All Questions
13,927 questions
2
votes
1
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118
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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 ...
- 60.5k
5
votes
1
answer
221
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How big is the class of all closed range bounded linear operator?
Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
- 63.9k
1
vote
1
answer
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth 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 curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 128k
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 60.5k
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
- 3,054
3
votes
1
answer
191
views
Extensions of bounded uniformly continuous functions
Let $X$ be a uniform space, $S\subseteq X$ and $f:S\to \mathbb{R}$ bounded uniformly continuous, then there exists a uniformly continuous extension of $f$ to $X$. (Katětov, 1951)
I am looking for ...
- 11.4k
1
vote
3
answers
359
views
For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?
Let $F$ be a tempered distribution on $\mathbb{R}^2$ and $n \in \mathbb{N}$ be a fixed natural number. I wonder what exactly it means by
$\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$ where $x,y \...
- 21
8
votes
0
answers
177
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Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
CommunityBot
- 1
0
votes
1
answer
123
views
Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
- 131
3
votes
0
answers
159
views
A question regarding weak Whitney embedding theorem
The weak Whitney embedding theorem states that any continuous map $f: D^n \to \mathbb{R}^{2n+1}$ (Let us focus on $D^n$ for this question) can be approximated (in $C^0$-norm) by embeddings. A counter ...
- 31
0
votes
0
answers
85
views
Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
2
votes
0
answers
84
views
Question about the Nemytsky operator on $L^p$ space
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
- 3,425
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
3
votes
0
answers
147
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 6,367
0
votes
0
answers
79
views
Is the Bures metric equivalent to the Euclidean one?
Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
- 1,334
3
votes
0
answers
119
views
The topological entropy of potential space filling curves on the unit interval
By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1$...
- 356
2
votes
1
answer
95
views
Specific distance between sets of points
Let us have closed curve without self-intersections,
initial point $O$ and curve parameter $t$, $0 \leq t \leq t_{\max}$ so $t(O) = 0 = t_{\max}$.
There are two sets of points on the curve, which are ...
- 512
1
vote
0
answers
90
views
Well-embedded type property for bounded functions
According to @Tyrone the term well-embedded set was first used in Measures on Metacompact Spaces by W. Moran.
In the article Extensions of Zero-sets and of Real-valued Functions by R. Blair and A. ...
- 1,211
11
votes
1
answer
227
views
Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$
Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
- 41
3
votes
1
answer
227
views
Algebraic and continuous duals of an inverse limit of finite dimensional vector spaces
I have been trying to understand the following section of a paper "Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique" by Gabriel Dospinescu and Arthur-César Le ...
- 12.9k
11
votes
2
answers
8k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
- 201
3
votes
1
answer
207
views
Embedding of a Banach space in $\ell_\infty(\Gamma)$ with a subspace embedded in a copy of $\ell_\infty$
In this question (which I think may be interesting in its own) I was asking if we can find a copy of $\ell_\infty$ between a separable subspace $Y$ contained in $\ell_\infty(\Gamma)$ and the whole ...
- 271
3
votes
1
answer
169
views
Copy of $\ell_\infty$ inside $\ell_\infty(\Gamma)$ containing given subspace
To complete a proof I need to know if the following is true:
Given a non-empty set $\Gamma$ and a separable subspace $Y$ of $\ell_\infty(\Gamma)$, there exists a subspace $A$ of $\ell_\infty(\Gamma)$ ...
- 271
1
vote
1
answer
170
views
Mean of probability distribution
I have a probability distribution defined by the following density function:
$f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
- 188k
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
1
vote
0
answers
98
views
$(\lambda I-A)^{-1}-(\lambda I-B)^{-1}$ compact implies $\sigma_\text{ess}(A)=\sigma_\text{ess}(B)$
Suppose $H$ is a Hilbert space and $A$, $B$ are two adjoint operators on it (not necessarily bounded), satisfying $D(A)=D(B)$.
Question: If $\exists \lambda\in \rho(A)\cap\rho(B)$ such that $(\lambda ...
- 12.9k
2
votes
1
answer
236
views
A sensible topology on the space of continuous linear maps between Fréchet spaces
Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
- 12.9k
6
votes
2
answers
349
views
Mutual metric projection
Given a subset $S\subseteq \mathbb{R}^n$, the metric projection associated with $S$ is a function that maps each point $x\in \mathbb{R}^n$ to the set of nearest elements in $S$, that is $p_S(x) = \arg ...
- 2,830
2
votes
0
answers
71
views
How to naturally define an output space with certain properties
Consider the following regression problem $v=A(u) + \varepsilon$
for some operator $A:\mathcal{U} \rightarrow \mathcal{V}$ and some function spaces $\mathcal{U},\mathcal{V}$, mapping from $\mathcal{X}$...
- 51
2
votes
0
answers
60
views
Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$
However, in various literatures, I ...
- 57
18
votes
0
answers
323
views
The analogy between dualizable categories and compact Hausdorff spaces
Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{...
- 12.9k
1
vote
1
answer
132
views
Deriving a specific bound for functions in Hardy Space
Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space)
Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
- 24.2k
9
votes
2
answers
471
views
Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
- 4,713
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 63.9k
1
vote
0
answers
66
views
The derivative of semigroup in the weak sense imply strong sense
Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
- 775
9
votes
0
answers
258
views
Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...
- 2,826
7
votes
1
answer
184
views
Functional calculus on the Schwartz space instead of $L^2$?
As far as I know, functional calculus is typically carried out on Hilbert spaces with (possibly unbounded) self-adjoint operators.
However, I wonder if there is a way to do it on the space of test ...
- 76
7
votes
2
answers
534
views
Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
- 557
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
-1
votes
2
answers
251
views
$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 12.9k
13
votes
1
answer
329
views
Is there a metric compactification that doesn't create new paths?
Every separable metric space $A$ has a metrizable compactification, i.e. a compact metrizable space $X$ for which $A$ embeds topologically as a dense subspace of $X$. There are many approaches to ...
- 12.9k
0
votes
1
answer
142
views
"Locally compact"-ly generated topological spaces
Let $P$ be a property of topological spaces - here I am interested in "compact" and "locally compact".
A topological space $X$ is $P$-ly-generated if, for any topological space $Y$,...
- 42.4k
5
votes
2
answers
432
views
Does closedness of the image of unit sphere imply the closed range of the operator
Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
- 12.6k
1
vote
0
answers
63
views
Is there any other norms besides cut norm defined on graphon?
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
- 359
0
votes
0
answers
66
views
convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
CommunityBot
- 1
2
votes
0
answers
96
views
Isometric Schröder-Bernstein theorem for injective Banach spaces?
It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space.
Let $X$, $Y$ be two injective Banach spaces such that,
...
- 63.9k
7
votes
0
answers
131
views
Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
- 3,425
3
votes
0
answers
124
views
Injective envelope of B(H)
$B(\ell^2)$ is an injective operator system by a result of Arveson. However, $B(\ell^2)$ is not an injective Banach space, since it is not linearly isomorphic to a $C(K)$ space (for instance, $C(K)$ ...
- 2,605
3
votes
1
answer
646
views
Is the Hopf link a Brunnian link?
I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on links). Not wishing to write ...
CommunityBot
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