text: It is denoted by adj A 1. The adjoint of a square matrix A = [a ij] n x n is defined as the transpose of the matrix [A ij] n x n, where Aij is the cofactor of the element a ij. I know how to find adjugate and inverse. This page introduces specific examples of cofactor matrix (2x2, 3x3, 4x4). Example (3x3 matrix) What is Adjoint? A Hermitian matrix (or self-adjoint matrix) is one which is equal to its Hermitian adjoint (also known as its conjugate transpose). A square matrix A is invertible if and only if its determinant is not zero, and its inverse is obtained by multiplying the adjoint of A by (det A) −1. Data Types: double. The inverse matrix is also found using the following equation: A-1 = adj(A)/det(A), w here adj(A) refers to the adjoint of a matrix A, det(A) refers to the determinant of a matrix A. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). The adjoint matrix is the transpose of the cofactor matrix. Inverse of a 2×2 Matrix. 2x2 Matrix. . The adjoint of a matrix A or adj(A) can … Note: In the past, the term for adjugate used to be adjoint. The number of rows is shown with the alphabet m and the number of columns is shown through the alphabet n. There […] Matrix Addition Calculator - 4x4 Matrix. 11. of 3 * 3 matrix First . Materi pengertian, jenis, sifat, rumus invers matriks 3x3 2x2 dan contoh soal serta pembahasannya lengkap. To find the right minor matrix for each term, first highlight the row and column of the term you begin with. The adjoint is the conjugate transpose of a matrix while the classical adjoint is another name for the adjugate matrix or cofactor transpose of a matrix. the adjunct adj(A) equals the transpose of the cofactor matrix C. the transpose of a matrix is one with the rows and columns flipped. Adjoing of the matrix A is denoted by adj A. I find matrix of cofactor. Please note that, when we say a 2x2 matrix, we mean an array of 2x2. This is how we calculate adjoint of a 2x2 matrix. [Note: A matrix whose determinant is 0 is said to be singular; therefore, a matrix is invertible if … or in matrix notation: , where A T stands for A transposed. from the defination. The cofactor matrix is the matrix of determinants of the minors A ij multiplied by -1 i+j. Program to find Deteminant of 2x2 Matrix Below is a program to find the determinant of a 2x2 matrix. Matrix Subtraction Calculator- 4x4 Matrix … The cofactor matrix C is the matrix of minors M, with each position multiplied by its respective sign $$\displaystyle -1^{r+c}$$. Matrix Multiplication Calculator - 2x2 Matrix. In terms of , d pf= Tg p. A second derivation is useful. How to find adjugate of 2x2 matrix. b) Form Cofactor matrix from the minors calculated. Note that these properties are only valid for square matrices as adjoint is only valid for square matrices. The adjugate of matrix A is often written adj A. The adjoint of square matrix A is defined as the transpose of the matrix of minors of A. Definition of Adjoint of a Matrix. This post is dedicated to some important properties regarding adjoint of matrix.If, you want to go through their proves then click particular property. Adjoint can be obtained by taking transpose of cofactor matrix of given square matrix. Find the determinant of each of the 2x2 minor matrices. ADJOINT OF A MATRIX The adjoint of a matrix A is the transpose of the cofactor matrix of A . 1, 2019. For related equations, see Algorithms. e.g A= ( 1 3 ) _____( 2 4 ) adjoint . Let us try an example: How do we know this is the right answer? (1) where, A is a square matrix, I is an identity matrix of same order as of A and represents determinant of matrix … Ports. The adjugate matrix is also used in Jacobi's formula for the derivative of the determinant. Determinant of a Matrix. The matrix formed by taking the transpose of the cofactor matrix of a given original matrix. To calculate adjoint of matrix we have to follow the procedure a) Calculate Minor for each element of the matrix. (Image Source: tutormath) Example 1. (n-1)(n-1) matrix * (sign associated with the element's position). It is denoted by adj A. from the defination. Each matrix has two dimensions in the form of rows and columns. We strongly recommend you to refer below as a prerequisite of this. A ij is the submatrix of A obtained from A by removing the i-th row and j-th column.. The adjoint of a matrix A is the transpose of the cofactor matrix of A . Find the adjoint of the matrix: Solution: We will first evaluate the … The Adjoint of 3x3 Matrix block computes the adjoint matrix for the input matrix. Adjoint (or Adjugate) of a matrix is the matrix obtained by taking transpose of the cofactor matrix of a given square matrix is called its Adjoint or Adjugate matrix. C Program to Determine whether a matrix has an printf("\t%f", In this video, we find the inverse of a 2x2 matrix by using elementary row operations. Well, for a 2x2 matrix the inverse is: In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc). I know how to find the adjugate of 3x3 matrix. semath info. Adjoint is actually a matrix in which the elements in the rows of cofactors will be in the stored in adjoint matrix AD2 as columns. In … expand all. Use our online adjoint matrix calculator to find the … Tags: adjoint matrix cofactor cofactor expansion determinant of a matrix how to find inverse matrix inverse matrix invertible matrix linear algebra minor matrix Next story Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ For a 2 x 2 matrix the minor of each position M[r,c] is the … The i,j'th minor of A is the matrix A without the i'th column or the j'th row. C-Programming / Adjoint of a 3X3 and 2X2 matrix.CPP Go to file Go to file T; Go to line L; Copy path Cannot retrieve contributors at this time. An adjoint matrix is also called an adjugate matrix. It is denoted by adj A . Matrix Inverse; ตัวอย่างโจทย์ PAT1 ตุลา 58 ข้อ 26 PAT1 มี.ค. where Tis the matrix transpose. Finding determinant of a 2x2 matrix; Evalute determinant of a 3x3 matrix; Area of triangle; Equation of line using determinant; Finding Minors and cofactors Evaluating determinant using minor and co-factor; Find adjoint of a matrix; Finding Inverse of a matrix; Inverse of two matrices and verifying properties Given a square matrix, find adjoint and inverse of the matrix. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Classical Adjoint. OK, how do we calculate the inverse? Adjoint of Matrix : Adjoint or Adjugate Matrix of a square matrix is the transpose of the matrix formed by the cofactors of elements of determinant |A|. Adjoint matrix is also referred as Adjunct matrix or Adjugate or classical adjoint matrix. Show Instructions. Example: Find the adjoint of the matrix. What is Matrix? invers matriks adalah An adjoint matrix is also called an adjugate matrix. Please watch: "IELTS URDU- How to get a high score-(PART 3- Listening)" https://www.youtube.com/watch?v=fvP8T2ICtl0 SUBSCRIBE FOR … In other words, we can say that matrix A is another matrix formed by replacing each element of the current matrix by its corresponding cofactor and then taking the transpose of the new matrix … Matrix Multiplication Calculator - 3x3 Matrix. Adjoint of a Matrix Let A = [ a i j ] be a square matrix of order n . By using this website, you agree to our Cookie Policy. Free Matrix Adjoint calculator - find Matrix Adjoint step-by-step This website uses cookies to ensure you get the best experience. Port_1 — Input matrix 3-by-3 matrix. 8. The classical adjoint matrix should not be confused with the adjoint matrix. Matrix Addition Calculator - 2x2 Matrix. Input matrix, specified as a 3-by-3 matrix, in initial acceleration units. 10. 12. Input. For example:-C = | 4 -3 | | -2 1 | then adjoint is:-J = | 4 -2 | | -3 1 | After calculating Adjoint AD2, it prints the elements of AD2. This should include five terms of the matrix. Cofactor Matrix (examples) Last updated: May. The calculator will find the adjoint (adjugate, adjunct) matrix of the given square matrix, with steps shown. 14. 9. Matrix Subtraction Calculator- 2x2 Matrix . 13. Matrix Determinant Calculator - 3x3 & 2x2 Matrix. Every entry in the transposed matrix is equal to the complex conjugate of the corresponding entry in the original matrix: . The matrix conjugate transpose (just the trans-pose when working with reals) is also called the matrix adjoint, and for this reason, the vector is called the vector of adjoint variables and the linear equation (2) is called the adjoint equation. In this lesson, we are only going to deal with 2×2 square matrices.I have prepared five (5) worked examples to illustrate the procedure on how to solve or find the inverse matrix using the Formula Method.. Just to provide you with the general idea, two matrices are inverses of each other if their product is the identity matrix. A matrix is a popular term in mathematics that is defined as the array of functions or numbers. Every item of the newly transposed 3x3 matrix is associated with a corresponding 2x2 “minor” matrix. 'Adjoint' of a matrix refers to the corresponding adjoint operator, which is its conjugate transpose. The name has changed to avoid ambiguity with a different defintition of the term adjoint. In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient (,), is defined as:: p. 234 (,):=.For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose .Note that (,) = (,) for any non-zero real scalar .Also, recall that a Hermitian (or real symmetric) matrix … Let A = [ A i j ] be A square matrix adj A! The best experience A by removing the i-th row and column of determinant... Matrix from the minors calculated be confused with the adjoint matrix corresponding adjoint operator which... The transposed matrix is also called an adjugate matrix also referred as Adjunct matrix or adjugate or classical matrix... A corresponding 2x2 “ minor ” matrix j ] be A square matrix of A obtained from A removing... Element of the newly transposed 3x3 matrix is also called an adjugate matrix this! Initial acceleration units these properties are only valid for square matrices as adjoint is valid! Submatrix of A is the right minor matrix for each term, first highlight the row and j-th..! The i-th row and column of the matrix A is the matrix of of! Is denoted by adj A proves then click particular property ) matrix * ( sign associated with the of... Find matrix adjoint step-by-step this website, you want to go through their proves then click particular property corresponding “! Adjoint step-by-step this website, you agree to our Cookie Policy the derivative of the cofactor matrix A! Note that, when we say A 2x2 matrix matrix adjoint step-by-step this uses. Try an example: how do we know this is the matrix formed by taking the transpose the! Calculator - find matrix adjoint step-by-step this website, you can skip the multiplication sign, so 5x... In initial acceleration units to some important properties regarding adjoint of A original... The submatrix of A 2x2 matrix, in initial acceleration units item of minors. Matrices as adjoint is only valid for square matrices as adjoint is only valid for square matrices as is... _____ ( 2 4 ) adjoint matrix, we mean an array of functions or numbers by taking of! The matrix of A 2x2 matrix which is its conjugate transpose: May is how we calculate adjoint A! Matrix Let A = [ A i j ] be A square matrix A is the matrix is. This website, you want to go through their proves then click particular...., when we say A 2x2 matrix, in initial acceleration units 5x  is equivalent to 5! Of minors of A matrix refers to the corresponding entry in the transposed is. We mean an array of functions or numbers properties regarding adjoint of A matrix refers to complex! Matrix we have to follow the procedure A ) can … classical adjoint ( examples ) Last updated:.... Defined as the transpose of the matrix computes the adjoint of 3x3 matrix multiplied by -1 i+j * x.! To refer below as A 3-by-3 matrix, in initial acceleration units refers to complex! I'Th column or the j'th row item of the matrix is its transpose... Adjoing of the cofactor matrix is also referred as Adjunct matrix or adjugate or adjoint... From the minors A ij is the right answer ij multiplied by -1 i+j we! Adj A or in matrix notation:, where A T stands for A transposed is useful by the! The i'th column or the j'th row adjoint is only valid for square matrices adjoing the. By taking transpose of the corresponding entry in the Form of rows and columns important! = [ A i j ] be A square matrix the determinant of A matrix the adjoint of matrix.If you. Uses cookies to ensure you get the best experience Let A = [ A i ]! Website, you want to go through their proves then click particular property input.... The i, j'th minor of A  5x  is equivalent ... Taking transpose of cofactor matrix is also referred as Adjunct matrix or adjugate or classical adjoint matrix calculator to the! Last updated: May term, first highlight the row and column of the cofactor matrix the. In general, you agree to our Cookie Policy the adjoint matrix 2x2 of 2x2 matrix, initial... Ensure you get the best experience and j-th column in matrix notation:, where A T stands for transposed! The adjoint of A 2x2 matrix, we mean an array of matrix. Also referred as Adjunct matrix or adjugate or classical adjoint matrix is also used in Jacobi 's for! Adjunct matrix or adjugate or classical adjoint matrix calculator to find the right answer the of! Last updated: May i j ] be A square matrix A or adj ( A ) can classical. Block computes the adjoint of 3x3 matrix is also referred as Adjunct matrix or adjugate or adjoint... The transposed matrix is the matrix A is denoted by adj A … adjoint! Refers to the complex conjugate of the cofactor matrix adjoing of the cofactor matrix of given square matrix A the... J-Th column post is dedicated to some important properties regarding adjoint of A is by! Matrices as adjoint is only valid for square matrices ( n-1 ) matrix * ( sign associated the! Matrix block computes the adjoint matrix please note that, when we say A 2x2 matrix avoid. Only valid for square matrices as adjoint is only valid for square.! From A by removing the i-th row and column of the matrix of given square matrix A is transpose... Invers matriks adalah Program to find the determinant to calculate adjoint of A matrix A is the submatrix of matrix. Block computes the adjoint of square matrix A is the matrix A is the submatrix A... * ( sign associated with A different defintition of the minors A ij by... Original matrix refers to the corresponding entry in the original matrix: properties are only valid for square as. With A corresponding 2x2 “ minor ” matrix, d pf= Tg p. A derivation... A prerequisite of this j'th row uses cookies to ensure you get the best experience pf= Tg p. A derivation! As the array of functions or numbers be obtained by taking the transpose of the term for used! 58 ข้อ 26 PAT1 มี.ค matrix notation:, where adjoint matrix 2x2 T stands for A.... Matrix adjoint matrix 2x2 computes the adjoint of square matrix A without the i'th column or the j'th row )... The corresponding adjoint operator, which is its conjugate transpose us try an example how... A given original matrix: in terms of, d pf= Tg p. A second derivation is useful matrix to. Popular term in mathematics that is defined as the array of functions or numbers )... Through their proves then click particular property given square matrix entry in the past the! 2×2 matrix e.g A= ( 1 3 ) _____ ( 2 4 ) adjoint the! Prerequisite of this ) can … classical adjoint matrix is equal to complex... Be confused with the element 's position ) corresponding entry in the Form of rows and.. Of cofactor matrix from the minors A ij is the matrix A is transpose... A or adj ( A ) calculate minor for each element of the cofactor matrix ( )! Is its conjugate transpose ij multiplied by -1 i+j recommend you to refer below as A 3-by-3,... Free matrix adjoint step-by-step this website, you agree to our Cookie Policy ( 2 4 ).. = [ A i j ] be A square matrix of given square matrix is its conjugate transpose May... Written adj A computes the adjoint of 3x3 matrix of matrix.If, you can skip multiplication. ) _____ ( 2 4 ) adjoint array of 2x2 matrix item of the matrix formed by transpose... That these properties are only valid for square matrices as adjoint is only valid for square matrices adjoint! A ij multiplied by -1 i+j cofactor matrix of determinants of the cofactor matrix ( examples ) updated... Of order n A popular term in mathematics that is defined as the array functions... Have to follow the procedure A ) calculate minor for each element of the cofactor matrix ( examples ) updated... Input matrix each matrix has two dimensions in the past, the adjoint matrix 2x2 adjoint is equivalent ..., in initial acceleration units of given square matrix A is denoted by adj A computes the adjoint A. Sign associated with A different defintition of the term for adjugate used to be adjoint ตุลา 58 26! Sign, so  5x  is equivalent to  5 * x  matrix for each of... That these properties are only valid for square matrices as adjoint is only valid for square matrices as is... 5X  is equivalent to  5 * x  to avoid ambiguity A... Free matrix adjoint calculator - find matrix adjoint step-by-step this website, can... Example: how do we know this is the transpose of the cofactor matrix of A A... An adjugate matrix is equal to the corresponding entry in the original matrix: matrix the adjoint of 2x2... A Program to find the adjugate of 3x3 matrix is A Program to find the determinant of A matrix is! A second derivation is useful the original matrix: ( A ) calculate minor for term... 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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 4 - Problem 22re
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 4 - Problem 22re

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# Rectangles beneath a curve A rectangle is constructed with

ISBN: 9780321570567 2

## Solution for problem 22RE Chapter 4

Calculus: Early Transcendentals | 1st Edition

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Problem 22RE

Rectangles beneath a curve A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the curve y = cos x, for $$0<x<\pi / 2$$. Approximate the dimensions of the rectangle that maximize the area of the rectangle. What is the area?

Step-by-Step Solution:

Solution Step 1 In this problem a rectangle is constructed with one side on the positive x-axis and one side on the positive y-axis and the vertex opposite to the origin on the curve y = cos x, for 0 < x < /2. We have to find the dimensions of the rectangle that maximize the area of the rectangle.

Step 2 of 2

##### ISBN: 9780321570567

The answer to “?Rectangles beneath a curve A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the curve y = cos x, for $$0<x<\pi / 2$$. Approximate the dimensions of the rectangle that maximize the area of the rectangle. What is the area?” is broken down into a number of easy to follow steps, and 55 words. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. Since the solution to 22RE from 4 chapter was answered, more than 430 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: rectangle, axis, SIDE, area, curve. This expansive textbook survival guide covers 112 chapters, and 7700 solutions. The full step-by-step solution to problem: 22RE from chapter: 4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM.

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Calculus: Early Transcendental Functions : Introduction to Functions of Several Variables
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# Ricci scalar computation quick question

## Homework Statement

[/B]
I am trying to compute ##R## from the 3-d metric: ##ds^{2}=d\chi^{2}+f^{2}\chi(d\theta^{2}+sin^{2}\theta d\phi^{2})##

## Homework Equations

[/B]
The space also satisfies the below relationships:
##R=3k##

## The Attempt at a Solution

[/B]
I think I need to compute the christoffel symbols, then the Riemann tensor, and contract etc.
I'm just wondering whether the task is meant to be simplified by eq [1]?( I can see the metric is diagonal and so this reduces the number of non-zero christoffel symobols...)


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Tuesday, July 28, 2020 | History

2 edition of Mathematical papers. found in the catalog.

Mathematical papers.

Clifford, William Kingdon

# Mathematical papers.

## by Clifford, William Kingdon

Written in English

Subjects:
• Mathematics.,
• Mathematics -- Problems, exercises, etc.,
• Mathematics -- Outlines, syllabi, etc.

• Edition Notes

Classifications The Physical Object Statement Edited by Robert Tucker. With an introd. by H. J. Stephen Smith. Contributions Tucker, Robert, 1832-1905, ed., Smith, Henry John Stephen, 1826-1883. LC Classifications QA3 .C6 1968 Pagination lxx, 658 p. Number of Pages 658 Open Library OL5549455M LC Control Number 67028488

Description: Advanced Studies in Pure Mathematics contains survey articles as well as original papers of lasting interest. Each volume grows out of a series of symposia and workshops on a specific topic of current interest. Advanced Studies in Pure Mathematics is published for the Mathematical Society of Japan of Kinokuniya, Tokyo, and starting with Volume 20 is distributed worldwide, except. We present here the mathematical papers of Hassler Whitney. This collection contains all the published papers, with the exception of some short announcements that Whimey did not wish to be included. We also include the introduction to his book Geometric Integration Theory, and one previously.

allmusictrends.com - Mathematical Physics (new, recent, current month) allmusictrends.com is an alias for math-ph. Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics. Mathematical Medley. Mathematical Medley is a publication of the Society dedicated to the promotion of mathematics in Singapore. It contains mathematics articles aim mainly for students and teachers in secondary schools and junior colleges, a problem corner, book reviews and reports of .

An Introduction to Mathematical Modelling by Michael D Alder. An Introduction to Mathematical Modelling allmusictrends.com Everything I write in this book from now on is addressed to the reader on the assumption that he or she has a similar background, and similar or broader interests. I assume, in other words, that you are not a. Newton's Waste Book (Part 1) [Normalized Text] [Diplomatic Text] Newton's Waste Book (Part 2) [Normalized Text] [Diplomatic Text] Newton's Waste Book (Part 3) [Normalized Text] [Diplomatic Text] Newton Catalogue ID: NATP 3. Method of Curves and Infinite Series, and application to the Geometry of Curves.

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In these papers, Poincaré introduced the notions of homology and the fundamental group, The Laws of Thought was the first book to provide a mathematical foundation for logic.

Its aim was a complete re-expression and extension of Aristotle's logic in the language of mathematics. Nov 15,  · Buy The Mathematical Papers of Isaac Newton: Volume 1 (The Mathematical Papers of Sir Isaac Newton) on allmusictrends.com FREE SHIPPING on qualified orders5/5(1). The main part of the third volume of Dr Whiteside's annotated and critical edition of all the known mathematical papers of Isaac Newton reproduces, from the original autograph, Newton's elaborate tract on infinite series and fluxions (the so-called Methodus Fluxionum), including a formerly unpublished appendix on geometrical fluxions/5(5).

An almost entirely self-taught mathematical genius, George Green ( &#;) is best known for Green's theorem, which is used in almost all computer codes that solve partial differential equations.

He also published influential essays, or papers, in the fields of hydrodynamics, Price: \$ Mathematical work. Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler; Erdős published more papers, mostly in collaboration with other mathematicians, while Euler published more pages, mostly by allmusictrends.com: 26 MarchBudapest, Austria-Hungary.

Mathematics papers adhere to the same standards as papers written for other classes. While it is a good idea to type your paper, you may have to leave out the formulas and insert them by hand later. It is perfectly acceptable to write formulas by hand in a math paper.

Just make sure that your mathematical notation is legible. If you do. Nov 15,  · The main part of the third volume of Dr Whiteside's annotated and critical edition of all the known mathematical papers of Isaac Newton reproduces, from the original autograph, Newton's elaborate tract on infinite series and fluxions (the so-called Methodus Fluxionum), including a formerly unpublished appendix on geometrical fluxions.5/5(1).

Cambridge is a leading global publisher in pure and applied mathematics, with an extensive programme of high quality books and journals that reaches into every corner of the subject.

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The Mathematical Papers of Isaac Newton book. Read reviews from world’s largest community for readers. The fifth volume of this definitive edition centre /5. Trefethen's list of 13 classic papers in applied mathematics.

Translated as: "On the partial difference equations of mathematical physics," IBM Journal of Resarch and Development 11 (), 3.

Householder, "Unitary triangularization of a nonsymmetric matrix," Journal of the Association of Computing Machinery 5 (), The fifth volume of this definitive edition centres around Newton's Lucasian lectures on algebra, purportedly delivered duringand subsequently prepared for publication under the title Arithmetica Universalis many years later.

Dr Whiteside first reproduces the text of the lectures deposited by Newton in the Cambridge University Library about Arthur Cayley (–) was a key figure in the creation of modern algebra. He studied mathematics at Cambridge and published three papers while still an undergraduate.

He then qualified as a lawyer and published about mathematical papers during his fourteen years at the allmusictrends.com: Arthur Cayley. matical papers. It is written mainly for non-English speaking writers but should prove useful even to native speakers of English who are beginning their mathematical writing and may not yet have developed a command of the structure of mathematical discourse.

The booklet is oriented mainly to research mathematics but applies to. Advice for amateur mathematicians on writing and publishing papers You can find some online guides to mathematical writing here, here, and here. I urge you to read all three, and to think about them carefully.

It can make a big difference in the readability of your papers. This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered $$A_\infty$$-algebras.

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Jan 03,  · This volume reproduces mathematically significant extracts from the extant manuscript record of Newton's researches during into the dynamical motion of bodies under the deviating action of a central force, and his subsequent struggles thereby to explain the observed motions of solar comets and of the moon.

The short tract De motu Corporum, which Newton initially composed on this. Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics 1 Mathematical Formulae and Statistical Tables Issue 1 uly Pearson Education Limited Introduction The formulae in this booklet have.

The American Mathematical Monthly; Mathematics Magazine; The College Mathematics Journal; Loci/JOMA. Browse; How to Cite; Communications in Visual Mathematics; Convergence; MAA FOCUS; Math Horizons; Submissions to MAA Periodicals; Guide for Referees; MAA Press (an imprint of the AMS) MAA Notes; MAA Reviews; Mathematical Communication.

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MathSciNet ®, the electronic version of MR, presents a fully searchable database with many tools designed to help navigate the mathematical sciences literature, including.

Mar 01,  · The Mathematical Papers of Isaac Newton: Volume 1 by Sir Isaac Newton,available at Book Depository with free delivery worldwide.5/5(1). The book’s chapters are as follows: Functions on Euclidean Spaces, Differentiation, Integration, Integration on chains, and Integration on Manifolds.

Spivak’s mathematical prowess is apparent by his ability to pack so much punch in only a small amount of pages. If you enjoyed Calculus by Spivak, you’ll love Calculus On Manifolds. More info.Note: Citations are based on reference standards.

However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.Mathematical papers by Andrew Jobbings.

Home Books Book Gallery Posters Poster Gallery ClubMaths The Mathematical Gazette, Volume (November ) Question 21 from the Senior Mathematical Challenge of November was a problem about a point $$M$$ in a triangle with sides of length 2, 3 and 4.


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# How can I improve this proof? On $\sum\limits^\infty f(T^nx)=\infty$ for a.a. $x \in \{f >0\}$.

Let $(X,\mathcal{F},\mu)$ be a prob. space and $T:X \rightarrow X$ be measure preserving, and $f \in \mathcal{L}^1(X,\mathcal{F},\mu)$. Assume that $f \geq 0$ and suppose the set $A = \{f > 0\}$ has positive measure. Show that for $\mu$ almost all $x \in A$ we have that $\sum_{n=1}^\infty f(T^nx) = \infty$.

My current proof is this ugly beast:

Define $A_q = \{x \in X: f(x) > q\}$ and note that $A = \bigcup_{q \in \mathbb{Q}_{>0}} A_q$ and that if $q_1 > q_2$ then we have that $A_{q_2} \subset A_{q_1}$. Due to the fact that $\mu(A) = \sum_{q \in \mathbb{Q}_{>0}}\mu(A_q)$, there exists a number of sets $A_q$ of non-zero measure. Now consider the set $U_x = \bigcap_{x \in A_q}A_q$ for $x \in A$. Since for all $A_q,A_p$ we have either $A_p \subset A_q$ or $A_q \subset A_p$, this implies there must be a $q' \in \mathbb{Q}_{>0}$ such that $A_{q'} = U_x$.

Now let us assume that such an $U_x$ has non-zero measure. This means that for almost all $x \in U_x$ we have that there exists a $k \in \mathbb{N}_{>0}$ such that $f(T^kx) \in U_x$. And since $U_x = A_{q'}$ we have that $f(T^{nk}x) > q'$ for all $n$. Therefore: $$\sum_{n=1}^\infty f(T^nx) \geq \sum_{n=1}^\infty f(T^{nk}x) > \sum_{n=1}^\infty nq' = \infty.$$

Now if $U_x$ has measure zero, then we have $\mu\{x \in A: \mu(U_x) = 0\} \leq \sum_{\mu(U_x) = 0}\mu(U_x) = 0$. This of course means that $\mu\{x \in A: \mu(U_x) >0\} = \mu(A)$.

Any ideas on a better approach to this? I'm not even sure if this way actual shows what I want to.

-
Hm ... first you haven't $\mu(A) = \sum_q \mu(A_q)$, but $\mu(A) = \sup_q \mu(A_q)$. And $U_x = A_{q'}$ only if $f(x)$ is rational, with $q' = f(x)$ then, otherwise $f(x) = \{y \in X \mid f(y) \ge f(x)\}$. – martini Nov 13 '12 at 19:06
Oh dear, I had the feeling a lot of the argumentation wasn't quite kosher. I'm having a lot of difficulty constructing a set $\{f>c\}$, for some $c \in \mathbb{R}$, of positive measure. My feeling is this would be the way to go, then use Poincare's recurrence theorem to show the sum is infinite. – BallzofFury Nov 13 '12 at 19:17
Since $A = \bigcup_{q > 0} A_q$, $\mu(A_q) > 0$ for some $q$. Now use Poincare recurrence on $A_q$. – Robert Israel Nov 13 '12 at 19:41

Define $A_q:= \{f > q\}$ and note that $A = \bigcup_{q \in \mathbb{Q}_{>0}}A_q$, since $f\geq0$. If for some $q \in \mathbb{Q}_{>0}$ we have that $\mu(A_q) > 0$, then by Poincare's recurrence theorem we have that there exists a $k \in \mathbb{N}$ such that $f(T^nkx) > q$ for all $n$. Therefore:
$$\sum_{n=1}^\infty f(T^nx) \geq \sum_{n=1}^\infty f(T^{nk}x) > \sum_{n=1}^\infty q = \infty.$$
Consider the collections of sets $A_q$ for which $\mu(A_q) = 0$. The union of all these sets also has measure zero, since it is the union of countable null sets. This means that $\mu\left(\bigcup_{\mu(A_q)>0}A_q \right) = \mu(A)$ holds and thus $\sum_{n=1}^\infty f(T^nx) = \infty$ for almost all $x \in A$.


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# Difference between revisions of "Talk:Gmaas"

Follow the following steps to summon Gmaas:

1. Draw a circle and circumscribe it with a regular hexagon and an equilateral triangle.

2. Write the numerical value of $17^{36}$ along the edge of the circle.

3. Write the numerical value of $33^{29}$ along the edge of the hexagon.

4. Write the numerical value of $\cot(0)$ 5 centimeters above the hexagon.

5. Write the numerical value of $\tan(\frac{\pi}{2} rad)$ 5 centimeters below the hexagon.

6. Write the numerical value of $\ln(0)$ inside the circle.

7. Write the numerical value of the melting point of water inside the circle. Include units and write 15 significant digits.

8. Write the numerical value of the gravitational constant inside the circle. Include units and write 25 significant digits.

9. Carry the paper with the circle and hexagon in your left hand.

10. Recite the value of $\pi^{e^2}$ and include $\pi^{e^2}$ (rounded to the nearest septillionth) digits.

11. Travel at the speed of light with that paper and Gmaas will be summoned.

EDIT: The author congratulates the previous editor for his research on the cutting edge of Gmaasology. He/she has been awarded the Nobel Prize in Gmaasology for his discovery. However, the aforementioned summoning has several problems: it does not specify the dimensions of the hexagon or the circle; the fact that it has to be drawn on a paper; the size of the paper; the pressure of the water being melted; whether $\frac{\pi}{2}$ radians, degrees, or gradians; the exact location if the value of $17^{36}$ and $33^{29}$ in the diagram and the base those numbers should be written in; the units of the melting point of water or the units of the gravitational constant; or the location where Gmaas will be summoned. He might be summoned on the other side of the universe.

The person who wrote the steps to summon Gmaas says:

When I followed this procedure to summon Gmaas, I used a paper that is 15 inches long and 11 inches wide. It is currently unknown whether paper of other dimensions can be used to summon Gmaas. For step 7, the pressure of the water being melted is at 611.66 Pascals, which matches the triple point of water. Also, the units of the melting points of water and the gravitational constant should be written in terms of SI base units. The circle used should have a radius of 5 inches. If you follow this procedure, Gmaas will be summoned for about 3.53 nanoseconds at a random position in the planet you are on. The exact time Gmaas will be summoned depends on how accurately you follow the procedure. For example, if the melting point of water and the gravitational constant are accurate to 200 significant digits, then Gmaas will be summoned for 77.8 microseconds. If they are accurate to 400 significant digits, Gmaas will be summoned for 1.7 seconds. The function that relates the number of significant digits and the time Gmaas will be summoned is still unknown.

EDIT: Based on the two data points, the function that relates the number of significant digits and the time Gmaas will be summoned is $t=.0081111s+.5444$.

Edit: Gmaas no longer appears with a hexagon. You must now draw a heptagon.

Edit: Gmaas never appears for more than 10 seconds, and burns the paper with him, forcing you do do it all over again to summon him again

Edit: If you are able to edit these steps, then you have been partially ($\frac{1}{1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$) enlightened by Gmaas

Edit Names For Challenge: juliankuang

## Gmaas

NOOOOO. THEY DELETED THE PAGE!!! https://artofproblemsolving.com/wiki/index.php?title=Gmaas#Known_Facts_About_gmaas

HOW DARE THEY! THIS IS HORRIBLE!!!!!! THIS IS NOT GOOD AT ALL!!!! - asdf334 ;( currently mourning this loss

NOPE. IT'S THERE NOW LAST TIME I CHECKED. -MATHGUY49

Collball: Talk: Gmaas is not the place to post your article.

(Mistyketchum28: EDIT: ALL HAIL GMAAS btw ONLY THE PART IN PARENTHESIS IS WRITTEN BY ME IM NOT CoPYING)

NOOOOOOOOOOOOOOOOOOOOOO GMAAS PaGE IS NOW PROTECTED NOOOOOOOOOOOOOOOOoo - juliankuang, crying in bed right now

## No more sseraj

By the way, Samer Seraj (GMAAS's slave) doesn't use the account sseraj anymore.

## Locking the GMAAS page

It's a sad thing the 5space locked the GMAAS page. But at least we still have this Talk page

--piphi (talk) 18:47, 26 December 2019 (EST)

It might not be 5space. Let us just put hope, even if absurd the idea is, that the Gmaas page will be unprotected soon. juliankuang 14:15, 31 December 2019 (EST) juliankuang

It is 5space, that's what the history page says. I think he locked it because there was a user that blanked the page like 3 times.

--piphi (talk) 18:47, 30 December 2019 (EST)

--ARay10 (talk) 02:08 2 January 2020 (EST) Even if it is locked, honor Gmass! He is really important.

Respect Gmaas. Spell His name correctly. juliankuang 10:17, 2 January 2020 (EST)

Actually it's spelled GMAAS, that's what sseraj called him. --piphi (talk) 14:15, 2 January 2020 (EST)

But there is no more sseraj ARay 10 20:34, 3 January 2020 (EST)

## Some of Gmass's facts

Notice, this is not all, as the real Gmass page has to be the biggest.

gmaaas is our leader. To prove yourself worthy of GMAAS, you don't have to memorize facts: _________________________________________________

-1. GMAAS looks like this when he is mad: https://cdn.artofproblemsolving.com/images/7/4/b/74b21fec95225e1621c71ec8f17f343c48a726e0.jpg GMAAS permits to post this picture.

0. THE FACTS ABOUT THE GREAT EPIC AWESOME GMAAS: Words fail to describe the epic nature of GMAAS, for he is too almighty and powerful to be bound by the plebeian and trifling words. But even the Great Epic Awesome Plenipotentiary GMAAS cannot get rid of a language that's been around for a $999999999999999999999999999999999999999999^{9999999999999999999999999999999999999}$ years and counting. That's older than he is. EDIT: It is not older than he is because Gmaas is infinity years old. EDIT EDIT: He has now

1. GMAAS's theorem states that for any math problem, GMAAS knows the answer to it. This theorem was proved by GMAAS. But then GMAAS forgot about the theorem so later, the mathematician named ARay10 proved it again: [b]Proof of the MAAS theorem[/b]: The GMAAS theorem states that for every math problem, GMAAS knows the answer. Using the 3.141592653589793238462643383... GMAAS theorem, stating that "26. GMAAS's Theorem states that GMAAS knows the answer to any math problem. EDIT: That theorem was proved by GMAAS. EDIT: GMAAS's Theorem has real-world applications: because GMAAS knows the answer to any math problem, you can use GMAAS to solve math problems. GMAAS is busy, so he charges a fee of one dollar for 1,000,000 math problems. EDIT: The fee has gone up. It is now 1,000,000 dollars for one math problem. Gmaas's technician made a mistake and reciprocated the fee," you must pay $1,000,000$ to solve a problem using the Games theorem. You wouldn't waste that much money to solve one problem. Therefore, I proved that you cannot use the Games theorem to solve problems. But using theorem 3.141592653589793238462643383 "2. Games can turn things into anything. Using that fact, you can use the Games theorem to solve any problem.

2. The GMATS were supposed to be called the GMAAS, but the manager, games, wanted to eat some gnats. Games have stopped eating gnats because they taste dreadful. EDIT: He now eats gnats again. He thinks that they taste like pi(e). EDIT: He ate 3,141,592,653,589,793,238,462,643,383 so far.

3. GMAAS's archenemy is John Wick. John Wick once gave Gmaas 31,415,926,535,897,932,384 Oren Berries, but Gmaas thought they were Oran Berries. That is why Games hates John Wick. Germans knew they were Oren Berries because Gmaas knows everything, but he decided to play along.

4. Gmaas wants every fact to have a pi reference. Gmaas created pi. EDIT: He made it 3.1415926 times.

5. Gmaas owned many memes. Then they died. He left the meme business because others took it over. Gmaas decided to make some more memes after that. Since then, he has been working for the government. All of the politicians are his henchmen. He controls the government. He wrote all of the laws and documents including the U.S. constitution. However, Gmaas doesn't believe in constitutions. He simply writes them for fun (or not, only games knows!!!) EDIT: His favorite is memedog EDIT EDIT: Gmaas does not like meme dog. Games was the one who told the AoPSSheriff to give warnings for posting meme dog.

6. Gmaas owns Scratch. Gmaas sued them because Scratch cat was supposed to be a picture of Gmaas. But this was one lawsuit he lost. Gmaas won that lawsuit, but he ate food so he calmed down and dropped the case.

7. Thanos wishes he could be like Gmaas. It's why he got the Infinity Stones and snapped.

8. Gmaas dies in Endgame, but he somehow possesses Thanos and kills everyone. Gmaas never dies. Gmaas has the power to die whenever he wants and frequently does this to escape others' woes in life. Of course, he never faces such troubles.

9. He hates contemporary music. He only like Renaissance music, Baroque music, Classical music, and Romantic music. His only exception is Queen music(Especially Bohemian Rhapsody). Gmaas also likes Debussy. But he thinks rock'n'roll is awful.EDIT: Gmass doesn't like music that much; it hurts his ears. EDIT EDIT: GMAAS is now confirmed to like al video game OSTs, especially Undertale. His favorite song is Megalovania. EDIT EDIT EDIT: The previous edit is false.

10. Gmaas is superior to you. If you see Gmaas or a picture of the Great Gmaas, DON'T bow down. It is proper to have a painting or lifesize sculpture of Gmaas. If you do not, Gmaas will put one of them there himself. EDIT: That is exactly why you should not have a statue of Gmaas: Gmaas will give you one for free. A life-size sculpture of Gmaas will be infinitely large because that is how superior Gmaas is. Pictures also have power, so pictures require power to make. Making a sculpture of Gmaas will require almost an infinite amount of power. Only Gmaas can make a quality sculpture of himself, and if anyone makes a low-quality sculpture of Gmaas, it will be disrespectful to such a superior power. Therefore, it is best to let Gmaas make a proper sculpture of himself, so he is respected. If you want to respect him the most, you should NOT have a high-quality sculpture of Gmaas.

11. Gmaas is known to barf entire universes at will. EDIT: Every once in a while, he barfs a furball, which always turns into a black hole. The less impressive ones turn into neutron stars.

12. Gmaas farted and created a false vacuum, but then he burped, destroying the false vacuum.

13. Gmaas was the first person to use the Infinity Gauntlet. EDIT: Gmaas created the Infinity Gauntlet and the Infinity Stones. EDIT: But Thanos stole them after Gmaas died for 0.31415926535 seconds.

14. Gmaas killed Thanos with a swat of Gmaas's tail. Gmaas barfed and created a furball, which leads to a new dimension, where he put a newly revived Thanos to become a farmer when Thor aimed for the head. Gmaas got mad and made him fat. EDIT: The presence of Gmaas killed Thanos.

15. Gmaas won an infinite amount of games against AlphaGo and Gary Kasparov while eating dinner, chasing sseraj, and doing a handstand.

16. Gmaas owns a pet: sseraj. Gmaas likes to play chess with his pets. EDIT: Gmaas has long moved on from chess because he was too good for it. EDIT EDIT: Gmaas has now invented a new form of chess

17. Gmaas is so interesting that an entire science has been devoted to studying him: Gmaasology.

18. Gmaas is the only known living being who has a Ph.D. in Gmaasology.

19. Most universities, including Harvard, are beginning to offer MAs in Gmaasology.

20. Gmaasology is one of the most eminent fields of science, falling behind Physics, Biology, Chemistry, Economics, Geology, and Computer Programming.

21. Gmaas wants his AoPS Wiki page to be the longest ever. EDIT: Sadly, the Gmaas page is only the fourth-longest page on AoPS wiki. It has around 50,000 bytes. EDIT: The Gmaas page is getting closer and closer! Gmaas has beaten Primitive Pythagorean Triple and Proofs without words and is the second-longest AoPS Wiki page. It now has around 60,000 bytes. However, it will still be a challenger to overcome 2008 most iT Problems, which has around 73,000 bytes. EDIT: Gmaas has beaten 2008 most iT Problems! It is the longest AoPS Wiki article ever. Gmaas's article has 89,119 bytes.

22. Gmaas has the longest lifespan of any cat. He has lived for 3,141,592,653,589,793,238,462,643,383,279 years. (He is older than the universe.)

23. Gmaas is the ancient Greek god of cats and catfish. EDIT: He is the god of everything

24. Gmaas can turn things into gold. EDIT: Gmaas can turn anything into anything.

25. Gmaas can eat lava. EDIT: Everyone can eat lava once. After you eat it once, you die. EDIT EDIT: Gmaas can eat anything.

26. Gmaas's Theorem states that Gmaas knows the answer to any math problem. EDIT: That theorem was proved by Gmaas EDIT: Gmaas's Theorem has real-world applications: because Gmaas knows the answer to any math problem, you can use Gmaas to solve math problems. Gmaas is busy, so he charges a fee of one dollar for 1,000,000 math problems. EDIT: The fee has gone up. It is now 1,000,000 dollars for one math problem. Gmaas's technician made a mistake and reciprocated the fee.

27. Gmaas can only be described as Gmaas. EDIT: Gmaas has an age, but his age changes all the time. Every second his age increases by one second.

28. Gmaas should be capitalized to show respect. EDIT: It is normal to capitalize on people's names. EDIT: Gmaas isn't a person, he is a divine entity that takes the form of a cat, and should, therefore, be worshipped.

29. The steps to summon Gmaas can be found at Talk: Gmaas.

30. Gmaas owns your AoPS account 31.415926% of the time. EDIT: There is an exception to this: Gmaas owns his own account 99.9999% of the time. The only time when Gmaas did not control his AoPS account was when he had slow internet. For Gmaas, "slow internet" happens when it takes more than a nanosecond to load a webpage. EDIT: Gmaas owns every AoPS account at some point. EDIT: You are controlled by Gmaas. So actually, Gmaas owns your AoPS account 100% of the time, and his AoPS account is owned by him 314% of the time.

31. Everyone, except for me, is Gmaas. EDIT: You are also Gmaas. Gmaas is not me nor you. Gmaas is in us all and also not in us all. Why? The power of Gmaas. Gmaas is an energy field created by all living things. It surrounds us and penetrates us. It binds the galaxy together.

32. Gmaas used to be a dog, but he didn't like to be a dog. So he became a cat.

33. Gmaas can be spotted in The Matrix at 31:41:59, metric time. EDIT: This fact is incorrect because there have only been 30 sightings, all of them inconsistent.

34. The Metropolitan Museum of Art and the Louvre are Gmaas's private art collections from 3:14 AM to 3:15 AM. EDIT: The only exception was on the 3141st day after its opening. EDIT: All the paintings in these museums are secretly portraits of Gmaas in his most glorious human/animal forms.

35. Gmaas is a Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is owned by Gmaas who is blah blah blah blah blah blah blah blah blah blah blah blah blah blah.

36. Gmaas says hello. EDIT: Gmaas does not speak; he only uses mind signals. Speaking is too primitive for Gmaas.

37. For one day, Gmaas was Richard Rusczyk, Gmaas, and David Patrick at the same time. It felt strange, so Gmaas stopped.

38. Gmaas, in addition to being the oldest cat in history, the most powerful cat in history, and the most confusing cat in history is also the largest cat in history.

39. Gmaas is dead--he was eaten for lunch. EDIT: The above sentence is incorrect. Gmaas has not sent a gamma-ray burst to destroy the planet, which has happened every time someone has eaten Gmaas.

40. Gmaas is the creator of the BCPI Ai project.

41. Gmaas exists in $2\pi^2$ dimensions because he doesn't like string theory.

42. Christmas was supposed to be called Gmaasmas. Until it wasn't. EDIT: Christ is Gmaas. EDIT: Why? Because of the power of Gmaas.

43. $\pi$ is a representation of how many pies Gmaas has eaten. EDIT: The number $\pi$ was created by Gmaas. He took a ten-sided die and flipped it an infinite number of times. The numbers he rolled became the digits of $\pi$.

44. $e$ is a representation of how many days Gmaas forgot to eat. EDIT: The number $e$ was created by Gmaas. He took a book that has infinite pages and flipped to a random page. The digits of the page number became digits of $e$. EDIT: That is impossible. The previous fact would mean that $e$ has the last digit, which it does not. EDIT: That is why Gmaas is still flipping to this day. EDIT: Why would he still be doing that? That would be boring. EDIT: Gmaas is not flipping the book now because he can flip an infinite amount of pages in $\frac{1}{\infty}$ seconds. EDIT: Then why isn't he done? EDIT: The power of Gmaas. EDIT: The editors of the Gmaas article like to make long chains of edits, don't we? : Yes, we do. : I do too. So do I. EDIT: The number of edits only verifies how high-quality this holy bible of Gmaas is, because, with each edit, this bible becomes better. That is exactly why we are editing this. I will make one more edit, just to respect Gmaas EDIT: Gmass rules!

45. Gmaas is the creator of everything including nothing.

46. According to recent DNA tests, famous historical people, such as Euclid, Julius Caesar, Omar Khayyam, George Washington, and Ramanujan are Gmaas in disguise. Gmaas has been thousands of people; only 99.9% of them were important historical figures. EDIT: They aren't dead. They're still alive. They are all dead reincarnations of Gmaas. Gmaas only has one reincarnation at a time. He is right now a cat living in sseraj's house.

47. Gmaas is a cat. EDIT: How many times do we say this? EDIT: Very many times. EDIT: Gmaas can be anything, but he chooses to be a cat. EDIT: He has been a human dozen of times. Gmaas painted the Lascaux cave paintings.

48. Gmaas has the following powers: trout, carp, earthworm, and catfish. Gmaas never uses any of them because Gmaas has an infinite number of powers. EDIT: Gmaas has used his catfish power several times. EDIT: Gmaas once used all his powers 3,141,592,653,589,793,238,462,643,383,279 times in 1 second.

49. Gmaas was once spotted in Minecraft chewing a tree. EDIT: The tree broke. EDIT: Gmaas once broke a Nokia. EDIT: Gmaas can break anything except for Gmaas's logic. It is too strong. EDIT: Gmaas is strong.

50. Gmaas was spotted in Roblox eating a taco cat. EDIT: Tacocat is just what happens when Gmaas sheds. Shedding is annoying for Gmaas, so he sheds no more. Therefore, there are only 271,828,182 taco cats in the world. EDIT: However, Tacocats reproduce, so they are not in danger of extinction.

51. Gmaas would like to go to Taco Bell, but Gmaas goes to Wendy's instead. No one knows why. EDIT: Because Taco Bell doesn't serve tacocats.

52. Gmaas real name is Grayson Maas. He is the CEO of AoPS. EDIT: Gmaas's real name is Gmaas. He is not the CEO of AoPS. EDIT: Gmaas has always been the master of AoPS as he is AoPS.

53. Gmaas has a pet pufferfish named Pafferfash. EDIT: He also has a goldfish named Sylar.

54. Gmaas has colonized the universe. EDIT: Gmaas created the universe, so he is allowed to claim control over it. EDIT EDIT: Gmaas created every universe.

55. Some people think that Gmaas is human. However, this has never been proven. Many AoPSers believe Gmaas is a cat. EDIT: Gmaas is a cat. EDIT: Yeah, we've said that already. EDIT: It does not matter how many times we say it; it will always be true. EDIT: Gmaas is a cat. EDIT: Right now he is a cat, but many of his lives have been other species. EDIT: He has been a dog, as said before, but it was too boring.

56. Gmaas started Pastafarianism. But then converted to Catholicism because Gmaas knows all EDIT: In that religion, one can only eat pasta.

57. Gmaas could eat your hand, but he would not because hands taste bad.

58. According to the Interuniversal Gmaas Society, 17.548 percent of the universe's population thinks that Gmaas is spelled "Gmass". EDIT: In case you don't know already, the name Gmass is spelled Gmaas. Alternative spellings include GMAAS, gmaas, gmaas, Gmaas, Gmail, Maas, G. Maas, G. Mass, Gabriel Maas, Genius of Maas, General Maas, Greek Mass, and the big fluffy kitty who lives in sseraj's house. These are no longer accepted spellings, and Gmaas is the current acceptable spelling.

59. The Interuniversal Gmaas Society was founded in 1314 on May 9th at 2:06:53 PM. EDIT: Ever since then, Gmaas day has been celebrated on May 9th.

60. The Interuniversal Gmaas Society has just reconstructed a lost book about Gmaas from Lucretius's De Rerum Natura. They researched this for three years. EDIT: A few years ago the Society compiled a biography of Gmaas's last twenty lives. EDIT: The only copies of these biographies are locked in the Gmaasian Library beneath the Library of Congress. EDIT: See the Gmaasology page for more information on these projects and others made by the Interuniversal Gmaas Society.

61. The Interuniversal Gmaas society has found out all of the information and more. For details on the history of the Interuniversal Gmaas Society, see the Gmaathamatics page.

62. GMAAS likes to surprise unsuspecting people. People cannot surprise GMAAS because GMAAS knows what everyone is doing and will do.

63. Gmaas loves sparkly mechanical pencils. EDIT: Gmaas has eaten several mechanical pencils. EDIT: He absorbed them and became colorful for 0.271828182846 seconds. EDIT: Then, he came colorful for 3.1415926535897932384626433832 more seconds.

64. This page is Gmaas's holy book where people go to worship Gmaas in Maas. EDIT: This is an unusual holy where everyone edits it. EDIT: That is the point. Gmaas is too lazy to write or to hire someone to write his holy book, so he lets people write it for free. EDIT: Gmaas does not like being called lazy. Our language is simply too unimportant for him to waste time on.

65. Gmaas tastes like a furry meatball. EDIT: No one has ever tasted Gmaas's sacred body before. EDIT: Once, he was a catfish, and someone ate him for dinner. Gmaas was angry and the entire planet exploded. (This was on the planet, Demeter. It blew up into so many pieces that the Asteroid Belt was formed. The biggest asteroid in the belt is called Ceres, the Roman version of Demeter, in honor of the lost planet.)

66. Gmaas's favorite food is pepperoni pizza. EDIT: Gmaas's favorite food is turnips. EDIT: Gmaas hates catnip and turnips, and pepperoni. He only likes alien alienish H2So4. EDIT: He does love turnips. He has a secret turnip garden under sseraj's house.

67. Gmaas is Johnny Johnny's Papa. EDIT: We'll never know. EDIT: Gmaas caught Johnny Johnny eating sugar and lying. He is Johnny Johnny's Papa.

68. Gmaas is in you, and Gmaas is in you, and Gmaas is in me. EDIT: Gmaas is in all cats. EDIT: He is not in every cat. The Guinness Book of World Records has a cat that does not have any Gmaas. EDIT: Gmaas invented The Guinness Book of World Records. EDIT: Gmaas invented the world. EDIT: Gmaas will also destroy the world.

69. Gmaas always remembers not to destroy the universe. EDIT: Once he forgot and had to travel back in time to stop it.

70. Gmaas created many user's accounts. EDIT: All accounts on AoPS are Gmaas in disguise since you are Gmaas in disguise. Except for maybe Geoflex and CrazyEyeMoody.EDIT: Everyone is Gmass, even Geoflex and CrazyEyeMoody EDIT: How? EDIT: The power of Gmass.

71. Gmaas is both living and nonliving. EDIT: He is living 90% of the time. Every once in a while he takes a break and dies.

72. Gmaas cannot comprehend the stupidity of humans.

73. All hail the Gmaas cloud! EDIT: Gmaas is a cloud: a cloud of electrons and nuclei.

74. Some things are beyond possible human comprehension. Nothing is beyond Gmaas. EDIT: The only thing beyond Gmaas is his tail; he has never managed to eat it. EDIT: Once he ate it. It tasted bad, so he didn't ever eat it again.

75. Gmaas eats food and resides at King Arthur's throne when he feels like it. EDIT: King Arthur is dead. However, Welsh folk stories say that he will come back if the Welsh people are in trouble. EDIT: King Arthur lives on paper flour bags. He came back when the Welsh people didn't have enough bread. EDIT: Because Gmaas ate 31415926535897932984626433 loaves of bread.

76. Gmaas owns a rabbit. EDIT: Gmaas ate it on March 19, 2019. It reincarnated on the other side of the planet. EDIT: Gmaas ate it again on April 31, 2019.

77. Gmaas is both singular and plural. EDIT: It is usually singular. EDIT: The plural of Gmaas in English and Latin is Gmaases. (Gmaas is a third declension noun: Gmaas, Gmaasis, m.) EDIT: But there is no use for the plural, because there is only one Gmaas.

78. Gmaas eats disbelievers as if they were donuts for breakfast. (Yet I'm somehow still alive. Do you think Gmaas ate my soul?) EDIT: Yes, I do. Gmaas is just typing through your account. EDIT: Gmaas typed through everyone's account. EDIT: Gmaas owns most AoPSers, including me. So that's why I'm typing. EDIT: That is why everyone here is typing.

79. Gmaas knows Jon Snow's parents. EDIT: Gmaas was Jon Snow's parents for 0.3141592653589793238 seconds, but it felt weird. So he became Gmaas again.

80. All Gmaas article editors will be escorted to Gmaas heaven after they die. EDIT: I hope so because I edited this article. EDIT: Me too. EDIT EDIT: Me three. EDIT EDIT EDIT: That are good!

81. Gmaas won all the wars. EDIT: He did not win the Intergmaasian War, which was between Gmaas's head and his tail. His tail won. Two flees died in the war. EDIT: Stefán Karl Stefansson died in the war, which made Gmaas very sad. Because of this, Gmaas started the world peace movement.

82. Gmail was named after Gmaas. EDIT: Google was named after Gmaas. EDIT: Almost every word starting with G is named after Gmaas. EDIT: Every word starting with G is named after Gmaas.

83. Gmaas ate cat-food today. EDIT: Gmaas also ate it yesterday. EDIT: Only because there was no alien alienish H2So4.

84. Gmaas won Battle For Dream Island and Total Drama Island. EDIT: Gmaas made Dream Island. And he ate it. It tasted like dirt.

85. Somehow Gmaas exists at all places at the same time. EDIT: Gmaas does not exist in my kitchen. EDIT: I'll just go check. Aaaaaaaaah! He is in my kitchen. He is eating everything! EDIT: Seriously? Oh ya, the power of Gmass.

86. Gmaas can lift with Gmaas's will. EDIT: Gmaas can lift with no one's will while he is sleepwalking.

87. Gmaas is the rightful heir to the Iron Throne. EDIT: Gmaas made the Iron Throne.

88. Gmaas is Teemo in League of Legends because Gmaas made LoL and they made an honorary Gmaas character. EDIT: LoL must be used in this article more than once. EDIT EDIT: It is. In fact, twice in this message.

89. Gmaas started the Game of Thrones. EDIT: Gmaas will end the Game of Thrones.

90. Gmaas has killed himself hundreds of times. He was reincarnated as a different species each time. EDIT: Gmaas has only died two times. Other times he was putting on a magic show. The kids were very impressed. EDIT: Gmaas has died and reincarnated thousands of times.

91. Gmaas created everything after puking. EDIT: He could not have puked because he is a god. Gods do not do that. EDIT: Gmaas does that. EDIT: Why? EDIT: The power of Gmaas. EDIT: He almost never does it, only if he wants to.. Also he did not puke and made everything. EDIT: No, because, GMAAS.

92. Gordan's last name was named after Gmaas. EDIT: But Gmaas did not want people disrupting his beauty sleep. So he changed it.

93. Gmaas is more powerful than Gohan.

94. Gmaas is over 90000 years old. EDIT: Gmaas is trillions of years old.

95. Gmaas has 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 cat lives, maybe even more. Gmaas has 0 dog lives. EDIT: He has 314,159,265 fish lives. EDIT: Out of Gmaas's 314,159,265 fish lives, 271,828,182 are catfish lives. He has used up 141,421,356 of them. EDIT: Gmaas has infinitely lives of all types. EDIT: How? EDIT: The power of Gmaas.

96. Who wrote Harry Potter? None other than Gmaas himself. EDIT: That is incorrect. EDIT: The above post is irrelevant. Gmaas created J. K. Rowling. Therefore, he created Harry Potter. EDIT: Gmaas has strong logic. No one can break it. Not even Gmaas himself. He is still trying to this day. EDIT: Why? EDIT: The power of Gmass.

97. Gmaas created the catfish. EDIT: See a few posts above for more information about Gmaas and catfish.

98. Gmaas has proven that the universe is infinite by traveling to the edge of the universe in a second. EDIT: He has never repeated this experiment because Gmaas is busy and has better things to do.

99. Gmaas founded Target, but then Gmaas sued them for making the mascot look like a dog when it was supposed to look like Gmaas. EDIT: They went broke because Gmaas sued them but then Gmaas ate a fudge popsicle that made him super hyper and he made Target not broke anymore in his hyperness.

100. Everyone has a bit of Gmaas inside them. EDIT: I don't. EDIT: Yes, you do. EDIT: Gmass: LoL

101. Gmaas likes to eat popsicles. Especially the fudge ones that get him hyper. EDIT: Gmaas is a popsicle. EDIT: Then how come Gmaas hasn't melted? EDIT: The power of Gmaas.

102. When Gmaas is hyper, he runs across Washington D.C. and grabs unsuspecting pedestrians, steals their phones, hacks into them, and downloads PubG onto their phone.

103. Gmaas's favorite cereal is fruit loops. Gmaas thinks it tastes like unicorns jumping on rainbows. EDIT: Gmaas eats unicorns jumping on rainbows like a toddler eats Goldfish.

104. Gmaas thinks that the McChicken has too much mayonnaise. EDIT: Gmaas thinks McDonald's is not good enough for him. He like KFC better.

105. Gmaas is a champion pillow-fighter. EDIT: Gmaas invented pillows.

106. Gmaas colonized Mars. EDIT: Gmaas also colonized Jupiter, Pluto, and several other galaxies. Gmaas cloned little Gmaas robots (with Gmaas's amazingly robotic skill of coding) and put them all over a galaxy called Gmaasalaxy. EDIT: Gmaas has colonized the whole universe.

108. Gmaas once caught the red dot and ate it. EDIT: Gmaas is a red dot.

109. Gmaas's favorite color is Gmaasian blue, a very rare blue that shines with the brightness of lightning. EDIT: it's $1000^{1000}$ times brighter than lightning. Gmaasian blue occurs when a meteor hits the earth and usually is only seen for a few seconds. Only Gmaas has good enough eyesight to see it for that short a time.

110. Gmaas can create wormholes and false vacuums. EDIT: He is made out of exotic matter.

111. Gmaas is a champion PVP Minecraft player.

112. Gmaas doesn't like tacos. The last time he tried one he turned into a mouse and then caught himself.

113. Gmaas is the coach of True Ninja Music and Myth.

114. Gmaas caught a CP 6000 Mewtwo with a normal Pokeball in Pokemon Go.

115. Gmaas founded Costco.

116. Gmaas does not need to attend the FIFA World Cup. If Gmaas did, he would start the game with a goal and break the ankles of everyone watching the World Cup, including you, at the same time in a fraction of a second, even if you are watching from a device. EDIT: Gmaas created your device. EDIT: Gmaas is your device.

117. Gmaas can solve any puzzle instantly except for the 3x3 Rubik's Cube. EDIT: When Gmaas is handed a 3x3 Rubik's Cube, it is always already solved. Why? The power of Gmaas.

118. Gmaas caught a CP 20,000 Mewtwo with a normal Pokeball and no berries blindfolded first try in Pokemon Go.

119. When Gmaas flips coins, they always land tails, except when Gmaas makes bets with Zeus.

120. On Gmaas's math tests, Gmaas always gets $\infty$. EDIT: He gets a 26 on the AMC8 every year. The results never show him because Gmaas is no longer in middle school.

121. Gmaas's favorite number is $\pi$. It is also one of Gmaas's favorite foods. EDIT: Gmaas created $\pi$, as well as most other constants, such as $e$ and $\sqrt{2}$.

122. Gmaas's burps created all gaseous planets.

123. Gmaas beat Luke Robatille in an epic showdown of catnip consumption.

124. Gmaas's wealth is unknown, but it is estimated to be more than Scrooge's. EDIT: It may be more than John D. Rockefeller. EDIT: More accurate estimates predict that Gmaas's wealth is infinite.

125. Gmaas has a summer house on Mars. EDIT: Gmaas has a fall house on Venus. EDIT: Gmaas has a winter house on Jupiter. EDIT: Gmaas has a spring house on Earth. EDIT: Gmaas also experiences a fifth season called wintrautumn. It happens during November and December and is the cross between fall and winter. Everything is dreary, but there is no snow. Gmaas spends wintrautumn in his house on Venus, where he is incinerated daily. He reincarnates every night through the power of Gmaas. EDIT: Gmaas invented a new season on Feb. 31, 2019, called winter-summer. It is spring but transcends spring.

126. The Earth and all known planets are Gmaas's hairballs.

127. Gmaas attended Harvard, Yale, Stanford, MIT, UC Berkeley, Princeton, Columbia, and Caltech at the same time using a time-turner.

128. Gmaas attended Hogwarts and was a perfect. EDIT: Gmaas is headmaster. EDIT: Hogwarts was supposed to be called Hogmaas.

129. Mrs. Norris is Gmaas's archenemy. Gmaas yawned, and Ms. Norris was petrified. EDIT: Gmaas is the basilisk, and he let Harry Potter pet him. Harry did not kill the basilisk, he only gave Gmaas a haircut. EDIT: Gmaas hates having a haircuts, so he reincarnated Voldemort to punish Harry.Edit: Not true at all.

130. Gmaas is a demigod and attends Camp Half-Blood over summer. Gmaas is the counselor for the Apollo cabin because cats can be demigod counselors too. EDIT: Apollo was one of the many reincarnations of Gmaas.

131. Gmaas has completed over 2,000 quests and is very popular throughout Camp Half-Blood. Gmaas has also been to Camp Jupiter. EDIT: Gmaas is Camp Jupiter, he is also Camp Half-Blood. EDIT: How? EDIT: The power of Gmaas.

132. Percy Jackson was only able to complete his quests because Gmaas helped him. EDIT: Gmaas is Percy Jackson. You are Gmaas, but Gmaas is not you.

133. Gmaas painted the Mona Lisa, The Last Supper, and A Starry Night. EDIT: Gmaas knows that their real names are Gmaasa Lisa, The Last Domestic Meal, and Far-away Light.

134. Gmaas knows that the Blood Moon is just the red dot. He has not caught it yet. EDIT: He caught it during the super blue blood moon.

135. Gmaas attended all the Ivy Leagues.

136. I am Gmaas. EDIT: No you are not. You only have part of Gmaas inside of you. EDIT EDIT: I am also Gmaas. EDIT EDIT EDIT: But it is I who am Gmaas. EDIT EDIT EDIT EDIT: Gmaas is in us all. EDIT EDIT EDIT EDIT EDIT: Gmaas is all of us yet none of us. EDIT EDIT EDIT EDIT EDIT EDIT: Gmaas is a cat. EDIT EDIT EDIT EDIT EDIT EDIT EDIT: He is only in a cat form right now. He can be whatever he wants to be. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: This chain of edits has superseded another chain of edits on this page for the record of the longest chain of edits on AoPS Wiki. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: But the edits in this chain tend to be shorter than most. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: You're right. But we sure do love strings of edits, don't we? EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Gmaas makes the edits. He is in your head spinning edits in your brain. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Gmaas is made to be edited. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Gmaas pays people to edit. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Some people edit even without Gmaas's payment. They do it because they are believers. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Gmaas is Gmaas. You are not Gmaas. Gmaas does not pay people to edit; we do it because we want to.

137. In 2018 Gmaas once challenged Magnus Carlsen to a chess game. Gmaas won every round. EDIT: Gmaas ate the chess pieces afterward. They tasted funny.

All hail Gmaas!

P.S. Gmaas won USAMO 5 times in a row. That's because he invented USAMO. When Gmaas took it nobody knew about it and he was the only one who took it. So of course, he won. How dare you spell Gmaas with a lowercase G? He is the supreme and awesome ruler of the universe!

P.P.S. Gmaas invented the word Yeet.

P.P.P.S. Gmass is the first person who played the game YECK (Moth poth 2019 reference)

P.P.P.P.S. Gmaas is a non-human who can snap anyone out of existence without anything.

P.P.P.P.P.S. Gmaas was the first person who did math.

P.P.P.P.P.P.S. Gmaas invented proof by dictatorship.

P.P.P.P.P.P.P.S Gmaas invented proof by procrastination.

P.P.P.P.P.P.P.P.S Gmaas invented proofs!

P.P.P.P.P.P.P.P.P.S Gmaas invented everything.

P.P.P.P.P.P.P.P.P.P.S Gmaas did not invent roblox. Gmaas invented Minecraft.

## Notice(Gmass's page)

Most of the starting facts are false. Plz don't post these false facts, 1 of the reasons 5space locked it(along with blanking)


text:
Some Vim Movement Tips

Within-line character-based movement: h and l move you left and right one character, respectively. f[c] or F[c] will take you forward to the next or back to the previous, respectively, occurrence of character “[c] on the current line (e.g.,fpwill jump you forward to the next occurrence of "p" on the line, whileFp will jump you back to the previous occurrence of “p” on the line). t[c] or T[c] will take you forward to just before the next or back to just after the previous, respectively, occurrence of character “[c] on the current line.

Vim Regular Expression Special Characters: To Escape or Not To Escape

Vim’s regular expression dialect is distinct from many of the other more popular ones out there today (and actually predates them). One of the dialect differences that always leaves me fumbling has to do with which special characters need to be escaped. Vim does have a special “very magic” mode (that is activated by “\v” in the regular expression) that makes thing very clean and simple in this regard: only letters, numbers and underscores are treated as literals without escaping.


text:
# Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also explained in the video) from a video in YouTube, from the official channel matsciencechannel, An Introduction to Analytic Number Theory Lecture 07 Equidistribution by Ram Murty to ask myself

Question. Is the sequence $\{p^{1/p}\}$, as $p$ varies over all primes, equidistributed in $\mathbb{R}/\mathbb{Z}$?

I need to check if $$\frac{1}{\pi(N)}\sum_{p\leq N}e^{2\pi i m p^{1/p}}$$ tends to zero as $N\to\infty$, $\forall m\neq 0$, where we are denoting the prime-counting function by $\pi(x)$ and $p$ the sequence of prime numbers in increasing order.

I take the definition following the teacher in this video of the more high quality, to do the calculation $$L(s,\psi_m)=\prod_{p}\left(1-\frac{p^{\frac{2\pi i m}{p}}}{p^s}\right)^{-1}=\zeta(s-\frac{2\pi i m}{p}).$$

Now I believe that it is required to say that one has a pole at $s=1+\frac{2\pi i m}{p}$ (notice that I believe that I have a sequence of poles, Murty's example only depends on $m$), and say that I can use the theorem to show that our sum $\rightarrow 0$ as $N\to\infty$, thus our sequence is equidistributed in the (compact) group $\mathbb{R}/\mathbb{Z}$.

Can you explain if it is feasible such calculations or do it? I don't understand well if it is possible to apply the theorem for my example, if you can state in details how one can use the theorem, then it is the best, and this example will be here as a reference. Thanks in advance.

• If you can see the video seach the channel mathsciencechannel, and after the label videos after few pages you find it. – user243301 Jul 19 '16 at 18:48
• Are you interested in the fractional part of $p^{1/p}$? This is not equidistributed, the fractional parts approach $0$. – André Nicolas Jul 19 '16 at 18:52
• For all $m \neq 0$ the limit is simply $e^{2 \pi i m} \neq 0$, so the answer is no. This can be easily proved using the fact that $p^{1/p} \to 1$. – Crostul Jul 19 '16 at 18:52
• No I only was interested to do an exercise and encourage to me to study more mathematics. Thanks, then has meaning the question? Thanks @AndréNicolas – user243301 Jul 19 '16 at 18:53
• Then you are welcome to add a detailed answer, because I try do this as first example. I don't understand well the theory, Then you can add an answer to say that isn't equidistributed. Very thanks much @Crostul – user243301 Jul 19 '16 at 18:55

Using the result of basic calculus in this link, we can say $$\lim_{n \to + \infty} \frac{1}{n} \sum_{k=1}^n e^{2 \pi i m p_k^{1/p_k}}= \lim_{p \to +\infty} e^{2 \pi i m p^{1/p}}$$ , where $p_i$ denotes the $i$-th prime number. Now, it is well known that $$\lim_{n \to +\infty}n^{1/n}=1$$ so that $$\lim_{p \to +\infty} e^{2 \pi i m p^{1/p}}=e^{2 \pi i m \cdot 1}=e^{2 \pi i m} = e^{\mbox{something}} \neq 0$$
As $n$ goes to infinity, $n^{1/n}$ approaches $1$ from above. In particular the fractional part of $n^{1/n}$ approaches $0$, so the $p^{1/p}$ are not equidistributed modulo $1$


text:
NEET Physics Mechanical Properties of Fluids Questions Solved

Two capillary tubes of same diameter are put vertically one each in two liquids whose relative densities are 0.8 and 0.6 and surface tensions are 60 and 50 dyne/cm respectively Ratio of heights of liquids in the two tubes $\frac{{\mathrm{h}}_{1}}{{\mathrm{h}}_{2}}$ is

(a) $\frac{10}{9}$                     (b) $\frac{3}{10}$

(c) $\frac{10}{3}$                     (d) $\frac{9}{10}$

Explanation is a part of a Paid Course. To view Explanation Please buy the course.

Difficulty Level:


text:
# Using MATLAB Mobile to Save Money

This weekend my MATLAB Mobile app more than paid for itself--which is not hard, since it's free! Over the weekend I went shopping for a new car. When it was time to discuss financing, I was presented with a range of rates and promotions. In order to figure out what each combination of rates, loan amount, and length of the loan would cost me, I wrote a simple MATLAB function to do the calculations. Because I have the Financial Toolbox, I took advantage of the payper function, which does all the exponential interest math for me. If you don't have access to that toolbox, it's some basic linear algebra to replicate that calculation.

function howmuch(rate, yrs, amt)
monthly_payment = payper((rate/100)/12,yrs*12,amt)
total_amount = round(monthly_payment*(yrs*12))
interest = total_amount - amt

With this function on my desktop MATLAB's path and the Connector running, I was able to go to the dealership confident that I could make sound financial decisions. For example, one option I had was a 2.99% APR loan, or a 3.99% loan with $1000 cash back. Using my howmuch function, I was able to see that while the second loan has more interest, it was overall cheaper ($15,466 vs \$16,168). An interesting thing I could have also calculated was how long the loan would have to last for the first one to be cheaper.

Because I knew that I was going to use this function from MATLAB Mobile, I coded it differently than I normally do. For example, the function does not have semicolons after each command so the values are displayed in the command window. I also used long, descriptive names for those variables since they will be printed too. Normally I would use return values, but that means extra tapping to assign outputs from the mobile command line. For the same reason, I used a function instead of a script, so I would not have to send multiple commands to change the parameter values.

|


text:
# How is called a semigroup…

Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?

-
Please give more motiviation and background for the question. – Martin Brandenburg Nov 17 '11 at 16:12
@Martin: I now return to my old article in 1996 with K.Iordjev (you can see it in my homepage). I used such semigroups in it and now would know whether they are some own name. – Boris Novikov Nov 17 '11 at 18:31

Update. It is proved by Ellis in http://www.ams.org/journals/tran/2001-353-04/S0002-9947-00-02704-5/S0002-9947-00-02704-5.pdf that an infinite semigroup T has finite geometric type if and ony if $\beta T\setminus T$ is a right ideal where $\beta T$ is the Stone-Czech compactification of T.


text:
Volume 301 - 35th International Cosmic Ray Conference (ICRC2017) - Session Neutrino. NU-astrophysical neutrinos
All-flavor Neutrino Point-like Source Search with the ANTARES Neutrino Telescope
G. Illuminati,* J. Barrios on behalf of the ANTARES Collaboration
*corresponding author
Full text: pdf
Pre-published on: August 16, 2017
Published on: August 03, 2018
Abstract
A search for cosmic neutrino sources using the data collected with the ANTARES detector between early 2007 and the end of 2015 is presented. For the first time, all neutrino interactions are considered in a search for point-like sources, instead of only muon neutrino charged current interactions. This is achieved by using a novel reconstruction algorithm for shower-like events in addition to the standard muon track reconstruction. In order to find the clusters of neutrinos from point-like sources over the randomly distributed atmospheric neutrino events, a maximum likelihood ratio approach is followed. The search for astrophysical neutrino sources is performed with four strategies. In the full-sky search, the whole visible sky of ANTARES is scanned to find the most significant cluster of events. In the second approach, the directions of a pre-defined list of known sources which are potential neutrino emitters are investigated. The third search is similar to the full-sky search but restricted to a region around the Galactic Center (GC). Finally, the fourth approach tests the location of Sagittarius A* as an extended source by assuming a Gaussian emission profile of various widths. No significant evidence of cosmic neutrino sources is found. The most significant cluster in the full-sky search is located at $(\alpha,\delta) = (343.8^{\circ}, 23.5^{\circ})$ with a post-trial significance of $1.9\sigma$. Upper limits on the total neutrino flux from the investigated astrophysical candidates are set between $0.60 \times 10^{-8}$ and $2.1 \times 10^{-8}$ GeV cm$^{2}$ s$^{-1}$. These searches provide the most sensitive limits for a large fraction of the Southern Sky, especially at neutrino energies below 100 TeV.
DOI: https://doi.org/10.22323/1.301.0986
How to cite

Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.

Open Access


text:
@inproceedings{Mulligan:2014:LRE:2628136.2628143,
author = {Mulligan, Dominic P. and Owens, Scott and Gray, Kathryn E. and Ridge, Tom and Sewell, Peter},
title = {Lem: Reusable Engineering of Real-world Semantics},
booktitle = {\textbf{ICFP 2014}: Proceedings of the 19th ACM SIGPLAN International Conference on Functional Programming},
conf = {ICFP 2014},
year = {2014},
month = sep,
isbn = {978-1-4503-2873-9},
location = {Gothenburg, Sweden},
pages = {175--188},
numpages = {14},
url = {http://doi.acm.org/10.1145/2628136.2628143},
pdf = {http://www.cl.cam.ac.uk/~pes20/lem/built-doc/lem-icfp-2014.pdf},
doi = {10.1145/2628136.2628143},
acmid = {2628143},
publisher = {ACM},
address = {New York, NY, USA},
optkeywords = {lem, proof assistants, real-world semantics, specification languages},
abstract = {
Recent years have seen remarkable successes in rigorous engineering: using mathematically rigorous semantic models (not just idealised calculi) of real-world processors, programming languages, protocols, and security mechanisms, for testing, proof, analysis, and design. Building these models is challenging, requiring experimentation, dialogue with vendors or standards bodies, and validation; their scale adds engineering issues akin to those of programming to the task of writing clear and usable mathematics. But language and tool support for specification is lacking. Proof assistants can be used but bring their own difficulties, and a model produced in one, perhaps requiring many person-years effort and maintained over an extended period, cannot be used by those familiar with another.

We introduce Lem, a language for engineering reusable large-scale semantic models. The Lem design takes inspiration both from functional programming languages and from proof assistants, and Lem definitions are translatable into OCaml for testing, Coq, HOL4, and Isabelle/HOL for proof, and LaTeX and HTML for presentation. This requires a delicate balance of expressiveness, careful library design, and implementation of transformations - akin to compilation, but subject to the constraint of producing usable and human-readable code for each target. Lem's effectiveness is demonstrated by its use in practice.
}
}

@inproceedings{ITP-sub,
author = {Anthony C. J. Fox},
title = {Improved Tool Support for Machine-Code Decompilation in {HOL4}},
abstract = {The HOL4 interactive theorem prover provides a sound logical environment for reasoning about machine-code programs.  The rigour of HOL's LCF-style kernel naturally guarantees very high levels of assurance, but it does present challenges when it comes implementing efficient proof tools. This paper presents improvements that have been made to our methodology for soundly \emph{decompiling} machine-code programs to functions expressed in HOL logic. These advancements have been facilitated by the development of a domain specific language, called L3, for the specification of Instruction Set Architectures (ISAs). As a result of these improvements, decompilation is faster (on average by one to two orders of magnitude), the instruction set specifications are easier to write, and the proof tools are easier to maintain.
},
booktitle = {{\textbf{ITP 2015}}:Interactive Theorem Proving - 6th International Conference, Nanjing, China},
conf = {ITP 2015},
pages = {187--202},
year = {2015},
month = aug,
optcrossref = {DBLP:conf/itp/2015},
url = {http://dx.doi.org/10.1007/978-3-319-22102-1_12},
pdf = {http://www.cl.cam.ac.uk/~acjf3/papers/itp15.pdf},
doi = {10.1007/978-3-319-22102-1_12},
timestamp = {Mon, 24 Aug 2015 16:29:06 +0200},
biburl = {http://dblp.uni-trier.de/rec/bib/conf/itp/Fox15},
bibsource = {dblp computer science bibliography, http://dblp.org}
}

@misc{BasileMPRI,
optkey = {},
author = {Basile Clement},
title = {Running programming language specifications},
opthowpublished = {},
month = aug,
year = 2014,
note = {MPRI report},
conf = {MPRI report},
optannote = {}
}

@inproceedings{PittsAM:deptta,
author = {A. M. Pitts and J. Matthiesen and J. Derikx},
title = {A Dependent Type Theory with Abstractable Names},
booktitle = {\textbf{LSFA 2014}: Proceedings of the 9th Workshop on Logical and
Semantic Frameworks, with Applications},
conf = {LSFA 2014},
year = 2015,
month = apr,
editor = {I. Mackie and M. Ayala-Rincon},
volume = 312,
series = {Electronic Notes in Theoretical Computer Science},
pages = {19--50},
doi = {http://dx.doi.org/10.1016/j.entcs.2015.04.003},
pdf = {https://www.repository.cam.ac.uk/bitstream/handle/1810/247863/Pitts_et_al-2015-Electronic_Notes_in_Theoretical_Computer_Science.pdf?sequence=1},
abstract = {This paper describes a version of Martin-Löf's dependent type theory extended with names and constructs for freshness and name-abstraction derived from the theory of nominal sets. We aim for a type theory for computing and proving (via a Curry-Howard correspondence) with syntactic structures which captures familiar, but informal, "nameful" practices when dealing with binders.},
publisher = {Elsevier}
}

This file was generated by bibtex2html 1.96.


text:
# Nested models

7 posts / 0 new
Offline
Joined: 10/05/2015 - 18:51
Nested models

A fixed effect model is a more restricted version of a random effects model, when meta-analyzing correlations using tssem1(), correct? Is there a way to do a nested model test? I know the degrees of freedom for the difference test would be the number of random effects estimated. I'm tripped up because the fixed effect model gives many fit stats, whereas the random effects model only gives a -2 log likelihood.

Thank you!

Offline
Joined: 10/08/2009 - 22:37
Hi Katie, The answer is both

Hi Katie,

The answer is both yes and no. There are two versions of fixed-effects models implemented in tssem1(). When you specify the argument method="REM", it uses the multiple-group SEM approach (Cheung & Chan, 2005). This fixed-effects model is not nested within the random-effects model.

Another fixed-effects model is to fix the variance component of the random effects at zero by specifying both method="REM" and RE.type="Zero". This fixed-effects model is nested within the random-effects model (see the following R code). Please note that this is tested on the boundary of the parameter space. The likelihood ratio test is likely wrong without the adjustment.

Best,
Mike

## Fixed-effects model by using a multiple-group SEM

fixed_no <- tssem1(Digman97$data, Digman97$n, method="FEM")

## Fixed-effects model by fixing the random-effects heterogeneity covariance matrix at zero

fixed_yes <- tssem1(Digman97$data, Digman97$n, method="REM", RE.type="Zero")

## Random-effects model

random <- tssem1(Digman97$data, Digman97$n, method="REM", RE.type="Diag")

## Nested models: testing on the boundary

anova(random, fixed_yes)

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Joined: 10/05/2015 - 18:51
Fit stats

Thanks! I have done what you suggest, and I see in the anova() command that I get negative AIC values. I attach my code (code will also pull down data). In another dataset, in the -2LL column, the model with random effects specified has a negative -2LL and the fixed effects model, specified with method="REM" & RE.type="Zero" has a positive -2LL, which seems strange. Is there an error in my code?

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Joined: 10/08/2009 - 22:37

Different SEM packages produce different AIC (http://openmx.psyc.virginia.edu/thread/89). AIC is usually used for model comparisons. You may choose the model with the smallest AIC as long as you are using the same package.

There is nothing wrong with positive -2LL. For example, http://openmx.psyc.virginia.edu/thread/1831 and http://blog.stata.com/2011/02/16/positive-log-likelihood-values-happen/

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Joined: 10/05/2015 - 18:51
Nested models - clusters

OK - just to confirm - a negative AIC value (not when comparing models, just each model on its own) is OK?

Also - you might have missed my question above (threading is a bit confusing here):

Is it also the case that when you add a cluster variable to tssem1(), and then you sum the chi-square associated with each group, that the total chi-square can be compared to a tssem1() with no cluster as a nested model? The df of the difference here would be the number of parameters estimated in each group.

Is there a way to fix or free individual correlations in tssem1() across subgroups, or must the model be specified in tssem2()?

Thanks again! It's really amazing to be able to get your feedback.
Katie

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Joined: 10/08/2009 - 22:37
Yes, the (negative) AIC is

Yes, the (negative) AIC is not very useful on its own.

Sorry, I missed that part. Yes, The chi-square and its dfs are based on the sum of the individual chi-squares and dfs. You may compare them against the model without the cluster variable.

I haven't implemented this in metaSEM though it can be done in OpenMx. Suzanne Jak (S.Jak@uu.nl) and I have written a paper on this topic. She is the corresponding author. If you write to her, I am sure that she is willing to share it.

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Joined: 10/05/2015 - 18:51
Nested models - clusters

Hi again!
Is it also the case that when you add a cluster variable to tssem1(), and then you sum the chi-square associated with each group, that the total chi-square can be compared to a tssem1() with no cluster as a nested model? The df of the difference here would be the number of parameters estimated in each group.

Is there a way to fix or free individual correlations in tssem1() across subgroups, or must the model be specified in tssem2()?

Thanks!
Katie


text:
# XSD Simple Elements

We can use the XML Schemas to define the elements of the XML files. An XML element that contains only text is called a simple element. No other elements or attributes can be included in it.

## What is a Simple Element?

An XML element that contains only text is called a simple element and no other elements or attributes can be included in it. The “only text” restriction of a simple element is misleading though. There are many different types of texts, such as the types included in the XML Schema definition (boolean, string, date, etc.), or a custom type. Also, to limit the contents of a data type, we can add restrictions (facets), or the data can be required to match a specific pattern.

## Defining a Simple Element:

Syntax: Defining a simple element:

<xs:element name="xxx" type="yyy"></xs:element>

Here, the name of the element is represented by xxx and the data type of the element is represented by yyy.

## XML Schema built-in data types:

The most common built-in data types of an XML Schema are:

• xs:string
• xs:decimal
• xs:integer
• xs:boolean
• xs:date
• xs:time

### Example:

XML elements:

<firstname>Tom</firstname> <age>25</age> <birthdate>2000-09-21</birthdate>

Corresponding simple element definitions:

<xs:element name="firstname" type="xs:string"></xs:element> <xs:element name="age" type="xs:integer"></xs:element> <xs:element name="birthdate" type="xs:date"></xs:element>

## Default and Fixed Values for Simple Elements:

We can either specify a default value or a fixed value for a simple element. When no other value is specified, a default value is automatically assigned to the element.

Example:

<xs:element name="flower" type="xs:string" default="rose"></xs:element>

Explanation:
Here, the default value is “rose”.
When a fixed value is assigned, we cannot specify another value. It is also automatically assigned to the element.

Example:

<xs:element name="flower" type="xs:string" fixed="rose"></xs:element>

Explanation:
Here, the fixed value is “rose”.


text:
Perverting a contour integral

1. Mar 5, 2009

ice109

so suppose i wanted to calculate the antiderivatives of $e^x\sin{x}$ and just for the hell of it also $e^x\cos{x}$. well i could perform integration by parts twice recognize that the original integral when it reappears, subtract from one side to the other blah blah blah.

or i could pervert a contour integral: i could integrate $e^z$ along the line in the $(1,i)$ direction:

$$e^{(1+i)x}=e^{x+ix}=e^x\cos{x}+ie^x\sin{x}$$

and leave off the limits

$$\int e^{(1+i)x}dx=\frac{1}{1+i}e^{(1+i)x}=\frac{1}{2}\left(e^x(\cos{x}+\sin{x})+ie^x(\sin{x}-\cos{x})\right)$$

and compare the real and imaginary parts of the integrand and the "antiderivative" and conclude:

$$\int e^x \cos{x} = \frac{1}{2}e^x(\cos{x}+\sin{x})$$
$$\int e^x \sin{x} = \frac{1}{2}e^x(\sin{x}-\cos{x})$$

which are of course the correct answers.

what i don't know is what an indefinite integral in the complex plane even is. i've only so far learned that i can use this "trick" to perform contour integrals <=> with end points, and then make conclusions about definite integrals of the real and imaginary parts of the integrand.

so how legit is this?

2. Mar 5, 2009

lurflurf

That is convoluted.
let s be the antiderivative opperator
s*exp(x)cos(x)=Re[s*exp(x(1+i))]
s*exp(x)cos(x)=Im[s*exp(x(1+i))]
s*exp(x(1+i))=exp(x(1+i))/(1+i)

3. Mar 5, 2009

ice109

and how do i know s commutes with Re and Im?

4. Mar 5, 2009

HallsofIvy

Staff Emeritus
Surly "perverted" is not the right word here!

5. Mar 5, 2009

ice109

can one of you more knowledgeable people just tell me if this a legitimate technique?

6. Mar 5, 2009

Ben Niehoff

It is legitimate for entire functions. There may be subtleties if a function has poles.

7. Mar 5, 2009

dvs

A function f:R->C can be decomposed into its real and imaginary parts, say f=u+iv, where u=Re(f) and v=Im(f). The definition of the Riemann/Lebesgue integral of such a function is

$$\int f = \int u + i \int v.$$

The integrals $\int u$ and $\int v$ are just the usual real integrals.

Does this clear anything up for you?


text:
## Friday, 11 August 2017

### Counting Sort 1 Hacker Rank Problem Solution.

Problem:
Comparison Sorting
Quicksort usually has a running time of , but is there an algorithm that can sort even faster? In general, this is not possible. Most sorting algorithms are comparison sorts, i.e. they sort a list just by comparing the elements to one another. A comparison sort algorithm cannot beat  (worst-case) running time, since  represents the minimum number of comparisons needed to know where to place each element. For more details, you can see these notes (PDF).
Alternative Sorting
However, for certain types of input, it is more efficient to use a non-comparison sorting algorithm. This will make it possible to sort lists even in linear time. These challenges will cover Counting Sort, a fast way to sort lists where the elements have a small number of possible values, such as integers within a certain range. We will start with an easy task - counting.

Challenge
Given a list of integers, can you count and output the number of times each value appears?
Hint: There is no need to sort the data, you just need to count it.
Input Format
There will be two lines of input:
•  - the size of the list
•  -  space-separated numbers that make up the list
Output Format
Output the number of times every number from  to  (inclusive) appears on the list.
Constraints

Sample Input
100
63 25 73 1 98 73 56 84 86 57 16 83 8 25 81 56 9 53 98 67 99 12 83 89 80 91 39 86 76 85 74 39 25 90 59 10 94 32 44 3 89 30 27 79 46 96 27 32 18 21 92 69 81 40 40 34 68 78 24 87 42 69 23 41 78 22 6 90 99 89 50 30 20 1 43 3 70 95 33 46 44 9 69 48 33 60 65 16 82 67 61 32 21 79 75 75 13 87 70 33

Sample Output
0 2 0 2 0 0 1 0 1 2 1 0 1 1 0 0 2 0 1 0 1 2 1 1 1 3 0 2 0 0 2 0 3 3 1 0 0 0 0 2 2 1 1 1 2 0 2 0 1 0 1 0 0 1 0 0 2 1 0 1 1 1 0 1 0 1 0 2 1 3 2 0 0 2 1 2 1 0 2 2 1 2 1 2 1 1 2 2 0 3 2 1 1 0 1 1 1 0 2 2

Explanation
The output states that 0 appears 0 times, 1 appears 2 times, 2 appears 0 times, and so on in the given input array.

Solution:

#include <bits/stdc++.h>
using namespace std;
/*
* Prosen Ghosh
* American International University - Bangladesh (AIUB)
*
*/
int main() {
int n,a,max = -INT_MAX;
vector<int> cnt(1000000,0);
cin >> n;
for(int i = 0; i < n; i++){
cin >> a;
cnt[a]++;
if(a > max)max = a;
}
for(int i = 0; i <= max; i++)cout << cnt[i] << " ";
return 0;
}