All Questions
13,944 questions
-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
8
votes
0
answers
115
views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
-1
votes
1
answer
86
views
how take weak derivative of norms in hilbert spaces?
Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$.
Let $u∈L^2 ([0,T];V); ...
- 206
1
vote
0
answers
72
views
How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
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
$$
\varphi (s) :=
\begin{cases}
0 &\text{if} \quad s =0 , \\
s \...
- 128k
4
votes
1
answer
96
views
On the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$.
Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$?
Remark 1: A numerical experiment suggests that $...
- 109k
0
votes
0
answers
96
views
Sufficient condition for weak convergence in Banach spaces
The question is quite elementary but nonetheless no proof or counter example comes to mind immediately.
Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
1
vote
1
answer
130
views
Existence of solutions to a series of integral equations
I am trying to solve the following integral equation analytically:
$$
\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
1
vote
0
answers
55
views
Characterizing one-sided M-projections on real C*-algebras
Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map
$$
v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
- 11
45
votes
5
answers
3k
views
An "analytic continuation" of power series coefficients
Cauchy residue theorem tells us that for a function
$$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
1
answer
203
views
Hyperplane separation of a concave functional and a point, in domain theory
Problem:
Let $D$ be an $\omega$-BC domain, and $[D\to[0,\infty]]$ be the space of Scott-continuous nonnegative functions on $D$, equipped with the obvious ordering and the Scott-topology.
EDIT: I don'...
CommunityBot
- 1
14
votes
4
answers
1k
views
$L^p$ norm means
Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
3
votes
1
answer
375
views
Dimensionality reduction for total variation
Let $P_i,Q_i$, $i\in[n]$,
be distributions on a finite set $\Omega$.
We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions.
For each $i\in[n]$, define the
dimensionally-...
- 6,541
2
votes
1
answer
75
views
How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$
Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$.
Assume ...
- 3,425
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
7
votes
2
answers
824
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
CommunityBot
- 1
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
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 5,474
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
18
votes
1
answer
2k
views
Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain
Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
CommunityBot
- 1
7
votes
2
answers
248
views
Subspaces of $\ell_\infty^3$
Let $a,b\in\mathbb C$ be suc that $\max\{|a+b|,|a-b|\}\leq 1$ but $|a|+|b|>1.$ According to this paper by Arias, Figiel, Johnson and Schechtman https://www.jstor.org/stable/2155206?origin=crossref#...
- 31.5k
2
votes
0
answers
152
views
Riesz transform of constant function
My one-line question would be, what is the Riesz transform of the constant function, identically equal to 1 on $\mathbb{R}^2$?
But more fundamentally, my question stems from some confusion about the ...
8
votes
0
answers
103
views
Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
- 7,817
2
votes
1
answer
121
views
Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$
For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite,
$$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$
Using the triangle inequality $|\eta-\xi|\le |\eta|...
- 6,476
4
votes
0
answers
148
views
Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
2
votes
1
answer
474
views
Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?
Let
$$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
3
votes
2
answers
137
views
Non-complete space verifying uniform boundedness
Recently, I have seen the so-called uniform boundedness theorem, which says:
Let $(X,∥⋅∥)$
be a Banach space and $(Y,∥⋅∥)$
be a normed linear space. Let $A⊂B(X,Y)$
be a pointwise bounded family of ...
- 419
2
votes
1
answer
141
views
(Sub)Optimality of random transport
Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
- 6,001
2
votes
0
answers
40
views
Characterization of critical point of an integral operator
I have an integral operator and I wonder how I can characterize the critical point.
I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question.
...
- 115
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
3
votes
1
answer
219
views
Moment problem, ergodicity and spectral gap on the space of tempered distributions
Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$.
Just below formula [5] in p.122 of the Fröhlich paper, ...
- 3,477
0
votes
0
answers
42
views
Geometric alignment of adaptive models on evolving manifolds
Let $(M_t)_{t\in[0,T]}$ be a smooth family of compact $d$-dimensional Riemannian submanifolds of $\mathbb{R}^n$. Consider a function $f_t : \mathbb{R}^n \to \mathbb{R}$ evolving over time $t \in [0,T]$...
- 63.9k
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
CommunityBot
- 1
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
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
7
votes
0
answers
269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 460
3
votes
1
answer
287
views
Expectation comparison inequality for concave function of symmetric random variables
Suppose that $X_i$, $i\in[n]$ are
independent symmetric
random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
- 6,541
3
votes
0
answers
45
views
Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces
Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
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
250
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 ...
- 4,143
0
votes
0
answers
20
views
Decomposition of measures orthogonal to the algebra $R(K_1 \times \ldots \times K_n)$ - Can it be done via projection-preserving products of bands?
See "Measures orthogonal to tensor products of function algebras" by Marek Kosiek. Here, it is described for the two-dimensional case. It uses another, more general, approach to OB Bekken's ...
- 63
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