All Questions
744 questions
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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 ...
- 21
1
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1
answer
242
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Can (how) one distinguish germs of continuous functions by a countable set of params?
Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...
- 24.9k
1
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1
answer
239
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Reference request for weak solutions of an Elliptic PDE
Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one.
I want to find weak, non trivial, continuous, solutions of $$\...
- 698
1
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1
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82
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What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$
What is $\left\| f \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$.
I'm reading a paper
Chern-Yamabe flow
which said ...
- 809
1
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1
answer
212
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Lipschitz aspect of a projection on the boundary of a convex
Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that
\begin{...
- 449
1
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1
answer
176
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A condition on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge 0$
Assume that $f:[0,1]\to [0,1]$ is an diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ on $[0,1]$. But no proof so far.
The ...
- 333
1
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1
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191
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Good upper bound for $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where $a,b \in (0, 1)$ and $N \ge 1$
Let $a,b \in (0, 1)$ and $N \ge 1$, and consider the incomplete gamma function $x \mapsto \Gamma(1-a,x)$.
Question
Is there a simple bound (involving 'simple function's) for the expression $\Gamma(1-...
- 6,853
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0
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184
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A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$
Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
1
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1
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162
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Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
1
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1
answer
186
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Existence of a smooth function that approximates a characteristic function of an interval with certain property
Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the ...
- 3,625
1
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1
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117
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On summation methods of divergent series
$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\CC}{\mathcal C}$This previous question introduced the following notion of a summability space.
Let $\N:=\{1,2,\...
- 128k
1
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1
answer
94
views
Differentiability of some function defined as the maximum
Let $d,n\ge 1$ be fixed integers. Given some compact subset $E\subset \mathbb R^d$, consider the function $f: E^n\ni (x_1,\ldots, x_n) \longrightarrow f(x_1,\ldots, x_n)\in \mathbb R$ defined by
$$f(...
- 1,399
1
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1
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87
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Oscillating sums
Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some ...
- 539
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0
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186
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Lipschitz continuity of an implicit function generated by a monotonic and Lipschitz multivariate function
Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ satisfying the following conditions:
$z=F(x,y)$ is Lipschitz continuous w.r.t. $(x,y)$;
Given $x$, $F(x,y)$ is non-...
- 601
1
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0
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922
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A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
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0
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92
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Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
1
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2
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140
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Extending a discrete singular kernel
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 1,879
1
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0
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67
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Solution to recurrence relation from integro-differential dynamical system?
Consider the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1}
\end{equation}
such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
- 79
1
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1
answer
193
views
Quantitative finite speed of propagation property for ODE (cone of dependence)
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 839
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0
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259
views
Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
- 969
1
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0
answers
76
views
Error estimates for orthogonal polynomial approximation
tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials?
There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
- 19
1
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1
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190
views
Proof of extended version of non-random "almost supermartingale"
In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
- 45
1
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1
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119
views
A non-polynomial Young function satisfying a power-like condition
This post asked, essentially, for an example of a "non-polynomial" invertible increasing function $f\colon[0,\infty)\to[0,\infty)$ such that $f(0)=0$ and
\begin{equation}
f(cu)f(t)\le f(...
- 128k
1
vote
2
answers
346
views
Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?
According to numerical simulation, the relationship
$$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$
where $\Gamma$ is the Gamma function seems to be true.
Do you ...
user365311
1
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1
answer
401
views
linear recurrence inequality of positive terms
This is a follow up on my previous linear recurrence inequality question.
I have some matrices which satisfy a linear recurrence formula of the form
$$
A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad n\...
- 145
1
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0
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416
views
When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 61
1
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1
answer
266
views
Constant bound for the 1 dimensional Besicovitch covering theorem on real line
I recently looked through the proof of the Gagliardo–Nirenberg Interpolation Inequality, see proof and it says that for real line $R$, there exists a sequence of open intervals $\{I_k\}$, which covers ...
- 65
1
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0
answers
71
views
Continuous injection of metric ball into Euclidean ball
This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...
- 5,405
1
vote
0
answers
123
views
Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
- 1,835
1
vote
1
answer
620
views
Smallest Lipschitz constant on non-convex domains
It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...
- 959
1
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0
answers
76
views
Geometric series involving the Laguerre polynomials
Let put $\alpha=5$ and $x=3$. Consider the following set given by
$$M=\lbrace \; n \in N, \; \; 0 < |L_{n}^{5}(3)| < 1 \; \rbrace$$
Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
1
vote
1
answer
224
views
Sum of negative roots of a $5^{th}$ degree monic polynomial
Let $f(x)$ be a $4^{th}$ degree monic polynomial say $f(x) = x^4 + a_1x^3+a_2x^2+a_3x+a_4$ with the property that $a_1<0, a_4>0$ and $a_2<a_3$. They by Descartes' rule of signs we can ...
- 199
1
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1
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201
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Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
1
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1
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368
views
Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?
The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions :
Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
- 128k
1
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0
answers
103
views
Real root isolation for exponential polynomials
Suppose we are given an exponential polynomial $f:\mathbb{R}\mapsto\mathbb{R}$
$$
f(t)=\sum_{i=1}^n p_i(t)e^{\lambda_i t}
$$
where $p_i(t)$s are polynomials with algebraic coefficients and $\lambda_i$...
- 1,503
1
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1
answer
367
views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
- 413
1
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2
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183
views
Convergence of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$.
Question 1: Can we ...
- 169
1
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1
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132
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Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?
Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians:
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
- 1,467
1
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1
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136
views
On a case of real-analytic interpolation
Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$.
In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
- 1,133
1
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1
answer
236
views
What does Landau symbol mean in an inequality?
I'm reading about subdifferentiable function at page 232 of Villani's Optimal Transport: Old and New.
Definition 10.5 (Subdifferentiability, superdifferentiability). Let $U$ be an open set of $\...
- 835
1
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1
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171
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Superharmonic extension 2
This question is a simplified version of the one in the MO post Superharmonic extension.
Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ ...
- 411
1
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1
answer
166
views
Question abouth Skorokhod representation of random variables (II)
This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...
- 1,835
1
vote
1
answer
307
views
Convexity of discrete Fourier transform
Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
- 595
1
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1
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518
views
Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
- 305
1
vote
1
answer
348
views
Baire class 1 and (uncountably many) discontinuities
Consider a function $f:[0,1]\to[0,1]$ which is continuous on a co-meager set $C\subset[0,1]$ and discontinuous on $D=[0,1]\setminus C$. Suppose that $D\cap I$ is uncountable for every open interval $I\...
- 4,459
0
votes
0
answers
63
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A maximisation problem : finite or not?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,399
0
votes
0
answers
124
views
Reference for the Hardy maximal function on the torus
I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
- 3,425
0
votes
1
answer
116
views
Integrable function [closed]
Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...
- 119
0
votes
1
answer
124
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Uniform estimation of an integral
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
- 339
0
votes
1
answer
143
views
Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$
$\mathcal{S}^{1/2}_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that
\begin{equation}
\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq ...
- 3,477