All Questions
5,659 questions
13
votes
2
answers
813
views
A dichotomy for everywhere differentiable eikonal functions
Let $f: \mathbb R^n \to \mathbb R$ be everywhere differentiable, with $|\nabla f| = 1$ almost everywhere. Is it true that $|\nabla f| = 0$ or $1$ everywhere?
- 2,155
3
votes
1
answer
198
views
Can gradient zero implies that a function is constant with Hörmander vector fields
Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
CommunityBot
- 1
0
votes
1
answer
71
views
Upper bound on higher order derivatives of $\frac{1}{v(t)}$
Suppose that $ v(t) >l>0$ and
$$
\vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}.
$$
Can we give an upper bound for
$$
(\frac{1}{v(t)})^{(k)}
$$
?
Attempt:
We first compute the first fourth order ...
- 128k
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
CommunityBot
- 1
1
vote
1
answer
69
views
Exhausting sequences contain a $\pi$ lift of a subset with a $(1-\delta)$ factor
Let $\pi : Y \to X$ be a measurable map between the $\sigma$-finite measure spaces $(Y, \mathcal{B}, \nu)$ and $(X, \mathcal{A}, \mu)$. Suppose there exists $c \in (0, \infty)$ such that for all $A \...
CommunityBot
- 1
-2
votes
0
answers
64
views
A Problem using Limits of Sequences of Functions
Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
- 1
3
votes
2
answers
153
views
On nowhere differentiability of functions that just barely fail to be Lipschitz
By Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. I am wondering how badly this pointwise differentiability fails for functions that “just barely” fail to be Lipschitz....
3
votes
0
answers
95
views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 312
0
votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 128k
1
vote
2
answers
117
views
If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 128k
1
vote
0
answers
39
views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
-1
votes
0
answers
132
views
Trig conjecture about square roots and Arcsin
Let $r(a,b)$ be a rational number depending on positive integers $a,b$ and $r(a,b)$ being nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$.
Let $C(b)$ be a squarefree positive ...
- 769
1
vote
1
answer
188
views
Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?
This question is related to This question.
When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
- 128k
0
votes
2
answers
364
views
Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
- 128k
8
votes
1
answer
258
views
Sequential colimit of iterated quotients of Cauchy sequences
We work in constructive mathematics.
The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
CommunityBot
- 1
3
votes
0
answers
98
views
Square Roots of Non-Negative Even Functions
I'm trying to study properties of maps between quotients of representations of compact Lie groups and I stumbled upon the following problem. Suppose you have a smooth function $f:\mathbb{R}\to\mathbb{...
- 31
4
votes
1
answer
214
views
Characterisation of Sobolev spaces using their Lipschitz approximations
Let $f \in W^{1, p} (\mathbb R^n)$. A classical approximation theorem (see for instance, the book by Evans and Gariepy) says that we can approximate $f$ by Lipschitz functions, in the sense that for ...
- 498
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
- 5,409
4
votes
1
answer
256
views
Approximating an $L^1$ function with Riemann sums
Note: Here all functions are genuine functions, i.e. pointwise defined measurable functions instead of defined only a.e.
Let $f: [0, 1] \to \mathbb R$ be an arbitrary $L^1$ function. Of course, $f$ is ...
- 6,155
-1
votes
0
answers
51
views
Existence of continuous crossection
Let $G$ be a second countable locally compact Hausdorff groupoid. Then $G= \sqcup_{u\in G^{0}}G^{u}$ where $G^{0}$ is the unit space of $G$ and $G^{u}=r^{-1}(u)$. Here $r:G \to G^{0}, r(x)=xx^{-1}$.
...
0
votes
1
answer
86
views
Lattice of functions and their minimal separating set upto topological equivalence
There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures.
Let's start with the set of all functions $F: \mathbb{R} \...
- 39
2
votes
0
answers
90
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 93
4
votes
1
answer
119
views
Proving Equal Set Sizes in Sequential Point Selection on a Real Interval with Variable-Length Intervals
I'm here as an engineer working on a point sampling algorithm and I've noticed that when I perform the algorithm on an ordered set of points in one direction it selects the exact same number of points ...
- 151
3
votes
1
answer
212
views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation:
$$
\begin{cases}
\nabla \times (F\times\mathbf v)=g, \\
\operatorname{div}(\mathbf v)=0,
\end{cases}\label{1}\tag{1}
$$ where $F$ and $g$ are given vector fields. The ...
- 12.9k
3
votes
1
answer
851
views
Convergence of a certain sum
Suppose $ g_i: [0, 1] \to \Bbb R$, $i\in\Bbb N$, are $C^1$ functions and that there is some $c > 0$ such that for every $0 < \epsilon < c$, the functions
$$
s(\epsilon)_i := \sum_{k=0}^i {\...
CommunityBot
- 1
0
votes
0
answers
44
views
Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
5
votes
1
answer
256
views
Does a special property hold if the Archimedean property for reals doesn't hold?
Suppose $\mathbb{R}^e=A \cup B$ in which $A \cap B=\varnothing$ and there exist real numbers $a_0$ and $b_0$ such that $a_0 \in A$ and $b_0 \in B$. My question is, can we construct $a \in A$ and $b \...
6
votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 105k
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
- 457
2
votes
1
answer
117
views
When is a $p$-th order stationary point of a polynomial actually a local minimum?
Definition: For integer $p\geq 1$, we say $x\in \mathbb{R}^d$ is a $p$-th order stationary point of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ if there exists a $C>0$ and an $\epsilon>0$ ...
CommunityBot
- 1
2
votes
1
answer
734
views
Is Weierstrass function nowhere pointwise Lipschitz?
Consider the classical Weierstrass function
$$
W(x)=\sum_{n=1}^\infty \frac{e^{i2^nx}}{2^n}.
$$
It is a well-known result that this function is nowhere differentiable (Hardy, TAMS 1916, Thm 1.31). In ...
- 5,407
1
vote
1
answer
112
views
Bounding a Riemann sum by its integral limit?
Let $M_{n}(\mathbb{C})$ denote the space of complex $n \times n$ matrices and, for $a>0$, $a \in \mathbb{R}$ fixed, let $A: [0,a) \to M_{n}(\mathbb{C})$ be a given function. I will write $A(t) = (...
- 6,155
6
votes
2
answers
772
views
Finiteness of an integral
In a paper I am reading, the following seems to be claimed:
Let $f:[0,\infty)\to [2,\infty)$ be a continuous, monotonically increasing function with $\lim_{x\to\infty}f(x)=\infty$ and let $\alpha>3/...
- 109k
3
votes
1
answer
247
views
Is the derivative of a Lipschitz function continuous a.e.?
Let $f:(a,b) \to \mathbb R$ be Lipschitz.
The derivative $f'$ exists on some set $D \subset (a,b)$ of full measure and is bounded (by Rademacher).
Is $f'$ continuous (or some representative) on the ...
- 6,155
0
votes
1
answer
224
views
Computing the limit of a kind of "mean value" iteration
Let $n$ be a positive integer and $x_1,\ldots,x_n$ be real number, and define a kind of "mean value" iteration like
$$
x_i^{(k+1)} = \frac{x_i^{(k)}+x_{i+1}^{(k)}}{2}, i=1,\dotsc,n,\quad x_{...
CommunityBot
- 1
5
votes
1
answer
534
views
Minimiser of a certain functional
Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \...
- 6,155
2
votes
1
answer
288
views
How to estimate an integral by the variation and upper bound of the integrand?
Suppose that $f$ is a continuous function on $\mathbb{R}$. I want to estimate the definite integral
$$ I:= \int_{0}^a [f(x)-f(0)]dx $$
by the upper bound $M = \sup_{x\in[0,a]}|f(x)|$ and the variation ...
CommunityBot
- 1
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
CommunityBot
- 1
7
votes
2
answers
331
views
Does every subset of $\mathbb N$ with full natural density contain arbitrarily long geometric progressions?
We use the standard definition of natural density. We say a subset of $\mathbb N$ has full natural density if it has natural density $1$.
Question: Does every subset of the naturals with full natural ...
- 7,013
4
votes
1
answer
1k
views
Does the Lebesgue Differentiation Theorem hold for regular polytopes?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 28k
0
votes
0
answers
57
views
Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
1
vote
1
answer
157
views
Is finding the CDF from the Laplace transform well-posed?
In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
- 128k
1
vote
0
answers
45
views
Existence of optimal entropic weights for empirical modeling
Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
- 111
61
votes
1
answer
5k
views
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...
- 12.9k
5
votes
0
answers
608
views
What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?
1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
- 6,367
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
0
votes
0
answers
68
views
Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.
If ...
2
votes
0
answers
116
views
Existence of a sequence of real numbers
Let
$$g_{c;k}(z):=\frac{2 (c-z-1)^{k+2}}{(k+1) (k+2)}+\frac{1}{2} (-c+z+2)^2 z^k+\frac{-2 c (k+2)+4 k+6}{(k+1) (k+2)}+\frac{2z}{k+1}.$$
Do there exist $c\in(1,3/2)$ and a sequence $(a_k)_{k=0}^\infty$ ...
CommunityBot
- 1
3
votes
1
answer
220
views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 498
2
votes
0
answers
189
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185