7
votes
0answers
99 views
+50

When might some a variable leave the basis?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
9
votes
3answers
199 views
+50

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

Find all integral solutions for the Diophantine Equations $$x^4 - x^2y^2 + y^4 = z^2$$ and $$x^4 + x^2y^2 + y^4 = z^2$$ I basically think that to solve these equations we need to use the fact ...
5
votes
1answer
78 views
+50

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
8
votes
1answer
128 views
+150

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
6
votes
0answers
129 views
+150

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
11
votes
1answer
111 views
+150

Surjective exponentials for algebraically closed fields

The existence of the exponential on $\mathbb{C}$ has a very basic, yet very strong consequence : $(\mathbb{C}^*,\cdot)$ is a quotient of $(\mathbb{C},+)$. This question is concerned with fields $K$ ...
13
votes
1answer
153 views
+50

if $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$, show that $(\frac{x_{n+1}}{x_{n}}) $ convergent

Let a real positive number sequence $(x_{n})$ such that $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$ and $\sqrt{x_{1}}\geq \sqrt{x_0+1}$. Show that $(\frac{x_{n+1}}{x_{n}}) $ convergent. ...
5
votes
1answer
275 views
+100

Help with proving a statement based on riemann sums?

Suppose we have a reimmen sum with no removed partitions which I call the "total sum". $$\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\left(\frac{b-a}{n}\right)i\right)\left(\frac{b-a}{n}\right)$$ And ...
3
votes
0answers
39 views
+50

Davenport's Q-method (Finding an orientation matching a set of point samples)

I have an initial set of 3D positions that form a shape. After letting them move independently, my goal is to find the best rotation of the original configuration to try to match the current state. ...
5
votes
2answers
488 views
+50

Analytic continuation of a real function

I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique. Now i am wondering if this is ...
13
votes
0answers
177 views
+50

Find a closed form formula for $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes.

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
5
votes
0answers
57 views
+100

Optimization of approximate functions using varying objective function

Let $g(\theta;x)$ and $f(\theta;x)$ be two convex functions such that $g$ asymptotically approximates $f$: $g(\theta;x)\approx f(\theta;x)$, specifically: $$ |g(\theta;x)-f(\theta;x)| \leq ...
2
votes
0answers
68 views
+300

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there some some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, ...
3
votes
0answers
23 views
+50

Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in ...
4
votes
0answers
123 views
+50

Find example of bch code

I want to show that the dual of a BCH code is not necessarily a BCH code, by considering for example a binary BCH code of length $9$. The parity matrix of a binary BCH code is of the following form: ...
5
votes
0answers
143 views
+50

Spliting subspaces and finite fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
4
votes
0answers
59 views
+100

Why is the Barycenter operation in Hadamard spaces Lipschitz continuous?

I am looking into exercise 9.2.22 of "A course in metric geometry" by Burago-Burago-Ivanov. For a Hadamard space $H$ (a complete simply connected metric space of nonpositive curvature in the ...
5
votes
0answers
74 views
+50

Number of $\mathbb{F}_q$-rational points on a smooth variety

From the proof of Weil's conjectures it follows that $|q^k - \# X(\mathbb{F}_{q^k})| = O(q^{k(n - \frac{1}{2})})$, where $X$ is a smooth variety over $\mathbb{F}_q$ and $n = \dim X$ (see for example ...
4
votes
0answers
61 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
3
votes
1answer
91 views
+50

Let $f:[-1,1]\to \mathbb{R}$ diferentiable, with $f'$ integrable, such that $\frac{\int_{-1}^{1}e^xf(x)dx}{f(1)-f(-1)}=2(e+e^{-1})$.

Let $f:[-1,1]\to \mathbb{R}$ diferentiable, with $f'$ integrable, such that $$\frac{\int_{-1}^{1}e^xf(x)dx}{f(1)-f(-1)}=2(e+e^{-1})$$ Prove that there exists $c\in (-1,1)$ such that ...
2
votes
0answers
55 views
+100

Regularity, Dirichlet form

I have a question about Dirichlet form. Let $\Omega$ be an Euclidean domain of $\mathbb{R}^{N}$ and $X=\bar{\Omega}$. The measure $m$ on the Borel $\sigma$ algebra $\mathcal{B}(X)$ is given by ...
1
vote
0answers
59 views
+100

Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb ...
3
votes
0answers
41 views
+50

Matrix representations of the generators of the full octahedral group

I want to find matrix representations of the generators of the full octahedral group which has the presentation $\{a,b,c|a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\} $ where a,b and c are the generators of the ...
9
votes
0answers
115 views
+50

Proof of an inequality in $\mathbb{C}$

Let $z\in \mathbb{C}, n \geq 2$. Show this complex inequality $$|z^n-1|^2\le |z-1|^2\left(1+|z|^2+\dfrac{2}{n-1}\Re{(z)}\right)^{n-1}$$ For $n=2$ the inequality is easy to prove: $$|z^2-1|^2\le ...
3
votes
0answers
23 views
+100

Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $S_n$ denote the discrete square $S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and ...
9
votes
3answers
1k views
+100

Splitting equilateral triangle into 5 equal parts

Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable from each other by a rigid motion) parts?
5
votes
0answers
123 views
+50

Which properties characterize $\sin, \cos$?

I know a few properties of $\sin$ and $\cos$, for example: $\sin^2+\cos^2=1$ $\sin (a+b) = \sin a\cos b+\cos a\sin b$. $\cos (a+b) = \cos a\cos b-\sin a\sin b$. $\sin (x+\delta) = \sin x$ for some ...
7
votes
3answers
152 views
+100

(Elegant) proof of an inequality: $h(x) \geq 1- (1-\frac{x}{1-x})^2$, where $h$ is the binary entropy function

I am looking for the most concise and elegant proof of the following inequality: $$ h(x) \geq 1- \left(1-\frac{x}{1-x}\right)^2, \qquad \forall x\in(0,1) $$ where $h(x) = x \log_2\frac{1}{x}+(1-x) ...
5
votes
1answer
71 views
+50

Why this set is dense in $C_0(\mathbb{R})$

Let $C_0=\{f~|~ f:\mathbb{R}\to\mathbb{R},f~is~continous,\lim\limits_{\vert x\vert \to\infty}f(x)=0\}$ $A=\{f~|~f(x)=p(x)e^{-x^2},p(x)~is~polynomials\}$ Why $A$ is dense in $C_0$. The topology ...
3
votes
0answers
81 views
+50

Is this property of continuous functions equivalent to anything familiar? If not, does it at least have a name?

If I understand correctly, every morphism of topological spaces $f : Y \leftarrow X$ factorizes uniquely into a composite of three morphisms $$f = c \circ b \circ a$$ such that $c : Y \leftarrow ...
6
votes
1answer
90 views
+50

$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
3
votes
1answer
53 views
+50

Integrating a probability density function that only depends on the norm

I have a probability density function $f$ on $\mathbb{R}^3$ which only depends on the norm of a vector (that is, it takes the same value for $x,y$ if their length is equal). Let me call a region of ...
5
votes
0answers
62 views
+50

Abstraction and Genaralization

This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me. Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote: from the items $p_{i}$, find a common ...
3
votes
2answers
47 views
+50

A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B|

I know that B would look something like this: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ -a_{12}+c & c &...&&a_{2n}+c \\ . \\ . \\ . \\ -a_{1n}+c & ...
1
vote
0answers
29 views
+100

Sufficient sample size for 1 out of X sets

I want to create a "game" that aims at supporting the decision making. The user has to consecutively select one out of two random items. Each item belongs to one of X sets. My question is how many ...
6
votes
1answer
38 views
+50

On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise. By ideals we will mean to include $\{0\}$ and $R$ also. Let us call an integer $n>1$ a "principal number" if any ring ...
2
votes
0answers
60 views
+50

Combinatorics problem involving n-dimensional space

Consider a set of more than $\frac {2^{n+1}} {n}$ points $(n>2)$, chosen from the $2^n$ points of the $n$-dimensional space which have the coordinates $\{ \pm1, \pm1, ..., \pm1 \}$. Show that ...
0
votes
1answer
36 views
+50

Geometric interpretation of adding dependence on a otherwise constant in a vector field.

So if i have $$ \vec{u} = \Omega r \vec{e_{\theta}}$$ Now if i take the curl $$\omega = 2\Omega\vec{e_z}$$ This is what we expect, we have a "rotating flow". Who's curl would be pointing in the z ...
2
votes
2answers
92 views
+50

Analytic solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations. Motivation: The source of the question is a very convinient method to create random matrices with special properties. ...
8
votes
1answer
157 views
+50

The three unsolved problems of antiquity

In Sidelights on the Cardan-Tartaglia Controversy (Apr., 1938) by Martin A. Nordgaard in the National Mathematics Magazine, Vol. 12, No. 7, pp. 327-364, it is written on the first page The ...
1
vote
0answers
27 views
+50

Number of paths between two points in the first Quadrant.

[Extension of this] We can move in 4-directions and we need to reach $(0,b)$ from $(a,0)$ in exactly $n$ steps keeping in the first quadrant ($x\ge0$ and $y\ge0$) [$a,b\ge0$] Similar to previous ...
4
votes
1answer
70 views
+100

About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
2
votes
1answer
52 views
+50

On the linear combination of $\pm 1$ random variables

Let $X_1,\dots, X_n$ be i.i.d symmetric $\pm 1$ random variables, i.e. $X_j$ takes values in $\{-1,1\}$ with $$\mathbb{P}(X_j=1)=\mathbb{P}(X_j=-1)=\frac{1}{2}.$$ Let $a_1,\dots,a_n\in\mathbb{Z}$, ...
0
votes
0answers
41 views
+50

Testing if a submodule is free

This is hopefully a very simple question. In "Groebner Bases in commutative algebra" by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, ...
3
votes
0answers
172 views
+100

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
1
vote
1answer
69 views
+250

How to solve the following system $\frac{\text{d}x}{\text{d} t} = -Ax + \frac{B}{y} - C$, $ \frac{\text{d}y}{\text{d} t} = -Dx + \frac{E}{y} - F$

Is there a way to analytically solve the following ODE system? $$ \frac{\text{d}x}{\text{d} t} = -Ax + B\left(\frac{1}{y} -1\right) \\ \frac{\text{d}y}{\text{d} t} = -Cx + D\left(\frac{1}{y} ...
0
votes
0answers
13 views
+100

Which probability distribution(s) $f(x)$ allow for a closed form solution to $\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$?

I'm trying to find if there is a specific probability distribution $f\left(x\right)$ (or many) such that the following integral $$\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$$ has a closed form ...
1
vote
1answer
78 views
+50

Do non-second-countable spaces have “small” non-second-countable subspaces?

If $X$ is any space which is not second-countable, can one find a subspace $Y \subseteq X$ with $|Y| \leq \aleph_1$ which is also not second-countable? (Recall that a topological space $X$ is ...
2
votes
0answers
21 views
+50

Application of Jacobi's Theorem in Box Principle

Today I was going through Problem Solving Strategies by Arthur Engel, and found this in the chapter Box Principle Before the question it says it "treats a theorem of Jacobi and its applications" ...
1
vote
1answer
48 views
+50

A better way of showing the set of all $m\times n$ matrix is an $R$ module.

Let $R$ be a ring and $m$ and $n$ be any positive integers. For any $a\in R$ and $A=(a_{ij})\in M_{m\times n}(R)$ define $aA=(aa_{ij})$. Then prove that $M_{m\times n}(R)$ is an $R$-module. Edited: ...

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