17
votes
5answers
481 views
+100

How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is ...
-1
votes
0answers
43 views
+50

supporting function and halfspace (definition)

we've defined the following: supporting function: Let $P$ be a convex polygon in $E^d$ (euclidean vector space). Then the supporting function is defined as $h_P: S^{d-1} \to \mathbb{R}$ by $h_P(u) :=...
2
votes
0answers
59 views
+50

Optimal convex hull that maximizes # points from set A and minimizes # points from set B

This problem arose in a computer vision hobby project. Say I have two sets of points in three dimensional Cartesian space: A and B. The problem I would like to solve is to find the convex hull V of ...
5
votes
0answers
105 views
+100

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
0
votes
0answers
243 views
+50

seating at two round tables

We have $n$ people sitting in two round tables like the picture. We randomly change the place of the people we decide two of them and change their places. We can only change the place of the ...
-1
votes
0answers
37 views
+50

Boundedness of the Fourier transform of a Battle-Lemarie scaling function

Could anyone please give a short and simple proof of the following proposition: the Fourier transform $\hat\phi$ of Battle-Lemarie scaling function (of arbitrary order) is bounded on $\mathbb{R}$, ...
6
votes
4answers
171 views
+50

How to determine the existence of all subsets of a set?

Given The definition of subset; The axiom of power set: for any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$ we know what a subset is and what a power set ...
4
votes
2answers
110 views
+50

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ||x||\...
17
votes
1answer
639 views
+50

The class of all classes not containing themselves

In ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox ...
3
votes
1answer
86 views
+50

Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
2
votes
0answers
60 views
+50

Two dimensional (discrete) orthogonal polynomials for regression

This question How to work out orthogonal polynomials for regression model and the answer http://math.stackexchange.com/a/354807/51020 explain how to build orthogonal polynomials for regression. ...
2
votes
2answers
63 views
+50

Distance function on complete Riemannian manifold.

Let $\left(M, g\right)$ be a complete Riemannian manifold. Let us fix a point $p \in M$ and consider the distance function $$ r(x) := \operatorname{dist}(x, p). $$ I would like to characterize the ...
1
vote
2answers
81 views
+50

Formula for proportion of entropy

Let's say we have a probability distribution having 20 distinct outcomes. Then for that distribution the entropy is calculated is $2.5$ while the maximal possible entropy here is then of course $-\ln(\...
1
vote
0answers
66 views
+50

Hailstone collatz max sequence length upper bound of $260.5+x^{.43}$?

Let the Collatz function be defined as if $x$ even $c(x)=x/2$, if $x$ odd then $c(x)=3x+1$ over the naturals. Each operation is defined as a step. For example $3$ goes $(3,10,5,16,8,4,2,1)$ and takes ...
1
vote
1answer
156 views
+100

Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
1
vote
0answers
23 views
+100

Minimum of $F$ over Finite Perimeter Sets in $\mathbb R^N$

Problem: Let $G$ be a bounded Borel set. Let $X$ be the set of finite perimeter sets in $\mathbb R^N$ and $F: X \to \mathbb R \cup \{+\infty\}$ defined as \[ F(E)= \begin{cases} Per(E) \hspace{1,...
2
votes
1answer
36 views
+50

Good book on Spherical Trigonometry

Possible approach/content: Modern Practical (Navigation/Geodesy) unifies with Euclidean/Hyperbolic Trigonometry
10
votes
0answers
196 views
+50

Solving $(n+1)(n+2)…(n+k)−k = x^2$

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
1
vote
1answer
78 views
+250

States of a Group Ring

Post-Self-Answer Edit: In an answer I THOUGHT I had found sufficient conditions for $f$ to be a state. The new question (for the bounty) was are the conditions necessary? However I have since ...
4
votes
1answer
184 views
+50

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
2
votes
1answer
50 views
+50

How to compute the integral of a renewal process?

Let $\{S_n:n=1,2,\ldots\}$ be a renewal process (with the convention $S_0\equiv 0$) with $\mathbb E[S_1]<\infty$ and $S_1$ absolutely continuous with density $f$. Let $\{N(t):t\geqslant0\}$ be the ...
0
votes
1answer
48 views
+50

Integrating an expression wrt to a variable which is a function of variables which appear in the expression?

Say I have an expression like this: $$\partial C/\partial a = \frac{(a - y)}{a(1-a)x}$$ where $a$ and $y$ are independent, but $a$ is a function of $x$ and possibly some other variables (i.e. $a = f(...
0
votes
0answers
81 views
+100

Applying drag to a collision prediction formula

I feel like this question might be below the minds of Math StackExchange, but I'll try anyway. (I can understand Math generally, but I'm probably not the caliber of people here.) I've been working on ...
1
vote
1answer
149 views
+50

Why it is impossible for primitive Pythagoras triplets in integers to be all as powerful numbers?

I had seen an elementary proof for Fermat's last theorem at Quora. I had checked all the steps (around one page only),where I couldn't catch any error, but I was confused about the last step only ...
1
vote
3answers
81 views
+50

Double Dirac delta integral

Let $I=\int_{-1}^0\int_{0}^1 \delta(x-y)dxdy$ where $\delta(t)$ is defined as the limit of a symmetric Gaussian pdf. The ranges overlap only on the zero-length range $x=y=0$. Is the result here $0$ ...
7
votes
1answer
119 views
+100

How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!
2
votes
0answers
85 views
+50

representation of a group and its center

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \...
2
votes
1answer
48 views
+50

Irrotational fields and divergence

Let $F,G$ be $C^1$ vector fields from $\mathbb R^n$ in itself. The condition $$\int_{\partial A}F\cdot \nu_A\ d\sigma=\int_{\partial A} G\cdot \nu_A\ d\sigma$$ for every bounded domain $A$ whose ...
2
votes
1answer
101 views
+50

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
2
votes
0answers
37 views
+150

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
1
vote
0answers
46 views
+100

How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
29
votes
1answer
680 views
+50

Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
6
votes
3answers
121 views
+50

A System of Infinite Linear Equations

Suppose that $\{a_{i}\}_{i=-\infty}^{\infty}$ with $\sum_{i=-\infty}^\infty a_{i} \lt \infty$ is known and that $\{b_i\}_{i=-\infty}^{\infty}$ is such that $$\sum_{i=-\infty}^\infty a_{i}b_{-i} =1,$$ ...
6
votes
2answers
157 views
+50

Prove or disprove that $2^n$ divides $T_{2^n}$ for $n > 2$.

The Tribonacci sequence satisfies $$T_0 = T_1 = 0, T_2 = 1,$$ $$T_n = T_{n-1} + T_{n-2} + T_{n-3}.$$ Prove or disprove that $2^n$ divides $T_{2^n}$ for $n > 2$. (I think $2^n$ divides $T_{2^n}$...
1
vote
0answers
24 views
+50

Polynomial Regression for Images instead of Data Points

My goal is to calculate the time of evaporation for water when this vase (below) is completely full. I understand that the simplest way would be to experimentally measure the evaporation time, however,...
5
votes
1answer
70 views
+50

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
1
vote
0answers
142 views
+50

Prerequisites to study cohomology?

Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I'd like to study it on my own. Before I dive into a book or two, I'd like to make ...
2
votes
1answer
69 views
+100

Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
5
votes
1answer
248 views
+50

Prove $\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4$

$x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove $$\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4$$ A natural though is that from the condition $x^2+y^2+z^2+xyz=4$, I tried a trig substitutions ...
0
votes
0answers
91 views
+50

Linear transformation $T$ such that for every extension $\overline{T}$, $\|\overline{T}\|>\|T\|$.

Let $E$ and $F$ be normed spaces such that $\dim F < \infty$, $G$ a subspace of $E$ and $T:G\rightarrow F$ a continuous linear map. I know that there exists a continuous linear extension $\overline{...
1
vote
0answers
55 views
+100

Functions with constant divergence of gradient-like field $\phi\nabla \phi/|\nabla \phi|$

I would like to classify functions $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $$ \nabla \cdot \left( \phi \frac{\nabla\phi}{|\nabla\phi|} \right) = \text{const}. $$ The only examples I ...
5
votes
1answer
246 views
+100

Conjecture about primes and the factorial

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
5
votes
0answers
72 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
2
votes
1answer
113 views
+50

Help with a nonlinear partial differential equation

let : $$\frac{\partial f}{\partial x}=f _{x}\;,\;\;\frac{\partial f}{\partial t}=f _{t}\;,\;\;\frac{\partial}{\partial t}\frac{\partial f}{\partial x}=f_{tx}\;, \;\;\ \frac{\partial}{\partial x}\frac{\...
0
votes
1answer
34 views
+50

A question about many-one reducibility of two sets

We want to show that $ \big\{x:W_{x}$ is finite }$=Fin \leq _m Cof=\big\{x : W_{x}$ is cofinite}. But I really have not any idea. Would be grateful for your help.
0
votes
1answer
55 views
+50

Rank and null space of a particular block matrix.

Let $D_1, D_2 \in \mathbb{R}^{N \times N}$ be diagonal matrices with diagonals that are linearly independent vectors. Let $A, B \in \mathbb{R}^{N \times N}$ be rank-deficient matries. Define $S = \...
0
votes
0answers
61 views
+50

Bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$.

Prove that there is a bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$ which identifies the summand $T^*M$ with the vertical vectors. If $\omega_{can}$ is the canonical symplectic ...
0
votes
1answer
37 views
+50

the symplectic version of Gram-Schmidt process and one application of it.

I am trying to understand the following proposition: cf. McDuff and Salamon, Introduction to Symplectic Topology(second edition), Corollary 2.4,page 40 One can assume $\omega_0 =\sum_i dx_i \...
0
votes
0answers
47 views
+50

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that $\gamma'_1(1)...
3
votes
0answers
69 views
+50

Future learning for a math graduate in applied mathematics references

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications ...

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