All Questions
49
questions with bounties
2
votes
1
answer
119
views
+50
Maximal domain of unbounded linear differential operator
Let's consider the following (unbounded) linear Operator. (So called Transport-Operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 127
5
votes
0
answers
142
views
+50
Automorphisms for direct products of finite commutative nilpotent rings.
Let $(R, +, \cdot)$ be an associative commutative nilpotent ring of cardinality $2^n$ such that
$$
r^2 = 0,
$$
for every $r\in R$. Also $(V, +)$ is a vector space over $\mathbb{F}_2$. Let $\...
- 1,314
2
votes
0
answers
81
views
+50
Evans Chapter 4 Problem 16, initial value problem to Schrödinger equation, convergence
Problem 16 states to discuss the sense in which $u(\cdot,t) \rightarrow g $ as $t\rightarrow 0^+$ defined by $$ u(x,t) = \frac{1}{(4\pi i t)^{n/2}} \int_{\mathbb{R}^n} e^{\frac{i |x-y|^2}{4t}}g(y)dy \...
- 739
1
vote
2
answers
122
views
+50
Euler Lagrange equation of a functional on the space of traceless symmetric matrices
Let $\mathcal{S}_0:\{Q\in \mathbb{R}^{3\times 3}:\text{tr}Q=0 \hspace{5pt}\text{and}\hspace{5pt} Q_{ij}=Q_{ji} \hspace{5pt} \text{for any $i,j=1,2,3$}\}$ and
$$ \widetilde{ \mathcal{E}}(Q)
=\int_{\...
- 567
1
vote
3
answers
1k
views
+50
What is the number of dimensions of a scalar?
Am I correct that a matrix has two dimensions, and a vector has one dimension?
What is the number of dimensions of a scalar? zero?
Thanks.
- 46.3k
34
votes
0
answers
1k
views
+50
Determining the kernel of a Vandermonde-like matrix
The kernel of a Vandermonde matrix can be determined using this formula.
The following type of matrix has a similar structure, and should also have a one-dimensional kernel.
$$V=
\begin{bmatrix}
...
- 353
3
votes
2
answers
162
views
+50
Counterfeit money, guaranteeing a profit
Bob and I found two 50 dollar bills out of nowhere. We know they're either both legitimate or both counterfeit. If they're legitimate, they're worth 50 dollars each, otherwise 0. I get one 50 dollar ...
- 293
1
vote
1
answer
227
views
+50
Addition formula for generalized Laguerre polynomials
For the Hermite polynomials, there is the following addition formula
Is there a similar formula for the generalized Laguerre Polynomials, in particular for $a=b=1/2$.
I.e. what is
$L^m_k(0.5x + 0.5 y)...
- 69
5
votes
1
answer
199
views
+200
Asymptotic of $_3F_2(1, \frac{1}{2}+d+n, 1+c+n; 1+2c, 2+2n;1)$ as $n\to \infty$
Let $c,d$ be in a small neighborhood of $0$; I think the limit
$$\begin{aligned}&\quad \lim_{n\to \infty} 4^{-n} n^{3c-d} {_3F_2}(1, \frac{1}{2}+d+n, 1+c+n; 1+2c, 2+2n;1) \\
&= \lim_{n\to \...
- 18.6k
0
votes
0
answers
91
views
+100
What is the dimension and nature of this variety?
Let $1 < N \in \mathbb{N}$ and $x, a \in \mathbb{C}^N$ with $a$ fixed; also, let $b \in \mathbb{R}_{\ge 0}^N$ be fixed (this last bit can be weakened to the extent it makes no difference). For $n \...
- 435
0
votes
1
answer
97
views
+50
intuitively Understanding meaning of a Combinatorics problem to reach solution
I have been recently taking course on Combinatorics and landed on following problem, Here is the formal statement of problem:
A room contains a single bulb and $(2^{2^{10}}+2^{2^9})$ identical ...
- 477
2
votes
1
answer
114
views
+50
Plane through feet of normals to ellipsoid
The problem
If $P,Q,R,P',Q',R'$ be the feet of six normals drawn from a point to the ellipsoid
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1=0$$
and the plane $PQR$ is represented by $lx+...
- 73
1
vote
1
answer
79
views
+50
Show that the existence of $y(x)$ of ODE by Peano existence theorem.
Following this question: An existence of global solution of differential equation of first order.
Consider the IVP: assume that $f$ is a continuous function from $R\times R^d \to R^d$, for $x\in R$ ...
- 1,271
1
vote
0
answers
50
views
+50
numerical integration of a function satisfying a ode
I need to numerically approximate an integral of the form $$\int_0^\tau f(X_t)\:{\rm d}t,\tag1$$ where $(X_t)_{t\ge0}$ is the solution of a SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag2.$$...
- 13.4k
3
votes
1
answer
48
views
+50
Confusion with Complex Gaussian process with Auto-covariance
I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
- 67
3
votes
0
answers
61
views
+100
Connection between Lame equation with Weierstrass and elliptic sine
Lame function is the solution of the following equation,
$$\frac{d^2w}{dz^2}+\left(A+B\wp(z)\right)w=0,$$
where $A$ and $B$ are constants and $\wp(z)$ is the Weierstrass elliptic function. Wiki says ...
- 486
8
votes
1
answer
139
views
+100
How can I smoothly connect two points on the surface of a sphere with a pair of arcs passing through the points in certain directions?
Given two points on the surface of a sphere $\vec a$ and $\vec b$ and directions tangent to the sphere at those points, $\vec {a'}$ and $\vec {b'}$ how can I connect them with a pair of circular arcs ...
- 295
2
votes
0
answers
77
views
+350
Proof about Markov kernels and absolute continuity
Assumptions:
$(\mathsf{X}, \mathcal{X})$ is a measurable space.
$M_n$ and $L_{n-1}$ are Markov probability kernels for $n=2, \ldots, P$.
$\mu_n$ be probability measures on $(\mathsf{X}, \mathcal{X})$ ...
0
votes
0
answers
52
views
+50
Rate Distortion Function of Binary Distribution where the Reconstruction Alphabet is $\mathbb{R}$
Consider a random variable $X \in \{-a,a\}$ with equally likely probaiblity. We are interested in finding a rate-distortion function for this distribution given by
\begin{align}
R(D) = \min_{P_{\hat{X}...
- 5,781
1
vote
0
answers
39
views
+50
Convergence of the Jacobi method
Consider this question
The function $u(x)=x(x-1), 0 \leq x \leq 1$, is defined by the equations $u^{\prime \prime}(x)=2,0 \leq x \leq 1$, and $u(0)=u(1)=0$. A difference approximation to the ...
- 2,070
1
vote
0
answers
57
views
+50
Clarification on what the question wants me to do
Consider
Let the Gauss-Seidel method be applied to the equations $A \boldsymbol{x}=\boldsymbol{b}$ when $A$ is the nonsymmetric $2 \times 2$ matrix
$$
A=\left[\begin{array}{cc}
10 & -3 \\
3 & ...
- 2,070
1
vote
1
answer
121
views
+100
Do large sets have this specific type of self-similarity?
Suppose $(a_n)_{n\in\mathbb{N}}$ is a strictly increasing sequence of
positive integers such that $\displaystyle\sum_{n\in\mathbb{N}}
\frac{1}{a_n}$ diverges, i.e. $(a_n)_{n\in\mathbb{N}}$ is "...
- 17.8k
4
votes
2
answers
234
views
+100
How to prove $\sqrt{\frac{24a+13}{24a+13bc}}+\sqrt{\frac{24b+13}{24b+13ca}}+\sqrt{\frac{24c+13}{24c+13ab}}\ge 3$
Problem. Let $a,b,c\ge 0: ab+bc+ca>0$ where $a+b+c=3.$
To prove that: $$\sqrt{\frac{24a+13}{24a+13bc}}+\sqrt{\frac{24b+13}{24b+13ca}}+\sqrt{\frac{24c+13}{24c+13ab}}\ge 3.$$
I've tried to use AM-GM, ...
- 1,017
4
votes
0
answers
69
views
+200
Sharp constant in the $L^p$ regularity estimate?
Problem: Let us denote $\mathbb{W}^{2,p}(\mathbb{R}^2)$ the space of Sobolev functions in the plane. Let us denote with $\Delta$ the classic Laplacian operator. We know that there exists a constant $C&...
- 2,706
8
votes
1
answer
256
views
+50
For which set $A$, Alice has a winning strategy?
Alice and Bob are playing a game. They take an integer $n>1$, and partition the set $\{1,2,...n\}$ into two non-empty subsets $A,B$. Alice takes the set $A$ and Bob takes the set $B$. They take a ...
- 171
3
votes
0
answers
95
views
+50
Fourier multipliers on $L^2(\mu)$
On $L^2(\mathbb{R}^d)$, we have $T_m$ defined $\widehat{T_m f} = m \widehat{f}$ is a bounded operator on $L^2$ if and only if $m \in L^\infty$.
What can be said about the same problem for more general ...
- 510
0
votes
1
answer
61
views
+200
How to evaluate employees based on the NPS of their sales?
Suppose that, as a manager, I am creating a campign to evaluate employees based on the NPS (Net Promoter Score) of their sales - the objective is to reward high perfoming employees. However, due to ...
- 570
0
votes
0
answers
69
views
+50
Uniform lower bound in the Landau Prime Ideal Theorem
Are there positive constants $\alpha,\beta,C$ such that for every number field $K$, the number of prime ideals of $\mathcal{O}_K$ of norm at most $x$ is at least $\alpha x^{\beta}$ for all $x\geq C$?
...
- 3,215
0
votes
0
answers
84
views
+50
Paraphrasing the definition of a complete measure
I am trying to paraphrase what it means for a measure to be complete, so that I can get a better grasp of the concept. Here are the relevant definitions from my textbook, Real Analysis by Folland:
...
- 1,003
2
votes
0
answers
171
views
+500
Sum of even binomial coefficients modulo $p$, without complex numbers
Let $p$ be a prime where $-1$ is not a quadratic residue, (no solutions to $m^2 = -1$ in $p$).
I want to find an easily computable expression for
$$\sum_{k=0}^n {n \choose 2k} (-x)^k$$
modulo $p$. ...
- 4,182
0
votes
0
answers
38
views
+50
Is the divergence of positive definite matrix function locally Lipschitz?
Let $a:\mathbb{R}^d\to S_d^{++}$ where $S_d^{++}$ is the set of $d\times d$ positive definite matrices. Suppose that $a$ is $C^1$. Then by a theorem of Phillips and Sarason (Rogers and Williams ...
- 1,247
21
votes
5
answers
562
views
+100
Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$
I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
- 111
10
votes
1
answer
1k
views
+200
Inequality constraints in calculus of variations
$\def\d{\mathrm{d}}$It turns out that Yuri's answer to my earlier question, whilst correct (and I thank him for his effort), was not quite what I desired. I had not posed the question properly, so I ...
- 54
3
votes
2
answers
91
views
+50
How can you make a decimal point for a number display on Desmos?
I have this Desmos graph which is a number display. You type a button and that number gets displayed. I'm thinking about turning it into a calculator, but first I need a decimal point. (Also the ...
- 130
2
votes
2
answers
516
views
+100
Transforming a single Gaussian into a mixture of Gaussians
Given a standard normal random variable:
$X\sim \mathcal{N}(0,1)$
and given two real positive numbers $\mu$ and $\sigma$, I am interested in finding a (non-random) function $g:\mathbb{R}\rightarrow\...
- 742
3
votes
4
answers
176
views
+500
Let $a+b+c=3,$ then prove$\sqrt[3]{\frac{a+b}{5ab+4}}+\sqrt[3]{\frac{c+b}{5cb+4}}+\sqrt[3]{\frac{a+c}{5ac+4}}\ge \sqrt[3]{6}$
Problem
If $a,b,c\ge 0: a+b+c=3,$ then $$\sqrt[3]{\frac{a+b}{5ab+4}}+\sqrt[3]{\frac{c+b}{5cb+4}}+\sqrt[3]{\frac{a+c}{5ac+4}}\ge \sqrt[3]{6}.\tag{1}$$
I came up with when trying to prove $$\frac{a+b}{...
- 309
1
vote
0
answers
42
views
+100
graphing hyperbolic geodesics on a 3D surface
Does anyone know of a relatively simple and easy-to-use graphics app (like GeoGebra) that can graph hyperbolic geodesics on a hyperbolic paraboloid? I'm trying to graph the geodesics on the top image ...
- 65
3
votes
0
answers
59
views
+50
$L^2$-ness of the finite difference operator
Is there a proof of the fact that the operator $\mathfrak{D}$ defined by $\mathfrak{D}[f](x,y):=\dfrac{f(x)-f(y)}{x-y}$ is continuous from $H^n(\mathbb{R})$ to $H^m(\mathbb{R}^2)$ or even $L^2(\mathbb{...
- 303
4
votes
1
answer
350
views
+50
Continuous dependence on an initial condition (SDE)
Let's say I have a (one-dimensional) diffusion process
$$dX=\mu(X_t)dt+\sigma(X_t)dW.$$
Assume we have fixed $\epsilon > 0$ and $t >0$
Under what conditions is $\mathbb{P}^x(X_t < \epsilon)$...
- 532
11
votes
2
answers
312
views
+50
Can $5^n+6^n+10^n$ be a perfect power?
Related to this question:
Can $f(n)=5^n+6^n+10^n$ with a non-negative integer $n$ be a perfect power ?
It is not a perfect power for $0\le n\le 10^5$
Analysis modulo $3$ reveals that a perfect ...
- 82.3k
3
votes
1
answer
69
views
+50
Checking consistency of minimizer given uniform convergence of its objective function
Suppose the true parameter is
$$\theta_0 = \arg\min_{\theta \in \Theta} E[q(w, \theta)]$$
and $\theta_0$ is the unique minimizer, i.e.,
$$E[q(w, \theta_0)] < E[q(w, \theta)]$$
for all $\theta \in \...
- 1,914
0
votes
0
answers
158
views
+100
Hess' Proof of Fuss' Formula
I am trying to understand the short proof of Fuss' Formula in the paper "Bicentric Quadrilaterals through Inversion" by Albrecht Hess which is available here:
https://forumgeom.fau.edu/...
- 1,418
3
votes
0
answers
159
views
+50
Is the following infinite sum familiar?
I have a sum that comes as a special case of my original sum. I do not want to show the original sum as it is too complicated. The specific one looks like the following:
$$ G(z) = \sum_{j = 0}^{\infty}...
- 67
1
vote
0
answers
60
views
+50
Spherical harmonics of order $l$ with smallest $L_\infty$ norm
Let ${\mathbb H}_l$ be a space of spherical harmonics of order $l$, i.e.
$${\mathbb H}_l = \{f: {\mathbb S}^{n-1}\to {\mathbb R} \mid \Delta_{{\mathbb S}^{n-1}}f=-l(l+n-2)f\},$$
where $\Delta_{{\...
- 341
11
votes
1
answer
763
views
+500
What is the connection between algebraic groups and topoi?
I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, ...
- 43k
2
votes
0
answers
44
views
+50
Two problems relating principal curvatures and circles contained in surface
Let $k_1=k_2=1$ be the principal curvatures of a regular surface $S$ at point $p\in S$ and assume that there is a circle $c$ of radius $1/2$ passing through $p.$ Prove that
the geodesic curvature of $...
- 5,235
6
votes
0
answers
117
views
+50
Is $10^n+n^{10}$ prime for some integer $n \ge 2$?
Is $$f(n):=10^n+n^{10}$$ a prime number for some integer $n\ge 2$ ?
Two necessary conditions :
$\gcd(n,10)=1$
Since $n$ must be odd , we must also have $11\mid n$ , otherwise $11\mid f(n)$
In ...
- 82.3k
5
votes
0
answers
141
views
+50
Can $!1+!2+!3+\cdots+!n$ be a perfect power?
Can $!1+!2+!3+\cdots+!n$ be a perfect power if $n\geq3$?
Note that $!n$ is a subfactorial.
I do know that $1!+2!+3+\cdots+n!$ is only a perfect power if $n=1, 3$, since when $n\geq9, 1!+2!+3!+\cdots+9!...
0
votes
0
answers
70
views
+100
How to show the quadratic integer ring O is not a UFD.
Let $R=\mathbb{Z}[\sqrt{−n}]$ where $n$ is a squarefree integer greater than 3. Prove that $R$ is not a UFD. Conclude that the quadratic integer ring O is not a UFD for $D\equiv 2, 3$ mod $4$, $D < ...
