All Questions
12,935 questions
1
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73
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Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759
2
votes
0
answers
86
views
Besov spaces containing piecewise linear functions
Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
2
votes
0
answers
57
views
Mappings that preserve local or global minimum
In the most general form, I'm interested in any non-trivial results of the following question.
Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
- 275
0
votes
0
answers
27
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Comparison Principle for Courant Nodal Domain Theorem
Recall that Courant's Nodal domain gives an upper bound on the number of nodal domains of the $k$-th eigenfunction. I was wondering if it is possible to deduce some relation between the number of ...
- 537
4
votes
0
answers
125
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Are all solutions to the linear heat equation $\partial_t u - \Delta u = 0, u(0,\cdot) = 0$ continuous at $t = 0$?
Consider a distributional solution $u(t,\cdot) \in C^0([0,T],\mathcal{D}'(\mathbb R^n))$ to the linear heat equation
$$
\left\{
\begin{align*}
u_t - \Delta u &= 0, \\
u(0,\cdot) &= 0
\end{...
- 233
2
votes
1
answer
244
views
Characterization of normed spaces based on violation of parallelogram law
For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied.
I would like ...
- 213
5
votes
0
answers
113
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Stability of perturbation of an elliptic problem into a parabolic PDE
Fix some $f\in H^1(\partial (0,1)^d)$.
Let $\eta\ge 0$ and for each such $\eta$ consider the solution $u^{\eta}$ to the solution PDE
$$
\begin{cases}
\Delta u & = u_t - \eta u_{tt} \mbox{ on } \...
- 5,405
1
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0
answers
34
views
Discrepancy between probability measures, tested against bounded functions of bounded variance
When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
- 801
2
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0
answers
37
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Definition of coercive boundary value problems
In Folland's Introduction to PDE,page 242, he defines what it means for a Dirichlet form to be coercive (a standard definition). Let $X$ be a closed subspace of $H_m(\Omega)$ with $H_m^0(\Omega)\...
- 263
6
votes
3
answers
282
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Extreme points of the dual unit ball of a Banach algebra
Let $A$ be a unital Banach algebra. Let $f\in A^*$, $\Vert f\Vert=1$ satisfy that there exists a maximal left ideal $L\subset A$ such that $L\subseteq\ker{f}$.
Question: Is $f$ an extreme point of ...
- 2,605
9
votes
2
answers
418
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Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 452
5
votes
1
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183
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Question about modular group (Modular theory in operator algebras, section 2.14)
Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20:
I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
- 175
3
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0
answers
165
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$S^{n}(V)$ is "approximately" $V$ when $n$ goes to infinity (in the setting of normed space)
Let $B$ be a (separable) Banach space. $(v_{i})_{i}, i\in\mathbb{N}$ being a family of linear independent vectors. $V$ being the span of $v_{i}$. I try to prove that $V$ is dense in $B$. I define a ...
- 523
3
votes
1
answer
176
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Question about Lebesgue Bochner spaces
Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.
I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
- 1,759
3
votes
2
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137
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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 ...
2
votes
0
answers
92
views
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
- 101
2
votes
1
answer
117
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Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
- 1,759
11
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342
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The diagonal operators and unconditionality
The following is well-known:
Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$.
Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with
the ...
- 986
2
votes
0
answers
85
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Dirichlet problem for an elliptic operator
consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator.
My question is there is a formula of the ...
- 21
6
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0
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159
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Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 721
3
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answers
58
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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
0
votes
0
answers
75
views
Does nice behavior near a singular point force solution to be in Frobenius series?
I have a pair of partial differential operators $\Delta_1$ and $\Delta_2$ in $y_1, y_2$ formed from constants, multiplication by $y_1$ or $y_2$ and derivatives in the form $y_1 \frac{\partial}{\...
- 151
4
votes
1
answer
158
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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
1
vote
2
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237
views
Calderón–Zygmund/$L^p$ estimates for the linear heat equation
Let $C_r$ denote the open cylinder
$$
C_r = \{(x,t) \in \mathbb R^{n+1} : |x| < r, -r^2 < t < 0\}
$$
and consider a classical $C^{2,1}_{x,t}(C_1)$-solution to the linear heat equation
$$
\...
- 233
5
votes
0
answers
104
views
Convolution of a bounded function and measures
Given a function $f\in L^\infty(\mathbb{R}^n)$ and a family of Radon measure $\mu_\alpha$, under what condition do we have $f*\mu_\alpha$ equi-continuous?
One condition I know is if $\mu_\alpha$ has a ...
- 375
0
votes
0
answers
59
views
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
0
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0
answers
37
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Finding a particular solution to a Lax pair on $[0, \infty)$ to solve $q_t + q_{xxx} + u(x)q = 0$ using the Fokas Method
I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here).
This post is a follow up to my first that ...
- 547
0
votes
0
answers
40
views
Multivariable Variation of Parameters $\psi_1(x, k)\int^{(x,t)}e^{ik^3(t-\tau)}[M_1(\xi,k)q(\xi,\tau)d\xi-X_1(\xi,\tau,k)d\tau]+\dots$
I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here). For greater clarity, I split my question up into two ...
- 547
6
votes
1
answer
248
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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
4
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0
answers
139
views
Understanding a "straightforward" application of the method of stationary phase for proving a trace formula on compact hyperbolic surfaces
I'm reading the paper 'Uniform distribution of eigenfunctions on compact hyperbolic surfaces' by Steven Zelditch (Duke Mathematical Journal, 55, pp. 919-941 (1987), MR916129, Zbl 0643.58029) and am ...
- 383
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
2
votes
1
answer
113
views
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
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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
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0
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225
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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
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0
answers
220
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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
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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
5
votes
1
answer
139
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Dispersion of random walk with scaled step sizes
Let $Y_j$ be a sequence of independent Gaussian random variables with mean zero and unit variance ($\mathbb{E} Y_j = 0$ and $\mathbb{E} Y_j^2 = 1$) and let $\sigma:\mathbb{R}\to [1,2]$.
We define the ...
- 452
2
votes
0
answers
142
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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
2
votes
1
answer
141
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Existence of first variation
I am trying to compute the first variation of the functional
$$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$
where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...
- 21
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
6
votes
1
answer
228
views
Question about Bochner measurability
When I study parabolic pde's I often came across the following type of Bochner spaces $L^p([a,b];L^{q}(\Omega),\ W^{1,p}([a,b];L^{q}(\Omega))$ and $L^{q}([a,b];W^{1,p}(\Omega))$ where $p,q\geq 1$ and $...
- 1,759
0
votes
0
answers
29
views
On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
- 765
2
votes
0
answers
95
views
On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
6
votes
1
answer
197
views
On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
- 1,429
3
votes
1
answer
86
views
$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line
I consider the following problem on the half real line
$$
\begin{cases}
u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm]
u|...
- 43
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