All Questions
5,628 questions
4
votes
1
answer
250
views
Does a generic linear map admit a vector whose iterates span $V$?
We say a linear map $T$ on a finite dimensional vector space $V$ admits spanning vectors if there exists some vector $v \in V$ whose iterates $v, Tv, T^2 v, \dots$ under $T$ span $V$.
Question: ...
19
votes
2
answers
3k
views
Solutions-set first order ODE's without uniqueness
In short: What can we say about the set of all solutions of an ordinary differential equation (ODE) when we there is no uniqueness?
Consider the ODE
$f:D\to \mathbb{R}, \quad D\subseteq \mathbb{R}^2,$
...
- 3,574
6
votes
0
answers
108
views
Archimedean ordered field in which every function is smooth
In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \...
- 2,624
0
votes
0
answers
22
views
An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
0
votes
0
answers
618
views
The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
- 305
6
votes
1
answer
345
views
Characterization of sums of periodic functions over the real line
Is there any known characterization of the functions $\mathbb{R \to R}$ that can be written as a sum of (a finite family of) periodic functions? Not assuming any regularity condition (not even ...
- 156
4
votes
2
answers
296
views
Implicit function theorem for subdifferentiable convex functions
I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
- 29
3
votes
2
answers
429
views
Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 109k
1
vote
1
answer
162
views
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,669
0
votes
1
answer
192
views
A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 48.8k
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 $X$ ...
- 128k
1
vote
0
answers
48
views
How to derive a lower bound of a MinMax inequality?
Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$.
The goal
For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
- 405
3
votes
0
answers
171
views
Generalized family of Hölder inequalities
Is the "only if" direction of the following fact known?
For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^...
- 4,219
3
votes
1
answer
185
views
Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 21.5k
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
$$\lVert x\rVert_p \...
- 128k
1
vote
1
answer
234
views
Zeroes of elementary polynomials without involving closed-form solutions
Consider the following two polynomials, where $n$ is an integer:
$$
p_n(x) = x^3-\frac1nx-\frac2n, \\
q_n(x) = x^2-\frac2n.
$$
For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
- 1,440
9
votes
0
answers
1k
views
How complicated can an elementary antiderivative get?
I asked this question on MSE here.
I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
- 541
1
vote
1
answer
160
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
- 125
1
vote
0
answers
148
views
Rational solutions to $\cos(\lambda \pi) = \cos^2(a\pi) - \cos(b\pi) \sin^2(a\pi) $, with $a,b \in \mathbb{Q}$
In a similar vein to this question, I am trying to understand the occurrence of rational solutions $\lambda$ to the following equation $$\cos(\lambda \pi) = \cos^2 (a\pi) - \cos ( b\pi ) \sin^2 \left(...
- 51
7
votes
2
answers
567
views
Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...
- 21.5k
3
votes
0
answers
179
views
Maximum of an integral
Assume that $a>0$ and $r\in[0,1)$. How to prove that the function $$f(p)=\int_{-\pi}^\pi \left (1+r^2+2 r \cos x\right)^{a/2} |(2+a) \cos(x+p)-a r \cos(p)| \, dx$$ attains its maximum for $p=\pi/2$...
2
votes
0
answers
95
views
Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
- 657
3
votes
1
answer
146
views
Behaviour of the solution of a second order ODE
I am currently studying the following second order ODE
\begin{cases}
\ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\
y(0) = 0\\
\dot y(T) = c
\end{...
- 60.5k
5
votes
2
answers
352
views
Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$
I am trying to observe the behavior of $x_n \in (0,1)$ defined such that the function
\begin{equation}
f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)
\end{equation}
attains its maximum inside the ...
- 1,440
1
vote
1
answer
66
views
Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?
Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
- 128k
3
votes
1
answer
493
views
A strange condition of convexity?
During my research, I come across this question.
Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$.
Is it true that $\forall x \in \mathbb R, f''(x) \...
- 24.2k
5
votes
5
answers
1k
views
What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$
FYI: I asked this question here couple of days ago but got no answer yet.
$n$ is an integer
We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
- 113
2
votes
2
answers
307
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
1
vote
0
answers
96
views
Regularity of Feynman-Kac formula for a simple diffusion
Let consider the diffusion process given by:
$$dX_t = \alpha(X_t) dW_t$$
where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
- 393
1
vote
0
answers
162
views
Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
- 43.2k
1
vote
2
answers
167
views
Asymptotic estimation of an integral
I have an integral of the form
$$
I = \int\limits^{1}_{0} \exp\left(\dfrac{vt}{(v+1)^2 + v^2} - vt\right) dv
$$
and I want to prove that $I\leq c t^{-1}$ for the sufficiently large $t$, where $c$ is a ...
- 1,440
9
votes
1
answer
1k
views
Integration by parts formula for the double Riemann-Stieltjes integral
In my research the following integration by parts formula for the double Riemann-Stieltjes integral
$$\int\limits_{[a,b]\times[c,d]}f(x,y)\,dg(x,y)=f(b,d)g(b,d)-f(a,d)g(a,d)-f(b,c)g(b,c)+f(a,c)g(a,c)...
- 4,713
3
votes
1
answer
161
views
Equivalent definition for Skorokhod metric
I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$:
$$
d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
- 128k
-3
votes
2
answers
317
views
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
- 1,126
7
votes
2
answers
724
views
Sturm chain analogue for exponential polynomials?
I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real).
My first question is: is there an algorithm for ...
- 1,446
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
3
votes
1
answer
499
views
Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?
The integral is
$$\DeclareMathOperator{\dm}{d\!}
\int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}\dm x.
$$
Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational ...
- 6,367
7
votes
2
answers
592
views
Prove that the following function is positive
Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...
- 6,035
9
votes
2
answers
584
views
Does this integral condition characterize $L^\infty$?
Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition ...
- 109k
3
votes
1
answer
132
views
Existence of a density
Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite ...
1
vote
1
answer
113
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 6,035
1
vote
1
answer
106
views
Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$
Let $\alpha \in (0, 1)$ and $\psi : \mathbb R^d \to \mathbb R^d$ be a $C^\infty$-diffeomorphism such that $\|\nabla \psi\|_\infty + \|\nabla \psi^{-1}\|_\infty < + \infty$. Let
$$
I (t) := \sup_{x \...
- 128k
1
vote
1
answer
170
views
fourth-order multivariate Gaussian integral
I am struggling with an integral of form
$$ \int_{\mathbb R^n} y\otimes y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y). $$
I assume that it will involve the trace of some product of $R$ and $\...
0
votes
0
answers
122
views
Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
0
votes
0
answers
36
views
Convergence of numerical scheme for HJB equation
Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...
- 393
0
votes
0
answers
53
views
Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 6,367
10
votes
2
answers
1k
views
Does a conditionally convergent sum with random signs converge almost surely?
Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
- 29.8k
8
votes
1
answer
1k
views
When can a sum be re-signed to converge to any limit?
Let $a_n$ be a sequence of positive real numbers with $\sum a_n < \infty$. What are the necessary and sufficient conditions for the following to hold?
For any $S \in \mathbb R$ with $-\sum a_n \...
- 60.5k
1
vote
1
answer
137
views
Inequality with convolution
I have some troubles with the following problem:
A definition
Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian ...
- 128k
2
votes
1
answer
152
views
Growth rate of elementary sequences
We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
- 24.2k