0
votes
1answer
64 views
+50

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
4
votes
1answer
165 views
+50

How prove this sequence $u_{m}=v_{m}$

Question: Assume that $m$ is a positive integer, define the sequence $$\{u_{k}\},\{v_{k}\},u_{0}=v_{0}=u_{1}=v_{1}=1$$ and for any real number $a_{i},i=\{1,2,\cdots,m-1\}$, $$\begin{cases} ...
4
votes
1answer
51 views
+50

What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
6
votes
1answer
111 views
+50

Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
13
votes
1answer
246 views
+50

How fat is a triangle?

The slimness factor of a geometric shape in 2 dimensions is the ratio between the side-length of its smallest containing square and its largest contained square. This is an important factor in ...
4
votes
1answer
198 views
+100

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
5
votes
2answers
124 views
+100

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
8
votes
0answers
142 views
+100

How to estimate $Pr[vr_i=ur_i]$ in the presence of rotations

Suppose we want to compute the probability that for two different random vectors (with elements that are $0$ or $1$), denoted by $v$ and $u$, multiplying them with the rotations of a random vector $r$ ...
5
votes
2answers
281 views
+50

How prove this inequality $\sum\limits_{cyc}\frac{1}{a+3}-\sum\limits_{cyc}\frac{1}{a+b+c+1}\ge 0$

show that: $$\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}-\left(\dfrac{1}{a+b+c+1}+\dfrac{1}{b+c+d+1}+\dfrac{1}{c+d+a+1}+\dfrac{1}{d+a+b+1}\right)\ge 0$$ where $abcd=1,a,b,c,d>0$ I ...
2
votes
0answers
201 views
+300

Inequality involving Pochhammer symbols

Let $m,S$ be integers satisfying $2\leq m\leq S$. I would like to show that $$h_1\left(x\right) h_3\left(x\right) \leq h_2^2\left(x\right)$$ for all $x\geq 0$ where $$h_k\left(x\right) \equiv ...
21
votes
2answers
314 views
+100

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
8
votes
1answer
68 views
+200

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
9
votes
1answer
125 views
+50

Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
4
votes
1answer
148 views
+50

Area of ​​the intersection of two discs: Integral solution?

Here is the problem : We consider two circles that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
12
votes
2answers
309 views
+200

A topological function with only removable discontinuities

I've posted similar questions here and here, but no one has answered them to my satisfaction. Suppose that $f:\mathbb{R} \to \mathbb{R}$ is such that $\lim_{y\to x}f(y)$ exists for all $x$, that is, ...
33
votes
5answers
813 views
+200

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
6
votes
2answers
236 views
+50

If $a_1a_2\cdots a_n=1$, then the sum $\sum_k a_k\prod_{j\le k} (1+a_j)^{-1}$ is bounded below by $1-2^{-n}$

I am having trouble with an inequality. Let $a_1,a_2,\ldots, a_n$ be positive real numbers whose product is $1$. Show that the sum $$ ...
2
votes
2answers
56 views
+100

Regression with error coming from rounding

I am looking at the following model: $c$ is a fixed vector in $\mathbb{R}_+^n$ and for any $x \in \mathbb{R}_+^n$ we obtain a value $y =[c^Tx]$, i.e. rounding $c^Tx$ to the nearest integer. I want ...
21
votes
3answers
543 views
+500

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

Please help me to evaluate this integral: $$ I={\large\int}_{0}^{\infty}{\ln\left(x\right) \over 1 + x}\, \,\sqrt{\,x + \sqrt{\,1 + x^{2}\,}\, \over 1 + x^{2}\,}\,\,{\rm d}x.\tag1 $$ Mathematica could ...
1
vote
1answer
59 views
+50

Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = ...
8
votes
0answers
129 views
+200

Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
1
vote
2answers
86 views
+50

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
2
votes
1answer
51 views
+50

Without using Stokes theorem, compute surface and line integrals

Let S be the surface in $R^3$ given by the part of the sphere $x^2+y^2+z^2=2$ that lies above the plane $z=1$. Let v(x,y,z) be the vector field given by $v(x,y,z)=(z,x,z)$. a) Without using Stokes' ...
4
votes
0answers
59 views
+100

'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
1
vote
0answers
63 views
+50

Proof verification: Munkres exercise 10, section 23

Can someone please verify my proof or offer suggestions for improvement? I'm thoroughly confused by this question, and I'm sure there's a mistake somewhere in my proof. Let $\{X_\alpha\}_{\alpha ...
0
votes
0answers
28 views
+50

Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs: $$\hat{L}u(x,t)=0$$ with the constraint $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$. During evolution, the ...
3
votes
1answer
86 views
+50

Base of logarithm decrease when variable count increase

I run a large online platform where users submit articles and earn points. I am working on an algorithm where the more comments they submit, the higher score they will receive. In its simplest ...
0
votes
0answers
25 views
+50

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
15
votes
0answers
294 views
+100

Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
6
votes
3answers
100 views
+100

Splitting $\Phi_{15}$ in irreducible factors over $\mathbb{F}_7$

I have to split $\Phi_{15}$ in irreducible factors over the field $\mathbb{F}_7$. It has been a while that I did this kind of stuff, and to be the honest, I've never really understood this matter. I'd ...
1
vote
0answers
64 views
+50

Is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
3
votes
1answer
42 views
+50

Rewriting a continuously differentiable function

I have the following $i$-th regressor function: $\phi_i(x)$ in which $x$ is a vector with elements $x_1, \ldots, x_n$. I cite from an article: Let $e_i = \hat{x}_i - x_i$ and note that, since ...
0
votes
0answers
27 views
+50

Changing a queueing processes

I am wondering if there are any general results related to how queue behavior changes if one is allowed to repeatedly make one-off behavior changes. Situation Consider a general queueing system ...
4
votes
1answer
63 views
+150

Alternative proof of Girard's theorem

I am looking for an alternative proof of Girard's theorem. The standard proof, which is almost trivial, relies too much on visualizing spherical triangles on the sphere. Is there a more algebraic ...
0
votes
0answers
26 views
+50

Calculating the constants in the general solution of second order homogeneous ODE reduced from Riccati equation

I am writing a code to simulate a kind of volumetric flow, and I have encountered the non-linear Riccati equation in its general form near the end of my calculations. I am having trouble finding the ...
-1
votes
0answers
38 views
+50

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
2
votes
0answers
79 views
+50

Verifying an antiderivative found in any integral table

If $a > 0$, and $0 < b < c$. \begin{equation*} \int \frac{1}{b + c\sin(ax)} \, {\mathit dx} = \frac{-1}{a\sqrt{c^{2} - b^{2}}} \, \ln\left\vert\frac{c + b\sin(ax) + \sqrt{c^{2} - ...
2
votes
0answers
89 views
+50

Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups. Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups. Definition: ...
1
vote
0answers
11 views
+50

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
1
vote
0answers
75 views
+100

Expected frequency of most frequent die roll

Suppose we have an fair $m$-sided die, and we roll it $n$ times. What is the expected frequency $E(n, m)$ of the most frequently rolled face? If we fix $n$ we can calculate $E(n,m)$ like so. Let ...
0
votes
0answers
67 views
+50

Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
2
votes
0answers
37 views
+100

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
2
votes
2answers
49 views
+50

Distributed Convex Optimization Algorithm

Consider the convex optimization problem $$ \min_{x_1, \cdots, x_N, y} \sum_{i=1}^{N} f_i(x_i,y) $$ $$ \text{subject to: } x_i \in X_i \ \ \forall i, \ \ y \in Y, \ \ y = \sum_{i=1}^{N} x_i $$ ...
3
votes
0answers
49 views
+50

How prove this $\prod_{i=1}^{r}\left(1+\frac{1}{x_{i}}\right)\le \frac{2^{2^r}-1}{2^{2^r-1}}$

Let $x_{1},x_{2},\cdots,x_{r}$ be positive integers such that $$1\le x_{1}\le x_{2}\le \cdots\le x_{r}$$ and $$\prod_{i=1}^{r}\left(1+\dfrac{1}{x_{i}}\right)<2.$$ Show then that ...
1
vote
0answers
34 views
+50

Trick for Jordan-Matrix and transformation of basis

some time ago I found a 'trick' for getting a basis-transformation-matrix for jordan. I'd like to understand it, but at a certain point I stuck. Maybe you can help me? Given is a matrix A: ...