Examples of mathematical writing
... In a discussion with last year’s MSc students, the following strengths and weaknesses were mentioned. As an exercise, read the paper and see if you agree with them: there is a plenty of room for debate, particularly about the perceived weaknesses. Strengths: well motivated, lots of interesting real- ...
... In a discussion with last year’s MSc students, the following strengths and weaknesses were mentioned. As an exercise, read the paper and see if you agree with them: there is a plenty of room for debate, particularly about the perceived weaknesses. Strengths: well motivated, lots of interesting real- ...
Answers to Even-Numbered Homework Problems, Section 6.2 20
... so that {u, ṽ} is an orthonormal set. (Note that u is already a unit vector.) 26. A set of n nonzero orthogonal vectors must be linearly independent by Theorem 4, so if such a sets spans W , it is a basis for W . Since W is therefore an n-dimensional subspace of Rn , it must be equal to Rn itself. ...
... so that {u, ṽ} is an orthonormal set. (Note that u is already a unit vector.) 26. A set of n nonzero orthogonal vectors must be linearly independent by Theorem 4, so if such a sets spans W , it is a basis for W . Since W is therefore an n-dimensional subspace of Rn , it must be equal to Rn itself. ...
7.5 ASA - Van Buren Public Schools
... Included Side The side between two consecutive angles of a triangle. ...
... Included Side The side between two consecutive angles of a triangle. ...
Name: Math 490, Fall 2012: Homework #1 Due
... right triangle whose side lengths are consecutive natural numbers? If so, can you find MORE than one such example? Why or why not? ...
... right triangle whose side lengths are consecutive natural numbers? If so, can you find MORE than one such example? Why or why not? ...
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.