All Questions
5,850 questions
0
votes
1
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261
views
Second order ODE
I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?
$$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$
$C$ ...
user37929
0
votes
1
answer
259
views
How to perturb a function to separate points
Consider two smooth functions $f,g\in C^\infty(\Omega)$ with $\partial \Omega$ smooth and $\Omega\subset \mathbb{R}^3$. Assume that $f=g$ on $\partial \Omega$.
For any given $\varepsilon>0$, how ...
- 35
0
votes
1
answer
250
views
Equation in integers of irrational degree
Are there any algebraic irrational numbers in $\{log_xy|x,y\in\mathbb{N},x,y\geq2\}$?
- 7,377
0
votes
1
answer
1k
views
Proper Group action on a metric space
Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {...
- 1
0
votes
1
answer
1k
views
Global Implicit Function Theorem
Let $F:\mathbb R^2\rightarrow\mathbb R$ be a measurable function. Under what conditions on $F$ does there exist a function $\theta:\mathbb R^2\rightarrow\mathbb R$ such that
$F(x,\theta(z,x))=z$ for ...
- 13
0
votes
1
answer
155
views
Ratio of eventually close sequences
Let $a_n$,$b_n$ with $b_n>0$ be two bounded sequences which are eventually close to, respectively, two other sequences $\bar a_n$,$\bar b_n$ with $\bar b_n>0$, that is, for every $\epsilon >0$...
0
votes
1
answer
612
views
Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 213
0
votes
1
answer
265
views
H\"older spaces
In Gilbarg and Trudinger, they have an example where a function is in $C^1(\bar\Omega)$ but not in $C^\alpha(\bar\Omega)$ where $\alpha<1$. $\Omega$ is bounded and is defined as follows
$\Omega:= ...
- 3
0
votes
1
answer
126
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,153
0
votes
1
answer
95
views
A question on finite Fourier series
Let $\mathcal F(N)$ denote the space of finite Fourier series up to frequency $N > 0$, i.e. $f\in \mathcal F(N)$ if and only if it can be written as
$$f(x) = \sum_{k=0}^N a_k\cos(kx+\theta_k)$$
for ...
- 71
0
votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
0
votes
1
answer
139
views
Proving negativeness of function involving $-\log t$
I have been trying to solve the following function is non-increasing with respect $\theta$
\begin{equation}
h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
- 23
0
votes
1
answer
128
views
Characterizing the integral as a function of $n$
Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
- 25
0
votes
1
answer
80
views
Orthogonal space of polynomials
Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...
0
votes
1
answer
100
views
Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
- 31
0
votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
80
views
Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
- 434
0
votes
1
answer
235
views
Does this property implies Lipschitz continuity?
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be such that, for $x,y,z \in \mathbb{R}^{n}$, we have that
$$|f(z) - f(x)| \leq |f(z) - f(y)| \Rightarrow \|z-x\| \leq \|z-y\|$$
Can I say that this ...
0
votes
1
answer
140
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
0
votes
1
answer
58
views
Integration algorithm and analytic property
This question is the continuation of the previous one.
In the article about the integration of analytical polynomial - time computable function $f(x)$ with the Taylor series $$f(x) = \sum_{n=0}^{\...
- 81
0
votes
1
answer
148
views
Equi-coercivity of functionals on a metric space
Definition: A family of functionals $\{F_n: X\to\bar{\mathbb R}\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ ...
- 1,101
0
votes
1
answer
150
views
Property of $p$-norm in the $n$-simplex
Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that
$$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$
implies that
$$\lVert x\rVert_p \...
0
votes
1
answer
252
views
Reference request: log Sobolev inequality for uniform measure (uniform distribution over discrete set)
Suppose that $N \in \mathbb N_+$ is fixed and denote by $\mu = (\mu_0,\ldots,\mu_N)$ the uniform distribution on the set $\{0,1,\ldots,N\}$ (i.e., $\mu_n = \frac{1}{N+1}$ for each $0\leq n\leq N$). I ...
- 730
0
votes
1
answer
106
views
The sequence has a stationary accumulation point
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that
(a) $\{x^k\}$ has an ...
- 21
0
votes
1
answer
130
views
Explanation for Tauberian theorems for Laplace transform
I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow :
Let $F : [0,\infty) \to \mathbb{R}$ of ...
- 393
0
votes
1
answer
167
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
0
votes
1
answer
270
views
Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
user516435
0
votes
1
answer
129
views
Sequence of functions converges pointwise to identity [closed]
Let
For $n\in \mathbb{N}$ and $k\in \{0, 1, 2, ..., 2^{n}-1 \}$ is defined
$$I_{k}^{n}=\left[\frac{k}{2^{n}}, \frac{k+1}{2^{n}}\right)$$
and $f_{n}:[0, 1) \rightarrow \mathbb{R}$ is defined by
$$f_{n}(...
- 251
0
votes
1
answer
67
views
Multivariate random variable problem [closed]
I'm stuck on this problem:
Let a random vector $X$ be given in $\mathbb{R^{10}}$ with the standard scalar product. It is known that $\mathbb{E}[XX^T] = 5I_{10}$, $I_{10}$ – identity matrix of order 10....
- 69
0
votes
1
answer
98
views
Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\...
- 825
0
votes
1
answer
118
views
For any smooth function $f$ on $[0,1]$, do we have $\lVert P_N f \lVert_{1} \leq \lVert f \rVert_1$ for $P_N$ defined by Fourier expansion?
Let $C^\infty[0,1]$ be the space of periodic smooth functions on $\mathbb{R}$ with the period $1$. It is well-known to be a Frechet space with the uniform convergence of all derivatives.
Also, $\{ e^{...
- 3,477
0
votes
1
answer
139
views
A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$
Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, ...
- 1,295
0
votes
1
answer
206
views
Series involving sine and cosine
Let $(a_n)_n$ be an increasing real sequence with $a_n=O(\sqrt n)$.
Is it true that there exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $$\lim \left|\sum\limits_{k=1}^{\phi(n)}\...
- 4,074
0
votes
2
answers
166
views
Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions
Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by
$$
F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}.
$$
It is clear that $F$ is strictly ...
- 6,853
0
votes
1
answer
163
views
Lower bounds on eigenvalues of sum of two matrices (one of them is symmetric)
Let $L$ be a matrix with eigenvalues($\lambda$ $\geq$ 0). If I add a constant value (say $a$) to all the elements of $L$, what can we say about the minimum eigenvalue of this perturbed matrix?
Note: $...
0
votes
1
answer
66
views
Vector recurrences (asymptotic property)
Fix $m\in \mathbb{N}.$
For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that
$$X_{n+1}=A_n X_n+B_n,$$
$$\lim_{n\rightarrow ...
- 3
0
votes
1
answer
104
views
Show that $\frac{1}{n} \sum_{i=1}^n a_i \operatorname{erf} \left( \frac{b_i-x}{\sqrt{2}} \right) \to x$ for some sequence $\{a_n\}$ and $\{b_n\}$
Consider the following function
\begin{align}
f_n(x)=\frac{1}{n} \sum_{i=1}^n a_i \operatorname{erf} \left( \frac{b_i-x}{\sqrt{2}} \right)
\end{align}
where $\operatorname{erf} $ is the error ...
- 671
0
votes
1
answer
270
views
Determine if an integral expression is in $L^2(\mathbb{R})$
Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...
0
votes
1
answer
296
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 5
0
votes
2
answers
176
views
How to compute the unique disintegration w.r.t. the first coordinate?
Set $\pi=\frac{1}{4}(\delta_{(1,0)}++\delta_{(1,3)}+\delta_{(1,1)}+\delta_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.
How to get the disintegration of $\pi$ with respect to $\mu$?
- 288
0
votes
1
answer
165
views
Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicative up to an factor?
Related to this question.
For $x_+ \in (0,\infty)$, $a \in \mathbb{R}$ let $F\colon[0,x_+] \to [a,\infty)$ be a twice continuous differentiable (in $(0,x_+)$) function with $f := F'$, $f(x) > 0$, ...
- 15
0
votes
1
answer
55
views
Kernels of sequences of operators
Let $\left( S_{n}^{1}\right) $ and $\left( S_{n}^{2}\right) $ two sequences
of operators in $\mathcal{L}(E_{1},F_{1})$ and $\mathcal{L}(E_{2},F_{2})$
where $E_{i},F_{i},i=1,2$ are Hilbert spaces such ...
- 617
0
votes
1
answer
191
views
Calculating derivatives of arbitrary-order at an operator's root
Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of ...
- 583
0
votes
1
answer
253
views
$f'(x)>f(f(x))$ implies $f(f(f(x)))\leq0$ for nonnegative $x$
If $f\in C^1(\mathbb R)$ satisfies $f'(x)>f(f(x))$ for all $x\in\mathbb R$, then $f(f(f(x)))\leq0$ for all $x\geq0$.
I have some trouble to prove this. I wonder if there's some relations between ...
- 1,219
0
votes
1
answer
79
views
Proof of yet another extension of deterministic variant of "(Almost) Supermartingale" convergence theorem
In this question, there is a proof for deterministic version of "Almost Supermartingale"
Question: Can we extend [1] as following? If yes, can we prove it?
Let the non-negative sequences be ...
- 45
0
votes
2
answers
215
views
Asymptotic for eigenvalues for the following ode?
Consider the following Sturm-Liouville problem,
$$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$
where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$
I ...
- 537
0
votes
1
answer
91
views
Asymptotic moment of a multivariate normal distribution
Let the pdf of a multivariate normal distribution be
\begin{equation}
p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2).
\end{equation}...
- 550
0
votes
1
answer
103
views
How to show the following inequality $_2F_1(5.5, 1, 5;-|x|^2]>0$? [closed]
Consider the function $_2F_1(5.5, 1, 5;-|x|^2]$ for $x\in \mathbb{R}^n.$ I want to show that this function is positive. I checked that it does not have any roots so can I conclude the inequality by ...
- 537
0
votes
1
answer
171
views
How to compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{[-1,1]^n}\exp[2\pi i(\theta_1 v_1.x+\theta_2v_2.x)]d^nx d\theta_1d\theta_2$
Let $\mathbf{v}_1, \mathbf{v}_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral:
$$\int_{-\infty}^{ \infty} \int_{-\infty}^{\infty}
\int_{[-1,1]^n}
e(\...
- 3,625
0
votes
1
answer
70
views
Properties of superposition operators in $W^{1, p}$
Let $\Omega$ be a bounded open subset of $\mathbb R^n$, and $T: \mathbb R \to \mathbb R$ an absolutely continuous function with sublinear growth, in the sense that
$$|T(x)| \leq 1 + |x|, \forall x \in ...
- 6,233