All Questions
5,909 questions
-2
votes
1
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92
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An inequality between two real-valued concave functions
Can anyone help me prove the following inequality? Thanks!
- 26.3k
1
vote
1
answer
162
views
Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
3
votes
1
answer
154
views
Solving Matrix/Operator Equation $H P X + X P H + HQ = 0$
This problem arises when minimizing the operator equation $X P X^* + X Q + R$ over positive $X$ with respect to the positive cone on a Hilbert space $\mathcal{H}$.
The (reduced) task:
Given $P$ and $...
CommunityBot
- 1
1
vote
1
answer
104
views
Is there a Darboux function like the one described below?
Does there exists Darboux function which is continuous at only one point and discontinuous at all other points in the interval on which it is defined?
- 32.5k
2
votes
2
answers
122
views
Convexity inequality
Let $E$ be a subset of $\mathbb{R}^n$ such that $\mathbb{R}^n \setminus E$ is convex.
Let $x,y$ be in $\mathbb{R}^n$. Is it true that for $t\in [0,1]$, we have: $$d(tx+(1-t)y,E) \geq td(x,E) - (1-t)d(...
- 23.7k
7
votes
0
answers
221
views
integrality of a Riccati-type equation
The following is a problem we were unable to prove and left stated in the paper
"Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846.
Define ...
- 43.2k
7
votes
2
answers
340
views
Sign-oscillations for power series with random coefficients
Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<...
- 96.4k
1
vote
2
answers
256
views
Monotonicity of the sequences of the lower and upper Darboux sums
Consider the following lower and upper Darboux sums
$$ s_n(x)\ :=\ \sum_{k=1}^n\frac 1{n\cdot x+k} $$
and
$$ S_n(x)\ :=\ \sum_{k=0}^{n-1}\frac 1{n\cdot x+k} $$
for every real $\ x>0\ $ and ...
- 489
6
votes
1
answer
273
views
Avoiding equal distances
Is the following consistent?
There exists $X \subseteq [0, 1]$, such that $X$ does not have measure zero and for every $Y \subseteq X$, if $Y$ does not have measure zero, then there are $y_1 < y_2 ...
- 9,641
5
votes
2
answers
1k
views
real analyticity, Fourier coefficients [duplicate]
Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
- 105k
4
votes
2
answers
519
views
Closed-Form solution for system of simple nonlinear equations
I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...
CommunityBot
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4
votes
1
answer
1k
views
For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
CommunityBot
- 1
4
votes
1
answer
1k
views
Indefinite integral of squared hypergeometric function
I am trying to compute the indefinite integral
$$
\int_0^u {}_2F_1\left(\frac{1}{4},\frac{5}{4},2,1-v^2\right)^2 dv
$$
for $0<u<1$. Using Clausen's formula for the square of the hypergeometric ...
CommunityBot
- 1
33
votes
1
answer
2k
views
How quickly can the derivative of an everywhere differentiable function change sign?
Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
- 9,641
15
votes
3
answers
2k
views
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I've been trying to find an asymptotic expansion of the following series
$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$
and
$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
CommunityBot
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3
votes
0
answers
223
views
Does the divergent solution of this equation :$f'=e^{f^{-1}}$ of Gevrey type and could be Borel summation applied for it?
This question was asked here in MO by someone seeking for the solution of the functional -differential:$f'=e^{f^{-1}}$ not exactly an O.D.E, and again here seeking for the growth rate of it solution ...
CommunityBot
- 1
0
votes
0
answers
308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 6,259
1
vote
3
answers
845
views
$H^{-1}$ conservative gradient flow and $L^2$ projection
Consider Cahn-Hilliard (see this) equation hich is known as the $H^{-1}$ gradient flow of Cahn-Hilliard energy functional, also it is easy to verify that this equation is mass preserving i.e. measure ...
- 2,641
16
votes
7
answers
6k
views
Understanding Gibbs's inequality
Short version
Gibbs's inequality is a simple inequality for real numbers, usually
understood information-theoretically. In the jargon, it states that
for two probability measures on a finite set, ...
- 8,512
9
votes
1
answer
1k
views
Limit formula for the second derivative
Suppose that $f$ is a real-valued function which is twice differentiable in the interval $(-1,1)$. Does the following hold?:
$$\lim_{h \to 0} \frac{f(h) - 2f(0) + f(-h)}{h^2} = f''(0)$$
If $f''(x)$ ...
- 25.8k
13
votes
3
answers
2k
views
"Values" of divergent integrals
Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
CommunityBot
- 1
18
votes
1
answer
3k
views
How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
2
votes
0
answers
73
views
A question on groupoids and measurable fields of Hilbert spaces
Suppose that we have the following data:
$ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and
range maps denoted by $ s $ and $ r $ respectively.
$ (\lambda^{x})_{x \in \...
- 1,396
-1
votes
1
answer
519
views
Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
CommunityBot
- 1
11
votes
3
answers
3k
views
Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
CommunityBot
- 1
1
vote
1
answer
130
views
Resolvent difference of absolute values!
Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.
Is there a way to write
$$(\left\lvert ...
CommunityBot
- 1
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 462
2
votes
1
answer
102
views
Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 16.5k
4
votes
1
answer
223
views
Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
CommunityBot
- 1
8
votes
1
answer
353
views
$E_n(\ell^\infty)=SL_n(\ell^\infty)$?
Let $R$ be a commutative unital ring $R$ with unit element $1$.
For $n\in \mathbb{N}=\{1,2,3,\cdots\}$, let $SL_n(R)$ be the group of all $n\times n$ matrices with entries from $R$ having ...
- 7,893
2
votes
2
answers
169
views
A general question on comparison of integrals and a specific problem
When working on an applied math topic, I have come across the following general problem.
Let $f(x_1, x_2, ..., x_n)$ be a real function of $n$ real variables $x_1, x_2, ..., x_n$ which is ...
- 21
3
votes
1
answer
174
views
How does the function g(x) behave as x tends to 1?
Suppose $g:[0,1)→R$ is a continuous function satisfying
$g(x^2)=x−g(x)$
for every x on interval $[0,1)$.
How does the function g(x) behave as x tends to 1?
- 53.3k
5
votes
0
answers
411
views
Partition of the unit interval into uncountably many sets of full outer measure
Is it possible to construct an uncountable partition $(A_\delta)_{\delta\in[0,1]}$ of the unit interval $[0,1]$ such that $\mu (A_\delta)=1$ for each $\delta\in[0,1]$? ($\mu$ stands for the outer ...
1
vote
0
answers
71
views
Proving an Algorithm that generates minimal $\|x\|_0$ for the underdetermined system $Ax=b$
Let $A \in \mathbb {F}^{m \times n}$ with $m< n,$ $b \in \mathbb{F}^m$ and let $x$ be unknown in $\mathbb{F}^n.$ Assume $0<p<1.$ Then $$\arg \min\limits_{x: Ax=b} \|x\|_0 = \lim\limits_{p \to ...
- 259
5
votes
1
answer
249
views
If $\mathcal R_j f\in L^1$ then $\widehat{\mathcal R_j f}=-i\frac{\xi_j}{|\xi|}\widehat{f}(\xi)$
For any $f\in L^1(\mathbb{R}^n)$ and $1\le j\le n$, recall that the Riesz transform $\mathcal{R}_jf\in L^{1,\infty}(\mathbb{R}^n)$ is given by
$$ \mathcal{R}_jf:=c_n\lim_{\epsilon\to 0}\left(\frac{x_j}...
- 3,146
12
votes
1
answer
927
views
On an Inequality of Lars Hörmander
Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha},
\end{equation}
where as usual ...
CommunityBot
- 1
5
votes
2
answers
359
views
Proof without distributions
I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...
- 23k
2
votes
1
answer
250
views
Density in the Space of absolutely convergent Fourier series
It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : https://math.stackexchange.com/questions/67334/...
CommunityBot
- 1
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
CommunityBot
- 1
2
votes
1
answer
91
views
Question about optimizing a given function by optimizing an approximation
Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''...
- 12.7k
0
votes
1
answer
1k
views
Proper Group action on a metric space
Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {...
- 31.2k
1
vote
1
answer
91
views
Design measure, which cannot be factorized as a product of measures
Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that
$$
0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y =...
- 24.8k
2
votes
0
answers
226
views
degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
1
vote
0
answers
180
views
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 ...
CommunityBot
- 1
3
votes
0
answers
280
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
1
vote
3
answers
653
views
When does $f^{-1}=\frac{1}{f}$ with $f$ a function mapping $\mathbb{R}^{*}$ to $\mathbb{R}$?
In mathematics, an inverse function is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ ...
- 4,713
11
votes
2
answers
1k
views
Two divergent series conspiring?
Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus
$$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad
\sum_{n\geq0}\frac{a_n}{2n+...
- 105
3
votes
1
answer
196
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
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