All Questions
5,878 questions
6
votes
1
answer
816
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,155
4
votes
1
answer
249
views
Does this functional admit an absolute minimizer?
This is a close relative of the following problem.
Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
- 6,155
7
votes
1
answer
346
views
Mean Cauchy sequences
Let $X$ be a complete metric space. Suppose a sequence of elements $x_n$ is Cauchy in mean, in the sense that
$$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N} \sum_{...
- 6,155
9
votes
3
answers
696
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 7,817
2
votes
1
answer
309
views
Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
CommunityBot
- 1
7
votes
1
answer
224
views
Does the decomposability of $\mathbb{R}$ imply analytic LLPO?
By "BISH" I mean constructive mathematics without axiom of countable choice.
By $\mathbb{R}^f$ I mean real numbers as fundamental sequences of rational numbers and by $\mathbb{R}^d$ I mean ...
- 32.5k
7
votes
2
answers
706
views
Poisson binomial conjecture
Let $X_i\in\{0,1\}$
be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$
and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$,
for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=...
- 148k
2
votes
1
answer
315
views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions?
I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 60.5k
3
votes
1
answer
240
views
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and
$$
[f]_{\frac{2}{\...
- 7,817
4
votes
1
answer
110
views
Scaling of stopped Hölder norm of Brownian motion
I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.
For fixed $T>0$, self similarity ...
- 5,474
0
votes
1
answer
153
views
Lebesgue measure of the level set of sum of two nonnegative functions
Let $f, g:\mathbb{R}^n\to \mathbb{R}$ be nonnegative functions such that $g$ is a strictly positive homogeneous function. As commented by Fedor Petrov below, one may not have that for any $\lambda>...
- 109k
4
votes
1
answer
340
views
Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?
Let $S^1$ be the circle, let us consider a function $f(x,t): S^1 \times [0,\infty) \to \mathbb{R}$ such that
\begin{equation}
\int_0^T \int_{S^1} \lvert f(x,t) \rvert dxdt <\infty
\end{equation}
...
CommunityBot
- 1
3
votes
2
answers
614
views
Should coffee machines be placed at the region's boundary?
This is a continuation of Should coffee machines be deconcentrated?
Recall that some region is denoted by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the ...
- 128k
-4
votes
1
answer
302
views
A Question in Fourier Analysis proposing a conjecture
Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
- 698
1
vote
1
answer
125
views
Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
CommunityBot
- 1
7
votes
1
answer
179
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 105k
6
votes
2
answers
492
views
Does this polynomial have a real zero less than or equal to $1/2$?
Is the smallest root $x$ of
$$
10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\
+2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
- 105k
9
votes
3
answers
2k
views
Smallest root of a degree 3 polynomial
Is it true that the smallest root $t$ of the polynomial
$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
- 105k
-1
votes
1
answer
122
views
Divergent summation [closed]
Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series:
$$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
- 21
1
vote
1
answer
127
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
CommunityBot
- 1
2
votes
1
answer
231
views
Is Boltzmann entropy well-defined for arbitrary probability density function?
$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by
$$
\varphi (s) :=
\begin{cases}
0 &\text{if} \quad s =0 , \\
s \...
- 128k
4
votes
1
answer
96
views
On the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$.
Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$?
Remark 1: A numerical experiment suggests that $...
- 109k
1
vote
1
answer
130
views
Existence of solutions to a series of integral equations
I am trying to solve the following integral equation analytically:
$$
\sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T],
$$
where $(f_n(t))_n$ is the unknown ...
CommunityBot
- 1
11
votes
2
answers
425
views
Maximization of a cubic form over the $14$-dimensional sphere
For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number.
Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$
the sum
$$\sum_{1\le i<...
- 109k
45
votes
5
answers
3k
views
An "analytic continuation" of power series coefficients
Cauchy residue theorem tells us that for a function
$$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$
the coefficient $a(k)$ can be extracted by an integral formula
$$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-...
- 6,367
1
vote
0
answers
125
views
Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$
Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
- 63.9k
2
votes
0
answers
58
views
$L^2$ approximation of delta functions on real algebraic varieties and asymptotic bounds
Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Consider a probability measure $\mu$ on $X(\mathbb{R})$, absolutely continuous with respect to the Lebesgue measure induced ...
- 63.9k
0
votes
0
answers
100
views
Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube
A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if
\begin{equation*}
\dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K,
\end{equation*}
for any $x,y\in \...
- 69
0
votes
0
answers
71
views
Nearest integer to fractional power series
Let $k$ be a positive integer. Let
$$\displaystyle f_0(x) = a_n x^{\frac{n}{k}} + \cdots + a_1 x^{\frac{1}{k}} + a_0 + \sum_{h \geq 1} a_{-h} x^{-\frac{h}{k}}$$
be a Laurent series in the variable $x^{...
- 26.9k
2
votes
0
answers
152
views
Riesz transform of constant function
My one-line question would be, what is the Riesz transform of the constant function, identically equal to 1 on $\mathbb{R}^2$?
But more fundamentally, my question stems from some confusion about the ...
8
votes
0
answers
103
views
Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
- 7,817
2
votes
1
answer
474
views
Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?
Let
$$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$
be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
- 5,409
9
votes
1
answer
378
views
Does “on average” Hölder continuity imply Hölder continuity?
Let $\Omega$ be a smooth, bounded, connected open subset of $\mathbb R^n$.
A function $f: \Omega \to \mathbb R$ is said to be Hölder continuous on average of order $\alpha$, for $0 < \alpha < 1$ ...
- 920
7
votes
0
answers
313
views
Did Lebesgue like non-measurable set or not?
I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:
Vitali's nonmeasurable set, ...
- 3,065
1
vote
0
answers
46
views
Optimal transport and the geometry of singular measures on fractal Sets
Let $K$ be a self-similar fractal set in $\mathbb{R}^n$ with Hausdorff dimension $d < n$, equipped with a self-similar measure $\mu$ supported on $K$. Let $\mathcal{P}(K)$ denote the space of ...
- 11
3
votes
0
answers
45
views
Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces
Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...
10
votes
1
answer
518
views
Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions
Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
CommunityBot
- 1
3
votes
1
answer
187
views
Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
2
votes
0
answers
65
views
Construct a differentiable function whose gradient has a prescribed modulus of continuity
$\newcommand{\bR}{\mathbb{R}}$
Let $\alpha := e^{-(1 + \sqrt{2})}$. We define the following modulus $\psi : \bR_+ \to \bR_+$ of continuity
$$
\psi (x) :=
\begin{cases}
0 &\text{if} \quad x =0 , \\
...
- 835
7
votes
0
answers
249
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
2
votes
0
answers
65
views
Generalized Fourier transforms associated to Schroedinger operators
Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
- 4,135
1
vote
0
answers
42
views
Approximation of the function $f(z)=z^2/|z|$ by $C^1$ immersions
Let $D$ denote the unit disk in $\mathbb C=\mathbb R^2$. We consider the function $f:D\rightarrow\mathbb C $ defined by $$f(z):=\frac{z^2}{|z|}.$$ Then as proved in Global invertibility (p324 Remark 4)...
- 435
2
votes
0
answers
120
views
On mollifiers acting between $L^2$ and Sobolev spaces
(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.)
Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by
$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
- 6,367
20
votes
1
answer
2k
views
How rich is the richest person in a society satisfying the Pareto principle?
The Pareto Principle roughly states that in many societies, the top 20% of people hold over 80% of the wealth. Suppose we had a society that satisfied this principle in every stratum of society - how ...
- 6,367
5
votes
0
answers
190
views
Number of discrete Lipschitz functions with given Lipschitz constant
Fix $T, K, N \in \mathbb Z_+$. How many distinct Lipschitz functions $f: \{0, \dots, T\} \to \mathbb Z$ are there with Lipschitz constant $K$, and supremum norm at most $N$ satisfying $f(0) = 0$?
In ...
- 6,155
9
votes
1
answer
366
views
Can the canonical Eudoxus-real representatives be defined easily?
(See e.g. here for background on the Eudoxus reals, which motivates this question.)
Let $\mathcal{Z}=(\mathbb{Z};+,<)$. Say that a Eudoxus function is an $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such ...
- 148k
0
votes
0
answers
101
views
A special Hamel basis and a special additive function
On mathstackexchange I recently
asked
whether for an irrational number $a$ a special Hamel basis of type $\bigcup_{i\in I}\{x_i,y_i,ay_i\}$ exists,
where $x_i, y_i$ and $ay_i$ are $\mathbb Q$-...
- 687
2
votes
1
answer
127
views
Density of smooth functions in weighted Sobolev space
Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
- 3,425
0
votes
0
answers
52
views
References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
CommunityBot
- 1
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) ...
- 128k