0
votes
0answers
284 views

Change of Basis and Coordinate system question

My apologies in advance: I am not very proficient in the field of mathematics. If my question seems discombobulated, it probably is, and I would greatly appreciate your help cleaning it up! Alright, ...
2
votes
5answers
530 views

Prove that if $ab \equiv 1 \pmod{p}$ and $a$ is quadratic residue mod $p$, then so is $b$

Prove that if $ab \equiv 1 \pmod{p}$ and $a$ is quadratic residue mod $p$, then so is $b$ where $p$ is odd prime, and $(a,p) = (b,p) = 1$. Besides $b$ is the inverse of $a$, what else does ...
4
votes
2answers
187 views

Laplace transform of $[f''[x]]^n$

Can anyone help me get this Laplace transform, $$ L[(f''(x))^n] $$ where $f'(0)=0$ and $f''(0)=0$ and $n$ is power of $$f''(x)$$?
0
votes
1answer
114 views

Further thoughts on the energy estimate

I put my question about the energy estimate two days ago. And finally I can get $$\frac{d}{dt}\|x\|^2=2\|x\|\frac{d}{dt}\|x\|=\frac{d}{dt}\langle x,x \rangle=2Re\langle x, Ax\rangle$$ If I have the ...
9
votes
3answers
353 views

Is there experimental evidence that people ever play mixed Nash equilibrium in real games?

Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game?
1
vote
3answers
284 views

references for the spectral theorem

Recently, I am thinking about the question in spectral theory. And I finally found that I need help with the properties of unitary operator. Its a consequence of the spectral theorem for the normal ...
2
votes
1answer
133 views

strong separation of sets

I'm going through some of the practice questions in my textbook, and one has me stumped, on account of seeming almost too straight-forward. We're instructed to show that two sets, $A$ and $B$ of ...
7
votes
1answer
270 views

When can we say that a theorem has been proven?

I'm taking a Data Structures and Algorithms course for a CS program. The introductory material was all mathematics, mostly a series of formulas that we are to remember. I can work through the formulas ...
4
votes
2answers
327 views

Rotate a 2D subspace in 4 or more dimensions

Two non-co-linear vectors define a 2D subspace that passes through the origin. In 3D, you can represent a 2D subspace by its Normal. It's very easy to define the angle between the two planes: $\theta ...
14
votes
4answers
1k views

What are the polar coordinates of the origin?

In polar coordinates, the origin has $r = 0$, but $\theta$ is not unique. what sort of problems does this create, and how can I resolve them? For example, suppose an ant is wandering around a plane. ...
1
vote
2answers
492 views

Modular equations system

I have the following task - I have to find all a for which the following system has a solution: $x \equiv 1\pmod 2$ $x \equiv 2\pmod 3$ $x \equiv a\pmod 5$ I ...
2
votes
2answers
370 views

probability distribution of coverage of a set after `X` independently, randomly selected members of the set

I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
3
votes
1answer
161 views

Relation: pairwise and mutually

Suppose we can define a relation $R$ over the sets $X_1, …, X_k$ for any natural number $k$, note not specified for a particular $k$. I was wondering if there is some definition or conditions ...
5
votes
2answers
517 views

How can I find a basis for the kernel of a homomorphism on a free abelian group?

Let $\phi\colon F\to G$ be a homomorphism of finitely generated abelian groups. If $F$ is free, then $\ker(\phi)$ is also free and thus admits a basis. Question: Is there a general procedure to find ...
4
votes
2answers
271 views

interpretation of independence events

$\{A_i, i \in \mathbb{N} \}$ are defined to be independent, if $P(\cap_{k=1}^{n} A_{i_k}) = \prod_{k=1}^{n} P(A_{i_k}) $ for any finite subset of $\{A_i, i \in \mathbb{N} \}$. We know ...
3
votes
3answers
821 views

Area's of rectangle and circle

If a string with length of 20 cm was to create a rectangle and circle, would area of these objects be the same?
3
votes
2answers
541 views

What non-symmetric matrices satisfy $x^TAx>0,\forall x\neq 0$

Let $A\in R^{n\times n}$ be a matrix. It is positive definite if and only if $A$ is symmetric and $x^TAx>0,\forall x\in R^n$. My question is: if $x^TAx>0,\forall x\in R^n$ but $A$ is not ...
4
votes
2answers
171 views

Character theory questions

I am following the text by Isaacs on character theory and I have a few questions. From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a ...
2
votes
1answer
150 views

Counting number of ways a sports season can be played

Howdy all - I think I have the answer to this question, but I wanted to make sure. Given a sports season of, say 12 teams and 12 games, how would you calculate how many possible ways the season can be ...
0
votes
1answer
211 views

runge kutta 4 question

I have a quick question that I'm having a little trouble with, it seems simple enough but i just need a bit of clarification. If I had a system of ode's and I was to choose the runge kutta 4 method ...
3
votes
3answers
504 views

Finding probability of an unfair coin

An unfair coin is tossed giving heads with probability $p$ and tails with probability $1-p$. How many tosses do we have to perform if we want to find $p$ with a desired accuracy? There is an obvious ...
1
vote
1answer
166 views

Nash equilibrium for 2 players' game

Consider a game with two players P1 and P2. For P1 the set of strategies is $x_1,...,x_m$ and $y_1,...,y_n$ for P2, gains are $f_1(x_i,y_j)$ for P1 and $f_2(x_i,y_j)$ for P2. Define mixed strategies ...
0
votes
2answers
56 views

Equation for simple transform

I have an ordinal list that I am trying to represent mathematically. The list is as follows: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, ...
2
votes
1answer
92 views

All functions with the property $ k \mid f(m+n) \iff k \mid f(m)+f(n)$

Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.
0
votes
4answers
1k views

Capital Pi Simplification

I have the following, which I have arrived at after a series of calculations. $\prod_1^n 6.\bigg\{2 - \frac{5}{6}\Big(n\Big)\bigg\}.$ My problem is my maths is a little rusty...Could someone explain ...
2
votes
1answer
349 views

Number of solutions to a set of homogeneous equations modulo $p^k$

Let p be a prime number and k be a positive integer. How do I determine the number of solutions to a set of equations in variables $0<=x_1,...,x_n<p^k$? All equations are of the form ...
5
votes
0answers
144 views

Invariance of Wall's self-intersection under the regular homotopy

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
2
votes
1answer
543 views

Confused about “Solve $5\cos\theta = 3\cot\theta$”

I recently got this question only half correct: "Solve for values of $\theta$ the equation $5\cos\theta = 3\cot\theta$, in the interval $0 \leq \theta \leq 360$" My solution was: $$5 \cos\theta = 3 ...
2
votes
0answers
64 views

Iterative Compounded Growth Calculation

Hello I would like to develop a quick algorithm for computing compounded change over an arbitrary period $T$. I'll illustrate with an example. Suppose I have $N$ data points as follows: $$(x_0, t_0, ...
0
votes
1answer
329 views

Closedness of the image of the closed unit ball under a linear operator from a reflexive Banach space to an arbitrary Banach space

Let $V$ be a reflexive banach space. If $W$ is a Banach space and if $T$ is in $L(V,W)$, show that $T(B)$ is closed in $W$ where $B$ is closed unit ball in $V$, the problem is in the chapter of weak ...
2
votes
1answer
321 views

Exact sequence of sheaves

Let $X$ be a scheme and let $Y$ be a closed subscheme with ideal sheaf $I$. Let $F$ be a coherent sheaf on $X$. Is the sequence $$ 0 \to I\otimes F \to F \to F \otimes O_Y \to 0 $$exact? This is ...
4
votes
1answer
127 views

$H_p(\mathbb{R}P^3 \times \mathbb{R}P^2)$

I'm working through an example of the Kunneth formula in my book. Without showing any working it states that for $X = \mathbb{R}P^3 \times \mathbb{R}P^2$ $$H_p(X)=\begin{cases} \mathbb{Z} & ...
2
votes
0answers
107 views

Question on the transversality between sections

Let $M^n$ be a differentiable manifold and $\pi\colon E\to M$ is $n$-dimensional vector bundle over $M$. We have a zero section $s\colon M\to E$ of $\pi$. How can I make a section $s'$ which is ...
2
votes
0answers
107 views

Question on the transversality

Let $f\colon N^n\to M^{2n}$ be an immersion. Then, we can extend $f$ to $\bar{f}\colon E(\nu_g)\to M$ of the total space of the normal bundle. Let $s_0\colon N\to E(\nu_g)$ be a zero section and ...
6
votes
5answers
3k views

Prove $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ if $\gcd(a,b) = 1$

I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$ $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ holds. I think the first step should look something like this: $d = \gcd(a+b, a-b) = \gcd(2a, a-b)$ ...
3
votes
2answers
116 views

The equation $F(x) \equiv 0 \pmod m$ has integer solution for x

Let $F(x)=(x^2-17)(x^2-19)(x^2-323)$ and let $m$ be a positive integer. How can one show that the equation $F(x) \equiv 0 \pmod m$ has an integer solution?
6
votes
2answers
167 views

$(7a+1)x^3+(7b+2)y^3+(7c+4)z^3+(7d+1)xyz=0$ does not have integer solutions

Let $a,b,c,d$ be integers. How can I prove that the equation $$(7a+1)x^3+(7b+2)y^3+(7c+4)z^3+(7d+1)xyz=0$$ Does not have an integer solution $(x,y,z)$ such that $\gcd(x,y,z)=1$?
2
votes
1answer
427 views

All subgroups of a finite abelian p-group

Given a finite abelian p-group: $G = \displaystyle\prod_{i=1}^n p^{k_i}\mathbb{Z}_{p^k}$ for some integers $k,k_1,...,k_n$. Regarding elements of G as tuples $(x_1,...,x_n) \in G$, I can get ...
0
votes
1answer
2k views

How do I calculate weighted mean with negative weights?

This might be very trivial for you guys. If : I buy 2 oranges for 5 dollars each then I buy 3 oranges for 6 dollars each and then I sell 3 oranges for 4 dollars each How do I calculate the average ...
1
vote
2answers
432 views

Simple Maths Question - Capital Sigma/Pi

I haven't studied math in a long time and am trying to solve a simple first order non-homogeneous recurrence relation. This approach uses the general formula: $$ f(n) = \prod_{i=a+1}^n b(i) \bigg\{ ...
1
vote
2answers
457 views

Find out which faces of a 3D polygon cause it to be concave

I have a 3D polygon based on a set of faces. Each face lies in a single plane, and I know the normal for each face (as well as the points that create the face). I also know that each normal points ...
3
votes
2answers
236 views

When to write that $\sqrt{x} = \pm y$?

Here is the textbook solution to a simple trigonometry problem I just completed. The Exercise was to rewrite the LHS in terms of $\cot\theta$. ...
1
vote
2answers
367 views

Integral of the fractional part of $\frac1x$ multiplied by $x$ on interval $(a,b), a\ge 0$.

I'm interested in finding the value of the integral of $\left\{\frac{1}{x}\right\}\cdot x$ (the fractional part of $\dfrac{1}{x}$ multiplied by $x$) on the interval $(a,b), a\ge 0$ the integral of ...
2
votes
1answer
298 views

A proof of Sylow theorem

Having proved the Sylow theorem for general linear group over finite field, how to prove it for any finite group?
3
votes
3answers
102 views

Write $\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$ in a simpler form, where $F_n$ is the n-th element of the Fibonacci sequence?

The exercise asks to express the following: $\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$ in a simpler form, not necessarily a closed one. The previous problem in the set was the same, with a different ...
2
votes
3answers
943 views

Prove that with 2 parallel planes, the one in between is given by

I would like to prove that having two planes $$ax+by+cz+d_1 = 0 \quad\text{and}\quad ax+by+cz+d_2 = 0$$ you can automatically have a plane with equal distance from each plane that looks like this: ...
8
votes
1answer
493 views

the localization of a ring is an integral domain iff the annihilators of zero divisors are comaximal ideals

I would appreciate any help with the following problem. let $R$ be a commutative ring (with $1$). I need to show that the following are equivalent i) for every prime ideal $P$, the localization $R_P$ ...
0
votes
1answer
517 views

Singular Value Decomposition (SVD) of a three dimensional Array

The Singular Value Decomposition (SVD) of a matrix is $$A_{m\times n} = U_{m\times m}\Lambda_{m\times n} V_{n\times n}'$$ where $U$ and $V$ are orthogonal matrices and $\Lambda$ has (i, i) entry ...
5
votes
1answer
601 views

spectrum of the “discrete Laplacian operator”

In numerical analysis, the discrete Laplacian operator $\triangle$ on $\ell^2({\bf Z})$ can be written in terms of the shift operator $\triangle=S+S^*-2I$ where $S$ is the right shift operator. ...
6
votes
1answer
117 views

How do I figure out the coproduct on graded algebras?

I have to figure out the duals to a couple of graded algebras. This requires a comultiplication (also called a coproduct in Hatcher). Hatcher's book shows what form the comultiplication must take ...

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