvalue of CC'.
is either a least-squares estimate
of [beta] or an initial guess for the parameter values.
Here we use the incidence data starting on April 18 and continuing past April 30 to obtain a least-squares estimate
of the minimum point [t.
The best least-squares estimate
for the parameter Y of the nonexistent laboratory-effects model (1) is the weighted mean [x.
1] = the least-squares estimate
of the population regression coefficient for X
But its least-squares estimate
would almost certainly be biased or inconsistent, even if the form of the equation were correct, becasue the bond rate is almost certainly not exogenous and hence not independent of the equation's disturbances.
Restricted least-squares estimates
of age composition from length composition.
Weighted least-squares estimates
are not possible if P [is less than or equal to] 0 or P [is greater than or equal to] 1.
In particular, if the individual-specific error component is correlated with X, then least-squares estimates
of [Beta] from cross-sectional data will be inconsistent.
rt] are contaminated by outliers, these least-squares estimates
are unduly susceptible to sampling variation and may be quite unstable.


A 2014-T6 aluminum alloy compression member with an effective length of 1.25 m has the T cross section shown in Fig. P9-72.
a. Determine the maximum axial compressive load P permitted by Code 2.
b. Use the allowable stress method to determine the maximum bending moment M in the yz-plane that can be applied as shown when the column is supporting a 175-kN axial load.
2014- T6 aluminum alloy – T section
Effective length of the column,
Area of cross section,
Radius of gyration along x axis,
Radius of gyration along y axis,


Today I read a paper titled “An analysis of a random algorithm for estimating all the matchings”
The abstract is:
Counting the number of all the matchings on a bipartite graph has been transformed into calculating the permanent of a matrix obtained from the extended bipartite graph by Yan Huo, and Rasmussen presents a simple approach (RM) to approximate the permanent, which just yields a critical ratio O($n\omega(n)$) for almost all the 0-1 matrices, provided it’s a simple promising practical way to compute this #P-complete problem.
In this paper, the performance of this method will be shown when it’s applied to compute all the matchings based on that transformation.
The critical ratio will be proved to be very large with a certain probability, owning an increasing factor larger than any polynomial of $n$ even in the sense for almost all the 0-1 matrices.
Hence, RM fails to work well when counting all the matchings via computing the permanent of the matrix.
In other words, we must carefully utilize the known methods of estimating the permanent to count all the matchings through that transformation.


Lecturer: Professor James A. Sellers, Ph.D.
Multiplying and dividing large numbers. Simplifying fractions and converting percentages. Handling square roots and exponents. These and other skills are the veritable foundation on which all of mathematics rests. To master them is to unlock the door to more advanced areas of study—such as algebra, geometry, and calculus—and to discover new levels of confidence in dealing with the math of everyday life.
increase your chances of success in high-school and college math classes;
prepare you for a career in a field that requires a strong foundation in math, such as economics, engineering, medicine, and the building trades;
strengthen your everyday critical thinking skills; and
help you handle with confidence everyday tasks such as shopping and planning a personal budget.
Yet despite how basic this kind of math may seem, the mechanics of mathematics remains a mystery to many of us because we’ve been taught to focus solely on our answers. But in the opinion of award-winning Professor James A. Sellers of The Pennsylvania State University, a true understanding of basic math involves more than just arriving at the right solution. It involves
properly understanding the nature of numbers and mathematical concepts,
paying close attention to the step-by-step processes behind different calculations, and
thinking about what you’re solving for—and why you’re solving for it in a specific way.
This more well-rounded approach to the basics of mathematics is a surefire way to strengthen your current knowledge or to gain new skills for more deftly and confidently approaching and dealing with math. And it’s all available to you in Professor Sellers’ engaging course, Mastering the Fundamentals of Mathematics. Using the same inspirational teaching skill and experience he’s brought to his other popular Great Courses in math, Professor Sellers reveals the secrets behind all the key math topics you need to know. In 24 lectures packed with helpful examples, practice problems, and guided walkthroughs, you’ll finally grasp the all-important fundamentals of math in a way that truly sticks.
Explore All the Essential Areas of Basic Math
Designed for lifelong learners of all ages, Mastering the Fundamentals of Mathematics zeroes in on topics that everyone needs to know:
Adding, subtracting, multiplying, and dividing whole numbers, fractions, negative numbers, and decimals
Converting between fractions, decimals, and percentages
Solving real-world problems involving ratios and proportions
Working with whole-number exponents and square roots
With each topic, Professor Sellers shows you how to approach, understand, and solve problems of varying complexity. And, later in the course, he offers brief introductions to more advanced areas of math and prealgebra, including
elementary number theory, and
basic probability and statistics.
And whether he’s discussing the order of operations or introducing you to methods for plotting points on a coordinate plane, Professor Sellers shows that the key to facing down more intimidating math problems is by tapping into basic concepts and calculations you’ve already mastered. Like an inspirational instructor who only has your success in mind, he reveals how basic math comes together—and even works together—to help you solve problems such as finding the area of a circle or breaking down a complex word problem involving statistics.Learn Tricks and Shortcuts for Solving Problems
To help you solve problems with greater ease, Mastering the Fundamentals of Mathematics is packed with tips, tricks, techniques, and shortcuts. Here’s just a small sampling of what you’ll find in this course.
Reducing fractions to their lowest terms: When dealing with fractions in math, you’ll often be required to express your answers in the lowest terms to make the fractions easier to understand. So how can you tell when a fraction has been reduced to its lowest term? You’ll know because the only divisor or factor that the numerator (top number) and denominator (bottom number) share is 1. For example, the fraction 4/8 is not in its lowest term because both numbers share a factor of 2.
Adding numbers with different signs: What’s a less complicated way to solve an addition problem such as 7 + (- 3) without using a number line? First, figure out which number has the larger absolute value (7). Then, subtract the other absolute value from this one (7 – 3 = 4) and attach the sign of the number that had the larger absolute value (4). That’s it!
Lining up decimal points: Sometimes, performing calculations with large decimals (such as 153.46 + 5343.3) can be tricky, but the important point is knowing when to align your decimal points. In addition and subtraction problems, it’s essential to line up corresponding digits in a right-justified way to get the correct answer; with multiplication and division, however, this alignment is unnecessary.
Calculating tips in your head: Do you always find yourself unsure of how much of a tip to leave? Knowing how to work with percentages and decimals makes it easy. To calculate a 15% tip, take 10% of the bill just by moving the decimal point one place to the left (example: $12.00 would be $1.20). Then, add half of that number ($0.60) to that amount and you’ve got the answer ($1.80). If you want to leave a 20% tip, take 10% of the bill ($1.20) and just double it ($2.40).
An Interactive, Engaging Way to Learn Math
An added feature of Mastering the Fundamentals of Mathematics is its interactive nature. At specific points in a given lecture, Professor Sellers gives you a problem and invites you to pause the course, try your best to solve the problem, and then continue the course to check youranswer alongside his and chart your personal progress. Oftentimes, Professor Sellers roots his practice problems in everyday scenarios in which you’re likely to find yourself, such as paying for groceries and tipping at restaurants.
Plus, he’s crafted a free, comprehensive workbook with a complete answer key to go along with his course—one that comes filled with additional practice problems on each topic he covers in the course.
Yet even with its wealth of practice problems and exercises, what makes this course so rewarding is ultimately Professor Sellers himself. As Director of Undergraduate Mathematics at The Pennsylvania State University, he’s in the unique position of knowing the specific areas math students have trouble with—and the specific ways to help them over these common hurdles. Calm and clear, this winner of the Teresa Cohen Mathematics Service Award is a constantly encouraging presence who refuses to let you give up and helps you prove to yourself that you can be successful in math.
So whether you’re just setting out on your mathematical journey or whether you simply want to rediscover what you’ve forgotten, you’ll find Mastering the Fundamentals of Mathematics to be an invaluable guide to an invaluable subject.


the bike costs $277.5 I hope this helped
to solve you would find the cost of the bike then multiply by the percentage $350*0.35=$122.50 that is how much you get off then you subtract that from the total cost of the bike $350-$122.50=$227.50
So the answer is the bike would cost $227.50 on sale
0.0076, if you meant 4.125 divided by 0.03125 it would be 132
first you have to move the decimal in 0.03125 over until its behind the 5 which is 5 places over. now the number is 3125
then you have to move the decimal in 4.125, 5 places over. now the number is 4,112,500.
now you divide like you would normally would and get 0.0076


Help your students reason with shapes and their attributes using double bubble maps. This download includes graphic organizers for comparing two 2-D shapes, two 3-D shapes, 2-d to 3-D shapes, and a blank template for students to choose their own shapes.
Meets the following Common Core Standards:
K.G.B.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
1.G.A.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.


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In this number 4 worksheet, students count and color 4 eggs. Students practice writing the number 4 and the word four. Students trace and then try on their own.
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A vertical pole AB measuring 5 meters snaps at point C. The pole remains in contact at C and the top of the pole touches the ground at point T, a distance of 3 meters from A.
Find the length AC, in meters, the point where the pole snapped.
The broken pole forms a RIGHT triangle with the ground
the slanted portion we will cal 'h' (as in 'hypotenuse' )
the vertical portion is 5 - h
and the other leg of the triangle (the ground ) = 3
(5-h)^2 + 3^2 = h^2
h^2 -10h+25+9 = h^2
-10h = -34
h = 3.4 the pole broke 3.4 meters from the top .... AC is thus: 5-3.4 = 1.6 meters


Quantities such as entropy and dimension, suggested by information theory, give only rough characterizations of cellular automaton behaviour. Computation theory suggests more complete descriptions of self-organization in cellular automata (and other systems). Sets of cellular automaton configurations may be viewed as formal languages, consisting of sequences of symbols (site values) forming words according to definite grammatical rules.
The set of all possible initial configurations corresponds to a trivial formal language. The set of configurations obtained after any finite number of time steps are found to form a regular language. The words in a regular language correspond to the possible paths through a finite graph representing a finite state machine. It can be shown that a unique smallest finite graph reproduces any given regular language (see ref. ). Examples of such graphs are shown in Fig. 8. These graphs give complete specifications for sets of cellular automaton configurations (ignoring probabilities). The number of nodes in the smallest graph corresponding to a particular set of configurations may be defined as the `regular language complexity' of the set. It specifies the size of the minimal description of the set in terms of regular languages. Larger correspond to more complicated sets. (Note that the topological entropy of a set is given by the logarithm of the algebraic integer obtained as the largest root of the characteristic polynomial for the incidence matrix of the corresponding graph. The characteristic polynomials for the graphs in Fig. 7 are , and
The regular language complexity for sets generated by cellular automaton evolution almost always seems to be nondecreasing with time. Increasing signals increasing self-organization. may thus represent a fundamental property of self-organizing systems, complementary to entropy. It may, in principle, be extracted from experimental data.
Cellular automata that exhibit only class 1 and 2 behaviour always appear to yields sets that correspond to regular languages in the large time limit. Class 3 and 4 behaviour typically gives rise, however, to a rapid increase of with time, presumably leading to limiting sets not described by regular languages.
Formal languages are recognized or generated by idealized computers with a `central processing unit' containing a fixed finite number of internal states, together with a `memory'. Four types of formal languages are conventionally identified, corresponding to four types of computer:
Examples are known of cellular automata whose limiting sets correspond to all four types of language (L. Hurd, in preparation). Arguments can be given that the limit sets for class 3 cellular automata typically form context-sensitive languages, while those for class 4 cellular automata correspond to unrestricted languages. (Note that while a minimal specification for any regular language may always be found, there is no finite procedure to obtain a minimal form for more complicated formal languages; no generalization of the regular language complexity may thus be given.)


By this time I’m sure everyone whose curiousity was piqued by the statement of the Grothendieck-Riemann-Roch theorem has read it themselves. Nevertheless, in case you haven’t, I will proceed to outline the steps of the surprisingly “easy” proof. It is “easy” in the sense that the most is made of a relatively simple computation on projecive space. Last time we saw that it is enough to prove the formula separately for an injection and a projection. We’ll see here how to carry these two steps through and how the first may be reduced to the inclusion of a divisor. Though last time I said that I wanted to go into each step in more detail, I realized that 1) probably very few people are (still?) following along, 2) for those who are, they will get more by seeing an outline and reading the paper or looking at Fulton’s Intersection Theory themselves, and 3) this way we can illustrate the power of the theorem with some applications.
Step 1: Projective Space.
Let be n-dimensional projective space and let be a coherent sheaf on . The first step will be to show that the so-called Hirzebruch-Riemann-Roch formula is valid: . As discussed previously in the series, this is the GRR formula in the special case of a map from projective space to a point.
To verify this formula, let be a hyperplane and let be its class. The Chern polynomial of is and so .
The sheaf corresponds to a graded module over . By Hilbert’s syzygy theorem, admits a finte length resolution by free graded modules. This implies that in , the sheaf is equal to a linear combination of sheaves of the form and by linearity, it suffices to show the theorem holds for . We know everything we could hope to about this sheaf, in particular, and . We are reduced to verifying the formula
and this may be rewritten in terms of residues as
We’ll leave this statement as an exercise with the hint that the change of variables will be quite helpful.
Step 2: A projection
Let be n-dimensional projective space and consider the projection map . We have the following fact, which is not too difficult to prove:
The natural map is surjective.
We won’t prove this fact, but will only say that the proof procedes by induction on n, and uses a homotopy property of the K-groups that is similar to the corresponding property in the Chow ring (the corresponding properties for divisors are proved in Hartshorne’s Algebraic Geometry, II.6.5 and II.6.6).
The fact that GRR is true for the map now follows from Step 1 and from the last Lemma stated in the previous post.
Step 3: A closed immersion of a divisor
Suppose that is the inclusion of a smooth subvariety. We will denote by the normal bundle of and the sheaf of ideals. We have the exact sequence
and the identity .
The GRR formula for this case then reads for . Indeed, this follows by the projection formula and the identity .
We now will prove this formula when is a divisor in and for .
The left hand side of the equation becomes . Here denotes the line bundle associated to the divisor (note: restricting to gives the normal bundle ). The first equality is an application of the projection formula. The second equality follows from the exact sequence above. Because is a ring homomorphism, this then reads .
We now analyze the right hand side of the equation. This reads . Here the second equality is another application of the projection formula and the formula finally follows because .
Corollary 1 GRR is true for the injection for the map for a fixed point .
By the Lemma from last time, the proof of this corollary reduces to the case when is a point. Then the statement follow from the previous argument when and in general by induction. (Warning – that barely constitutes a proof sketch).
Corollary 2 If the statement above is true when , then it is true in general.
This follows from the first Lemma from last time and Corollary 1 because we may compose with an injection for a sufficiently large dimensional projective space.
Step 4: Reduction to the case of a divisor; i.e., end of proof.
Part of the beauty of the proof of GRR is that the absolute most is made out of the computation of projective space over a point and the inclusion of a divisor. We now reduce the general case of an inclusion of a subvariety to that of a divisor using the blow-up construction.
By Corollary 2 above, we may assume that (the inclusion) is of codimension . Let be the blowup of X along Y. Denote by the exceptional divisor , the inclusion and the projection. Recall that if is the normal bundle of in , and is the pullback to , then constains the restriction of the line bundle associated to restricted to , which we will denote . We will set , a bundle of rank .
We have the following facts about the behavior under such a blowup.
A. and so is the identity map.
C. For we have .
D. modulo as long as .
Recall that of a bundle means we take the alternating sum of the wedge products of that bundle, interpreted as an element in K-theory. We have the following further
Fact E: for a bundle of rank .
We may now conclude the proof of GRR:
By the same computation as in the divisor case, we must verify formula
Because the codimension of is large, we may apply Fact D to know that is equivalent to zero modulo . Then this class is in the image of (exercise) and so by the previous case, we may apply GRR to this class and the map . This gives:
We will then verify that pushing forward this equation by gives equation (1).
By Fact C, we have that is equal to and then the LHS of our verification follows from Fact A.
Using the fact that is multiplicative and Fact E, we compute that
Since , we get the formula
By Fact B then, the image of the two sides of this equation by is equal to and the RHS of the verification follows from the fact that . Whew!
I’ll leave the Facts as exercises or inspiration to learn more about blowups. Facts A and B are pretty easy and Fact E is a straightforward compuation. Facts C and D require much more work – but at the moment I can’t see a great reason for going through their proofs. Applications to follow (more quickly than this post did I hope).


sorry if this is a dumb question, but I was doing some homework and I noticed that whenever I solved a planar system which had complex eigenvalues, I would always end up with complex eigenvectors. I was wondering whether I could ever somehow get a totally real-valued eigenvector from a complex eigenvalue.
The short answer is no. Let's suppose that $A$ is a real matrix with a complex eigenvalue $\lambda$ and an associated (nonzero) real eigenvector $v$. Then by the definition of eigenvector, $Av = \lambda v$. So $\lambda v$ must be complex since $v$ has real entries but $\lambda$ is not real; on the other hand, $Av$ must be real since both $A$ and $v$ have real entries by assumption.
This is impossible. Now we can say that if $A$ is a real matrix with a complex eigenvalue, then any associated eigenvector cannot have only real entries.


This page is deficient and requires revision. Please see Talk:Integrability for further comments.
A vaguely defined yet very popular notion which may mean one of the following:
- For functions - existence of the integral in some sense (Riemann integrability, Lebesgue integrability, improper integrals etc.);
- For geometric structures and partial differential equations - conditions guaranteeing existence of solutions (Frobenius integrability condition
- For differential equations (both ordinary and partial) and dynamical systems:
- a possibility to find solution in a given class of functions (Darbouxian integrability, Liouville integrability, integrability in quadratures etc.) or just in some closed form;
- existence of one or more first integrals, functions which remain constant along solutions;
- preservation of some additional structures (e.g., Hamiltonian systems are sometimes called integrable to distinguish them from dissipative systems);
- complete integrability for Hamiltonian systems means existence of the maximal possible number of first integrals in involution.
Follow the links for more details.
Integrability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrability&oldid=25868


The NCHS report indicated that in 2002 the prevalence of cigarette smoking among American adults was 21.1%. Data on prevalent smoking in n=3,536 participants who attended the seventh examination of the Offspring in the Framingham Heart Study indicated that 482/3,536 = 13.6% of the respondents were currently smoking at the time of the exam. Suppose we want to assess whether the prevalence of smoking is lower in the Framingham Offspring sample given the focus on cardiovascular health in that community. Is there evidence of a statistically lower prevalence of smoking in the Framingham Offspring study as compared to the prevalence among all Americans?
Here we use the proportion specified in the null hypothesis as the true proportion of successes rather than the sample proportion. If we fail to satisfy the condition, then alternative procedures, called exact methods must be used to test the hypothesis about the population proportion.
Your lab report should probably be based around one or two central research questions (RQs). To start off with, caste a wide net and generate at least half a dozen possible RQs. You may want to write down all the variables in the study.
In order to test the hypotheses, we select a random sample of American males in 2006 and measure their weights. Suppose we have resources available to recruit n=100 men into our sample. We weigh each participant and compute summary statistics on the sample data. Suppose in the sample we determine the following:
Do the sample data support the null or research hypothesis? The sample mean of 197.1 is numerically higher than 191. However, is this difference more than would be expected by chance? In hypothesis testing, we assume that the null hypothesis holds until proven otherwise. We therefore need to determine the likelihood of observing a sample mean of 197.1 or higher when the true population mean is 191 (i.e., if the null hypothesis is true or under the null hypothesis). We can compute this probability using the Central Limit Theorem. Specifically,
Also try to develop some possible hypotheses for each of your RQs - this could be revealing - you might find that its difficult to establish hypotheses for some of your RQs.
The decision rule is a statement that tells under what circumstances to reject the null hypothesis. The decision rule is based on specific values of the test statistic (e.g., reject H0 if Z > 1.645). The decision rule for a specific test depends on 3 factors: the research or alternative hypothesis, the test statistic and the level of significance. Each is discussed below.
(Notice that we use the sample standard deviation in computing the Z score. This is generally an appropriate substitution as long as the sample size is large, n > 30. Thus, there is less than a 1% probability of observing a sample mean as large as 197.1 when the true population mean is 191. Do you think that the null hypothesis is likely true? Based on how unlikely it is to observe a sample mean of 197.1 under the null hypothesis (i.e.,
How likely it is to observe a sample mean of 192.1 or higher when the true population mean is 191 (i.e., if the null hypothesis is true)? We can again compute this probability using the Central Limit Theorem. Specifically,
There is a 33.4% probability of observing a sample mean as large as 192.1 when the true population mean is 191. Do you think that the null hypothesis is likely true?
Neither of the sample means that we obtained allows us to know with certainty whether the null hypothesis is true or not. However, our computations suggest that, if the null hypothesis were true, the probability of observing a sample mean >197.1 is less than 1%. In contrast, if the null hypothesis were true, the probability of observing a sample mean >192.1 is about 33%. We can't know whether the null hypothesis is true, but the sample that provided a mean value of 197.1 provides much stronger evidence in favor of rejecting the null hypothesis, than the sample that provided a mean value of 192.1. Note that this does not mean that a sample mean of 192.1 indicates that the null hypothesis is true; it just doesn't provide compelling evidence to reject it.
In hypothesis testing, we select a critical value from the Z distribution. This is done by first determining what is called the level of significance, denoted α ("alpha"). What we are doing here is drawing a line at extreme values. The level of significance is the probability that we reject the null hypothesis (in favor of the alternative) when it is actually true and is also called the Type I error rate.
Because 2.38 > 1.645, we reject the null hypothesis. (The same conclusion can be drawn by comparing the 0.0087 probability of observing a sample mean as extreme as 197.1 to the level of significance of 0.05. If the observed probability is smaller than the level of significance we reject H0). Because the Z score exceeds the critical value, we conclude that the mean weight for men in 2006 is more than 191 pounds, the value reported in 2002. If we observed the second sample (i.e., sample mean =192.1), we would not be able to reject the null hypothesis because the Z score is 0.43 which is not in the rejection region (i.e., the region in the tail end of the curve above 1.645). With the second sample we do not have sufficient evidence (because we set our level of significance at 5%) to conclude that weights have increased. Again, the same conclusion can be reached by comparing probabilities. The probability of observing a sample mean as extreme as 192.1 is 33.4% which is not below our 5% level of significance.


Makers of Mathematics S. Hollingdale (Penguin, 1989)
There are not many books on the history of mathematics which are pitched at a suitable level. Hollingdale gives a biographical approach which is both readable and mathematical. You might also try E.T. Bell Men of Mathematics (Touchstone Books, Simon and Schuster, 1986). Historians of mathematics have a lot to say about this (very little of it complimentary) but it is full of good stories which have inspired generations of mathematicians.
Alan Turing, the Enigma A. Hodges (Vintage, 1992)
A great biography of Alan Turing, a pioneer of modern computing. The title has a double meaning: the man was an enigma, committing suicide in 1954 by eating a poisoned apple, and the German code that he was instrumental in cracking was generated by the Enigma machine. The book is largely nonmathematical, but there are no holds barred when it comes to describing his major achievement, now called a Turing machine, with which he demonstrated that a famous conjecture by Hilbert is false.
The Man Who Knew Infinity R. Kanigel (Abacus, 1992)
The life of Ramanujan, the self-taught mathematical prodigy from a village near Madras. He sent Hardy samples of his work from India, which included rediscoveries of theorems already well known in the West and other results which completely baffled Hardy. Some of his estimates for the number of ways a large integer can be expressed as the sum of integers are extraordinarily accurate, but seem to have been plucked out of thin air.
Fermat’s Last Theorem Simon Singh
You must read this story of Andrew Wiles’s proof of Fermat’s Last Theorem, including all sorts of mathematical ideas and anecdotes; there is no better introduction to the world of research mathematics. You must also see the associated BBC Horizon documentary if you get the chance. Singh’s later The Code Book (Fourth Estate) is not so interesting mathematically, but is still a very good read.
The Colossal Book of Mathematics M. Gardner (Norton 2004)
Over 700 pages of Gardner for under 20 pounds is an astonishing bargain. You will be hooked by the very first topic in the book if you haven’t seen it before (and probably even if you have): a diophantine problem involving a monkey and some coconuts — can’t say more without writing a spoiler. At the beginning, about 60 other books by Martin Gardner are listed, none of which will disappoint.
To Infinity and Beyond Eli Maor (Princeton, 1991)
Not much hard mathematics here, but lots of interesting mathematical ideas (prime numbers, irrationals, the continuum hypothesis, Olber’s paradox (why is the sky dark at night?) and the expanding universe to name but a few), fascinating history and lavish illustrations. The same author has also written a whole book about one number (e The Story of a Number), also published by Princeton (1994), but not yet out in paperback.
How to Think like a Mathematician Kevin Houston (CUP, 2009)
This sounds like the sort of book that elderly people think that young people should read. However, there is lots of good mathematics in it (including many interesting exercises) as well as lots of good advice. How can you resist a book the first words of which (relating to the need for accurate expression) are: Question: How many months have 28 days? Mathematician’s answer: All of them.
Mathematics: a very short introduction Timothy Gowers (CUP, 2002)
Gowers is a Fields Medalist (the Fields medal is the mathematical equivalent of the Nobel prize), so it is not at all surprising that what he writes is worth reading. What is surprising is the ease and charm of his writing. He touches lightly many areas of mathematics, some that will be familiar (Pythagoras) and some that may not be (manifolds) and has something illuminating to say about all of them. The book is small and thin: it will fit in your pocket. You should get it.
The Pleasures of Counting T.W. K¨orner (CUP, 1996)
A brilliant book. There is something here for anyone interested in mathematics and even the most erudite professional mathematicians will learn something new. Some of the chapters involve very little technical mathematics (the discussion of cholera outbreaks which begins the book, for example) while others require the techniques of a first or second year undergraduate course. However, you can skip through the technical bits and still have an idea what is going on. You will enjoy the account of Braess’s paradox (a mathematical demonstration of the result, which we all know to be correct, that building more roads can increase journey times), the explanation of why we should all be called Smith, and the account of the Enigma code–breaking. These are just a few of the topics K¨orner explains with enviable clarity and humour.
Beyond Numeracy J. A. Paulos (Penguin, 1991)
Bite-sized essays on fractals, game-theory, countability, convergence and much more. It is a sequel to his equally entertaining, but less technical, Numeracy.
The New Quantum Universe T. Hey & P. Walters (CUP, 2003)
All you ever wanted to know about quantum mechanics, from fusion to fission, from Feynman diagrams to super-fluids, and from Higgs particles to Hawking radiation. With potted biographies, historical background, and packed with wonderful illustrations and photographs (including an electron microscope image of a midge). This is an excellent and unusual introduction to the subject. The same authors also wrote a splendid book on relativity (Einstein’s Mirror).
Was Einstein Right? C.M. Will (Basic Books, 1988)
Einstein’s theory of General Relativity is a theory of gravitation which supersedes Newton’s theory and is consistent with Special Relativity. The basic idea is that space-time is curved and you feel gravitational forces when you go round a curve in space, in the same way as you feel centrifugal force when your car goes round a bend. This book is about observational tests of the theory, all of which have been passed with flying colours. In particular, there is a binary pulsar which loses mass by gravitational radiation and, as a result, its period of rotation increases by 76 ± 2 millionths of a second per year; General Relativity predicts 75. There is much to be learnt here about physics, cosmology and astronomy as well as about Einstein and his theory.
The Accidental Universe P.C.W. Davies (CUP, 1982)
All the buzz-words are here: cosmic dynamics; galactic structure; entropy of the Universe; black holes; many worlds interpretation of quantum mechanics, but this is not another journalistic pot-boiler. It is a careful and accurate account by one of the best writers of popular science.
Advanced Problems in Mathematics S.T.C. Siklos (1996 and 2003)
These are selections of STEP–like problems complete with discussion and full solutions. (STEP is the examination normally used as a basis for conditional offers to Cambridge.) The problems are different from most A-level questions, being much longer (‘multi-step’ is the current terminology) and sometimes covering material from apparently unconnected areas of mathematics. They are more like the sort of problems that you encounter in a university mathematics course, although they are based on the syllabuses of school mathematics. Working through one or both of these booklets would be an excellent way of getting your mathematics up to speed again after the summer break. The 2003 booklet (Advanced Problems in Core Mathematics) is in a sense a prequel, since it is based on a less advanced syllabus (basically the A-level core plus some mechanics and probability).
Mathematical Methods for Physics and Engineering K F Riley, M P Hobson & S J Bence (Cambridge University Press 1998)
Most of A-level pure mathematics consists of what could be called ‘mathematical methods’ — i.e. techniques you can use in other areas (such as mechanics and statistics). The continuation of this material forms a basic part of every university course (and would count as applied mathematics!).
We are currently developing our very own Maths Library in the department to give our pupils more access to the titles on our reading list.
Check back soon for more information!


The Get Real Math videos showcase over 80 common core skills used in the real world. The videos, produced at manufacturing companies, serve as a capstone after a skill is learned in school. Math skills featured are 3rd grade through high school. The lesson plans are created by math teachers.
What does toilet paper have to do with math?In this video, you will calculate the area of a parent roll of tissue and the area of a roll of toilet paper. Using that information, you can calculate how many rolls of toilet paper can be made each day.
What do robots have to do with math? (Georgia-Pacific)In this video, you will learn about laser guided vehicles and how to program them to work in your company. You will need to calculate distances and angles.
What does a tank reducer have to do with math?In this video, a part needs to be welded onto an exterior part of a tank reducer. Exactly where the part is to be attached needs to be determined first, based on a blueprint. In this activity, students need to determine the correct length from the top of the tank to place the object.
What does rigging have to do with math?Understanding the weight of different things is very important for many reasons - safety being one of the main reasons. In this video, students learn how to calculate the weight of a table to determine the appropriate straps to use for rigging the table to move it properly and safely.
What does steel rectangular tubing have to do with math?In this video, you will need to determine the total amount of tubing (assume the top of the riser will be added later) needed to produce a riser that will be used in a robotic welding enclosure at KI in Green Bay. You will be challenged to understand a two-dimensional figure (blueprint) that represents a three-dimensional object. You will also look at the loss of material from cutting.
©2020 NEW Manufacturing Alliance


STEM subjects are dynamic and ever changing with constant new discoveries, inventions and theories – and it’s exciting to see this dynamism reflected in how we teach STEM subjects!
Massachusetts Institute of Technology in 2012, which sees young girls being tutored in mathematics – through ballet! The Shine for Girls is an after-school club, which uses dance and movement to help students understand quadratic equations, fractions, statistics, problem-solving and probability.
Researchers found that combining intellectual and physical activities helped students “more effectively understand and retain material”. By learning through dance, pupils become ‘active learners’ rather than passively absorbing information as they would in the traditional education settings.
Many of the girls who participated were struggling with the subject prior to starting the lessons, but after a 10 week course the programme leaders found that the pupils involved had an average improvement in maths scores of 184 per cent and an average increase in confidence of 148 per cent. Some girls’ academic performance increased by 600 per cent.
While using dance to teach maths isn’t a reality in most classrooms – we might have to stick to puzzles and games to encourage ‘active learning’ – the success of the Shine for Girls programme clearly shows the power of taking a different approach and using creativity in the classroom.


The Journal of Graph Algorithms and Applications found at this Web site is the electronic version of the scientific journal with the same name. It is a collection of research papers dealing with the "analysis, design, implementation, and applications of graph algorithms." The current volume consists of select papers presented at the 1999 Symposium on Graph Drawing, which have since been revised. Previous volumes are archived on this site as well, and they can be freely accessed. Almost any discipline requires some sort of graphical representation, and specific uses of graph algorithms in various fields are addressed in this journal.


.5 Middle School Math Coach
Oak Grove School
This is a new teaching position to assist 6th, 7th and 8th grade Math teachers with instructional strategies, use of new interactive white boards and common core standards. The position is part-time and days could be flexible.
Contact Principal Dave Karolewicz karolewicz2ogschool.org or Mark Clement, Curriculum And Learning email@example.com


TREND Numbers 0-10 Match Me Puzzle Game, Ages 3 - 6 Item # 923724
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Learning numbers can be fun
- Interactive learning game that teaches numbers 0-10. Helps children develop matching, memory and word/picture recognition skills.
- Easy-to-follow format provides 3 fun ways to play the game.
- Double-sided game boards allow children of multiple skill levels to play.
- For 1-8 players.
- Game includes eight two-sided game boards and 48 cards.
|age recommendation||3 - 6 years|
|Maximum Number of Players||8|
|Minimum Number of Players||1|
|Recommended Age||3-6 Year|
|Skill Learning||Matching; Memory; Number|
|What's Included||Two-Sided Game Boards; 48 x Game Pieces; Sturdy Storage Box|


Priced at around $1 million, the open source (Linux)- based supercomputer is housed at the Computational Mathematics Laboratory (CML) of TIFR, located at the Pune University campus.
The computer has a speed of one tera floating point operations per second (flops), equivalent to Param Padma, the fastest and most powerful in the Param series of supercomputers developed by the Centre for Development of Advanced Computing (C-DAC), and allows CML to run various complex algorithms with upto one billion variables.
The supercomputer has been operational since last week and has a dozen scientists from various fields working on it. Already it is delivering 80 per cent of its peak performance(0.8 tera-flops) .
To know about this more refer to:http://timesofindia.indiatimes.com/articleshow/msid-1083936,curpg-2.cms


posted by Haley .
A hypothetical planet has a radius 2.1 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface?
What equation should i use?
well, Force is proprotional to mass, which is the same, and inversly proportional to radius squared. That means the graviational field vector (g) is 9.8/2.1^2 N/kg or as you probably know it, 9.8/2.1^2 m/s^2, the acceleration do to gravity at the surface.


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National Council of Educational Research and Training (NCERT) Book Solutions for Class 8th
Chapter: Chapter 16 – Playing with Numbers
Class 8th Maths Chapter 16 Playing with Numbers NCERT Solution is given below.
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Symmetry is everywhere, but it's often hard to notice. This lesson explores a few ways to identify it. By folding and making mirror images for different shapes, students will discover a new world of symmetrical figures.
Students will learn to convert metric units by applying concepts in word problems. Exercises and worksheets incorporate length (meters), and optional practice offers more instructions on mass (grams) and volume (liters).
Your students will become junior math detectives as they hunt down the missing side of a rectangle by applying the area formula for rectangles. The only clues they have are the rectangle's area and the measure of one side.
Planning for a substitute teacher in the classroom has never been easier than with this week-long sub packet for fourth grade! Your substitute can supercharge learning with lessons that will educate and inspire your students!
The order of operations is a critical concept for fourth graders to grasp in preparation for algebra. This introductory lesson on the order of operations will teach your students about the helpful acronym PEMDAS.
Interjections, one of the eight parts of speech, can be the most exciting. They help students interject some fun and drama into their writing and conversations. In this lesson, they'll brainstorm how to use interjections in their sentences.
It's not enough to just memorize the multiplication table! It helps students to know how to explain their strategy to find the product too. Teach this lesson on its own or use it as support for the lesson Slap and Roll Timed Multiplication.


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Proper orthogonal decomposition for reduced order dynamic modeling of vapor compression systems
Published Web Locationhttps://doi.org/10.1016/j.ijrefrig.2021.09.016
A computationally efficient but accurate dynamic modeling approach for vapor compression systems is important for many applications. Nonlinear model order reduction techniques which generate reduced order models based on high fidelity vapor compression cycle (VCC) models are attractive for the purposes. In this paper, a number of technical challenges of applying model order reduction methods to VCCs are described and corresponding solution approaches are presented. It starts with a reformulation of a standard finite volume heat exchanger model for matching the baseline model reduction structure. Reduced order models for evaporator and condenser are constructed from numerical snapshots of the high fidelity models using Proper Orthogonal Decomposition (POD). Methodologies for system stability and numerical efficiency of POD reduced order models are described. The reduced order heat exchanger models are then coupled with quasi-static models of other components to form a reduced order cycle model. Transient simulations were conducted over a wide range of operating conditions and results were compared with the full order model as well as measurements. The validation results indicate that the reduced order model can execute much faster than a high-fidelity finite volume model with negligible prediction errors.


Math worksheets dynamically created math worksheets 196016. This is a comprehensive collection of free printable math worksheets for sixth grade organized by topics such as multiplication division exponents place value algebraic thinking decimals measurement units ratio percent prime factorization gcf lcm fractions integers and geometry.
4th Grade Worksheets With Math Exercises 4th Grade Math Worksheets Grade 5 Math Worksheets Math Practice Worksheets
You will then have two choices.
Free printable 6th grade math worksheets. Our printable grade 6 math worksheets delve deeper into earlier grade math topics 4 operations fractions decimals measurement geometry as well as introduce exponents proportions percents and integers. This will take you to the individual page of the worksheet. Math worksheets workbooks for sixth grade.
6th grade math worksheets printable free with answers collection. 6th grade math worksheets. Free math worksheets for grade 6.
6th grade math worksheets pdf printables to practice skills on math topics learnt in grade 6 these worksheets were created with teachers homeschool parents and sixth graders in mind. Sixth grade math worksheets free pdf printables with no login. Math worksheets dynamically created math worksheets 196015.


Course of Study
EMIL Learning Outcomes
Content Knowledge: EMIL professionals will know and understand deeply the mathematics of elementary school as well as how mathematics concepts and skills develop through middle school including specialized knowledge that teachers need in order to understand and support student learning of elementary mathematics as specified by the AMTE Standards for Elementary Math Specialists.
Pedagogical Knowledge for Teaching Mathematics: EMIL professionals will have a foundation in pedagogical content knowledge as specified by the NCATE/NCTM Standards for Elementary Mathematics Specialists.
Leadership Knowledge and Skills: EMIL professionals will be prepared to take on collegial non-evaluative leadership roles within their schools and districts. They will have a broad view of many aspects and resources needed to support and facilitate effective instruction and professional growth.
- MTH 611 Counting and Whole Number Operations: K-8 Learning and Teacher Practices (3)
- MTH 612 Fractions and Proportions: K-8 Learning and Teacher Practices (3)
- MTH 613 Geometry and Measurement: K-8 Learning and Teacher Practices (3)
- MTH 614 Statistics and Probability: K-8 Learning and Teacher Practices (3)
- ED 637 Advanced Content Pedagogy: Mathematics (3)
- ED 673 Elementary Mathematics Leadership Practicum I (3)
- ED 609 Elementary Mathematics Leadership Practicum II (3)
- MTH 615 Patterns and Algebraic Thinking K-8 Learning and Teacher Practices (3)
- MTH 616 Algebra and Functions: K-8 Learning and Teacher Practices (3)


A generous 12? An expansive 16?
Joel Haddley and Stephen Worsley laugh at your meagre efforts.
The mathematicians of the University of Liverpool have developed a ‘new’ technique for slicing your favourite Italian dish that can produce as many as 20, 28 or even 36 slices.
The mathematicians set out to see if they could construct monohedral tiles with a ‘trivial intersection’, breaking the usual pattern of two flat edges and an arch. Basically, by cutting the pizza into a “wobbly star” (the technical term), you’re able to produce a variety of shapes to spread a pizza further than before.
To find out more about the maths that’s being employed here, you can read Haddley and Worsle’s paper here.


The purpose of this paper is to show that an affine model which incorporates the condition of no arbitrage enables improvements in forecasting the term structure of interest rates in Mexico. The three factors of the yield curve (level, slope and curvature) used in the model are estimated by the method of principal components. The forecasting model is specified as a linear relationship between each of the interest rates and these factors, for maturities of 1 to 60 months. Affine model predictions are compared with four benchmark models: a forward rate, an AR(1), a VAR(1), and a random walk model. The main finding is that the affine model has a performance comparable to benchmark models for horizons of 12 and 18 months, except for the random walk model. However, improving its forecasting performance for the 24-month horizon, and especially for 60-month maturities.
Affine Model Forecasts Yield Curve Principal Components Condition of no Arbitrage


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I love hands-on activities for introducing math concepts. Spielgaben starts with hands-on activities and now has a math workbook series to add extension activities for anyone who owns the complete Spielgaben set.
I want to show some ideas for combining the workbook series with activities using the Spielgaben materials. Today, I’m focusing on work with left and right and giving directions (including clockwise and counterclockwise).
This is a sponsored post written by me on behalf of Spielgaben. All opinions are honest and my own.
Complete Spielgaben Set, Version 4
In case you’re not familiar with Spielgaben toys, they are exceptionally high-quality educational toys designed for ages 3-12. You can see from the photo that the cabinet is very high quality and fits in well in a room with traditional Montessori materials. I love the easily removable drawers in version 4.
The Spielgaben complete set comes with over 700 pages of full-color printed learning materials. These printed materials help you as a parent or teacher easily use Spielgaben toys educationally with your child and/or students.
Note: The yarn balls in set 1 are fine for babies and toddlers. The other sets should only be used with ages 3 and older unless your toddler no longer mouths objects or you are constantly supervising your toddler during an activity using the materials.
Spielgaben Math Workbook Series
If you own a complete Spielgaben Set, you can now order an optional extension resource: a set of math workbooks for Preschool/Kindergarten through Grade 6. These don’t have as many direct hands-on activities as the original printed learning materials (or digital materials if you have an earlier Spielgaben set). Some of the activities are hands-on, but this is a workbook series with mainly paper and pencil work illustrated with Spielgaben materials. Year 3 on up has answer pages in the back.
You can use the math workbooks in a homeschool however you prefer. You could work straight through them after using the original hands-on lessons, combine workbook activities with hands-on lessons from the original printed materials, or skip around, using the workbooks as needed to reinforce work from the hands-on materials. I like to skip around and review lessons from the original printed materials or use the hands-on materials for inspiration at the same time as I use a workbook page.
Spielgaben Materials Used in This Post
For this post, I used parts from Spielgaben cabinet drawers 3-6, 7, and 10. With younger children, I like to set up a tray or basket with the parts needed for a particular activity (as a control of error). For kindergarteners on up, I just have the child bring the drawers needed for the activities to the work area.
For the first part of the activity, I used one of the Spielgaben Nature Cards that come with the complete set. The full-color printed materials have lots of creative, hands-on activities for learning math concepts.
Nature Card Inspiration with Spielgaben Materials
My just-turned-6-year-old granddaughter, Zoey, has done lots of Montessori hands-on math work along with Spielgaben hands-on math activities. I chose a couple of pages from the Spielgaben Math Workbook, Year 1. The workbook typically uses Spielgaben creations as the illustrations, so they’re fun as building inspiration. I always encourage Zoey to build one of the images with the Spielgaben materials.
A fox built with materials from Spielgaben drawer 7 was one of the first images on the first workbook page we were doing. Because we’re on a woodland animal unit, I used the fox nature card as inspiration for Zoey.
The page shows a fox created with drawer 7 and a fox created with drawers 3-6 along with dots from drawer 10. Zoey had fun creating the foxes. Then she added features to the first fox, a hairbow to the first fox, and a bow tie to the second fox. After that, she created food for the foxes and had some creative dramatic play with the foxes. I love that the Spielgaben materials can be easily used for both learning activities and creative play!
Working with Left and Right and Giving Directions Using Spielgaben Math Book, Year 1
When Zoey was finished with the dramatic play, we did Math Wookbook, Year 1, lesson 10-3 about left and right. We hadn’t focused specifically on left and right on a page, and the workbook had clear examples. First Zoey had to turn a picture of a beetle built with Spielgaben materials right and left. Then she had to identify from a row of 3 foxes built with Spielgaben materials (like the first one Zoey built) which fox was on the left and which was on the right. It was a simple page but a clear way to introduce or review left and right.
Then we did Math Workbook, Year 1 lesson 29-3 about giving directions. Zoey had fun with this page because it was like a game. First, the page introduced clockwise and anticlockwise (which I said is a British or Australian way to say “counterclockwise”). It also showed backwards and forwards. Then Zoey had to use a map and the words to get a person from one point to another a few different times using backwards, forwards, clockwise, and counterclockwise. Zoey hadn’t focused on clockwise and counterclockwise before, and she enjoyed giving the directions.
Then she had to move a girl forward and backward to spell words on a row of letters. I had Zoey move a Spielgaben dot to illustrate the work on this page. It was a simple page to do but reinforced an important concept.
I like that the workbook covers a wide variety of first grade math concepts. It works well as an extension of the Spielgaben printed lessons. It’s also helpful for children who need extra work with a specific concept.
All My Spielgaben Posts
Many of these include links to free printables! (Click on an image to go to the related blog post. The gallery is ordered according to publication date with my latest post at the top.)
Version 4 of Spielgaben Educational Toys
I highly recommend Spielgaben as an amazing educational material for home or school. Even though the toys aren’t inexpensive, they’re definitely worth the price because of the high quality of materials and number of years your children or students can use them.
Special deal for my readers!
Learn more about my eBook Montessori at Home or School: How to. Teach Grace and Courtesy!
If this is your first time visiting Living Montessori Now, welcome! If you haven’t already, please join us on our Living Montessori Now Facebook page where you’ll find a Free Printable of the Day and lots of inspiration and ideas for parenting and teaching! And please follow me on Pinterest (lots of Montessori-, holiday-, and theme-related boards), Instagram, and YouTube. You can find me on bloglovin’ and Twitter, too.
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3 Unit Circle Trigonometry
In This Chapter
The shape of a plucked guitar string, fixed at both ends, can be described by trigonometric functions of a real variable.
A Bit of History The discussion in Section 2.4 leads directly to a more analytical approach to trigonometry where the cosine and sine are defined as the x-and y-coordinates, respectively, of a point (x, y) on a unit circle. It is this interpretation of ...


Your maths and English GCSE qualifications are your passport to a brighter future. You could have every qualification and skill under the sun, but without maths and English you may not be able to reach your destination.
To study at HRC it's necessary for you to have at least a Grade C/Level 4 in GCSE maths and English. However, if you don't have these when you arrive at college we will support you throughout your journey to gain them.
We do this by embedding maths and English both within your course timetable and within your study programme. So not only are you attending dedicated classes, you are also learning about how maths and English is relevant to your industry.
Every bricklayer needs to know how many bricks to order, every musician needs to know how much to charge for gig tickets and every business executive needs to know how to write a professional report. These are just small examples as to how important maths and English are to helping you achieve your career dreams.
At HRC we offer three levels of maths and English study that are matched to your current qualifications:
- Functional Skills: for students who have achieved a GCSE Grade E/Level 2 or below
Designed to help prepare you for GCSEs
- GCSE maths and English: for students who have achieved a GCSE Grade D/Level 3
Designed to help you achieve a Grace C/Level 4 or higher at GCSE
- Upskilling: for students who have achieved a GCSE Grade C/Level 4 or higher
Designed to help you improve your skills


A Coq formal proof of the LaxMilgram theorem
CPP, pp. 79-89, 2017.
The Finite Element Method is a widely-used method to solve numerical problems coming for instance from physics or biology. To obtain the highest confidence on the correction of numerical simulation programs implementing the Finite Element Method, one has to formalize the mathematical notions and results that allow to establish the soundne...More
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Possessing a functional understanding of mathematics is not simply a significant portion of a student's scholastic career, but it is a vital working aspect used in the day-to-day lives of nearly every person on the planet today. As a direct consequence of this fact, it is crucial that every child has a solid foundation in the basics of math. That foundation is needed to correctly utilize currency in order to buy and sell goods and services, work with modern technology, and to pursue many future academic goals. If you are the parent of a young child in grade school who is stressed with trying to comprehend the fundamental principles of math, consider having them study in a one-on-one setting where they can obtain the assistance they need. The educational directors at Varsity Tutors can help you meet with an independent skilled math tutor in Akron, NY, who is able to assist students in properly preparing for math.
With a professional private math tutor in Akron, your child could not miss anything. If the teacher covered the topic of fractions too quickly, or if your student does not have an understanding of the relationship between fractions and decimals, their tutor can take all the time they need to explain it in ways that are easily understood. Tutoring, unlike regular classroom learning, centers on a one-on-one relationship involving the instructor and your child, offering immediate feedback and constructive criticism that can't be found in a traditional academic setting. This fosters self-esteem and a greater confidence in the course material being presented, and it can expand to benefits outside of the classroom curriculum, such as improved deductive reasoning and time-management techniques. No matter what areas your student needs help with, their tutor will have a wealth of real-world experience in math and will be in a position to effectively answer their questions and efficiently solve their problems.


Parameter Estimation for Stable Distribution: Spacing based and Indirect Inference
- Author(s): Tian, Gaoyuan
- Advisor(s): Jammalamadaka, S.Rao
- et al.
Stable distributions are important family of parametric distributions widely used in signal processing as well as in mathematical finance.
Estimation of the parameters of this model, is not quite straightforward due to the fact that there is no closed-form expression for their probability density function. Besides the computationally intensive maximum likelihood method where the density has to be evaluated numerically, there are some existing adhoc methods such as the quantile method, and a regression based method. These are introduced in Chapter 2.
In this thesis, we introduce two new approaches: One, a spacing based estimation method introduced in Chapter 3 and two, an indirect inference method considered in Chapter 4.
Simulation studies show that both these methods are very robust and efficient and do as well or better than the existing methods in most cases.
Finally in Chapter 5, we use indirect inference approach to estimate the best fitting income distribution based on limited information that is often available.


Quantum Probability And Related Topics: Qp-pq (Volume Ix) - Qp-pq: Quantum Probability And White Noise Analysis 9 (Paperback)Luigi Accardi (editor)
- We can order this
Publisher: World Scientific Publishing Co Pte Ltd
Number of pages: 388
Weight: 540 g
Dimensions: 214 x 155 x 22 mm
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National Mathematics Day
December 22 was designated as National Mathematics Day (NMD) by Government of India in 1987. Since then this day is celebrated throughout the country with great enthusiasm. IPS Academy, IES, Department of Mathematics every year celebrates National Mathematics Day on 22nd December to commemorate the Birthday on Srinivasa Ramanujam.
Celebration of Mathematics Day on 22 December 2020


- Is 0.2 or 0.25 bigger?
- Is 0.25 or 0.05 greater?
- Is 0.7 the same as 7?
- Is 0.01 or 0.05 bigger?
- Is 0.7 or 0.07 smaller?
- What does 0.01 mean in math?
- Is 0.1 and 0.10 the same?
- Is 0.1 and 1 the same?
- Is 0.5 or 0.75 greater?
- Is 0.5 or 0.7 smaller?
- What does 0.01 mean?
- How do you know which decimal is smaller?
- Which fraction converts to a terminating decimal number?
- What is a decimal equivalent to?
- Is 0.5 and 0.05 the same?
- Is 0.5 or 0.25 bigger?
- What is .01 as a percent?
Is 0.2 or 0.25 bigger?
Answer: 0.25 is bigger than 0.2 because of an additional 0.05..
Is 0.25 or 0.05 greater?
0.25 is bigger than 0.05. Both are less than 1,So if you compare the decimal parts starting after the decimal point and working towards right; comparing the tenth:0.25 has 2 tenths and 0.05 has 0 tenth. So comparing both tenth ,0.25 is bigger than 0.05.
Is 0.7 the same as 7?
The right of the decimal point is like a fraction. … For example, 0.7 is in the tenths place and represents the fraction 7/10. In the number 0.07 the 7 is in the hundredths place and is the same as the fraction 7/100.
Is 0.01 or 0.05 bigger?
So 1% (0.01) is greater than 5% (0.05), in that we can more comfortably accept alternative hypotesis (there is difference, there is association or correlation).
Is 0.7 or 0.07 smaller?
A decimal number is just a fractional number. Comparing 0.7 and 0.07 is clearer if we compared 7/10 to 7/100. The fraction 7/10 is equivalent to 70/100 which is clearly larger than 7/100. … If each decimal place value is the same then the decimals are equal.
What does 0.01 mean in math?
Hundredth means 1/100. In decimal form, it is 0.01.
Is 0.1 and 0.10 the same?
For example, 0.1 (one tenth) is equal to 0.10 (ten hundredths). Let’s take a look at a model given below to understand equivalent decimals in a better way. From the model, it is clear that 0.1 is equal to 0.10.
Is 0.1 and 1 the same?
0.1 is a number, but a fraction is how much OF something which could vary. So . 1 could be one-tenth of 1 but it would not one-hundredth of 10. … It the thoughts are along these lines they’re not seeing fractions as numbers.
Is 0.5 or 0.75 greater?
0.75 first since it’s the greatest. 0.55 is next since it’s larger than 0.5 but, smaller than 0.75.
Is 0.5 or 0.7 smaller?
0.7 is larger than 0.5 by 0.2. 0.7， because both 0.7 and 0.5 are positive number and 0.5 is closer to 0 than 0.7.
What does 0.01 mean?
0.01 (point zero one) which is also equal to 1/100.
How do you know which decimal is smaller?
The greater a decimal is, the closer it is to one whole. The smaller a decimal is the farther it is from one whole. The first thing you need to look at is the digit number in each decimal. These each have two digits in them, so you can compare them right away.
Which fraction converts to a terminating decimal number?
Any rational number (that is, a fraction in lowest terms) can be written as either a terminating decimal or a repeating decimal . Just divide the numerator by the denominator . If you end up with a remainder of 0 , then you have a terminating decimal.
What is a decimal equivalent to?
In algebra, equivalent decimals are two decimal numbers that are equivalent, that is, they represent the same value or amount. Here, both the decimal numbers show the same amount of space taken.
Is 0.5 and 0.05 the same?
0.5 is greater then 0.05. As, 0.5=5/10 but 0.05=5/100.
Is 0.5 or 0.25 bigger?
0.5 because 0.5 is the same as 0.50, which is larger than 0.25.
What is .01 as a percent?
0.1%Decimal to percent conversion tableDecimalPercent0.0010.1%0.011%0.022%0.033%20 more rows


Define the following terms using mathematical notations. Also provide example for each term- Subset Universal set Power set
The Answer to the Question
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Here's the Solution to this Question
Let us define the following terms using mathematical notations and also provide example for each term.
We say that a set is a subset of a set if each elements belongs to For example, let Since impllies we conclude that is a subset of
A universal set is a set which contains all objects that may appear in some task or class of tasks. For example, if the problem lies in finding the real roots of quadratic equations, then for this task the universal set is the set of real numbers.
The power set (or powerset) of a set is the set of all subsets of , including the empty set and itself, that is For examle for the set its power set where is the empty set.


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This wiki page contains the complete problem set developed for the OLPC SoCon project.
Career & Technical Education > General Career & Technical Education > Technology Mathematics > General Mathematics > Algebra Mathematics > Equations Mathematics > Number Sense & Operations Mathematics > Problem Solving


You found Ms. Kivi!
Welcome to my webpage! I will be teaching Math 3 and Honors Math 3 for the 2023-24 school year and am looking forward to working with you. It is an honor to teach at Northwest!
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High speed and high resolution. Breakthrough ADC technology switches from 8 to 16 bits in the same oscilloscope.
Figure 2 shows a typical trace of the sound wave beating.
Q1. Estimate the beat frequency of the sound wave.
A1. Frequency is calculated using the frequency formula:
Frequency = (time taken for one complete cycle to pass)-1
From the graph we measured that a beat occurs every 40 ms.
Beat frequency = (40 ms)-1 = 25 Hz
Q2. Estimate the frequencies of the two sound waves from the signal generators.
A2. From the graph we measured 1 cycle in 2233 ms.
Frequency of the combined signal = (2233 ms)-1 = 448 Hz
The two frequencies can now be calculated using the combined frequency and the beat frequency. Frequencies of the initial waveform = Frequency of the combined waveform ± the beat frequency.
Frequency of initial sound waves = 448 Hz ± 25 Hz
f1= 423 Hz and f2= 473 Hz
The two frequencies were measured to be 423 Hz and 473 Hz using the frequency meter in PicoScope, this gave a beat frequency of 24 Hz (this should be 25 Hz, the error was probably due to the setting of the signal generators). Figure 3 below shows the output from the spectrum analyzer.
Q3. What would happen if the amplitudes of the two sound sources were not equal?
A3. If the two sound waves are at similar frequencies but of different amplitudes the two sound waves would not cancel each other out at the destructive interference points. This sound wave would look like the one shown in figure 4.
Figure 5 show the spectrum of the combined waveforms, showing the difference in magnitude of the two wave forms.
This experiment was written by Dick Robinson of Maesydderwen School.