All Questions
5,857 questions
7
votes
1
answer
233
views
Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
- 541
2
votes
1
answer
266
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
840
views
Integration of gaussian times absolute value of cosine
Is there a way to compute/estimate the following integral?
$\int_0^\infty e^{-(x/c)^2}\left|\cos{x}\right|dx$
where $c$ is a real constant. I would like to know if it is of order $e^{-c^2/4}$ like the ...
- 23
5
votes
3
answers
349
views
minimum of two probability densities
Consider a smooth probability density $\pi(x)$ on $\mathbb{R}^d$. I am looking for natural for the integral $\iint_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv$ to be finite. If $\pi$ is a radially ...
- 2,133
2
votes
1
answer
573
views
Extension of Dynkin's formula, conclude that process is a martingale
This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
user61522
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
1
answer
350
views
Brownian motion, crossing intervals, possible usage of second moment method?
This is a followup to my question here.
Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$...
user80013
6
votes
2
answers
2k
views
Regarding sub-additive sequences and Fekete's lemma
A non-negative sequence $\{a_n\}$ is sub-additive if $a_{m+n}\leq a_m + a_n.$ Fekete's lemma says that for any non-negative sub-additive sequence:
$$\lim_{n\to\infty} \frac{a_n}{n} = \inf_{n} \frac{...
- 1,269
12
votes
1
answer
239
views
Interval arithmetic with different definitions of intervals
Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}...
- 497
1
vote
1
answer
350
views
Strong convergence in reflecxive Banach space
Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f_n \rightarrow f$ weakly in $X$, and $\|f_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly ...
- 425
2
votes
0
answers
192
views
Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
- 623
1
vote
0
answers
100
views
Higher order derivative of negative power of cosine function
This is a question I encountered in my own research on Generalized
Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the
basis measure for this family is
$$L\left( \...
- 984
2
votes
1
answer
349
views
Polynomial with subset of critical points and values prescribed
Motivated by this question I wish to pose the following question:
Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...
- 1,049
6
votes
1
answer
380
views
Asymptotic value of a multivariate integral
The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems.
Define
$$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
- 37.7k
7
votes
1
answer
820
views
Has this generalization of a determinant (assigning multiplicities to the rows) been studied?
I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
- 1,509
4
votes
1
answer
660
views
Reconstructing a (unitary) matrix from the determinant of its sub-matrices
I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ ...
- 61
1
vote
0
answers
311
views
Estimating an integral involving Bessel functions
I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...
- 161
4
votes
1
answer
222
views
Is every regular Borel outer measure topologically additive?
If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive?
If so what is a proof or a counterexample?
Definitions:
Topologically Additive: $X$ is a topological space, $m$ ...
- 41
1
vote
1
answer
980
views
Extension of a smooth function from a convex set
Let $C\subset \mathbb{R}^{n}$, $C'\subset\mathbb{R}^{m}$ be two convex sets with a non-empty interior. A function $F\: : \: C\to C'$ is said to be differentiable at $x\in C$ if there exists a linear ...
- 1,424
3
votes
1
answer
480
views
To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$
I wants to understand the integrals of the form
$$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$
where $\mu(\alpha)$ is a non decreasing function such ...
- 1,051
3
votes
1
answer
293
views
Determinant Evaluation
Is there a closed form (something involving a ratio of products) for:
$$\det\left[\binom{a_i+c}{a_i-i+j}\right]_{1\leq i,j\leq t},$$
where $a_i,c$ are positive integers? I think with $c=0$ this is ...
- 4,952
4
votes
0
answers
122
views
Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?
I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those ...
user114642
9
votes
1
answer
451
views
Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive
Is the following function real analytic in $t>0$:
$$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$
where $-a$ and $b$ are positive, and $c\not=a$?
...
- 353
6
votes
1
answer
188
views
On continuous perturbations of functions of the first Baire class on the Cantor set
Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
- 42k
2
votes
1
answer
2k
views
Fourier transform of $sin(\frac{1}{x})$ for $x > 0 (x > 1)$
Please, give me the cue: does exist analytical representation of Fourier Transform of $sin(\frac{1}{x})$ for$ x>0$ (or $x>1$). Maybe exist an approximation of $FT(sin(\frac{1}{x}))$ by Bessel ...
4
votes
0
answers
144
views
Asymptotic expansion of a Gaussian integral and heat kernel
When considering the heat kernel of a Schr\"odinger operator
$$- \Delta + V(x) $$
where $\Delta$ is the standard Laplacian on ${\mathbb R}^n$ and $V$ is a nonnegative potential function that has ...
- 1,207
1
vote
0
answers
101
views
Non standard Lipschitz extension
Consider a ball B and let $f(x) \in L^1(B)$ such that $\int_B f(x) dx = 0$. Furtheremore, there exists a closed set $E \subset B$ such that $f|_E$ is Lipschitz. The standard Lipschitz extension ...
- 455
2
votes
0
answers
227
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
1
answer
161
views
Proof of Convergence + Identifying Probability Distribution
I'm trying to prove that the series below converges to 1 and I noticed it looked strikingly similar to a probability distribution I once saw. My question is twofold:
Can anyone identify the ...
- 113
0
votes
2
answers
122
views
Root of a special rational function with positive coefficients
During my research I came across the following problem:
I need to find a root of the following function:
$$\Gamma_{N}(x) = \sum\limits_{i=0}^{M}\left(\frac{\sum\limits_{n=0}^{n_F}n\ \alpha_{i,n} x^n}{...
- 237
8
votes
1
answer
2k
views
Does integrating with respect to a finitely additive measure respect addition?
Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we ...
- 3,830
9
votes
1
answer
1k
views
Determinantal formula for the nullspace of a singular matrix
In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...
- 82.7k
2
votes
1
answer
221
views
Are there good ways of relating a minor to the full determinant?
Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
- 1,893
0
votes
2
answers
139
views
On the existence of $ \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} $ under some constraints
I am considering a smooth-enough real-valued function $ f: (0,1) \to (0,\infty) $ such that
$ f $ is decreasing,
$\lim_{x\rightarrow0^{+}}f(x)=\infty $,
$ x \mapsto x^{2} f'(x) $ is decreasing,
$\...
- 301
0
votes
1
answer
139
views
Moment problem with wrong solution
I will write a problem with an answer that apparently is wrong. My question would be what is wrong with this solution.
Define $$B(s)=\sum_{i=0}^s{{2\,s-i-1\choose s-1}\frac {i}{s}{5}^{i}{2}^{s-i}}$$ ...
- 333
1
vote
3
answers
496
views
Decompose the Laplacian
Is there a way to write the negative Laplacian on the 2-sphere as a decomposition of an operator $A$ and its adjoint $A^*$? I am interested in finding such a decomposition, but I could not get one by ...
- 21
2
votes
1
answer
186
views
Equivalence of definitions of the space of test functions
Let $\Omega\subseteq\mathbb{R}^n$ be some nonempty open, and use the notation $U\Subset V$ to imply that $U$ is a compact subset of $V$. Then, for all $K\Subset\Omega$, we can define the space $$\...
- 1,247
2
votes
0
answers
142
views
Self-adjointness on Banach spaces
Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded ...
- 501
2
votes
3
answers
3k
views
Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set
It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...
- 777
2
votes
2
answers
232
views
Heat equation: impact of the diffusion coefficient on the Harnack constant
Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le a(x,...
- 632
2
votes
0
answers
60
views
Gronwall type inequality involving iterated integrals
Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have:
$$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$
where $\kappa, C >...
- 31
1
vote
0
answers
43
views
Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball
Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [...
- 13.1k
2
votes
1
answer
216
views
Ask for a special function related to the error function
I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...
- 1,649
0
votes
1
answer
535
views
a sum of ratios of quadratic forms
I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} \right]\mathbf{x}_k\mathbf{x}_k^H\...
- 61
9
votes
3
answers
4k
views
Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?
A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...
user2529
2
votes
1
answer
254
views
Sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$
Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish ...
- 6,132
2
votes
1
answer
228
views
Convergence of Discrete Geodesic
Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$.
Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence:
\...
- 5,405
7
votes
1
answer
503
views
Poincaré inequality for curl-integrable functions
Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
- 632
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
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