All Questions
5,659 questions
6
votes
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319
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Does approximately Fréchet differentiable imply approximately Gateaux differentiable?
In what follows, $\mu^n$ denotes $n$-dimensional Lebesgue measure, and $B_r(x)$ is the open ball with radius $r$ centered at $x$.
In elementary calculus, if we have a function $f : \mathbb{R}^n \...
- 700
6
votes
1
answer
268
views
Decomposition of non negative Radon measure into $L^1$ and $H^{-1}$ functions
What is a reference for the following result (which appears to be well-known in measure theory)?
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely continuous ...
- 61
6
votes
1
answer
916
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A problem on rate of decay of fill distance?
Let $X$ be a random variable with values in a closed compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ is has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and ...
- 698
6
votes
1
answer
321
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On Glaeser's result for the square-root of a smooth non-negative function
One of the results due to Georges Glaeser is the following: there exists a non-negative $C^\infty$ function $f$ on the real line, flat at its zeroes, such that $\sqrt{f}$ is not $C^2$. On the other ...
- 16.2k
6
votes
1
answer
223
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Asympotic density of a very simple sequence
Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more ...
- 5,103
6
votes
1
answer
314
views
Generators of a convex cone defined by a differential inequality
Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_{++})$ and $f\colon \mathbb R_{++}\to\mathbb R_{++}$) that satisfy
\begin{...
- 61
6
votes
1
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1k
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About the generating structure of Borel field
This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
- 8,071
6
votes
1
answer
489
views
Henstock, Differentiation under the integral sign
Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...
- 63
6
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1
answer
843
views
Orlicz Norm and A result on expectation
I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined:
Consider arbitrary, non-negative, convex function $\psi:\...
- 1,109
6
votes
1
answer
380
views
Asymptotic value of a multivariate integral
The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems.
Define
$$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
- 37.7k
6
votes
2
answers
425
views
orderings of the field R((x, y))
I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the ...
- 620
6
votes
1
answer
3k
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Proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone
Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...
- 283
6
votes
1
answer
369
views
Denominators in the solution to Hilbert's XVII
Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...
- 85.6k
6
votes
1
answer
803
views
Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set
For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
- 143
6
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1
answer
817
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,233
6
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1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 434
6
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1
answer
376
views
Lavrentiev phenomenon between $C^1$ and $C^2$
Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is
$$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
6
votes
1
answer
1k
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First eigenfunction of $p=3$-Laplacian of a square domain in $\Bbb R^2$ : reference for any work on this?
In the last few decades, lots of work on first eigenfunction of $p$-Laplace with Dirichlet and other boundary conditions. But I couldn't find much on periodic boundary conditions. I have computed the ...
- 698
6
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1
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2k
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Weak convergence in $H_0^1$ and strong convergence in $L^2$
I'm reading a hand-waving argument in a proof of Chapter 7 of Navier–Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly.
Let $\Omega\subset{\mathbb{R}^n}$ be ...
user14319
6
votes
1
answer
778
views
Inverse function theorem for DC-functions
I would like to have an inverse (or/and) implicite function theorem for DC-functions.
It seems that I have right definitions, but I fail to prove it...
Definitions:
Let $h:\mathbb R^n\to\mathbb R$ ...
- 45k
6
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0
answers
132
views
Do there exist strictly contracting eikonal functions on $\mathbb R^n$?
A function $f: \mathbb R^n \to \mathbb R$ is said to be a strict contraction if
$$|f(x) - f(y)| < |x - y|$$
for all $x \neq y$.
A function $f$ is said to be eikonal if it is differentiable ...
- 6,233
6
votes
0
answers
156
views
Generalized Rademacher theorem for fractional derivatives
It is known that if $f$ is $\alpha$ Holder and $\gamma<\alpha$ then $f$ is $\gamma$ fractional differentiable. See Theorem 14 in the paper by G. H. Hardy and J. E. Littlewood, "Some properties ...
- 1,904
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0
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431
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
- 2,836
6
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0
answers
632
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Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
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0
answers
220
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Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
6
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0
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108
views
Archimedean ordered field in which every function is smooth
In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \...
- 2,624
6
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0
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309
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Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
6
votes
0
answers
136
views
Injectiveness of a monotonic surjective mapping $\mathbb R^n \to \mathbb R^n$ with $\det J \neq 0$
Consider a surjective mapping $F \colon \mathbb R^n \to \mathbb R^n$, $F\in C^1$, $\dfrac{\partial F_i}{\partial x_j} > 0$, and $\det \left(\!\left( \dfrac{\partial F_i}{\partial x_j} \right)\!\...
- 243
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0
answers
213
views
Equivalent forms of Fourier restriction conjecture
this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow.
I'm reading Pertti Maattila's book ...
- 196
6
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0
answers
3k
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Top journals in mathematical analysis [closed]
How would you (broadly) rank the journals that specialize in mathematical analysis and related areas such as PDEs?
As far as I know, GAFA looks like it is the top one. But apart from that, how can I ...
- 393
6
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0
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405
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Using the Lorentz operators to build polynomials that converge to a continuous function
Questions
Let $f(\lambda):[0,1]\to (0,1)$ have a $\beta-\lfloor\beta\rfloor$)-Hölder continuous $\lfloor\beta\rfloor$-th derivative, where $\beta>0$.
Find explicit bounds, with no hidden constants,...
- 697
6
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0
answers
208
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Can every weakly converging sequence be made to converge strongly after taking a subsequence and rearranging?
Let $f_i: [0, 1] \to \mathbb R$ be functions in $L^1 \cap L^\infty$ with $\sup_i \|f_i\|_{L^\infty} < M$ for some $M > 0$.
Suppose $f_i$ converge weakly in $L^1$ to some $L^1$ function $f$ - ...
- 6,233
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0
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2k
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Do smooth cutoff functions analytically continue functions?
My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
- 1,730
6
votes
0
answers
151
views
Gap between consecutive Dirichlet eigenvalues
Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say
$$ -\Delta \phi_k = ...
- 4,153
6
votes
0
answers
283
views
Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
- 161
6
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0
answers
129
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Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
- 349
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0
answers
107
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Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
6
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0
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264
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Odd Steinhaus problem for finite sets
Call a finite subset $S$ of the plane with an even number of points an odd Jackson set, if there is an $A\subset \mathbb R^2$ such that $A$ meets every congruent copy of $S$ in an odd number of points....
- 19k
6
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0
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445
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Vector-valued interpolation for sublinear operators
Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem.
$\textbf{Theorem}$
Let $1\...
- 421
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0
answers
130
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ultrametric Rademacher theorem
The classic Rademacher theorem roughly states that Lipschitz continuous functions are almost everywhere differentiable. However, there are well-known ultrametric counterexamples, see Kobliz's classic ...
- 500
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0
answers
267
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Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
- 1,755
6
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0
answers
176
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Area-preserving map of punctured disk to itself
If $D_r = \{v\in \mathbb{R}^2 : 0 \lt |v| \lt r\}$, consider the map $f_r: D_r \to D_r$ given by:
$$f_r(x,y) = \frac{\sqrt{r^2-x^2-y^2}}{\sqrt{x^2+y^2}}\left(-y,x\right)$$
Geometrically, $f_r(v) \...
- 2,902
6
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1
answer
396
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Well definition of a function
I've edited, just skip the first attempt and go to the second one.
THE FRAMEWORK: let us consider a real topological vector space $V$.
We denote with $\mathscr C_k(V)$ the set of all continous ...
- 779
6
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0
answers
210
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Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
6
votes
0
answers
2k
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Interchange of integral and infimum
Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ U\...
- 61
6
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0
answers
396
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Recurrence Formula for Zernike polynomials
I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...
- 263
6
votes
0
answers
206
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Degree of Chebyshev polynomial necessary
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
- 13.9k
6
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0
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2k
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Are planar Lipschitz curves countable unions of graphs?
More precisely:
Question:
Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be Lipschitz. Do there exist Borel (or Suslin) sets $A_i \subset \mathbb{R}^2$ and directions $v_i \in \mathbb{R}^2$, for ...
- 3,270
6
votes
0
answers
223
views
Sum of product maximum
For which pairs of integers $(n,m)$ is the maximum of the following function $$f(x)=\sum_{i_1+\dots +i_n=m}\prod_{k=1}^n x^{i_k}_{k},\ \ x=(x_1,\dots,x_n), \|x\|=1$$ attained when $x_1=\dots=x_n$?
(...
- 259
6
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0
answers
8k
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Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385