All Questions
724 questions
3
votes
1
answer
623
views
Forwards Feynman–Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
- 33
3
votes
1
answer
158
views
Explicit eigenvalues of matrix?
Consider the matrix-valued operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
I am wondering if one can explicitly compute the eigenfunctions of that object on ...
- 536
3
votes
1
answer
201
views
"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients
In a lecture I once attended, I remember the speaker using a result of the following nature:
$``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form
$$P(n) ...
- 739
3
votes
2
answers
472
views
Regularity of lipschitz and derivable function
Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
- 4,074
3
votes
1
answer
296
views
Does this condition characterise intervals, among subsets of the real line?
For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$:
$\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
- 60.5k
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
3
votes
1
answer
299
views
Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)
Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align*}
|f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n.
\end{align*}
For $n \ge 2$, can we ...
- 1,731
3
votes
2
answers
447
views
Algebraic curve intersecting square-grid
Let us subdivide the unit square into square-grid cells with sidelength $w$. This will give us roughly $w^{-2}$ cells.
Formally
$$ g_{ij} = \{(wi, wj) + (x,y) : 0\leq x,y\leq w \},$$
for $i,j = 0,\...
- 479
3
votes
1
answer
125
views
Relation between the local maxima and the local minima for approximating the generalized Laguerre polynomial
I have already asked my question in the link below:
Minima approximation for Laguerre polynomials
I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, ...
3
votes
2
answers
2k
views
Expected gradient vs. gradient of expectation
Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in ...
- 205
3
votes
1
answer
88
views
More on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge0$
A previous question was as follows:
Assume that $f\colon[0,1]\to[0,1]$ is a diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ ...
- 128k
3
votes
2
answers
265
views
Can one realize this as an ergodic process?
Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.
We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$
In other words: For ...
user135480
3
votes
0
answers
232
views
When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
- 34.3k
3
votes
1
answer
442
views
Error of midpoint method for functions that are not twice-differentiable
All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...
- 19.7k
3
votes
1
answer
142
views
How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?
Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user478492
3
votes
0
answers
238
views
Move one element of finite set out from A in plane
Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
- 18.8k
3
votes
1
answer
251
views
Congruence modulo 2 for q-series
This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks.
I would like to ask:
QUESTION. Is this congruence true ...
- 43.2k
3
votes
1
answer
173
views
Weak Lebesgue spaces and an estimate for BV functions
Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that
$$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$
holds for a.e. $...
- 839
3
votes
1
answer
411
views
Continuation of a smooth function, whose every derivative is strictly monotonic
Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
3
votes
1
answer
383
views
"Nice" functions on infinite-dimensional space of germs of continuous functions at a point
Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
- 24.9k
3
votes
3
answers
427
views
Quantitative analytic continuation estimate for a function small on a set of positive measure
The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In ...
- 324
3
votes
1
answer
231
views
Under which conditions the domain of the surjective function $f:[a,b]\times[c,d]\to[0,1]^{2}$ can be split s.t. the restrictions are bijective?
This is a follow-up question to this.
Since it is not always possible to construct such partition, I would like to know if there are additional restrictions which we could impose so that the wanted ...
user127351
3
votes
1
answer
250
views
Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
- 1,835
3
votes
1
answer
496
views
Prove that these two definitions of "natural" integration constant coincide when both converge
These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...
- 10.1k
3
votes
3
answers
128
views
Detecting slow growth in a finite number of queries
The following question was asked at Can you solve this problem using a finite number of queries?
:
Let $g:[0,1]\to[0,1]$ be a continuous monotonically-increasing function. You can access $g$ using ...
- 128k
3
votes
2
answers
487
views
Integrating over the Intersection of Convex Regions
Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?
The ...
- 13.2k
3
votes
1
answer
193
views
Differentiability along hyperplanes
Definition. Let us say that a function $f\colon \mathbb R^d\to \mathbb R$ is differentiable along hyperplanes in the point $0\in \mathbb R^d$, if $f\circ \varphi\colon \mathbb R^{d-1}\to \mathbb R$ is ...
- 779
3
votes
0
answers
290
views
Does there exist a supersmooth non-polynomial function?
Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
3
votes
0
answers
305
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
3
votes
1
answer
496
views
"Square root" of multiplication operator on Sobolev space
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a non-negative, smooth, uniformly bounded function with uniformly bounded first derivative. Then $f$ defines a bounded operator on $L^2(\mathbb{R}^n)$ as ...
- 1,903
3
votes
1
answer
255
views
Is this constraint convex?
I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
- 31
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
3
votes
1
answer
115
views
Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?
Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$.
Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-...
- 6,741
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
2
votes
1
answer
142
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
- 23
2
votes
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
2
votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
2
votes
1
answer
206
views
Bound for zero-crossings of heat equation
I am considering the following problem.
Let $\mathcal{P}$ the classical heat-diffusion problem:
$$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
- 393
2
votes
1
answer
150
views
When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?
First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
- 850
2
votes
1
answer
105
views
Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define
$$
\begin{split}
K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\
M_n(a,t) &:= ...
- 6,853
2
votes
1
answer
186
views
Local equality of functions implies global equality?
The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
- 1,117
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
2
votes
1
answer
154
views
Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$
Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
- 835
2
votes
1
answer
255
views
On the infimal convolution of two norms on $\mathbb R^n$
$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation*}
K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1),
\end{equation*}
\begin{equation*}
M:=M_{n,...
- 128k
2
votes
1
answer
157
views
Inequality with decreasing rearrangement and non-decreasing function
This question is a continuation of the question here.
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$...
- 459
2
votes
2
answers
667
views
Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
- 39
2
votes
1
answer
143
views
Roots of rational function
Sorry, I asked a similar question yesterday which contained a mistake in the question posed, here is the real question.
Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property ...
- 73
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 835