All Questions
1,533 questions with no upvoted or accepted answers
3
votes
0
answers
169
views
Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?
Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$?
Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
3
votes
0
answers
172
views
Nekrasov Partition function and the leading term of Prepotential
I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...
- 425
3
votes
0
answers
504
views
Continuity of the conditional expectation
Consider the conditional expectation of $x$ given $y$,
$$
\mathbb{E}(x | y)
$$
where $x \in X$ and $y \in Y$ where $X, Y$ are Hilbert spaces (possibly infinite dimensional).
Question :
I am looking ...
- 252
3
votes
0
answers
240
views
About optimizing decay rate of Fourier transforms?
Suppose we have a density function $f(t)$ of a random variable and $f \in C^1(R)$. If characteristic function of $f$ is $\phi_f(x) \asymp O(x^{-\beta})$ and $f$ satisfies some restrictive conditions ...
- 31
3
votes
0
answers
137
views
Can monoids of "continuous words" be realized as initial monoid objects?
Whenever $X$ is a set, write $X^*$ for the monoid freely generated by $X$. The elements of $X$ are, of course, words in the letters $X$. When $X$ is finite, there also seems to be a great many ...
- 3,793
3
votes
0
answers
115
views
First order linear ODE with some decay condition
In Kronheimer [1, p.183], a certain statement is made of which I extract the following special case.
Let $\alpha:\mathbb{R}\to \mathrm{Mat}(n\times n,\mathbb{C})$ be smooth and suppose that there ...
- 31
3
votes
0
answers
155
views
Does one need Second Order Logic to do Calculus?
Second order Logic (SL) is required to define the Reals (otherwise they were at most countable). Based on this, SL is involved in the definition of the limit operator, as the 'core' of all Calculus.
...
- 39
3
votes
0
answers
92
views
Arithmetic progressions inside non meager sets
If $A \subseteq \mathbb{R}$ is non-meager Borel set, then $A$ contains arithmetic progressions of every finite length. I know that this is false if we do not assume that $A$ is Borel. In particular, ...
- 31
3
votes
0
answers
109
views
Weak convergence of series representing the log characteristic function
Disclaimer. I already asked this question on math.stackexchange.com without any answers or comments as of yet.
In which weak sense does the series representation of the log-characteristic function ...
- 101
3
votes
0
answers
77
views
Elliptic operator applied to the distance function
Let $\Omega$ and open subset of $\mathbb{R}^n$. Let us consider the following operator:
$$
\Delta_A (u)\, \, \colon= \text{div}(A \nabla u ), \qquad u \in C^{\infty}(\Omega)
$$
where $A(x)$ is a ...
- 823
3
votes
0
answers
63
views
Is the collection of Schur convex functions sequentially compact?
We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
- 185
3
votes
0
answers
148
views
When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
- 39
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 ...
user99666
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
3
votes
0
answers
214
views
Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
3
votes
0
answers
177
views
Interesting stipulation about completely monotone functions
This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
user78249
3
votes
0
answers
232
views
When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
- 34.3k
3
votes
0
answers
235
views
Is this "differentiation map" uniquely determined by these properties?
Let $A$ be the set of all real-valued functions having their domain a subset of $\Bbb R$ which are at least differentiable on an open set, and for $f \in A$, let $U_f$ be the largest open set on which ...
user101261
3
votes
0
answers
97
views
Dimension of a graph
Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?
If not, what extra condition do we ...
- 2,135
3
votes
0
answers
155
views
asymptotics of the largest real root
Suppose you have a family of polynomials
$$P_n(x)=\sum_{k=0}^n(-1)^ka_k^{(n)}x^k$$
for $n=0,1,2,\dots$.
Further assumptions:
(1) the coefficients $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)...
- 43.2k
3
votes
0
answers
1k
views
Concentration of Sub-exponential random Vectors
I was wondering if there is a similar definition of multivariate sub-exponential distribution as the sub-Gaussian case.
Specifically, a random vector $X \in \mathbf{R}^d$ is sub-Gaussian if
\begin{...
- 1,127
3
votes
0
answers
588
views
Time-dependent Sobolev spaces
Given the Sobolev space $H^1((a,b);H^2(\mathbb{R}))$ and a function $g$ in that space. Consider now another function $f \in C_c^{\infty}((a,b) \times \mathbb{R}).$ Then
for almost any $t \in (a,b)$ we ...
- 31
3
votes
0
answers
267
views
Link between standard convolution and Day convolution
There is a notion of convolution product between two functors called "Day convolution". (See here nlab for instance) I know that the definition of this notion is inspired by the discrete convolution $$...
- 1,017
3
votes
0
answers
160
views
integral with simple approximation. But why?
I have the following integral
$$g(x_0) = \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^{3/4}}\frac{1}{(1+(x+x_0)^2)^{3/4}}\exp\left(-\frac{2\pi i}{\lambda}\left[\sqrt{1+x^2}-\sqrt{1+(x+x_0)^2} \right] \...
3
votes
0
answers
848
views
Does a bounded convex domain has one smooth boundary point?
In the study of analysis and geometry of a bounded domain, its boundary regularity is important. For example, it is known that a bounded convex domain has Lipschitz bounday. This implies that a ...
- 285
3
votes
0
answers
198
views
Characterizing rational functions on $\mathbb{Q}$ in terms of smooth extensions to $\mathbb{R}$ and $\mathbb{Q}_p$
Consider a function $f$ from a cofinite subset of $\mathbb{Q}$ to $\mathbb{Q}$. As established here and here $f$ extending to a smooth function on a cofinite subset of $\mathbb{R}$ is not sufficient ...
- 12.5k
3
votes
0
answers
155
views
Dirichlet series decomposition of arbitrary function
Originally asked on MSE here: https://math.stackexchange.com/q/1780149/52694
Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the ...
- 4,897
3
votes
0
answers
306
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
3
votes
0
answers
161
views
Inverses of probability generating functions: positivity of derivatives
Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$.
So $G\in\mathcal{G}$ can be written $G(x)=\...
- 3,937
3
votes
0
answers
165
views
Extreme derivatives in Baire class 1
In the 1994 volume of "Differentiation of Real Functions" A. Bruckner poses the following problem (p.41):
"Find necessary and sufficient conditions on a continuous function $F$ that its Dini ...
- 277
3
votes
0
answers
373
views
An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only
The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
- 128k
3
votes
0
answers
131
views
What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?
For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...
- 111
3
votes
0
answers
689
views
"Nicely" strong measure zero sets
This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...
- 20.5k
3
votes
0
answers
74
views
Semi-continuity of the dimension of the null space
Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...
- 537
3
votes
0
answers
237
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
3
votes
0
answers
652
views
Derivatives of $O$-regular varying functions are $O$-regular varying functions?
The Monotone Density Theorem for regularly varying functions says, in essence:
Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ well-...
- 825
3
votes
0
answers
105
views
Can Mumford-Shah functional be adapted to lower $L^1$ space?
The well know Mumford-Shah functional functional
$$
F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1
$$
where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
- 679
3
votes
0
answers
166
views
Monotone version of one-dimensional Whitney extension theorem
Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...
- 29.1k
3
votes
0
answers
47
views
Limit Behavior of Iterated Curvature-Function
What can happen, if one defines an infinite sequence of functions as follows
$f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$
$f_{n+1}: \int_0^x \sqrt{1+(f_n'(t))^2}dt\mapsto\frac{f_n''(x)}{\...
- 13.2k
3
votes
0
answers
275
views
Maximizing the discrepancy in Jensen's inequality for a certain function
Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter.
Define
$$
D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}}
-\sum_{...
- 959
3
votes
0
answers
262
views
About the small set expansion conjecture
Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - \...
- 1,893
3
votes
0
answers
511
views
Two (strictly related) proofs by induction of inequalities
This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
3
votes
0
answers
187
views
An upper bound for a average of a function in $L_{p}([0,1))$
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j},~
1\leq j \leq 2^{n} \}
\right\} \right)_{n ...
- 103
3
votes
0
answers
290
views
Does there exist a supersmooth non-polynomial function?
Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
3
votes
0
answers
295
views
Density of function spaces
Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an ...
- 83
3
votes
1
answer
966
views
Continuity of minimizers to distance function from point to convex set
Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...
- 81
3
votes
0
answers
171
views
Generalized family of Hölder inequalities
Is the "only if" direction of the following fact known?
For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^...
- 31
3
votes
0
answers
119
views
Does the following inequality hold under Zygmund condition?
Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, \...
- 197
3
votes
0
answers
170
views
Is there such a matrix in $SO(n)$?
Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = \frac{B_{ij}}{\sqrt{B_{ii}B_{jj}}},...
- 131