All Questions
13,927 questions
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112
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Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
- 1,334
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0
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148
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About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
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84
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Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
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1
answer
139
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Build an explicit "small perturbation" of the identity satisfying some properties
How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant ...
user139844
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56
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Existence of minimal subset of dual ball such that the intersection of kernels is trivial
Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} ...
- 820
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198
views
eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
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176
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A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
- 227
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74
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Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
- 9
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165
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Compact embedding of Lipschitz continuous functions
Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
- 3,388
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41
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Selectively countable Boolean algebras of sets (terminology)
I am interested in the name for the following property of a Boolean algebra $\mathcal A$ of subsets of a set $X$:
$(\star)$ for any sequence $(A_n)_{n\in\omega}$ of pairwise disjoint nonempty sets in $...
- 41.9k
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109
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Generator problem for reduced group C*-algebra
(Not sure if it is appropriate or not, if no I will delete the post)
Recently I am concerned about the number of generator of $C^{*}_{r}(\mathbb{F}_{k})$, the reduced group algebra of the free group, ...
- 523
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1
answer
162
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Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
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199
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Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?
Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
- 111
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109
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The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
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129
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Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
- 111
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75
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Is second countability an extension property for non-Hausdorff spaces?
Let $G$ be an abelian topological group and let $H$ be a non-Hausdorff closed subgroup (so that $G/H$ is Hausdorff). If $H$ and $G/H$ are second countable, is $G$ second countable?
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213
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'Diamagnetic' inequality for negative Sobolev spaces
Let us look at the subspace of smooth complex functions of $L^2(\mathbb{R}^n,\mathbb{C})$, call $H^s$ the Sobolev spaces. By the diamagnetic inequality $\lvert \nabla \lvert\psi\rvert\rvert (x) \le \...
user140442
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72
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Does $L^{\infty}[0,1]$ admits infinitely many densely defined derivations in weak* topology?
To clarify the question. First we define what is densely defined derivation. A densely defined derivation $\delta:D(\delta):\rightarrow L^{\infty}[0,1]$ where $D(\delta)$ is a dense subalgebra( in ...
- 523
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96
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Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
- 101
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177
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On connectedness of the complement
In the application of Runge type theorems on the approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has ...
- 411
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1
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431
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Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$
$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\zbar-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$,...
- 11
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0
answers
133
views
Covering map preserved under homotopy equivalence
Given a $m-$sheeted covering map from $p:M^n\to N^n$, where $M,N$ are manifolds of dimension $n$. Suppose $M$ and $N$ are homotopy equivalent to finite CW complexes $X$ and $Y$ of same dimension $k$. ...
- 1,177
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197
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Link between a categorical and an algebraic characterization of (infinite-dimensional) Hilbert space
On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1 latest Arxiv version) offers a characterization of the ...
- 2,655
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85
views
How can we define $\chi_{\Omega}(A)$?
I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic ...
- 1,305
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78
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Pareto-optimal front $F$ in $m$-dimensional space can not have more than $\mathbf{H}_{m-2}(F)$ homology groups
I need to prove that a Pareto-optimal front $F$ in $m$-dimensional space (i.e. $m > 1$) can not have more than $\mathbf{H}_{m-2}(F)$ homology groups.
What it simply means that in a 2-dimensional ...
- 101
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95
views
Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
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1
answer
358
views
Continuity of point evaluation on space of Hölder functions with $L^p$ norm
Let $L>0$ and $\Omega \subset \mathbb{R}^n$ a bounded Lipschitz domain. Define
$$
B_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \...
- 319
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114
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Analysis, matrix exponential, infimum and limit
I was working in this problem for a long time and I didn't have success.
Someone could help me, please?
The problem:
Let $f: \mathbb{R}^{n^2} \times \mathbb{Z}^{n} \longrightarrow \mathbb{R}$ defined ...
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0
answers
85
views
An amenable operator algebra has the total reduction property
This is from
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
- 101
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109
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Operator algebra on an invariant subset
In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...
- 109
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291
views
Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
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0
answers
48
views
Surjectivity of the limiting operator
Consider the operator
\begin{eqnarray*}
K_{n} &:&L^{2}(0,1)\longrightarrow L^{2}(0,1)^{n}, \\
u(x) &\mapsto &A_{n}U_{n}(x)=A_{n}(u(\frac{x}{n}),u(\frac{x+1}{n}),...,u(%
\frac{x+n-1}{n})...
- 617
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101
views
Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?
I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
- 537
0
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0
answers
96
views
A property of the Hilbert transform involving the cotangent function
A lemma of a paper by T. Elgindi and I.-J. Jeong (Arch. Rational Mech. Anal. 235 (2020) 1763–1817, Lemma 2.2) states the following:
Let $g(z)=\operatorname{sgn}(z)k(|z|^\alpha)$ with $k$ smooth and $k(...
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72
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Li-Yorke sensitivity Vs Li-Yorke dense chaos
Let $X$ be a compact metric space, $X*X$ its cartesian product, and $A$ a subset of $X*X$.
Are the following two properties the same, or e.g. one is stronger than the other?
$A$ is dense and residual ...
- 141
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172
views
Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?
I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
- 617
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0
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79
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Compact operators and projective tensor space
I know that the space of all the bounded linear maps between two Banach spaces, denoted by $L(X,Y)$, has a relationship with the projective tensor space of $X$ and $Y$,
$$({X \widehat\otimes_{\pi} Y})...
- 11
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0
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72
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Di Perna-Lions theory for transport equations with an additional integral operator
I'm looking for a reference about some possible generalization of the well-known Di Perna-Lions theory for transport equations (say, on $[0,T] \times \mathbb{R}^d$) of the form
\begin{align}
\...
- 161
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161
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When does a positive operator preserve invertibility
Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...
- 175
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139
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Why the name 'regular' space?
It is well known that a regular space is a topological space $X$ with these two properties:
1)All one point sets are closed.
2)For every $x\in X$ and every closed set $B$ (such that $x\notin B$), ...
- 883
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120
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Are bounded maps determined by their images on the algebraic tensor product?
Let $\mathcal A,\mathcal B,\mathcal C$ be von Neumann algebras. Let $F:\mathcal A\otimes\mathcal B\to\mathcal C$ be a bounded linear map. Assume $F(a\otimes b)=0$ for all $a,b\in\mathcal A,\mathcal B$....
- 896
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63
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a lemma on interval translation map
Consider the map $S:T^1 \to T^1$ where $x \mapsto x+c_j$ , mod 1 where $c_j$'s are real numbers. We represent $T^1$ as a union of disjoint subsegments $M_j=[t_j,t_{j+1})$, $j=0 , \cdots ,n , t_0=t_n$....
- 145
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0
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106
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Extension of a Hilbert basis
The picture below is taken from this paper: http://real.mtak.hu/22877/.
The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
- 617
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votes
0
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154
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When is the heat semigroup Gibbs?
Defining the Laplacian on a region $Ω$ of $\mathbb{R}^d$ with Dirichlet boundary conditions, under what conditions on the region (or any other possible assumptions) is the semigroup it generates Gibbs,...
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0
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67
views
Dual of isometric copies into dual Banach spaces
Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens ...
- 1,879
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votes
0
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129
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Certain decompositions of decomposable Banach spaces
Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
- 1,453
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0
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144
views
Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
- 1,879
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0
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93
views
Regularity of semilinear parabolic PDE in the whole space
I need regularities in Holder space of the following parabolic PDE:
$$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
- 1,399
0
votes
0
answers
39
views
Topology on upper semicontinuous functions and Baire one functions
Which is the standard topology on upper semicontinuous functions and on functions of Baire class one, in both cases defined on a compact real interval?
0
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0
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247
views
Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93