All Questions
5,857 questions
-3
votes
1
answer
227
views
Equation $\ \binom xn'\ =\ \log(n)$ [closed]
Problem: Solve equation
$$ \binom xn'\ =\ \log(n) $$
Here prime stands for the derivative with respect to $x$.
Observe that:
$\quad$ for integer $n$ large,
the approximate solution is $\ ...
- 4,786
1
vote
0
answers
27
views
How does the principal value affects to the limit here?
In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
- 383
6
votes
1
answer
273
views
Avoiding equal distances
Is the following consistent?
There exists $X \subseteq [0, 1]$, such that $X$ does not have measure zero and for every $Y \subseteq X$, if $Y$ does not have measure zero, then there are $y_1 < y_2 ...
- 9,641
3
votes
0
answers
129
views
Function and distance on bounded set [closed]
Does there exist a bounded set $A \subset \mathbb{R}^ n$ (for some $n$) and a function $f:A\to A$ (not necessarily onto) such that $\forall x,y \in A$ then $\|x-y\| < \|f(x)-f(y)\|$?
If such a ...
- 131
1
vote
0
answers
61
views
How to see the divergence of a series is not faster than some order? [closed]
$$
\sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p}
$$
For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.
- 11
8
votes
2
answers
796
views
generalizations of Vandermonde matrix to high dimensions
Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix
the maps
$$
f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$
x\longmapsto (1,x,x^2,\cdots,x^{n-1})...
- 519
-1
votes
1
answer
195
views
Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 2,246
0
votes
0
answers
67
views
Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?
Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$.
I'm looking to construct some kind of dyadic cube decomposition or ...
- 1,179
2
votes
3
answers
260
views
Nonzero solutions of an infinite product
Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty ...
- 293
9
votes
3
answers
4k
views
Is there a reference for compact imbedding theory of Hölder space?
This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(...
- 1,399
2
votes
0
answers
80
views
Generalized definition of integrable condition on rough complex subbundle
Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...
- 938
2
votes
1
answer
573
views
Harmonic measure
Hi everyone: Let $ \omega $ be a bounded open set in $ \mathbb{R}^{q} $, $ q\geq 2 $, and $ E $ a subset of the boundary $ \partial\omega $ that has harmonic measure zero in $ \omega $. Let $ V $ be ...
- 411
3
votes
0
answers
97
views
Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?
I posted the following question on StackExchange a few months ago (https://math.stackexchange.com/questions/2898620/notions-of-beta-h%C3%B6lder-smoothness-when-beta-in-1-2-are-they-equivalent), but ...
- 31
8
votes
2
answers
2k
views
Do proper Zariski closed sets of algebraic sets have measure zero
This is a question related to another question I asked: here.
Say we induce a probability measure that is absolutely continuous with respect to to Lebesgue measure onto an irreducible real algebraic ...
- 81
1
vote
1
answer
325
views
On Riemann zeta function and Dirac delta function/distribution
Let $$I_{N} = \int_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N}(2\pi x)dx = \frac{2N-1}{2N} \int_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N-2}(2\pi x)dx $$ therefore (I think)
$$ I_N = \frac{(2N-1)!...
- 251
9
votes
2
answers
553
views
Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
1
vote
1
answer
178
views
Bochner integrability within a subspace
Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for ...
- 1,913
1
vote
1
answer
357
views
Does CLT hold for joint distribution of two dependent binomial variables?
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
- 111
2
votes
0
answers
881
views
Why does a convex function have to have a convex domain? [closed]
Other than convenience in convex optimization, is there a reason that the definition of a convex function includes the requirement for the domain to be a convex set?
- 121
9
votes
0
answers
979
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
10
votes
2
answers
6k
views
Who was the first to formulate the inverse function theorem?
Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$.
...
- 103
3
votes
1
answer
113
views
Asymptotic expansion of nonlinear Gaussian transformation in terms of covariance
I'm reading this paper and on page 8 the authors state without proof an asymptotic expansion of a multivariate Gaussian integral in terms of the covariance obtained by applying what they call the "...
- 617
5
votes
1
answer
3k
views
Equicontinuity and $L^2$ convergence imply uniform convergence
I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real ...
- 61
7
votes
3
answers
4k
views
Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 1,399
0
votes
0
answers
54
views
Existence of a solution for the Laplace equation with sub-linear non-linearity
At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for
$$
\begin{cases}
-\Delta u = u^p & \Omega \subset \...
- 371
4
votes
2
answers
562
views
Reference request: concave/convex envelope
I'm seeking the references concerning on the regularity analysis of concave envelopes, i.e. given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), ...
- 1,835
8
votes
1
answer
242
views
Does infinitesimal variance imply continuity?
Let $u:[0,1]\to\mathbb{R}^n$ be a bounded Borel function.
It is well-known that if, for any compact interval $I\subseteq [0,1]$,
$$ \int_I|u-u_I|^2\le C|I|^{1+\alpha} $$
for some $C,\alpha>0$ (here ...
- 3,146
3
votes
1
answer
1k
views
Showing a singular integral operator takes Hölder continuous functions to Hölder continuous functions of the same order
I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...
- 31
1
vote
1
answer
204
views
Why study the moment problem in one dimensional case( Hamburger moment problem)
I have been reading about moment problem and I have been curious about the following question.
What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?
I ...
- 125
0
votes
1
answer
165
views
Is there an example of a one to one and onto mapping between these two spaces?
Let $\Omega$ be a convex open subset of $\mathbb{R}^d$ with a smooth boundary. Is there an example of a one to one and onto mapping of the form $$L^{d+1}(\Omega) \to W^{1,d+1}(\Omega)$$
- 698
4
votes
0
answers
101
views
Injectivity of product functions on natural number sequences
Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...
- 749
2
votes
0
answers
116
views
A variant of the optimal transport
Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows:
$$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$
where the inf is taken ...
- 345
11
votes
1
answer
2k
views
Functions whose antiderivative behaves like xf(x)
I'm wondering if a classification of analytic functions, $f\,$ (it may be that $C^1$ is enough, but I'm not taking any chances, if you have a reason why I only need to consider a larger class of ...
- 1,049
6
votes
3
answers
481
views
Quantum Mechanics and bilinear optimal control theory
I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t)...
- 259
1
vote
0
answers
117
views
Estimation of the integral $\int_a^b e^{2\pi i f(x)} dx $
Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that
$$
\min_{x\in [a,b]} |f'(x)|>\lambda
$$
It is ...
- 675
2
votes
1
answer
84
views
How to choose function $\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$?
Can we expect to choose a function $f:\mathbb R \to \mathbb R$ (nonzero compactly supported) so that
$\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$ for all $x\in \mathbb R$ and $n\in \mathbb Z$?...
- 305
1
vote
0
answers
136
views
Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x↦x^{-x}$
The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...
1
vote
3
answers
125
views
Weighted sum of binomials with $r$-th power of lower index
Given $r\in(0,1),$ what is the best upper (asymptotic) bound for the following expression
$$S(n,r):=\sum_{k=0}^n{n\choose k}k^r?$$
Holder's inequality gives $S(n,r)\le 2^n(\frac{n}{2})^r$ but I guess ...
- 123
1
vote
2
answers
256
views
Monotonicity of the sequences of the lower and upper Darboux sums
Consider the following lower and upper Darboux sums
$$ 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} $$
for every real $\ x>0\ $ and ...
- 7,500
0
votes
0
answers
55
views
Smooth compactly supported function with good scaling with respect to the fractional Laplacian
Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user139845
8
votes
0
answers
210
views
Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
- 389
2
votes
1
answer
283
views
Why should these two functions approximate each other so well?
I am a physicist and during some of my research I realized that $\log_2 \sec(\pi x)$ is really well approximated by $\frac{6x^2}{1-x}$ for small but positive $x$. Is there any reason this should be so?...
- 607
4
votes
0
answers
673
views
Proofs of the second fundamental theorem of calculus
I am referring to the following version of the theorem, in the setting of the Lebesgue integral.
Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
- 18.7k
10
votes
1
answer
379
views
Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?
A function $f\colon \mathcal{P}(\mathbf{N})\to [0,1]$ is said to have the Darboux property whenever for all $X \subseteq \mathbf{N}$ and $y \in [0,f(X)]$, there exists $Y \subseteq X$ such that $f(Y)=...
- 1,511
2
votes
1
answer
497
views
Linear map of finite or infinite extreme points. Discuss injectivity and surjectivity
Before entering my problem, let me review some related results:
Suppose $\mathcal{S}$ is a convex hull of finite points: $\mathcal{S}=\operatorname{conv}(x_1,x_2,\ldots,x_m)$, then
By "https://...
- 345
3
votes
1
answer
432
views
the existence of a real polynomial satisfying the following property
It is easy to verify that
$$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2}
\quad \quad 0<t\leq1$$
I want to ask if there exist a real polynomial $h(t)$ such ...
- 1,997
6
votes
2
answers
2k
views
non-maximal prime ideal in the ring of continuous functions
Let $A=C(0,1)$ be the ring of continuous real valued functions on the open interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\...
- 7,609
4
votes
2
answers
261
views
Is Laplace's method applicable to this integral?
Consider the following integral
$$
pv\int_0^{\infty}e^{N(-2Ax+A\log x)}\frac{e^{-B\log x}}{1-2x}dx
$$
where $A,B>0$ and we take the Cauchy principal value at $x=1/2$. I am interested in obtaining ...
- 247
1
vote
0
answers
64
views
Regularity of superposition operator generated by function between Banach spaces
Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call
$$
\varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot))
$$
the ...
- 121
10
votes
1
answer
938
views
Why does this antisymmetric product factor out a determinant?
Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...
- 2,902