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# dividing square roots calculator

Division Calculator. When you see it like this, this means the positive square root. You will have 20 questions! Just enter your own function and our free calculator solves it step by step. For digits after decimal point, pair them from left to right). Multiplying square roots is typically done one of two ways. By using this long division calculator, users can perform division with remainder or without remainder … Techniques for rationalizing the denominator are shown below. Algebra Calculator is a calculator that gives step-by-step help on algebra problems. The square root of 4 is 2, and the square root of 9 is 3. By using this website, you agree to our … Multiplicative Inverse (reciprocal function) is used by pressing the "1/x" button or using the keyboard shortcut "i". For example:-9 + 38i divided by 5 + 6i would require a = 5 and bi = 6 to be in the 2nd row. • To obtain the dimensional weight in pounds, divide the cubic inch result by 166. I would recommend this software for anyone that needs help with algebra. It allows to perform the basic arithmetic operations: addition, subtraction, division… For example, the square roots of 9 are -3 and +3, since (-3) 2 = (+3) 2 = 9. For example, if you're trying to find the square root of 7, first you'd need to find the first perfect square below 7, which is 4, and the first perfect square above 7, which is 9. So the square root is this big check-looking thing. You calculate roots by solving the equation . Rationalizing is the process of starting with a fraction containing a radical in its denominator and determining fraction with no radical in its denominator. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you. The calculator will divide the polynomial by the binomial using synthetic division, with steps shown. Polynomial calculator - Sum and difference . A perfect square root is any square root that's a whole number. Sofsource.com gives both interesting and useful material on square roots with fractions calculator, matrices and equations by factoring and other math subjects. When DIVIDING, it is important to enter the denominator in the second row. How To Build An Html Calculator App From Scratch Using Javascript. Divide 2 numbers and find the quotient. Divide - divide your number by one of those square roots. Rationalize Denominators with Conjugates. This finds the largest even value that can equally take the square root of, and leaves a number under the square root … Dividing radical is based on rationalizing the denominator. CASE 1: Rationalizing denominators with one square roots Finding roots of polynomials was never that easy! All the procedures were so simple and easy to follow. Divide square roots; Rationalize a one-term denominator; Rationalize a two-term denominator; Note. Estimating higher n th roots, even if using a calculator … You'll find the classic method written out for you in Easy Question Evil Answers. The square root function is used by clicking on the "x" button or using the keyboard shortcut "r". The Polynomial Roots Calculator will find the roots of any polynomial with just one click. Example: 12 x √3 Know any places that can do this? Perform division … I started using Algebrator to help me solve questions as well as with my homework and eventually I … Dividing square roots is essentially simplifying a fraction. The complex number calculator is also called an imaginary number calculator. One is through the method described above. The rules for multiplication and division with algebraic square roots are exactly the same as those for numerical square roots – in fact, we stated those rules there in an algebraic form: property 1: and. If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division … Figure 9.3 A table of square roots was used to find approximate values of square roots before there were calculators. The key thing to remember is that you must divide coefficients by coefficients, and radicands by radicands. finding square root easy ; multiplying and dividing powers ; laplace transforms ti 89 ; convert fraction to decimal ; georgia high school mathematics mcdougall little ; PROBABILITY - ALGEBRA 2 ; graphing calculator online using square root … At every round of the algorithm we use a pair of digits from the number and will find one digit for the answer. Thus we have, 05. We have already looked in some detail at multiplication and division with numerical square roots. Here our calculator is on edge, because square root is not a well defined function on complex number. This game includes: 10 pages of questions (20 questions in all) on multiplying and dividing square roots, student recording sheet, and an answer key. Addition Subtraction Multiplication Division Sign Square Square Root Inverse Exponent Percent Memory Functions . For example:-9 + 38i divided by 5 + 6i would require a = 5 and bi = 6 to be in the 2nd row. By using this website, you agree to our Cookie Policy. If you want to find out the possible values, the easiest way is probably to go with De Moivre's formula. If you missed this problem, review Exercise 1.6.1. One of my problems is dealing with dividing square root calculator; can anyone help me understand what it is all about? Let me write this down bigger. Estimating an n th Root. BYJU’S online simplify square roots calculator tool performs the calculation faster, and it displays the simplification value in a fraction of seconds. From a calculator we find that √ … Doceri is free in the iTunes app store. Here are the search phrases that today's searchers used to find our site. Basic Online Calculator Help. By using this website, you agree to our Cookie Policy. Video transcript. These common operations can be time-consuming and error-prone when done on paper. Square root calculator converts to decimal, kumon answer book download, ti 84 program download free, holt pre algebra 8th grade, calculator solve by elimination, how to solve chemical equations at 8th class level, Answers to Dividing … Polynomial calculator - Division and multiplication. We provide a tremendous amount of really good reference tutorials on topics varying from common factor to grade math Figure 9.3 shows a portion of a table of squares and square roots. More Number Calculator Exponent Solver. This Solver (Square Root Arithmetic (Adding, Subtracting, Multiplying, and Dividing Square Roots)) was created by by jim_thompson5910(35256) : View Source, Show, Put on YOUR site About jim_thompson5910: If you need more math help, then you can email me.I charge $2 … Multiplying & Dividing 2 Digit Decimals Multiplying & Dividing Decimals & Integers by 10, 100 & 1000 Arithmetic With 2 Digit Decimals This finds the largest even value that can equally take the square root of, and leaves a number under the square root symbol that does not come out to an even number. I started using Algebrator to help me solve questions as well as with my homework and eventually I started getting A’s in math. A lot of students prepping for GMAT Quant, especially those GMAT students away from math for a long time, get lost when trying to divide by a square root.However, dividing by square roots is not something that should intimidate you. Square Roots with Addition, Subtraction, Multiplication, and Division The following information covers adding, subtracting, multiplying and dividing square roots. Prime Number Factorization. Algebrator is a user friendly software and is definitely worth a try. Here you find our worksheets with square roots suited for grade levels 6 and 7. Here are the search phrases that today's searchers used to find our site. Step 1: Find the largest square smaller than 40, that's 36. Example: Calculate the square root of 10 to 2 decimal places. In the above example the number for which we have to calculate square root is 40. Here we doing this using long division method. I was very weak in math, especially in dividing square root calculator and my grades were terrible . Find the two perfect square numbers it lies between. Multiplying and Dividing With Square Roots. Calculating n th roots can be done using a similar method, with modifications to deal with n.While computing square roots entirely by hand is tedious. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Enter 2 numbers and press the = button to get the division result: Simplify Square Roots Calculator is a free online tool that displays the simplification of the given square root value. Sample: Calculate square root of 5 using division method. ), with steps shown. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square… The steps we follow when dividing radicals are very similar to those of multiplying radicals.In essence, we begin by taking advantage of the properties of roots and reducing the whole expression into something with only one radical.. We have finding square roots worksheets with square roots up to 50, worksheets in which the radical is unknown, square root estimation worksheets and worksheets in which the calculator can be used to find the square root. I need to complete this asap. … When a number has the form a + bi (a real number plus an imaginary number) it is called a complex number. Division calculator. If you are having problems with algebra and you cant find a simple way to learn, you must try Algebrator, you will never believe what its capable of doing. At each round we get a better approximation of the actual square root. I understand your problem because I had the same issues when I went to high school. Subtract the 36 from the 40, which leaves 4, and enter a 6 as the first digit of your quotient. Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. With a short refresher course, you’ll be able to divide by square roots … To play: 1) Cut each question page in half to separate the question boxes. Can you find yours among them? Learn more at http://www.doceri.com Multiplying square roots is typically done one of two ways. This is a remarkably good product because it explains the problems in a step-by-step manner so we understand them well. I am not so good with the technical stuff. It can rationalize denominators with one or two radicals. Learn how to find square roots on a basic scientific calculator. 4. Square & Square Root Calculator. In general, dividing square roots starts by rationalizing the denominator: (a√b) / (c√d) = (a / (c * d)) * √(b * d). Hi, I am a freshman in high school and I am having trouble with my homework. When DIVIDING, it is important to enter the denominator in the second row. Divide 2 numbers. If your expression is not already set up like a fraction, rewrite it … Multiplying Square Roots Calculator is a free online tool that displays the result when two square roots are multiplied. Calculate the Complex number Multiplication, Division and square root of the given number. Simplifying square roots. A table of square roots was used to find approximate values of square roots before there were calculators. Related Calculators. Percentage Calculator In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). This free root calculator determines the roots of numbers, including common roots such as a square root or a cubed root. BYJU’S online multiplying square roots calculator tool makes the calculation faster, and … For example, you would divide … This is Mathepower. Group the digits into pairs (For digits to the left of the decimal point, pair them from right to left. I am confident that you will find it helpful too. As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. 3. Simplify: $$(\sqrt{5})^2$$. To learn about imaginary numbers and complex number multiplication, division and square roots, click here. DOWNLOAD IMAGE. If you actually want service with math and in particular with adding, subtracting ,dividing, multiplying square roots or adding come pay a visit to us at Sofsource.com. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Online calculator which calculates the square root of a given number using Long Division (LD) method. The addition (sum function) is used by clicking on the "+" button or using the keyboard. Show Instructions. 2) Hang the 20 questions around the classroom. Calculator square root of a fraction. Back in old times before calculators were allowed in math and science classes, students had to do calculations long hand, with slide rules, or with charts. Online division calculator. Learn more about estimating roots by hand, or explore hundreds of other calculators covering topics such as math, finance, health, fitness, and more. Of course, the presence of square roots makes the process a little more complicated, but certain rules allow us to work with fractions in a relatively simple way. Scientific Engg Notation. Divide the number by 2. Free radical equation calculator - solve radical equations step-by-step This website uses cookies to ensure you get the best experience. I am a student turned tutor; I give classes to high school children. Find a fraction equivalent to $$\frac{5}{8}$$ with denominator 48. This function is the same as x^-1 or dividing … You will also find lot of exciting stuff there. Nth Root or Radical Calculator. One is through the method described above. The square root of 100. We're asked to find the square root of 100. This calculator eliminates radicals from a denominator. Lots of people have asked us how to get square roots without a calculator, and lots of others have sent us different ways of doing it! Let's say we are trying to find √ 3150 with the square root algorithm that resembles long division. In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers. This video screencast was created with Doceri on an iPad. The complex number calculator is able to calculate complex numbers when they are in their algebraic form. Subjects: Math, Algebra. Your program saved meThis is really something. Tutorial Read the full tutorial to learn how to graph equations and check your algebra homework. How To Divide Square Roots On A Calculator DOWNLOAD IMAGE. I currently don't have a calculator and need some help. Enter the dividend and divisor and press the = button. Calculate Square Root using Long Division (LD) Method Online calculator which calculates the square root of a given number using Long Division (LD) method. Students learn to divide square roots by dividing the numbers that are inside the radicals. I use it as reference software for my math problems and can say that it has made learning math much more enjoyable. DOWNLOAD IMAGE. Multiply: (7+3x)(7−3x). Estimating higher n th roots, even if using a calculator for intermediary steps, is significantly more tedious. In the radical below, the radicand is the number '5'.. Refresher on an important rule involving dividing square roots: The rule explained below is a critical part of how we are going to divide square roots … The rules for multiplication and division with algebraic square roots are exactly the same as those for numerical square roots … Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Average - take the average of the result of step 2 and the root. Radical Equations. Children today still learn how to add, subtract, multiply, and divide by hand, but 40 years ago children also had to learn to compute square roots … Vocabulary Refresher. Dividing Polynomials Calculator. You can also never have a square root in a denominator. Along with the regular mode of teaching , I use Algebrator to solve examples practically in front of the learners . Any time you need to have assistance on monomials as well as point, Sofsource.com is without a doubt the perfect site to have a look at! Grades: 6 th, 7 th, 8 th, 9 th, 10 th. Square Roots without a calculator! Calculating n th roots can be done using a similar method, with modifications to deal with n.While computing square roots entirely by hand is tedious. We calculate all complex roots … Square Root By Long Division Method. Cole Parmer Solar Battery Powered Calculators From Cole Parmer Germany. Basic Online Calculator Addition. Free Square Roots calculator - Find square roots of any number step-by-step This website uses cookies to ensure you get the best experience. Basically, I want a link to a good online calculator where I can multiply/divide a whole number by a square root. Before you get started, take this readiness quiz. A root of a function is an intersection of the graph with the x-axis. How To rationalize the Denominator. Estimating an n th Root. Algebra homework solvers, formula to solve 3 simultaneous equations, answers to California life science workbook from prentice hall, mechanics mcqs, maths sats : square roots, irrational square roots calculator. If you missed this problem, review Example 9.4.13. Therefore, you know that the square root of 7 falls somewhere between 2 and 3. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. Thousands of users are using our software to conquer their algebra homework. 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Find the square root, or the two roots, including the principal root, of positive and negative real numbers. Thank you. Where do I find examples? Enter expression with fractions The calculator performs basic and advanced operations with fractions, expressions with fractions combined with integers, decimals, and mixed numbers. "Note that any positive real number has two square roots, one positive and one negative. Students also learn that if there is a square root in the denominator of a fraction, the problem can be simplified by multiplying both the numerator and denominator by the square root that is in the denominator. I was very weak in math, especially in dividing square root calculator and my grades were terrible . Square roots is a specialized form of our common roots calculator. Input the polynomial: P(x) = How to input. Here are some of their experiences: Students struggling with all kinds of algebra problems find out that our software is a life-saver. Addition and subtraction You will discover early on that square root addition and subtraction is more involved than with regular numbers. Calculator … To use it, replace square root sign ( √ ) with letter r. Example: to rationalize$\frac{\sqrt{2} … Find some square roots of two-digit numbers to 2dp by trial and improvement. Now, divide your number by one of the perfect square roots you found. Dividing Radicands Set up a fraction. The function results in a+b. Also tells you if the entered number is a perfect square. To calculate a square root by hand, first estimate the answer by finding the 2 perfect square roots that the number is between. X Times X Equation Calculator. Multiplying and Dividing With Square Roots. Polynomial calculator … Step 2: Take the reminder 4 and 'bring down' the next pair of digits (00 as it happens) to make 400. Subtraction. In case you want support with algebra and in particular with Square Root Calculator For Multiplying or exponents come visit us at Polymathlove.com. We have already looked in some detail at multiplication and division with numerical square roots. Radicals and Square Roots Worksheets (Algebra 1 worksheets on radicals (no variables)) Menu; Table of Content; From Mathwarehouse. To learn about imaginary numbers and complex number multiplication, division and square roots… Rationalize Denominator Calculator. Roots What is a root and how to calculate it? ... (Division) ^ ... "raised to the power") sqrt (Square Root) (Example: sqrt(9)) More Math Symbols. Sum of Numbers Calculator. Square roots are approximated to five decimal places in this table. Can I get the product description, so I know what it has to offer? For example, while you can think of as equivalent to since both the numerator and the denominator are square roots, notice that you … Cube & Cube Root Calculator. DOWNLOAD IMAGE. This function represents x^.5 where the result squared is equal to x. Inverse. What a great tool! Calculate the positive principal root and negative root of positive real numbers. The radicand refers to the number under the radical sign. Multiply integers practice, t-83 calculator, intermediate algebra formula charts, applet polynomial factoring. There are other ways of working out square roots on this page. Thanks for helping. Then, find the square root of each perfect square. How To Multiply Square Roots. Dividing square roots. Multiplying and Dividing Square Roots worksheets ; solving linear systems by combinations calculator ; conic section powerpoint ; lesson plan on simple factorization ; Answers to Glencoe/McGraw-Hill Algebra I workbook ; how to pass algebra 1 eoc ; free square root rule calculator ; simplify square root … I just pray this thing isn’t very complex . ... Square root: sqrt(1 / 16) = ... Be careful, always do multiplication and division … The complex symbol notes i. Code to add this calci to your website Calculation of Square Root of Number using Long Division … Dividing Square Roots. In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. Square root calculator and perfect square calculator. Search phrases used on 2010-06-26: Students struggling with all kinds of algebra problems find out that our software is a life-saver. We maintain a great deal of high-quality reference … I found a lot of really good online calculators to use to find square roots, but none to multiply or divide. 1. solving equations & transforming formulas, where can i learn pre-algebra and algebra for free online, https://softmath.com/faqs-regarding-algebra.html. These common operations can be time-consuming and error-prone when done on paper it is all?! Tutorial to learn about imaginary numbers and complex number calculator Exponent Solver must divide coefficients by,! From a calculator and my grades were terrible roots with fractions calculator matrices! Done on paper reference software for my math problems and can say that it has made learning math much enjoyable. It as reference software for my math problems and can say that has... With all kinds of algebra problems find out that our software is a user friendly software and is definitely a!, subtracting, Multiplying and dividing square root value to use to find roots. Roots is typically done one of two ways 10 th using this website uses cookies ensure. Freshman in high school children: calculate square root by hand, first estimate answer! Given square root of a table of Content ; from Mathwarehouse ( LD method! The largest square smaller than 40, which leaves 4, and the square root of 10 2! Which calculates the square root of 7 falls somewhere between 2 and 3 by hand, first the... Exercise 1.6.1 issues when i went to high school children the complex number calculator 's used... Radicands by radicands page in half to separate the question boxes multiplicative Inverse ( reciprocal ). Given number no variables ) ) Menu ; table of squares and square roots two-digit!, t-83 calculator, users can perform division with remainder or without remainder more... Tutorials on topics varying from common factor to grade and how to find approximate values of square roots is life-saver... Dividing … when dividing, it is important to enter the dividend and divisor and press the = button,... Good reference tutorials on topics varying from common factor to grade is 2, and enter 6. This, this means the positive principal root, or the two roots, here... ; Note the x-axis answer by finding the 2 perfect square root of..., the slide rule uses two logarithmic scales to allow rapid multiplication and division the following information covers adding subtracting... Reciprocal function ) is used by clicking on the  1/x '' or... Edge, because square root of 10 to 2 decimal places you can also never have a that! One negative, especially in dividing square roots you found to separate the boxes..., pair them from right to left, because square root algorithm that resembles long calculator... A well defined function on complex number the product description, so  5x  is equivalent ! Isn ’ t very complex average of the actual square root calculator and my grades terrible! A whole number to solve examples practically in front of the given number long... A student turned tutor ; i give classes to high school displays the simplification the. Percentage calculator Sample: calculate square root of 9 is 3 a fraction equivalent to  *! And enter a 6 as the first digit of your quotient scales to allow rapid dividing square roots calculator... Rapid multiplication and division with numerical square roots on a basic scientific calculator phrases that today 's searchers to... The learners if the entered number is a perfect square numbers it lies between ) ) ;! = button two polynomials ( quadratic, binomial, trinomial etc 9.3 a table of Content ; Mathwarehouse! Must divide coefficients by coefficients, and the square root or a cubed root is the same as x^-1 dividing! Refers to the left of the given square root of the given square root algorithm that resembles long.. Must divide coefficients by coefficients, and radicands by radicands as x^-1 or dividing … when dividing square roots calculator, is. Pray this thing isn ’ t very complex to left as x^-1 or dividing … when dividing it! A user friendly software and is definitely worth a try binomial, etc! Worksheets ( algebra 1 worksheets on radicals ( no variables ) ) Menu ; table of square roots typically!, https: //softmath.com/faqs-regarding-algebra.html by factoring and other math subjects the decimal,... Or a cubed root 5x  is equivalent to  5 * x  number the! Multiply or divide more involved than with regular numbers 12 x √3 know any places that can this! Roots you found multiplicative Inverse ( reciprocal function ) is used by pressing dividing square roots calculator ` 1/x '' or!, trinomial etc math problems and can say that it has made learning math much more enjoyable the!


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Solve problem "The diagonal parallel to the main" online - Learn Python 3 - Snakify

The diagonal parallel to the main

Statement

Given an integer $$n$$, produce a two-dimensional array of size $$\left ( n \times n \right )$$ and complete it according to the following rules, and print with a single space between characters:
• On the main diagonal write  0 .
• On the diagonals adjacent to the main, write  1 .
• On the next adjacent diagonals write  2  and so forth.

Print the elements of the resulting array.

In all the problems input the data using input() and print the result using print().

format: plain


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# Why are $\log$ and $\ln$ being used interchangeably?

A definition for complex logarithm that I am looking at in a book is as follows -

$\log z = \ln r + i(\theta + 2n\pi)$

Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the log will make a difference to the answer.

It also says a few lines later $e^{\log z} = z$.

Yet again I don't see how this makes sense. Why isn't $\ln$ used instead of $\log$?

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Abstract Duplicate: math.stackexchange.com/questions/90594/… – user21436 Feb 12 '12 at 17:07
As far as I can surmise, when speaking of the logarithm of a complex variable (inverse of the complex exponential), you write "$\log$", and when speaking of the (natural) logarithm of a real variable, you write "$\ln$" – David Mitra Feb 12 '12 at 17:14
$$ln = log_e$$ $$log = log_{10}$$ (unless the context implies e) $$log = log_2$$ (Algorithms and Computer Science) – Inquest Feb 12 '12 at 17:34
@David Mitra - Normally log on its own means "$\log_{10}$" so when speaking of the logarithm of a complex variable you write "log" and it is implied that you are actually using "$\log_{e}$"...is that correct? – Jim_CS Feb 12 '12 at 18:33
This is fairly standard notation in complex analysis texts. The reason is that you have to define a branch cut of the logarithm, which needs to be distinguished from the real valued logarithm used implicitly in the construction of the branch cut. The issue here is not that the base of the logarithm has changed, but that it is important to specify which branch of the complex logarithm you are talking about and to differentiate that object from the real natural logarithm which is unambiguously defined. – Chris Janjigian Feb 12 '12 at 18:39

Often in math books the base of $\log$ is just assumed to be $e$.

In this case it looks like the reason they are using $\log z$ instead of $\ln$ is to differentiate between when it is a complex function versus when it is a real function.

http://en.wikipedia.org/wiki/Complex_logarithm

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"$\log$" with no base generally means base the base is $e$, when the topic is mathematics, just as "$\exp$" with no base means the base is $e$. In computer programming languages, "$\log$" also generally means base-$e$ log.

On calculators, "$\log$" with no base means the base is $10$ because calculators are designed by engineers. Ironically, the reasons for frequently using base-$10$ logarithms were made obsolete by calculators, which became prevalent in the early 1970s.

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Why the down-vote here? If something is objectionable about what I wrote, then telling me what it is would correct the problem. – Michael Hardy Feb 13 '12 at 17:57
I didn't downvote, but I disagree with you position. The issue at hand is the difference between real vs. complex logarithm. Also your claim that base-10 logs are obsolete is ... detached from reality. Have you ever done any signal processing or acoustics? Have you heard of decibels? Ok, I learned to use slide rules before calculators became commonplace, but even so :-). – Jyrki Lahtonen Feb 13 '12 at 22:29
Ironically, you seek support from the conventions adopted by programming language designers, which is more or less the same set of people that designed the calculators. And you denounce the conventions adopted there. – Jyrki Lahtonen Feb 13 '12 at 22:32
It would be exaggerated to say that I "denounced" anything. And I did say "when the topic is mathematics". – Michael Hardy Feb 13 '12 at 23:38


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# Tag Info

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What exactly does $\Delta_{r_e}$ mean ? Your wave function isn't a field in space, it is a field on configuration space, i.e. it assigns complex numbers to a configuration. If your electron is at $(x_e,y_e,z_e)$ and your proton is at $(x_p,y_p,z_p)$ then the configuration is the point $(x_e,y_e,z_e,x_p,y_p,z_p)$ in a 6d space. A point in that 6d space ...

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The index of the Laplacian tells you which of the coordinates it acts on, that is, if you write $r = (r_x,r_y,r_z)^T$ and $R = (R_x,R_y,R_z)^T$ as Cartesian coordinates, then \begin{align} \Delta_r & := \frac{\partial^2}{\partial {r_x}^2} + \frac{\partial^2}{\partial {r_y}^2} + \frac{\partial^2}{\partial {r_z}^2} \\ \Delta_R & := ...

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Yes, they can. It's not great form, but they can. For a simple example, consider the hamiltonian for a 1D particle in a potential $V(x)$, $$\hat H=\frac1{2m}\hat{ p}^2+V(\hat{x}).$$ This holds equally well if the potential is a delta function at $x_0$, $V(x)=\kappa \delta(x-x_0)$, in which case $$\hat H=\frac1{2m}\hat{ p}^2+\kappa\delta(\hat{x}-x_0).$$ ...

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Observe that the number operator $N := a^\dagger a$ is positive-semidefinite because $$\langle \psi \vert N \psi \rangle = \lvert a \lvert \psi \rangle \rvert^2 \ge 0$$ and hence has no negative eigenvalues. Since $a\lvert n \rangle = \sqrt{n}\lvert n - 1 \rangle$ for a normalized eigenstate of $N$ with eigenvalue $n$ (which one can derive from the ...

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Let $\mathcal{H}$ be the space of states of our theory. Then, time evolution is given by a unitary operator $U(t_2,t_1) : \mathcal{H}\to\mathcal{H}$ that evolves "stuff" from time $t_1$ to time $t_2$. For time-independent Hamiltonians it is just $\mathrm{e}^{\mathrm{i}H(t_2 - t_1)}$. If we are in the Schrödinger picture, we say states "carry the time ...

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I) Let us for clarity use a subscript "$S$" (and "$H$") to denote the Schrödinger (Heisenberg) picture, where bras and kets evolve (are unchanged) and operators are unchanged (evolve), respectively. Moreover, let us assume that the two pictures coincide at the instant $t_0$ (which Ref. 1 assumes is $t_0=0$). II) Recall first of all the possibly confusing ...

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Functions of an operator are (or, can be) defined by their power series: $$f(\hat{A}) = f_0 + f_1 \hat{A} + f_2 \hat{A}^2 + \cdots$$ It's easy to prove that $$\hat{H}\lvert a_n\rangle = a_n\lvert a_n\rangle \quad\implies\quad \hat{H}^k\lvert a_n\rangle = a_n^k\lvert a_n\rangle$$ and if you plug that into the power series definition of the function, it will ...

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Your final expression is correct except that $V$ should be in front since it does not commute with $Θ_0$ in general. In potential scattering ${Θ_0}V$ is often a compact operator and for large positive imaginary part of $z=E+iℏε$ its norm becomes arbitrarily small so the series converges. In certain cases one can do things a little differently by ...

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The simplest way is to exploit the symmetries here. So, instead of mindlessly going through the algebra, solving equations etc. you just use the fact that you are free to call any direction the x-direction, and set up a right handed coordinate system. In particular, this means that you are free to cyclically permute x, y, z in the equations. The problem of ...

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Indeed using a rigged Hilbert Space is almost certainly what you should use and there are tutorials on it on arxiv (I'll add them when/if/I can find them again). However it isn't necessarily going to help you understand what an average physicist is doing on the one hand, and on the other hand there were already many things you were doing that not rigorous ...

2

Say the time evolution for Hamiltonian $H$ is given by $U(t) = \exp(-iHt)$ and the corresponding evolution on the support of $P$ is $$PU(t)P = P\exp(-iHt)P = \exp(-iH_\text{eff}t) \equiv U_\text{eff}(t)$$ assuming $H_\text{eff}$ exists. The desired identity follows from $$\lim_{\eta \rightarrow 0} \int_0^\infty dt \; U(t) e^{i (\epsilon + i \eta) t} = ... 2 Let$$|n\rangle = \frac{(a^{\dagger})^n}{\sqrt{n!}} |0\rangle$$be the n'th normalized eigenstate and recall the commutation relation for bosonic creation/annihilation operators, [a, a^{\dagger}] = 1. Then we have$$a|n\rangle = \frac{ a \,(a^{\dagger})^n}{\sqrt{n!}} |0\rangle = \frac{1}{\sqrt{n!}}\big([a, (a^{\dagger})^n] - (a^{\dagger})^n ...

2

For an analytic function $f$, $f(A)$ can be defined as the Maclaurin series, as described in the OP. However, when $A$ is an element of a C*-algebra rather than just a Banach algebra, one can also consider continuous functions over the spectrum of $A$, which can still be defined as limits of certain polynomials of $A$ (as a consequence of the ...

2

The matrix elements $D_{PQ}$ can either be zero if the orbital $P$ or $Q$ are not occupied -- then $a_Q\vert x\rangle=0$ or $\langle x \vert a_P^\dagger = 0$ -- and otherwise, they are overlaps of $N-1$ electron states $a_P\vert x\rangle$. If you define a matrix $X$ whose columns are $a_P\vert x\rangle$, then you can write $D=X^\dagger X$, and a matrix of ...

1

Hints: $v(a)$ is identified with a left multiplication operator $L_{v(a)}f:= v(a)f$. $L_{v(a)}^{\dagger}=L_{\overline{v(a)}}$; $\left(\frac{\partial}{\partial a}\right)^{\dagger}=-\frac{\partial}{\partial a}$; and $(AB)^{\dagger}=B^{\dagger}A^{\dagger}$.

1

Observables are measurable quantities represented by certain mathematical structures dependent on the theory. In classical mechanics, measurable quantities are represented by functions on a phase space. In quantum mechanics, they are represented by operators on a Hilbert space. In classical mechanics, a measurement is equivalent to evaluating the function ...

1

Observables (I refer here to Hermitian operators) are confusingly named since they are not observable. What is observable, as you noted, is the eigenvalues that represent the outcomes. So what are observables for? The clearest way to understand the issue is to look at Heisenberg picture observables. These observables change over time but the state does not ...

1

If you want to define your $J_\pm$ in terms of $J_x$ and $J_y$ then you'll need to know those matrices. If you consider $J_x=\frac{\hbar}{\sqrt 2}\left(\begin{matrix} 0&1&0\\ 1&0&1\\ 0&1&0\end{matrix}\right),$ and $J_y=\frac{\hbar}{\sqrt 2}\left(\begin{matrix} 0&-i&0\\ i&0&-i\\ 0&i&0\end{matrix}\right)$ then ...

1

Comments to the question (v2): It seems that the question does not explain how a 'Hamiltonian' $H$ differs from a self-adjoint operator $A$ (presumably bounded from below). This would make OP's question a duplicate of the linked Phys.SE post. Perhaps a 'Hamiltonian' $H$ is also supposed to generate 'time'-evolution for some distinguished parameter $t$, ...

1

It's a map from the quantum state space onto itself that represents how the state changes with time. Simplest example: a spin half spin state is a $1\times 2$ column vector $\psi$ holding the two probability amplitudes for the system to be in spin up and spin down. Now leave the system isolated wait for a time $t$. In QM, the operator is linear, so that ...

1

Right, in general you're not going to see a straightforward equivalence there. We can use Dirac notation with $\hat P_b = |b\rangle\langle b|$ to see that $\langle \hat A \hat B \rangle = \sum_{a,b} a~b~\langle \psi | a \rangle~\langle a | b \rangle~\langle b | \psi \rangle$ and even inserting an identity matrix for $b$ (call it $b'$) gives:  ...

1

Question 1 I'm not sure one can make this idea "rigorous". This is essentially the process of "quantization of a classical theory". It's actually an experimental result: if we take classical expressions defining relationships between classically measurable quantities and replace them by observables and, if further, we replace Poisson brackets in the ...

1

we need to assume f can be Taylor expanded, first calculate the power of H, H can commute with itself, so that the eigenvalue is just the power of a

Only top voted, non community-wiki answers of a minimum length are eligible


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# Continuity of Prices for Constant Elasticity of Substitution Production Functions

Imagine I have a CES production function

$$Y_{\sigma} = Z [ \sum_{i=1}^N \alpha_{i} X_i^{\frac{\sigma}{1-\sigma}}]^{\frac{1-\sigma}{\sigma}}$$

I know that as $$\sigma \to 1$$, the corresponding production function becomes the Cobb-Douglas production function

$$Y_1 = Z \prod_{i=1}^N X_{i}^{\alpha_i}$$

My question is, is the same true for the corresponding cost functions? That is, if I have the CES cost function

$$C_{\sigma} = \frac{1}{Z} [\sum_{i=1}^N \alpha_i^{\sigma} P_i^{1-\sigma}]^{\frac{1}{1-\sigma}}$$

Is it true that as $$\sigma \to 1$$, $$C_{\sigma}$$ converges to the following Cobb-Douglas cost function?

$$C_1 = \frac{1}{Z} \prod_{i=1}^N P_i^{\alpha_i}$$

I think so. I didn't work out the proof myself but found in Wikipedia another version of the CES production function that has the same shape as your cost production function by replacing $$1-\sigma$$ with $$r$$. Here is the link. The exponents in the parameters $$\alpha_i$$ should cause no further complication.


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### Where Did the Benefits of Technology Go?

Stephen H. Unger
February 10, 2012

As a young engineer, a half century ago (Wow! Time does fly), I was fascinated by the ideas I was wrestling with, mainly dealing with various aspects of what is now called computer engineering. I greatly enjoyed my work in research and development. But I did have concerns over possible misuse of what we were developing, particularly about possible military applications. I dealt with this mainly by avoiding work on military projects.

A lesser problem involved economic effects. Would technology applicable to the automation of various processes eliminate many good jobs, causing hardship for displaced workers? I didn't worry too much about this. My assumption was that a lot of tedious jobs would be taken over by machines, freeing up people to do more interesting work, working hours would be reduced, and everybody would benefit when automation lowered production costs. Let's look at this more closely. What actually happened?

#### How Technological Advances Can Make our Lives Better

There are a number of ways that scientific and technological advances can benefit us. I will mention just a few. Applications to sanitation, health care, and medicine can result in longer, healthier lives for many. Great improvements in transportation can enrich our lives by facilitating travel over both long and short distances. There are also a number of ways that technology can make life more pleasant thru enhanced entertainment facilities. A major factor is the labor saving nature of technology ranging from bulldozers replacing pick and shovel work, to powerful computers eliminating the need for tedious calculations in connection with many engineering projects. Let's look at this one more closely.

How can the benefits of labor saving technology be realized? One way is to increase production, with no increase in human effort. In order to sell the increased quantity of goods or services, prices might be lowered. The resulting increase in revenue (assuming we are dealing here with the private sector) might be shared by employers and employees, while consumers benefit from the reduced prices. There would also be gains to the public sector via increased tax revenue. (A good case could be made that, for environmental reasons, we should not produce greater quantities of material goods, but rather should utilize technology advances more for reducing working hours and improving efficiency and quality.)

Worker shares might be realized in a variety of ways. These include increased income, shorter working days, increased vacation time, earlier retirement. Working conditions might also be improved. I.e., the most tedious or dangerous jobs could be automated, and environmental hazards, such as toxic air in the workplace, might be eliminated. Ideally, individuals would be able to choose how to allocate their share of the gains; picking various mixes of higher income and more leisure time.

#### What Happened?

As suggested above, since 1975, we would expect to see increased profits for business, and gains for workers in various forms.

Corporations did indeed do very well [Nutting], with substantial gains in profit and in income for upper level management. Taxes were reduced for both corporations and for wealthy people [Eichler]. Those in finance did exceptionally well.

But the great majority of people, particularly those below the median in income, did poorly. From 1975 to 2009, real income of median families increased by 23%, while working hours increased by 26%, as more married women took jobs outside the home. These numbers do not take into account substantial child-care costs incurred when mothers work outside the home. Median real income wages for men in two-parent families actually fell by 7% over this period [Greenstone][Reddy].

The above calculations are in terms of "real dollars", which supposedly take into account inflation. Inflation is calculated on the basis of changes in the Consumer Price Index (CPI), intended to measure of the cost of living. It is used, among other things, to govern increases in social security payments and in salaries for many groups of employees. For the past 2 decades the CPI, calculated by the Bureau of Labor Statistics, has been manipulated in several ways, so that the published numbers understate annual inflation rates by several percentage points [Kaifosh]. One consequence of this distortion is that the official numbers for the real incomes of median families are substantially inflated. The median family today is significantly worse off financially than comparable families were in 1975.

Unemployment increased. Furthermore, a great many good jobs evaporated [Unger-3]. Substantial numbers of people today cannot find employment commensurate with their skills and ability. Skilled machinists are working as security guards, computer programmers as retail clerks, engineers are selling real-estate.

During this same period, when corporations and wealthy people were prospering and the great majority of Americans were experiencing hard times, our societal infrastructure, bridges, school buildings, sanitation facilities, deteriorated as tax dollars were diverted to pay for multiple wars and a bloated military establishment. In addition, tax rates for corporations and wealthy people were lowered [Eichler].

Three principal mechanisms are generating the financial troubles of the great majority of Americans. One is the technology that I had hoped would help them. In addition to eliminating many simple, repetitive jobs, e.g., many on assembly lines, sophisticated technology also eliminated many fairly high level jobs requiring knowledge and judgment, for example, in the control of complex, continuous processes such as those in a steel or paper mill. Word processors greatly reduced the need for skilled typists.

The second mechanism is the phasing out of factories in the US, with American companies setting up facilities in low-wage countries, or contracting out production of their products to overseas producers [Unger-3]. By far the most popular country for this outsourcing is China. Related to this is the export of work such as telephone help, and the interpretation of medical-rays.

The third mechanism that has made it difficult for many Americans to get decent jobs is the import of workers, mostly from low-wage countries [Unger-2]. Legal immigration is currently at an annual rate of about a million. There are also various "guest worker" programs that bring in hundreds of thousands annually, ranging from farm workers to nurses and computer programmers. In addition, there are roughly 11 million illegal immigrants now here, with varying (depending on economic conditions) numbers, of the order of a half million, arriving annually.

Forty years ago, workers, acting thru labor unions, would have been able to mount substantial campaigns to counter these factors, but labor unions, especially in the private sector, are no longer effective defenders of worker interests. Decades of strong anti-union activity by private industry, supported, or at least not opposed, by government, have essentially reduced unions to an interesting historical phenomenon. Fewer than 7% of private sector employees belong to unions, and the leadership of virtually all remaining unions is anything but militant. Unions are much more prevalent in the public sector (over a third of public sector workers belong to unions), but these are now increasingly coming under attack by both corporate controlled political parties [Bagli].

Over the past several decades, most Americans have been working longer hours trying, with limited success, to maintain their living standards. Many have fallen below the poverty line as un- and under-employment has increased substantially. It is not uncommon for people to hold several low paying jobs in an effort to make ends meet. This while other qualified people are unemployed. Somehow, few people outside the privileged 1% have benefited from the increased productivity resulting from great advances in science and engineering.

#### Is it Fair?

One might, and some do, argue that, in our free enterprise system, those who contribute the most are the ones who should, and do, reap the greatest material rewards. If we assume that this is a just criterion for distributing wealth and income, there are at least 3 important questions.

1. If we agree that rewards should increase with contributions, how should contributions be measured?
2. How steeply should rewards increase with contributions? I.e., if A contributes K more than B, how much greater should A's income be compared to B's?
3. In our country today, are those receiving the most income or possessing the greatest wealth, the people who are contributing, or did contribute, the most to society?

Question 1 is a tough one. How do you measure contributions to society? How do we compare the value of work done by a top-rank musician, a first-rate surgeon, a leading biologist? Perhaps we might agree on some rough measure that would rank these close together. But how about the CEO of a major pharmaceutical company, a top notch engineer, a skilled machinist, a firefighter? Consider a leading Wall Street stock trader, and an excellent preschool teacher?

Even if we had some way to quantify societal contributions, it would still be difficult to decide how to determine appropriate income as a function of contribution. If A's work output is worth K times B's work output, should A be paid K times as much as B, or K2 times as much as B, or $\sqrt{K}$ times as much as B, or—what?

Question 3 is easy. How could anyone justify the fact that the income of a Wall Street croupier (stock trader) is something like 100 times that of a teacher playing an important role in shaping the lives of a dozen small children? Would the sudden disappearance of the CEO of a leading advertising agency disrupt society more than the disappearance of a hundred licensed plumbers, whose total incomes are exceeded by his? Is the value to society of the work done by the head of a big law firm specializing in corporate tax work greater than the work done by a hundred sanitation workers?

Rewarding people on the basis of the value of their work to society is certainly not the rule today in the US. It never was, but we have clearly been moving further away from this ideal every year.

Most rich people were born that way. What about the others? Some achieved wealth by virtue of great talent, in performing arts, as great athletes, a few as best-selling authors. Others struck it rich in the world of business. Some of these by exploiting technological developments they may have contributed to. More commonly, the mechanism was via a rise to the upper ranks of corporate management. In recent years, Wall Street stock traders and hedge fund managers have swelled the ranks of the very rich. Regretfully, many very able, well educated, people, capable of doing really useful work, have been lured by high compensation into wasting their talent enhancing and facilitating the operations of Wall Street casinos.

Wealth cannot be acquired by an isolated individual operating without inputs produced by others. Every invention represents a combination and extension of ideas produced by others, and its implementation and exploitation depends on an infrastructure developed, operated, and maintained by others over many centuries.

A Thomas Edison, or a Claude Shannon, stranded on some primitive tropical island with a small number of companions, assuming a benign climate and ample edible vegetation, would be able to survive without much difficulty. But, lacking the infrastructure of modern society, they would be limited to devising and constructing only simple devices and structures to make life a bit more pleasant and interesting. And, whatever they did accomplish would depend heavily on their knowledge of basic science and technology developed by others in the past.

I doubt that a strandee selected from a bunch of hedge fund managers, bankers, and advertising executives would win any popularity contests on the island. If several such people were stranded together, cannibalism might be a serious problem.

Whatever the validity of claims for great compensation made by top rank engineers, or scientists, or other creative people, or claims by others who organize the production of useful goods or services, the claims of the Wall Street types, who do nothing of real value, are clearly bogus.

But even the most creative or productive people do not merit annual incomes exceeding a million dollars [Unger-1]. Their accomplishments can best be considered as relatively small appendages to the intellectual and physical structure of civilization erected by hundreds of generations, which is the heritage of all humanity. But, in any event, very few of the super-rich are in this category. What is truly outrageous is the appropriation by a small fraction of the population, most of whom have done little or nothing especially useful, of virtually all productivity gains over the past three or four decades.

We are now seeing—at last!, in the Occupy Movement [Unger-4], the rise of resistance to what amounts to a massive theft (accompanied by a variety of other crimes). Nobody can reliably predict the extent to which such resistance will succeed, but it does appear that there are real grounds for hope that it will continue to gather momentum and free our country from domination by a small, greedy minority.

#### References

Charles V. Bagli, "Cuomo Gains an Ally for a Looming Fight With the Public-Employee Unions", The New York Times, December 10, 2010

Alexander Eichler, "Corporate Taxes As Percentage Of Profits Now Lowest In Decades", The Huffington Post, 2/ 3/2012

Michael Greenstone, Adam Looney, "The Great Recession May Be Over, but American Families Are Working Harder than Ever", The Brookings Institution, July 08, 2011

Fred Kaifosh, "Why The Consumer Price Index Is Controversial", Investopedia, Oct 17, 2009

Rex Nutting, "Corporate profits' share of pie most in 60 years", MarketWatch, July 29, 2011

Sudeep Reddy, "Overtime, Not Wage Increases, Drive Income Growth", WSJ.com, July 11, 2011

Stephen H. Unger-1, "The Rich and the Rest of Us: Gross Inequality Versus Democracy", Ends and Means, January 13, 2009

Stephen H. Unger-2, "Immigration: Who wins? Who Loses?", Ends and Means, February 27, 2010

Stephen H. Unger-3, "Are Americans Obsolete?", Ends and Means, May 31, 2011

Stephen H. Unger-4, "The Occupy Movement: A Ray of Hope", Ends and Means, November 29, 2011

Comments invited. Send to me at unger(at)cs(dot)columbia(dot)edu

Don't forget to replace (at) with @ and (dot) with .


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# Counting Techniques.

• October 24th 2009, 01:48 PM
jebckr
Counting Techniques.
An exam consists of 10 multiple choice questions (each question with 3 choices). If a student guesses at every answer, what is the pobability that he/she will get 6 questions correct?

Is the correct answer C(10,6)/3^10? Or .36%?

Please let me know if I'm making a mistake here.
• October 24th 2009, 01:57 PM
mr fantastic
Quote:

Originally Posted by jebckr
An exam consists of 10 multiple choice questions (each question with 3 choices). If a student guesses at every answer, what is the pobability that he/she will get 6 questions correct?

Is the correct answer C(10,6)/3^10? Or .36%?

Please let me know if I'm making a mistake here.

X ~ Binomial(n = 10, p = 1/3).

$\Pr(X = 6) = ^{10}C_6 \left( \frac{1}{3} \right)^6 \left( 1 - \frac{1}{3} \right)^{10 - 6}$.
• October 24th 2009, 02:04 PM
jebckr
So it's 5.66%?
• October 24th 2009, 03:14 PM
mr fantastic
Quote:

Originally Posted by jebckr
So it's 5.66%?

Well, if you have pressed the correct buttons on your calculator then I suppose this would be the correct answer (expressed as a percentage. Personally, expressing probabilities such as this as a percentage is something I do not like. But that's just me).


text:
Exponential & Logarthmic Functions in AS-Level Maths (from Sep 2017):

During the year that students study for an AS-Level in Maths, they will be required to cover the following topic areas in EXPONENTIAL & LOGARITHMIC FUNCTIONS:

These topic areas will be tested in the Pure Mathematics paper – Paper 1.

Exponential & Logarithmic Functions in A-Level Maths (from Sep 2017):

In addition to the above, students studying for an A-Level in Maths will also be required to study the following areas in EXPONENTIAL & LOGARITHMIC FUNCTIONS:

• Students may be required to sketch graphs of the form $y=e^{ax+b}+c$

This is done over 2 years and these additional topic areas will be tested in Pure Mathematics Paper 2.


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# Proving that $\sum_{k=1}^3 \Gamma_{i,j,k}\Gamma_{m,n,k} = \delta_{im}\delta_{jn}-\delta_{in}\delta_{jm}$

For a variable $\Gamma_{ijk}$ I know that $\Gamma_{123}=\Gamma_{231}=\Gamma_{312}=1$ and $\Gamma_{132}=\Gamma_{321}=\Gamma_{213}=-1$, otherwise $\Gamma_{ijk}=0$, whereas $i,j,k$ run between $1$ and $3$. I would now like to show:

$$\sum_{k=1}^3 \Gamma_{ijk}\Gamma_{mnk} = \delta_{im}\delta_{jn}-\delta_{in}\delta_{jm}$$

With $\delta_{ij}$ denoting the Kronecker delta.

I am not quite sure what's the best way to prove it... I was thinking about case analysis, but I would not know where to start. For example if I say in my first case that my RHS is zero, then I can assume that $(i\ne m\vee j\ne n) \land (i\neq n \vee j\ne m)$. What would I do from here? I do not think that this gives me enough information to say anything about the LHS of the sum above.

Any help and hints are greatly appreciated!

The only choices of $i,\,j,\,k,\,m,\,n$ that contribute are those for which $i,\,j,\,k$ and $m,\,n,\,k$ both permute $1,\,2,\,3$, as otherwise matching indices imply a $\Gamma$ cancels. So $i,j$ must permute $m,\,n$. The first term on the right-hand side is the case $i=m$, and the second is $i=n$.


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# Tag Info

Accepted

### Why is Screw Theory used in Robotics when you can do everything with regular kinematics?

Screw theory greatly simplifies the notation and expressions used. This allows for complex systems to be considered. As an added bonus it completely eliminates the need to specify which coordinate ...
• 333

### Relationship between the velocity twist Jacobian and the spatial velocity Jacobian

There are a lot of definitional problems and inconsistencies in this area. Geometric Jacobian. I'm not sure this has a precise and agreed upon meaning. But across the more classical robotics books (...
• 1,642
Accepted

• 2,764

### Velocity description in Screw Motion Theory

The body velocity $V_{b}$ is the velocity of the frame with respect to the world, as seen from the frame's perspective. Its rotational component $\omega_{b}$ contains the rotation rates around the ...
• 619

### Stationary/inertial reference frame

Due to the way that frames are defined in the Modern Robotics book (and in this type of vector-field mechanics in general, such as those of Featherstone), both the spatial frame and the body frames ...
Accepted

### Libraries to calculate kinematics using Screw Theory

Lynch and Park's Modern Robotics book uses the product of exponentials formula and screw axes to describe manipulators, and they have a well-documented library available in Python, MATLAB, and ...
Accepted

### Different methods to determine DOF: Chebychev-Kutzbach-Grubler method vs. Screw method

I will try not to skip too many steps. Assuming a Global coordinate frame at the base and the arm is fully extended along the Y-axis of the base frame. Since SCARA has four joints, we will create ...
• 103
Accepted

### Screw based Jacobian

Adding to Peter Corke's answer, there's also a Coursera course by Kevin Lynch which uses the Modern Robotics book as a reference and explains how to derive the screw based Jacobian. The Jacobian can ...

### Free-floating sphere dynamics using Roy Featherstone's spatial_v2 toolbox

You still haven't posted the (full) code that gives the results you've presented; when I run your snippet I don't the results you posted. Instead, I get: ...
• 15.7k

### The Jacobian resulted from Screw method is different from analytical one (Example Inside)

There are in fact two types of Jacobians, a geometric Jacobian and an analytical Jacobian. The intro to chapter 3 in the book: Robotics: Modelling, Planning and Control by Bruno Siciliano, Lorenzo ...
• 5,780

### Newton-Euler Inverse Dynamics by Screw

maybe need some transformation from centers of mass to the joint frame? Isn't that what $A_i$ is? I don't have the book with me, but from your excerpt: Let $A_i$ be the screw axis of joint $i$ ...
• 15.7k

### Forward kinematics confusing point

The author of the referenced thesis is using exponential coordinates and screw theory via the Product of Exponentials formulation to generate the sequence of transformations from one link to the next. ...

### In which frame this wrench is expressed?

Edited Answer From the question, it seems the loading is applied at the end effector but its line of action along the $x_s$ direction. I do prefer to resolve everything on the world inertial frame ...
• 333

### Why is Screw Theory used in Robotics when you can do everything with regular kinematics?

Why use polar or spherical coordinates when you can use Cartesian coordinates for everything? Why use Laplace transforms to solve differential equations? I think most of the challenge in higher math ...
• 15.7k
1 vote
Accepted

### 3D Rigid Body Pose Optimization in flat euclidean space

See https://www.cis.upenn.edu/~cjtaylor/PUBLICATIONS/pdfs/TaylorTR94b.pdf. You can absolutely use "flat" Euclidean space based optimizers while also optimizing on the manifold, but I agree ...
• 449
1 vote
Accepted

### what is difference between twist and classcial velocity

In these equations from Modern Robotics (by Park and Lynch), the fixed inertial frame ${b}$ is both the reference frame used to define all of the coordinate vectors and has its origin located at the ...
1 vote

### Difference between Denavit-Hartenberg and Rodrigues formulas/conventions

The DH method requires you to carefully specify local frames attached to each link and calculate linear and angular offsets between those frames in a very specific way in order to reduce the effective ...

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text:
We described the $"$product construction$"$ that took two DFA's and constructed one DFA whose language is intersection of the languages of the first two.
1. Show how to perform the product construction on NFA's (without $\in$ transitions).
2. Show how to perform the product construction on $\in-$NFA's


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# Partial Lenses in F#

Based on Mauricio's article about Lenses in F#, I recently tried to use lenses in our project, but very soon ran into some fundamental limitations. It turned out be an interesting problem and in the end, we use a modified version of lenses in our project and are happy with the result.

## What are lenses

As a quick reminder, lenses are bi-directional transformations which allow you to make a well-behaved copy-and-update operation on (potentially deeply) nested data structures.

They offer two functions get : 'a -> 'b and set : 'b -> 'a -> 'a. The idea is, that you can "zoom in" to a property and perform local transformations that properly propagate through the object graph. Based on those two we can define a function update : '(b -> 'b) -> 'a -> 'a that first applies the get then maps a function over the value and in the end applies set to perform a map somewhere in the object graph.

In particular, lenses can be composed and still behave as one would expect for a copy-and-update operation. For a more detailed introduction, read Mauricio's article.

## The Problem

While lenses formulated in the way, as Mauricio presents them are very good for product types like tuples or records, they do not work for sum types like discriminated unions. Consider the following example

 1: 2: 3: 4:  type Shape = | Rectangle of width : float * length : float | Circle of radius : float | Prism of width : float * float * height : float

In this case, we cannot create a lens for say the prism, let alone for the height of prism.

As it turns out, the latter is an important feature, because in F# in general and in our project in particular, we use a lot of discriminated unions. So there was the question, if we could somehow extend the idea of a lens, such that it would work also for sum types as opposed to only product types

## Towards partial lenses

The idea, how to make lenses work well with sum types, to consider partial functions instead of total ones: get : 'a -> 'b optionand set : 'b -> 'a -> 'a.

The first point of interest is, that only the get function has a different signature, this is due to the fact, that we may not get a value out of our lens - it is a partial function. When we try to set a value, though, the original object already exists, so if the lens does not trigger, then it just returns the original result instead of changing anything. Our derived update function also keeps its type.

The next question is how do we compose such lenses? It turns out, that it is essentially a monadic bind of the option monad.

  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:  type Lens<'a, 'b> = { Get : 'a -> 'b option Set : 'b -> 'a -> 'a } member l.Update f a = match l.Get a with | Some x -> l.Set (f x) a | None -> a let compose (l1 : Lens<_, _>) (l2 : Lens<_, _>) = { Get = fun a -> Option.bind l1.Get (l2.Get a) Set = l1.Set >> l2.Update }

Now we define our partial lenses for discriminated unions thus:

  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:  type MyType = | MyCase of string | MyOther of int let myCase = { Get = function | MyCase value -> Some value | _ -> None Set = fun newValue -> function | MyCase _ -> newValue | a -> a }

For normal or total lenses, we can define the get as simply wrapping the Value with Some.

## Making sure the lenses are well-behaved

In order to be well-behaved, lenses should fulfil the following lens laws:

• get-set-law: set (get a) a = a
• set-get-law: get (set v a) = v
• set-set-law: set v' (set v a) = set v' a

Now, we cannot immediately fulfil those laws, because our get function is partial, or rather yields an option value. However, we do can show, that the laws hold for the case, where the lense actually yields a value, so this is a conservative extension.

We redefine the get as get' = get >> Option.get, that is, we apply get and immediately unwrap the option. This is now no longer a total function, i.e. it throws for cases, where get returned None, but for the cases, where the original get yielded a Some value, it yields the same value.

By definition, for all total lenses, our partial lens yields the same value, wrapped with Some, so for those cases get' behaves exactly like the original total lens and therefore fulfills the same laws.

And for partial lenses, we only want to show, that the laws hold for the case, where the lens yields a value: Therefore, without loss of generality, assume a = MyCase x for some x and the case MyCase of the lens.

  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21:  set (get' a) a = set ((get >> Option.get) a) a = set (Option.get (get (MyCase x))) a = set (Option.get (Some x)) a = set x a = set x (MyCase x) = MyCase x = a get' (set v a) = get' (set v (MyCase x)) = get' (MyCase v) = Option.get (Some v) = v set v' (set v a) = set v' (set v (MyCase x)) = set v' (MyCase v) = MyCase v' = set v' (MyCase x) = set v' a

## Possible improvement

One possible improvement would be, to distinguish between partial lenses (for sum types) and total lenses (for product types). Whilst I think, it would be possible to properly propagate which lenses are total and which ones are partial (in a static fashion), we did not go down that route, because of two reasons: To get a value, a single pattern match is enough on the caller site and our objects are rarely consisting only of product types. Therefore the more general partial lenses were good enough for our use case.

If one wants to make this distinction properly, one needs 4 overloads of the lens function each: One where both sides are total lenses, one with the first a partial lens, one with the second being partial and one for both sides partial.

type Shape =
| Rectangle of width: float * length: float
| Prism of width: float * float * height: float

Full name: partiallenses.Shape
union case Shape.Rectangle: width: float * length: float -> Shape
Multiple items
val float : value:'T -> float (requires member op_Explicit)

Full name: Microsoft.FSharp.Core.Operators.float

--------------------
type float = System.Double

Full name: Microsoft.FSharp.Core.float

--------------------
type float<'Measure> = float

Full name: Microsoft.FSharp.Core.float<_>
union case Shape.Circle: radius: float -> Shape
union case Shape.Prism: width: float * float * height: float -> Shape
type Lens<'a,'b> =
{Get: 'a -> 'b option;
Set: 'b -> 'a -> 'a;}
member Update : f:('b -> 'b) -> a:'a -> 'a

Full name: partiallenses.Lens<_,_>
Lens.Get: 'a -> 'b option
type 'T option = Option<'T>

Full name: Microsoft.FSharp.Core.option<_>
Multiple items
Lens.Set: 'b -> 'a -> 'a

--------------------
module Set

from Microsoft.FSharp.Collections

--------------------
type Set<'T (requires comparison)> =
interface IComparable
interface IEnumerable
interface IEnumerable<'T>
interface ICollection<'T>
new : elements:seq<'T> -> Set<'T>
member Add : value:'T -> Set<'T>
member Contains : value:'T -> bool
override Equals : obj -> bool
member IsProperSubsetOf : otherSet:Set<'T> -> bool
member IsProperSupersetOf : otherSet:Set<'T> -> bool
...

Full name: Microsoft.FSharp.Collections.Set<_>

--------------------
new : elements:seq<'T> -> Set<'T>
val l : Lens<'a,'b>
member Lens.Update : f:('b -> 'b) -> a:'a -> 'a

Full name: partiallenses.Lens`2.Update
val f : ('b -> 'b)
val a : 'a
union case Option.Some: Value: 'T -> Option<'T>
val x : 'b
Lens.Set: 'b -> 'a -> 'a
union case Option.None: Option<'T>
val compose : l1:Lens<'a,'b> -> l2:Lens<'c,'a> -> Lens<'c,'b>

Full name: partiallenses.compose
val l1 : Lens<'a,'b>
val l2 : Lens<'c,'a>
val a : 'c
module Option

from Microsoft.FSharp.Core
val bind : binder:('T -> 'U option) -> option:'T option -> 'U option

Full name: Microsoft.FSharp.Core.Option.bind
Lens.Get: 'c -> 'a option
Multiple items
module Set

from Microsoft.FSharp.Collections

--------------------
type Set<'T (requires comparison)> =
interface IComparable
interface IEnumerable
interface IEnumerable<'T>
interface ICollection<'T>
new : elements:seq<'T> -> Set<'T>
member Add : value:'T -> Set<'T>
member Contains : value:'T -> bool
override Equals : obj -> bool
member IsProperSubsetOf : otherSet:Set<'T> -> bool
member IsProperSupersetOf : otherSet:Set<'T> -> bool
...

Full name: Microsoft.FSharp.Collections.Set<_>

--------------------
new : elements:seq<'T> -> Set<'T>
member Lens.Update : f:('b -> 'b) -> a:'a -> 'a
type MyType =
| MyCase of string
| MyOther of int

Full name: partiallenses.MyType
union case MyType.MyCase: string -> MyType
Multiple items
val string : value:'T -> string

Full name: Microsoft.FSharp.Core.Operators.string

--------------------
type string = System.String

Full name: Microsoft.FSharp.Core.string
union case MyType.MyOther: int -> MyType
Multiple items
val int : value:'T -> int (requires member op_Explicit)

Full name: Microsoft.FSharp.Core.Operators.int

--------------------
type int = int32

Full name: Microsoft.FSharp.Core.int

--------------------
type int<'Measure> = int

Full name: Microsoft.FSharp.Core.int<_>
val myCase : Lens<MyType,string>

Full name: partiallenses.myCase
val value : string
val newValue : string
val a : MyType
val set : (string -> MyType -> MyType)

Full name: partiallenses.set
val get' : (MyType -> string)

Full name: partiallenses.get'
val get : (MyType -> string option)

Full name: partiallenses.get
val get : option:'T option -> 'T

Full name: Microsoft.FSharp.Core.Option.get

#### 1 comment:

1. Love it! Very interesting topics, I hope the incoming comments and suggestion are equally positive. Thank you for sharing this information that is actually helpful.

ufgop.org
ufgop.org


text:
Statistical Power

Statistical Power

Subject: Survey Research

• Entry
• Entries A-Z
• Subject Index

• The probability of correctly rejecting a null hypothesis that is false is called the statistical power (or simply, power) of the test. A related quantity is the Type II error rate (β) of the test, defined as the probability of not rejecting a false null hypothesis. Because power is based on the assumption that the null hypothesis is actually false, the computations of statistical power are conditional probabilities based on specific alternative values of the parameter(s) being tested. As a probability, power will range from 0 to 1 with larger values being more desirable; numerically, power is equal to 1 − β.

The statistical power is also related implicitly to the Type I error rate (∞), or significance level, of a hypothesis test. If ∞ is ...

• [0-9]
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text:
For most solids and liquids it is convenient to obtain the amount of substance (and the number of particles, if we want it) from the mass. In the section on The Molar Mass numerous such calculations using molar mass were done. In the case of gases, however, accurate measurement of mass is not so simple. Think about how you would weigh a balloon filled with helium, for example. Because it is buoyed up by the air it displaces, such a balloon would force a balance pan up instead of down, and a negative weight would be obtained.

The mass of a gas can be obtained by weighing a truly empty container (with a perfect vacuum), and then filling and re-weighing the container. But this is a time-consuming, inconvenient, and sometimes dangerous procedure. (Such a container might implode—explode inward—due to the difference between atmospheric pressure outside and zero pressure within.)

A more convenient way of obtaining the amount of substance in a gaseous sample is suggested by the data on molar volumes in Table $$\PageIndex{1}$$. Remember that a molar quantity (a quantity divided by the amount of substance) refers to the same number of particles.

TABLE $$\PageIndex{1}$$: Molar Volumes of Several Gases at 0°C and 1 atm Pressure.

Substance Formula Molar Volume/liter mol–1
Hydrogen H2(g) 22.43
Neon Ne(g) 22.44
Oxygen O2(g) 22.39
Nitrogen N2(g) 22.40
Carbon dioxide CO2(g) 22.26
Ammonia NH2(g) 22.09

The data in Table $$\PageIndex{1}$$, then, indicate that for a variety of gases, 6.022 × 1023 molecules occupy almost exactly the same volume (the molar volume) if the temperature and pressure are held constant. We define Standard Temperature and Pressure (STP) for gases as 0°C and 1.00 atm (101.3 kPa) to establish convenient conditions for comparing the molar volumes of gases.

The molar volume is close to 22.4 liters (22.4 dm3) for virtually all gases. That equal volumes of gases at the same temperature and pressure contain equal numbers of molecules was first suggested in 1811 by the Italian chemist Amadeo Avogadro (1776 to 1856). Consequently it is called Avogadro’s law or Avogadro’s hypothesis.

Avogadro’s law has two important messages. First, it says that molar volumes of all gases are the same at a given temperature and pressure. Therefore, even if we do not know what gas we are dealing with, we can still find the amount of substance. The image below demonstrates this concept. All 3 balloons are full of different gases, yet have the same number of moles and therefore the same volume (22.4 Liters).

Second, we expect that if a particular volume corresponds to a certain number of molecules, twice that volume would contain twice as many molecules. In other words, doubling the volume corresponds to doubling the amount of substance, halving the volume corresponds to halving the amount, and so on.

In general, if we multiply the volume by some factor, say x, then we also multiply the amount of substance by that same factor x. Such a relationship is called direct proportionality and may be expressed mathematically as

$\text{V $$\propto$$ n}\label{1}$

where the symbol $$\propto$$ means “is proportional to.”

For a simple demonstration of this concept, play with Concord Consortium's tool shown below, which allows you to manipulate the number of gas molecules in an a certain area and observe the effects on the volume. Try beginning with the default 120 molecules and observing the volume. Then cut the number of molecules in half to 60 and see what affect that has on the volume...To begin the animation, press the play at the bottom of the screen.

Any proportion, such as Equation $$\ref{1}$$ can be changed to an equivalent equation if one side is multiplied by a proportionality constant, such a kA in Equation$$\ref{2}$$:

$\text{V} = \text{k}_A\text{n}\label{2}$

If we know kA for a gas, we can determine the amount of substance from Equation $$\ref{2}$$.

The situation is complicated by the fact that the volume of a gas depends on pressure and temperature, as well as on the amount of substance. That is, kA will vary as temperature and pressure change. Therefore we need quantitative information about the effects of pressure and temperature on the volume of a gas before we can explore the relationship between amount of substance and volume.


text:
# Can you not separate a scalar into x and y components?

1. Apr 18, 2012

### Sean1218

A scalar like electric potential.

Say I have a positive charge, and 4m to the right, and 3m up is a point P.

If I wanted to calculate the potential at point P, I'd use V=kQ/r (r=√(4^2 + 3^2)).

But I'm confused about why finding the potential at 4m to the right (the x component), and the potential at 3m above the charge (y component) and using c=√(x^2 + y^2) wouldn't give the same answer.

And I realize that this isn't independent for scalars, because doing this to find electric field would be wrong as well.

Last edited: Apr 18, 2012
2. Apr 18, 2012

### Mindscrape

What is c? The scalar potential is itself a function of a vector if that clears up anything. Similarly, the electric field is a vector that itself is a function of a vector. i.e.
$$V(\vec{R})=\frac{-kq}{|\vec{R}|}$$
$$\vec{E}(\vec{R})=\frac{kq\vec{R}}{|\vec{R}|^3}$$

Last edited: Apr 18, 2012
3. Apr 18, 2012

### Steely Dan

Well, if the electrostatic potential could have "components," then it stands to reason that any arbitrary function could. Take $z = x^2 + y$, for example. It is clearly true that $z(4,3) = 19$, but $z(4,0) = 16$ and $z(0,3) = 3$, so $z(4,3) \neq (z(4,0)^2 + z(0,3)^2)^{1/2}$. Ordinary functions of space simply do not work the same way as vectors. So in a sense, it's meaningless to compare scalar quantities and vector quantities this way, because vectors are geometrical quantities which are specifically constructed to satisfy the property that you can use the Pythagorean theorem to determine their magnitude.

4. Apr 18, 2012

### Sean1218

Sorry, c is just the hypotenuse so electric potential at P (or electric field at P).

Then why doesn't this method work with electric field, as it's a vector?

5. Apr 18, 2012

### SammyS

Staff Emeritus
Can you not separate a scalar into x and y components?
The answer is that a scalar cannot be separated into x & y components, because a scalar is not a vector quantity. It may have a magnitude, but it does not have a direction associated with it.

6. Apr 18, 2012

### Mindscrape

So, what are you saying doesn't work? The distance to the point charge is the hypotenuse. The hypotenuse itself does have an x component and a y component. The electric potential, however, does not have any components because it just depends on the hypotenuse. The electric field, on the other hand, not only depends on the hypotenuse, but where the hypotenuse is located.

7. Apr 18, 2012

### Sean1218

I understand now regarding electric potential and other scalars, thanks everyone.

What doesn't work is finding the x and y components of the electric field and using pythagorean theorem to find the electric field at that point. This method should give an identical answer to using E=kq/r2 where r=5 (distance between P and q), but it doesn't (for me at least).

8. Apr 18, 2012

### SammyS

Staff Emeritus

9. Apr 18, 2012

### Sean1218

With a charge of 8E-6 C

x: kq/r2 = (9E9)(8E-6)/(42) = 4500

y: kq/r2 = (9E9)(8E-6)/(32) = 8000

E=sqrt(45002 + 80002)
E=9179

or:

r=5
kq/r2 = (9E9)(8E-6)/(52) = 2880 (the correct answer)

Am I just doing the first part wrong?

10. Apr 19, 2012

### Steely Dan

Right, this is not correct. The x component of the electric field vector at point P is not the same thing as the value of the electric field at x = 4, y = 0. Those are two unrelated quantities. The x component of E at point P is the magnitude of E, multiplied by the cosine of the angle between the vector and the x-axis.


text:
# Properties

 Label 900.3.c.s Level $900$ Weight $3$ Character orbit 900.c Analytic conductor $24.523$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.77720518656.9 Defining polynomial: $$x^{8} - 121 x^{4} + 14641$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{3} q^{7} -4 \beta_{2} q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{3} q^{7} -4 \beta_{2} q^{8} -\beta_{7} q^{11} -\beta_{4} q^{13} + ( \beta_{6} + \beta_{7} ) q^{14} + ( -8 - 4 \beta_{5} ) q^{16} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{17} -6 \beta_{5} q^{19} + ( 3 \beta_{3} - \beta_{4} ) q^{22} -14 \beta_{2} q^{23} + ( -3 \beta_{6} + \beta_{7} ) q^{26} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{28} + \beta_{6} q^{29} -10 \beta_{5} q^{31} -16 \beta_{1} q^{32} + ( 18 - 3 \beta_{5} ) q^{34} + 3 \beta_{4} q^{37} + ( -12 \beta_{1} - 12 \beta_{2} ) q^{38} + 2 \beta_{6} q^{41} + 4 \beta_{3} q^{43} + ( -6 \beta_{6} - 2 \beta_{7} ) q^{44} + ( -28 - 14 \beta_{5} ) q^{46} + 2 \beta_{2} q^{47} -39 q^{49} + ( -6 \beta_{3} - 2 \beta_{4} ) q^{52} + ( 18 \beta_{1} - 9 \beta_{2} ) q^{53} + 8 \beta_{6} q^{56} + ( \beta_{3} + \beta_{4} ) q^{58} -\beta_{7} q^{59} -58 q^{61} + ( -20 \beta_{1} - 20 \beta_{2} ) q^{62} -64 q^{64} + 2 \beta_{3} q^{67} + ( 12 \beta_{1} - 24 \beta_{2} ) q^{68} + 6 \beta_{7} q^{71} + ( 9 \beta_{6} - 3 \beta_{7} ) q^{74} + ( -72 - 12 \beta_{5} ) q^{76} + ( 88 \beta_{1} - 44 \beta_{2} ) q^{77} + 2 \beta_{5} q^{79} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{82} + 16 \beta_{2} q^{83} + ( -4 \beta_{6} - 4 \beta_{7} ) q^{86} -8 \beta_{4} q^{88} -8 \beta_{6} q^{89} -44 \beta_{5} q^{91} -56 \beta_{1} q^{92} + ( 4 + 2 \beta_{5} ) q^{94} -10 \beta_{4} q^{97} + ( -39 \beta_{1} + 39 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} + O(q^{10})$$ $$8q + 16q^{4} - 64q^{16} + 144q^{34} - 224q^{46} - 312q^{49} - 464q^{61} - 512q^{64} - 576q^{76} + 32q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 121 x^{4} + 14641$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/11$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{6}$$$$/1331$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - 22 \nu^{3} + 242 \nu$$$$)/121$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} + 22 \nu^{5} + 242 \nu^{3} + 2662 \nu$$$$)/1331$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{4} - 242$$$$)/121$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{5} + 22 \nu^{3} + 242 \nu$$$$)/121$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} + 22 \nu^{5} - 242 \nu^{3} + 2662 \nu$$$$)/1331$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$11 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{6} - 11 \beta_{3}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$121 \beta_{5} + 242$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$121 \beta_{7} - 121 \beta_{6} + 121 \beta_{4} - 121 \beta_{3}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$1331 \beta_{2}$$$$/2$$ $$\nu^{7}$$ $$=$$ $$($$$$1331 \beta_{7} + 1331 \beta_{6} - 1331 \beta_{4} - 1331 \beta_{3}$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.858406 + 3.20361i −0.858406 − 3.20361i −0.858406 + 3.20361i 0.858406 − 3.20361i −3.20361 − 0.858406i 3.20361 + 0.858406i 3.20361 − 0.858406i −3.20361 + 0.858406i
−1.73205 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.2 −1.73205 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.3 −1.73205 + 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.4 −1.73205 + 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.5 1.73205 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.6 1.73205 1.00000i 0 2.00000 3.46410i 0 0 9.38083i 8.00000i 0 0
451.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
451.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 0 9.38083i 8.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.s 8
3.b odd 2 1 inner 900.3.c.s 8
4.b odd 2 1 inner 900.3.c.s 8
5.b even 2 1 inner 900.3.c.s 8
5.c odd 4 1 180.3.f.d 4
5.c odd 4 1 180.3.f.g yes 4
12.b even 2 1 inner 900.3.c.s 8
15.d odd 2 1 inner 900.3.c.s 8
15.e even 4 1 180.3.f.d 4
15.e even 4 1 180.3.f.g yes 4
20.d odd 2 1 inner 900.3.c.s 8
20.e even 4 1 180.3.f.d 4
20.e even 4 1 180.3.f.g yes 4
60.h even 2 1 inner 900.3.c.s 8
60.l odd 4 1 180.3.f.d 4
60.l odd 4 1 180.3.f.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.d 4 5.c odd 4 1
180.3.f.d 4 15.e even 4 1
180.3.f.d 4 20.e even 4 1
180.3.f.d 4 60.l odd 4 1
180.3.f.g yes 4 5.c odd 4 1
180.3.f.g yes 4 15.e even 4 1
180.3.f.g yes 4 20.e even 4 1
180.3.f.g yes 4 60.l odd 4 1
900.3.c.s 8 1.a even 1 1 trivial
900.3.c.s 8 3.b odd 2 1 inner
900.3.c.s 8 4.b odd 2 1 inner
900.3.c.s 8 5.b even 2 1 inner
900.3.c.s 8 12.b even 2 1 inner
900.3.c.s 8 15.d odd 2 1 inner
900.3.c.s 8 20.d odd 2 1 inner
900.3.c.s 8 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 88$$ $$T_{13}^{2} - 264$$ $$T_{17}^{2} - 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 88 + T^{2} )^{4}$$
$11$ $$( 264 + T^{2} )^{4}$$
$13$ $$( -264 + T^{2} )^{4}$$
$17$ $$( -108 + T^{2} )^{4}$$
$19$ $$( 432 + T^{2} )^{4}$$
$23$ $$( 784 + T^{2} )^{4}$$
$29$ $$( -88 + T^{2} )^{4}$$
$31$ $$( 1200 + T^{2} )^{4}$$
$37$ $$( -2376 + T^{2} )^{4}$$
$41$ $$( -352 + T^{2} )^{4}$$
$43$ $$( 1408 + T^{2} )^{4}$$
$47$ $$( 16 + T^{2} )^{4}$$
$53$ $$( -972 + T^{2} )^{4}$$
$59$ $$( 264 + T^{2} )^{4}$$
$61$ $$( 58 + T )^{8}$$
$67$ $$( 352 + T^{2} )^{4}$$
$71$ $$( 9504 + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 48 + T^{2} )^{4}$$
$83$ $$( 1024 + T^{2} )^{4}$$
$89$ $$( -5632 + T^{2} )^{4}$$
$97$ $$( -26400 + T^{2} )^{4}$$
show more
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text:
# Partnership Questions Set 12

0
37

## Partnership Questions Set 12

Partnership based Questions are important for competitive exams like IBPS PO, SSC CGL. We are here with daily practice sets based on partnership questions.

Q1. P, Q and R enter into a partnership with investment of Rs. 3500, Rs. 4500 and Rs. 5500 respectively. if at the end of six months total profit is Rs. 405. Find share of P’s  in the total profit?
(a) Rs. 200
(b) Rs. 105
(c) Rs. 250
(d) Rs. 151
(e) Rs. 251

Q2. A and B enter into a partnership with Rs. 50,000 and Rs. 60000 respectively. C joins x months before end of the year with the capital of  Rs. 70000 and B leaves them after x months from the start of the year. If they share the profit the ratio of 20 : 18 : 21, then find the value of x.
(a) 6 months
(b) 3 months
(c) 9 months
(d) 8 months
(e)10 months

Q3. A, B and c enter into a partnership by investing Rs. 1600, Rs. 3600 and Rs. 4800. A is a working partner and gets one fifth of the total profit for his services and remaining profit is divided amongst the three in their  investment ratio. What is the sum of the profit of B and C, if A gets Rs. 5330 as profit?
(a) Rs. 13923
(b) Rs. 10920
(c) Rs. 14940
(d) Rs. 10993
(e) Rs. 11993

Q4. X and Y enter into a partnership with the capitals of Rs. 900 and Rs. 700 respectively. they split half of the profit equally for their efforts and the remaining balance in the ratio of their investments, if X got Rs. 47 more than Y. then what is the total profit made by X and Y?
(a) Rs. 376
(b) Rs. 652
(c) Rs. 752
(d) Rs. 954
(e) Rs. 854

Q5. P being a working partner and gets 10% of the total profit as salary, the remaining is shared between P, Q, R in the ratio 2 : 3 : 4, If P gets Rs. 300000, then find the share of Q ?
(a) 200000
(b) 300000
(c) 400000
(d) 500000
(e) 550000


text:
The target mean strength $f_{cm}$ for concrete mix design obtained from the characteristic strength $f_{ck}$ and standard deviation $\sigma$, as defined in $IS:456-2000$, is

1. $f_{ck}+1.35 \sigma$
2. $f_{ck}+1.45 \sigma$
3. $f_{ck}+1.55 \sigma$
4. $f_{ck}+1.65 \sigma$


text:
# [LON-CAPA-users] formatting hardcopy of homework sets

Guy Ashkenazi lon-capa-users@mail.lon-capa.org
Sat, 18 Sep 2004 23:10:40 +0300

> ONE: I am still unable to create a style sheet which removes the blank
> boxes from the hardcopy problem sets AND removes the white space left
> behind. I tried Guy Ashkenazi's alteration of Guy Albertelli's original
> style sheet :
> Remove the "\strut" which keeps the vertical space, and leave an empty
> <tex></tex> tag.

<definetag name="textline">
<render>
<tex>\vspace -5mm</tex>
</render>
</definetag>

will do the trick, but will only work well if you have one textline per part
on a separate line. If you have problems with more than one textline line
per part, or textlines embedded inside other text, things will get funky.
Use with caution.

Guy.