All Questions
3,628 questions with no upvoted or accepted answers
1
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113
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Hardy $Hp$ norm of similar function
Let $f(z)=\sum_{n=0}^{\infty} \frac{c_n}{n+1}z^n$, where sequence $c_n \in S^1=\{z:|z|=1\}.$ We observe $H^p$ norm $\|f\|_{H_p}$, where $H^p$ is Hardy space, $1 \leq p < \infty$.
Question: For the ...
- 259
1
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0
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82
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Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$
Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
- 1,959
1
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0
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99
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Question on norms on tensor product and algebra
Question 1
Let $V,W$ be normed spaces and $\iota:V\times W \rightarrow V\otimes W$ be the canonical (algebraic) bilinear map.
It can be easily shown that for any normed space $X$ and a continuous ...
- 337
1
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0
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159
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Construction of the quadratic variation process in infinite dimensions
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167
1
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0
answers
53
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Is it possible that a convex cone and its closure both induce vector lattices?
Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field.
Suppose that $P$ satisfies positive element stipulations.
(1) $X=P-P$.
(2) $P\cap-P=...
- 8,071
1
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0
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186
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Using continuity + commutativity to define "limit"
Let $S$ be the set of real sequences, and $S_0\subset S$ be the set of convergent real sequences. For each $f:\mathbf R\to\mathbf R$, we define $\bar{f}:S\to S$ by $$\bar{f}(\{x_n\})=\{f(x_n)\}.$$
It ...
- 1,630
1
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0
answers
105
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Variational Problems with Subsidiary Conditions
I am studying from
Gelfand, I.M.; Fomin, S.V., Calculus of variations. Transl. from the Russian and edited by Richard A. Silverman., Mineola, NY: Dover Publications. vii, 232 p. (2000). ZBL0964....
- 11
1
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0
answers
114
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Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?
In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
- 11
1
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0
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45
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Is research of the Hyers-Ulam stability of this functional equation already conducted?
The functional equation in question is of the type $f(g(x))=g(f(x))$, where $f$ is the unknown function. Are there existing research already conducted on the Hyers-Ulam stability of this generalized ...
- 21
1
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0
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215
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Minimal projections and pure states in scattered $C^*$ algebras
The question I have came up when reading a paper of Ghasemi and Koszmider https://arxiv.org/abs/1611.00221 on scattered $C^*$ algebras and the non-commutative Cantor-Bendixson derivatives. Throughout ...
- 56
1
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0
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336
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Existence of solution for Poisson equation in Markov chain
Consider $X_n\in \mathcal{X}$ a controlled Markov chain taking value in a compact set $\mathcal{X}$ with action $a\in \mathcal{A}$, where the action set $|\mathcal{A}|$ is finite.
(In particular, we ...
- 185
1
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110
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Are almost positive functionals close to positive functionals?
This is a bit of an open-ended question... Let $S$ be an operator algebra (or an operator system) and consider a functional $\nu:M\to \mathbb{C}$
that satisfies
$$\vert \nu(a)\vert \ge -\varepsilon \...
- 19
1
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0
answers
105
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Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
- 623
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0
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202
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Space of analytic function and sequence space $l^p$
Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function:
$N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|...
- 259
1
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0
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183
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Trace theorem for boundary value problem
Consider the inhomogeneous boundary value problem on the infinite strip $(x,y)\in \mathbb{R}\times [0,1]$ defined by
$$\begin{cases}\partial_{x}u + \partial_{y}v=f & {(x,y)\in \mathbb{R}\times (0,...
- 873
1
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0
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87
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Linear evolution equation $u'(t)=A(t,\omega)u(t)$ with time-dependent random operator
I have had some previous knowledge on evolution equations in a Banach space of the form $$u'(t)=Au(t),$$ where $A$ generates some strongly continuous operator semigroup. Now I am looking at a problem ...
- 151
1
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0
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87
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Trace of sobolev space induced by vector fields
Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition
$$ \text{rankLie}[X_{1},\cdots,X_{m}]=n $$
at every ...
- 287
1
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0
answers
180
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Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
- 115
1
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0
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118
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On the boundedness properties of the mapping $f\to |f|^{\frac12}$ on homogeneous Sobolev space
Suppose that $\Delta f\in L^1(\mathbb{R}^n)$, where $\Delta$ denotes the Laplacian. Then I’m asking whether or not we have the following estimates
$$
\|\nabla|f|^{\frac12}\|_{L^2}\leq C\|\Delta f\|_{...
- 879
1
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0
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304
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$C^*$-algebras with non-trivial center
The center $Z(A)$ of an algebra $A$ is the set of all those elements that commute with all other elements. If $A$ is the algebra of compact operators on a Hilbert space $H$, then $A^{**}$ is the ...
- 4,461
1
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124
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Singular value decomposition in two spaces (reference in Russian paper?)
Let $H$ be a Hilbert space and $X$ be a Banach space such that $H \cap X$ is dense in both.
Now, let $T$ be an operator such that $T: H \rightarrow H$ and $T:X \rightarrow X$ exists in the sense that ...
1
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0
answers
151
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Continuity of the spectrum under weaker notions of convergence
Let $T:X\to X$ be a linear operator on a Banach space $X$.
We know that the spectrum of $T$ is an upper semicontinuous
function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a ...
- 757
1
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0
answers
94
views
Riesz transform on almost periodic functions?
It is well established that the Riesz transform is well-defined for $f\in L^p(\mathbb{R}^d)$ via$$
\mathcal{R}_jf(x) = c_d\lim_{\epsilon\to 0}\int_{|x-y|>\epsilon}\frac{(x^j-y^j)f(y)}{|x-y|^{d+1}}\,...
- 663
1
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0
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96
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Infimum of equivalent measures
Suppose I have a functional of the form
$$
F(\mathbb{P})\triangleq \int_{\mathbb{R}^d} \int_{\Omega}f(x,\omega)\mathbb{P}(d\omega)m(dx),
$$
where $m$ is the Lebesgue measure and $\mathbb{P}$ is a ...
- 173
1
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0
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148
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Fourier–Stieltjes as the dual space of the full group algebra
I know that this fact is classical, but I can't find the proof of it. How to proof that $B(G)=(C^*(G))^*$? As I understood, I can take a functional $F: \ell_1(G) \to \mathbb{C}$, and there is one-to-...
- 631
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0
answers
48
views
Existence of arbitrarily small invariant neighborhood
Let $A$ be a bounded linear operator from a Banach space $M$ to itself.
Suppose that $\rho(A)<1$ where $\rho(A)$ is the spectral radius of $A$.
For any $\varepsilon>0$, does there exist an open ...
- 31
1
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0
answers
119
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Does $u\in H^{3/2}(\Omega)$ imply continuity of $\nabla u\cdot\overrightarrow{n}$ across an interior interface?
When investigaing the regularity of certain functions, I encountered this problem:
if $u\in H^{3/2}([0,1]\times [0,1])$,
what can we say about the continuity of $\nabla u\cdot\overrightarrow{n}$
...
- 398
1
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0
answers
80
views
What is the character space of $\mathcal P(K)$?
Let $K$ be a compact subset of $\Bbb C$. Let $\mathcal P(K)$ be the closed algebra generated by the complex polynomials on $K$.
What is the character space $\Phi_{\mathcal P(K)}$ of $\mathcal P(K)$?...
- 1,245
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154
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Lyapunov stability for nonlinear PDEs
Where can I find a theorem about Lyapunov stability for the equation in Hilbert space?
$$
y' = Fy,
$$
where $F : V \to V'$ is a nonlinear operator , $y' \in L^2(0,T,V')$, $V$ is a Hilbert space.
...
- 243
1
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0
answers
220
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About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
1
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0
answers
109
views
Two tensor product norms inducing different topologies on the space of simple tensors
Are there two Normed spaces $V,W$ for which the algebraic tensor product $V\otimes W$ admits two different norms, both satisfying $\parallel x \otimes y \parallel= \parallel x \parallel. \...
- 356
1
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0
answers
50
views
Verifying general assumption for parabolic PDE
I've got some problems verifying an assumption for a parabolic PDE. Namely, let $(V,H,V^*)$ be a Gelfand-Triple, $u_0 \in V$, $\psi\colon V \to \mathbb{R}$ convex and lower-semicontinuous and $a\colon ...
- 187
1
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0
answers
84
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Existence of a minimizer of a functional involving a power q
I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2052565/existence-of-a-minimizer-of-a-functional-involving-a-power-q) but did not get any answers so I am trying ...
- 11
1
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0
answers
66
views
How to define spectral multiplier for −Δ?
Put $e_{n}(t)=e^{int}=\prod_{j=1}^{d}e^{in_{j}t_{j}}$, $t\in \mathbb T^d, n\in \mathbb Z^d.$ ($t=(t_{1},..., t_d), n=(n_1,..., n_d)$)
We note that $\{e_{n}\}_{n\in \mathbb Z^d}$ forms an orthonormal ...
- 233
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0
answers
66
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Characterization of the weak completion of $L^2(\mathbb{R}^d)$
The completion $\overline{L^2_w(\mathbb{R}^d)}$ of $L^2_w(\mathbb{R}^d)$ (i.e. the completion of $L^2(\mathbb{R}^d)$ endowed with the $\sigma(L^2(\mathbb{R}^d),L^2(\mathbb{R}^d))$ topology) is ...
- 488
1
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0
answers
105
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compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
1
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0
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194
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Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
1
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0
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331
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Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
- 1,350
1
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0
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128
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determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 31
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0
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154
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Kasparov's descent homomorphism for higher KK groups
I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism
$$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$
but what still ...
- 19
1
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0
answers
123
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Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
- 1,835
1
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0
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127
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What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
- 41
1
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0
answers
74
views
Reference for a text book on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$
We need basic reference on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$. Usually text books cover much more subtle case of $T:l^{\infty}(\mathbb Z_+,...
- 11
1
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0
answers
94
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Space spanned by pointwise squares of basis functions
Consider the Hilbert space $L^2(\Omega)$ over some Euclidean domain $\Omega$.
Let $F=\{f_i;i\in\mathbb N\}$ be an orthonormal basis of this space consisting of functions in $L^2(\Omega)\cap L^4(\Omega)...
- 8,086
1
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0
answers
205
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Essential self-adjointness of tensor product of operators
Operators $A_i, i=1,2,\ldots,5$ are considered on Hilbert spaces $\mathcal{H}_i$. Operator $A_i$ is essentially self-adjoint on $D^{ess}(A_i) \subset \mathcal{H}_i$.
Consider operator
\begin{equation}...
- 111
1
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0
answers
256
views
Significance of Tikhonov matrix
I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
1
vote
0
answers
136
views
heat kernel for powers of some degenerate elliptic operators
Let $\Omega$ be a bounded open domain in $R^{n}$ with smooth boundary and $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of real smooth vector fields defined on $\Omega\subset \mathbb{R}^{n}$. If $X$ ...
- 287
1
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0
answers
90
views
Monotone operator subgradient
I am trying to solve a PDE of the form $\mathcal{A}u'(t) + \partial\Psi[u(t)] \ni 0$ where $\mathcal{A}$ is a skew-symmetric, linear, monotone operator, $\Psi$ is convex, and $\partial \Psi$ is the ...
- 11
1
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0
answers
238
views
Continuation of strictly monotone function in $R^n$
Let $f(x_1,...,x_n)$ be $C^0$ continuous function $R^n\to R$ defined on a compact domain $A\subset R^n$. Let $f$ be strictly monotonously increasing w.r.t. every argument in the domain of definition. ...
- 11
1
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0
answers
83
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What is optimal distance between inverse of convolution operator?
I am looking for a measure to find the optimal distance measure between inverse of an convolution operator $A$ and say another convolution operator $B$. I want my measure to be sharp that mean when $B$...