All Questions
5,849 questions
0
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1
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109
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Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...
- 261
0
votes
3
answers
211
views
Hausdorff convergence in bounded set preserves the volume
I was wondering if Hausdorff convergence relates to the volume of the converging sets. In particular, let $(C_n)$ be a sequence of closed sets contained in a bounded, closed set $Q$. Assume that $|C_n|...
- 1
0
votes
1
answer
142
views
Showing that a Gaussian achieves equality in a logarithmic Sobolev inequality
Consider the following Logarithmic Sobolev inequality on page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14): for $f\in H^1(\mathbb R^n),$ and $a>0$ any positive number,
$$
\frac{a^2}{...
- 1,755
0
votes
1
answer
345
views
Embedding of fractional Sobolev space into BMO
Is it true that $$\Vert u \Vert_{BMO(\mathbb R^2)} \lesssim_{s} \Vert u \Vert_{\dot H^s(\mathbb R^2)},$$
for $s \in (0,1)$, where $\dot H^s(\mathbb R)$ is the homogeneous fractional Sobolev space?
- 99
0
votes
1
answer
241
views
Dense sub-algebra of $C_{b}((0,1))$ which is not smooth
I am looking for a dense sub-algebra $B$ in $C_{b}((0,1))$ in uniform topology such that it satisfy following requirements:
$B\cap C^{\infty}_{b}((0,1))=\mathbb{R}$ (No polynomial, no bump function).
...
- 523
0
votes
1
answer
326
views
Domain of the fractional Laplacian operator
If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$
but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-...
- 109
0
votes
1
answer
110
views
Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 131
0
votes
1
answer
141
views
Solve $(x-a)^{\alpha +1} - \lambda*(b-x)^{\alpha + 1} = C(\frac{a+b}2 - x)^{\alpha}$ over $\mathbb R$ [closed]
I have been having trouble solving the following equation. Any help would be appreciated.
Let a,b,C,$\alpha$,$\lambda$ be real numbers with $C < 0$, $0 < \alpha < 1$, $\lambda > 1$. We ...
- 357
0
votes
1
answer
124
views
Uniform estimation of an integral
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
- 339
0
votes
1
answer
769
views
Square root of a continuous function
Is it square root of a real $\alpha-$Holder continuous function $f$ defined on $[0,1]$ a $\alpha/2$ Holder continuous, provided $\sqrt{f(x)}$ it exists and is continuous, i.e. whether $|f(x)-f(y)|\le ...
- 49
0
votes
1
answer
208
views
Tight upper bound on $\sum_{1\leq k\leq m} \sqrt{(n/k)+b}$
I apologise if this is obvious or off-topic.
Let $n$ be large and fixed, $b>0,$ and $m \in [\log n,n),$ say. This sum seems to be hard to evaluate/upper bound analytically (in closed form). ...
- 10.4k
0
votes
1
answer
163
views
Is the following function Lipschitz?
Given a vector $Q \in \mathbb{R}^{S\times A}$ where $S$ and $A$ are sets of finite cardinality, for $0<\gamma<1$ define the function $H_{w}:\mathbb{R}^{S\times A} \rightarrow \mathbb{R}^{S\times ...
- 53
0
votes
1
answer
281
views
Problem regarding Lebesgue measure in $\mathbb{R}^2$
Let $P=A_1\times A_2,$ where $A_1,A_2\subset \mathbb{R}$ are set of positive Lebesgue measure, and $Z\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue ...
- 123
0
votes
1
answer
322
views
Injectivity of analytic functions
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:
Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
- 431
0
votes
1
answer
102
views
Is the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ locally lipschitz on the space $C^2 [0 ,T] $?
Let the function $\Lambda : [0,T] \times \mathbb{R^n} \times \mathbb{R^n} \to \mathbb R$ be continuously differentiable. Then the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)...
- 369
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
0
votes
1
answer
335
views
On partial sum estimate on the series $S(p,q;s)=\sum_1^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$ and other Generalizations
Consider the following sum :
$$S(p,q;s)=\sum_{n=1}^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$
Here , $p$ is a variable w.r.t which we are going to analyse the sum.
$s$ is another parameter with ...
- 375
0
votes
1
answer
164
views
Restriction of non-metrizable topology to dense subset is non-metrizable
Let $(X,\tau)$ be a non-metrizable topological space which is not first-countable and let $\emptyset \neq Y\subset X$ be a proper dense subset. Is it possible for $(Y,\tau_Y)$ (where $\tau_Y$ is the ...
- 5,405
0
votes
1
answer
121
views
Extension of superharmonic functions
Let $V$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $W$ be an open neighborhood of the boundary $\partial V$ of $V$. If $u$ is superharmonic on $W$, is there a way to extend $u$ to a ...
- 411
0
votes
1
answer
135
views
Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$
I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
- 11
0
votes
1
answer
359
views
Dual norm of a max function [closed]
I am attempting to find the dual norm of
$$\|(x,y)\|_K=\max\{|x|,|y|,|x-y|\}.$$
I have obtained $\|(x',y')\|_K^* = |x'|+|y'|$, but don't think that this is correct.
I obtained this as follows :
$$K = ...
- 9
0
votes
1
answer
363
views
A question about Riesz decomposition theorem
Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions", vol. ...
- 411
0
votes
1
answer
298
views
Implicit inequality
Let $A,B,d\ge 1$ and suppose that $x\ge0$ satisfies
$$ x^{\frac{d+1}{d}}
\le
Ax+B.
\qquad(*)
$$
I can show that $(*)$ implies the bound
$$
x< d(A^d+B).
\qquad(**)
$$
Questions: (1) Can a ...
- 6,541
0
votes
1
answer
133
views
Product of sets with the Radon-Nikodym Property (RNP)
I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...
- 1,245
0
votes
1
answer
177
views
Denominator approximation sequence of a real number
For any positive integer $[n]$, let $[n]=\{1,\ldots,n\}$. Let $r\in\mathbb{R}$. We define for every positive integer $n\in\mathbb{N}$ the minimal difference from a rational with denominator $\leq n$ ...
- 48.9k
0
votes
1
answer
200
views
Is the function $f(x=\sum a_n/2^n)=\sum na_n/2^n$ continuous and nowhere differentiable? [closed]
Let $x=\sum_{1<=n<\inf}a_n/2^n$ be the binary expansion of a real number in [0,1]. Assume that infinitely many of a_n are 0 so that the expansion is unique.
Define $f(x)=\sum_{1<=n<\inf}...
0
votes
1
answer
115
views
Infinite norm of two randomly picked points [closed]
Let X and Y be points in the 4000-dimensional unit cube, picked at random with uniform distribution, which means from I what I understand that all locations in the cube are equally likely. $X \in [0,1]...
- 139
0
votes
1
answer
244
views
Proving that $\|\mathbf{T}^n\|^2=\sum_{g\in \mathbf{G}(n,d)}\|\mathbf{T}_g\|^2\,$
Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.
For ${\bf A} = (A_1,...,A_d) \in \mathcal{B}(F)^d$, the norm of ${\bf A}$ is given by
$...
- 724
0
votes
1
answer
419
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
1
answer
350
views
Uniformly Bounded (updating)
Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~
(1+t+z)^{a_3}}\exp\big\{-\frac{...
- 119
0
votes
1
answer
375
views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
- 213
0
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1
answer
137
views
Given these conditions, can a function be defined that is well defined a.e.?
I have two functions, and I want to combine them to define a certain function.
Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a....
- 2,069
0
votes
1
answer
119
views
Are these conditions enough to ensure joint measurability?
Suppose $f(x, e): \mathbb{R} \times (0, \infty)\to [0,\infty]$ is right continuous in $x$, and monotone increasing in $e$. Is $f$ jointly measurable?
- 2,069
0
votes
1
answer
168
views
Does differentiating an integro-differential equation results in equivalent stability of the solution?
I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
...
0
votes
2
answers
132
views
Dirichlet problem for capillary equation over convex domain
Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary.
Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function.
Let $L$ be a quasilinear elliptic ...
- 823
0
votes
2
answers
235
views
An inequality on length of two curves [closed]
I am looking for a proof, reference, comment of an inequality as follows:
If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that:
$f(a)=g(a)$ and $f(b)=g(b)$
$(...
- 1,776
0
votes
1
answer
87
views
Differentiablity of certain composite function
Let $I_1$ and $I_2$ be two closed bounded intervals.
Suppose $W(x,y)$ is a smooth function whose support is contained inside $I_1 \times I_2$.
Suppose I have $\Phi= (\Phi_1(x,y), \Phi_2(x,y)) : \...
- 3,625
0
votes
1
answer
166
views
Can this result be proven by using only two (or maybe three) results listed below? [closed]
I wan to show that there is no continuous real-valued function of a real variable that sends rationals to irrationals and irrationals to rationals by using only $1)$ and $2)$:
$1)$ We can use the ...
user114642
0
votes
1
answer
186
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
0
votes
1
answer
662
views
A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
- 309
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
0
votes
1
answer
138
views
Moment problem with wrong solution
I will write a problem with an answer that apparently is wrong. My question would be what is wrong with this solution.
Define $$B(s)=\sum_{i=0}^s{{2\,s-i-1\choose s-1}\frac {i}{s}{5}^{i}{2}^{s-i}}$$ ...
- 333
0
votes
1
answer
122
views
Nilpotent infinite binary matrices
Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix multiplication for $A, B\in \text{Mat}(\mathbb{N},\{0,1\}$) and $m,n\in\mathbb{...
- 48.9k
0
votes
1
answer
126
views
Comparing tails of polynomial functions
Suppose that $P(x) = a_m x^m + \dots + a_0$ and $Q(x) = b_n x^n + \dots + b_0$ are two polynomials, with $m > n > 1$ and $a_m > b_n > 0$. Suppose that $P$ has $m$ distinct real roots $y_1&...
- 225
0
votes
1
answer
434
views
Long term behavior of a certain discrete time dynamical system on graphs
Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$.
Assume that each vertex has an $\textit{...
- 2,441
0
votes
1
answer
55
views
On 1-iso maps and subsets of the unit circle
Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
- 97
0
votes
1
answer
843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
0
votes
1
answer
94
views
A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?
Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...
- 109
0
votes
1
answer
557
views
Is the limsup or liminf of n-wise independent events independent?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
0
votes
1
answer
482
views
Complement of a finite union of convex sets
Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...
- 2,933