All Questions
10,447 questions
3
votes
0
answers
58
views
Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
- 31
4
votes
1
answer
158
views
Is the image of a complemented subspace complemented?
This question has been crossposted from mathstackexchange:
Let $X, Y$ be two Banach spaces and $T:X\to Y$ a continuous surjection. Assume $Z$ is a complemented subspace of $X$ and that $T(Z)$ is ...
- 655
5
votes
1
answer
175
views
Orthogonal projection onto cones in inner product spaces
Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$,
$$A=\sum_i \lambda_i x_ix_i^*,$$
one can define the positive and negative ...
- 4,484
0
votes
0
answers
59
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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
- 63
3
votes
1
answer
177
views
Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
- 131
6
votes
1
answer
248
views
The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients
Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by
$$Lu = \partial_i(a^{ij}...
- 91
15
votes
1
answer
601
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
- 32.5k
10
votes
3
answers
455
views
When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
2
votes
1
answer
113
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Showing $\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}\leq \inf _{a \neq 0} \frac{\|\hat{a}\|_{\infty}}{\|a\|}$ in a commutative banach algebra
Suppose $A$ is a commutative Banach algebra, and let $u=\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}$, $v=\inf _{a \neq 0} \frac{r(a)}{\|a\|}$ ($r(a)$ is the spectral radius of $a$). I need to ...
- 775
3
votes
1
answer
203
views
Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
- 1,374
10
votes
0
answers
225
views
Can the trace be computed in any Schauder basis?
I'm cross-posting this question from Math.SE, as it didn't get much attention there.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
- 233
2
votes
0
answers
220
views
Ultraviolet divergences of entanglement entropy in QFT
I've often read that entanglement entropy in quantum field theory is ill-defined because local algebras are generally of type III, which implies that a trace doesn't exist. For a normal state $\omega_{...
- 424
0
votes
0
answers
34
views
Locally compact groupoid with range map restricted to isotropy groupoid is open
Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋
a locally compact space is such that isotropy subgroups of H are isomorphic to each other.
Can this be an example of a ...
3
votes
0
answers
116
views
On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
- 131
2
votes
0
answers
142
views
A $C^*$-algebra with the bidual $B(H).$
Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} \equiv B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$...
2
votes
0
answers
126
views
Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
- 307
3
votes
0
answers
69
views
Perturbation of one-parameter groups of unitary operators
Let $H$ be a Hilbert space and let $h$ be a fixed, densely defined, possibly unbounded, self-adjoint operator on $H$. Letting $B(H)$ denote the space of all bounded operators on $H$, it is well ...
- 2,263
0
votes
0
answers
29
views
On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
- 765
2
votes
0
answers
95
views
On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
3
votes
1
answer
79
views
How to deal with singularities in thin plate splines?
Follow up from this question Thin-Plate-Spline understanding and solution.
In the general case of $\mathbb{R}^N$ the following problem (interpolant which minimizes the Thin Plate Energy, specifically ...
- 115
2
votes
1
answer
179
views
Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$
Let us consider the heat equation
$$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$
where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
- 452
1
vote
2
answers
164
views
Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 35
1
vote
0
answers
88
views
Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
1
vote
2
answers
209
views
Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
- 29
1
vote
1
answer
152
views
SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
- 151
3
votes
0
answers
98
views
Bounding the norm of a sum of fourth-order Gaussian vectors
Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$, where $\Sigma = \operatorname{diag}(\lambda_1, \cdots, \lambda_k)$ and
$\lambda_1 \ge \cdots \ge \...
- 97
1
vote
0
answers
248
views
Solving functional analysis problems by using Algebraic geometry
I am thinking about some open problems in nonlinear functional analysis and I just wanted to know if there are any problems that have been solved by using Algebraic geometry techniques in these fields....
2
votes
0
answers
82
views
What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$
Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping
$$
f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X,
$$
with $f(0)=0$
is a radial and maps ...
- 5,327
6
votes
1
answer
256
views
Example/Existence of Positive Linear Functional which is NOT Hermitian
We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
- 173
0
votes
0
answers
93
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,706
0
votes
0
answers
56
views
What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?
I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay.
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$
is $n$ (thanks to this ...
- 113
1
vote
1
answer
151
views
Some operators on spheres
Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
- 1,879
2
votes
1
answer
89
views
The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$;
see this for instance.
It seems that we can deduce the time derivative of the heat ...
- 121
5
votes
2
answers
258
views
Boundary value of Sobolev space
Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
- 177
0
votes
0
answers
118
views
Find the maximum of an expression under the logconcave assumption
Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
1
vote
0
answers
30
views
Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
2
votes
1
answer
118
views
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 ...
- 765
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
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 $$\lim_{t\to\infty}f(ta,...
- 765
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
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\...
- 1,759
7
votes
1
answer
253
views
Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 515
8
votes
0
answers
177
views
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
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 ...
- 1,759
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 ...
- 91