All Questions
10,049 questions
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206
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About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
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votes
0
answers
184
views
Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?
$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...
- 365
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votes
0
answers
216
views
Bound on integral of elliptic theta function
I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely $\theta_3(0,\mathrm{e}...
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74
views
Weak convergence of 4-th degrees
Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, $...
- 243
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0
answers
65
views
Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
- 103
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votes
1
answer
552
views
Teaching profession:Differential Equations and Mean Value Theorems
Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...
- 1,422
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83
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Comparison between operators
I have found the following two concepts:
$\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The
operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$
and for any $\varepsilon>0$,...
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answers
210
views
Weak derivatives and dual of Hölder functions
Let $0<\alpha<1$ and $f \in C^{\alpha}$ be a Hölder function (either with compact support on $\mathbb R^n$ or on a closed Riemaniann manifold).
From what I understand, the derivative of $f$ in ...
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428
views
Given an even function how to obtain the most close odd function and vise versa?
Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference $|...
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answers
178
views
$2$-normed Spaces
Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper ...
- 1
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474
views
Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
- 85
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votes
2
answers
180
views
A basic question about JL Lions' transformation of a Stefan problem
In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...
- 23
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answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
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answers
537
views
matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
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0
answers
724
views
"correlation" between two subspaces
It is frequently important to determine the coherence of a matrix by finding the maximum pairwise correlation between all its column vectors. Similarly, when working with a union of subspaces of a ...
- 223
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votes
2
answers
168
views
Let f:J→R be an absolutely continuous and f'\in...?
Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous.
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$
\Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- \...
- 111
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votes
0
answers
405
views
Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
- 21
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votes
0
answers
533
views
Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$
Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X \...
user24451
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answers
156
views
Can a function be constructed from the direction of its gradient?
Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation
$Du=J|Du|$, $u|_{\partial \Omega}=f$
has a solution in a ...
- 23
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votes
1
answer
2k
views
Second proof of Jordan-Von Neumann theorem
I am looking for a second proof of Jordan-Von Neumann theorem that characterizes inner product in normed spaces. The book "Inner Product Structures: Theory and Applications" talks about a second proof ...
- 113
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0
answers
362
views
Banach-Mazur distance estimate finite-dimensional $\ell_p$ spaces
Hey my fellow Banach space guys. Sorry for another elementary question, and yes I have looked for the past hour and a half to see if I can find it on Google or in my books at home.
Fix $n\in\mathbb{...
- 1,591
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votes
0
answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
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515
views
If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?
Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...
user24451
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0
answers
100
views
Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
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0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91
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0
answers
113
views
Reference Search for a Functional Minimization Problem
Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is
$$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
- 151
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0
answers
146
views
How to bound Haar coefficients in terms of total variation?
I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says:
We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ such ...
- 7,633
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votes
1
answer
347
views
Dual space of Bochner space: is there an easier proof to show they're isometric?
It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$.
If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are isometric? The proof that I know for the general case ...
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0
answers
137
views
$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$
Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
- 113
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0
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93
views
Infinite limit in all points
Do there exist a Banach space (possibly nonseparable) $X$ and a mapping $F: X\to X$ such that
$$
\lim_{x\to a} \|F(x)\| = +\infty \quad \forall a\in X\quad?
$$
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0
answers
152
views
Need help determining whether a certain map is a $C^\ast$ homomorphism
Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
- 365
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0
answers
402
views
separable spaces-QM vs. functional analysis
In Nouredine Zettili's QM book, Hilbert space $H$ is said to be separable when: There exists a Cauchy sequence $\psi_n \in H$ ($n=1,2,\ldots)$ such that for every $\psi$ of $H$ and $\varepsilon > 0,...
- 1
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0
answers
166
views
Harnack's Inequality and (hypo)elliptic PDE
Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
- 1
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
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votes
0
answers
223
views
functional equation, how to solve
Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
$$F(x, y) = \frac{x\circ Ay}{x^TAy}$$
$$G(x, y) = \frac{x\circ By}{x^TBy}$$
where $A$ and $B$ are ...
- 13
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answers
160
views
Is this function in the weighted Sobolev space $H^{2,-s}$?
I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order $...
- 1
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1
answer
656
views
Gel'fand Yaglom functional determinant of non-diagonal operator?
Introduction:
As a quick reminder, the Gel'fand Yaglom theorem uses the generalized zeta-function approach to compute functional determinants of differential operators. Given a differential operator ...
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0
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241
views
Continuity of a function
Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$:
$$ F(z)=\bigg(\alpha-i\...
- 71
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0
answers
189
views
functional maximization
Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these ...
- 221
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0
answers
214
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Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
- 101
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1
answer
302
views
An interpolation inequality.
For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such ...
- 45
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0
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335
views
A basis of the space of continuous function of countable ordinals $C(\alpha) = C [0, \alpha]$
A basis of the space of continuous function of countable ordinals $C({\alpha}) = C [0, {\alpha}]$, which consist of characteristics functions of clopen subsets of $C({\alpha})$, in some order. But can ...
- 1
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0
answers
80
views
relationship between different function classes
I was wondering if there is a survey of relationship between several different well-studied function classes ?
ps - The question may be vague but I am looking for something along the lines of - http:/...
- 1
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0
answers
218
views
Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
- 421
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0
answers
231
views
Pure greedy algorithm
I study pure greedy algorithms in different basises. I am interested in 1 one question: is there such a Riesz basis $D$ in Hilbert space and $f\in H$ such that
$\|f-G_m(f,D)\|>Cm^{-1/2}\lvert\{f}\...
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
0
votes
1
answer
177
views
Laurent series with analytic coefficients
Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc.
I consider the function $f$:
$$f (t)=\sum f_{i}t^{i} \in A[[t]]$$
I suppose that the $t$-adic valuation of it is less or ...
- 3,472
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
0
votes
1
answer
142
views
A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 17
0
votes
0
answers
186
views
Properties of Eigenfunctions of a Kernel
I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.
I've and Kernel function $K(x,y)$
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
- 17