All Questions
5,857 questions
0
votes
1
answer
270
views
Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
user516435
3
votes
3
answers
550
views
Solving interval problems without outer measure
Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ?
Problem 1
If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
3
votes
0
answers
137
views
On the continuity with respect to the increasing convex order
For $p\ge 1$, let $\mathcal P_p(\mathbb R)$ be the set of probability measures on $\mathbb R$ of finite $p^{\rm th}$ moment. Denote by $W_p$ the Wasserstein metric of order $p$ and by $\preceq$ the ...
- 1,399
0
votes
0
answers
32
views
Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions
For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
1
vote
1
answer
300
views
Convergence of concave/convex function
Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
- 393
1
vote
1
answer
108
views
If all mixed partials of a $C^1$ function exist and are continuous, is the function $C^2$? [closed]
For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be a $C^1$ function such that the mixed partial derivatives $\partial_i \partial_j f$ exist and are continuous for all $i \neq j$. Is it true that $f$...
- 6,223
33
votes
2
answers
2k
views
What is the smallest set of real continuous functions generating all rational numbers by iteration?
I recently came across this problem from USAMO 2005:
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
- 4,862
9
votes
1
answer
764
views
Does the family of fat Cantor sets contain a measurable rectangle?
Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$.
...
- 6,223
11
votes
2
answers
532
views
Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$
I'm interested in the asymptotics of
$$\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$$
as $x\to\infty$. I expect the results to behave similarly to $e^{x^2}=\sum_{k\ge 0}\frac{x^{2k}}{k!}$. However, I'...
- 870
20
votes
3
answers
2k
views
Convergence of convex functions
I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...
- 28k
15
votes
2
answers
1k
views
Converse of mean value theorem
Note: This is an attempt to narrow down conditions under which the conjecture stated in this previous post is true. As stated, it is false as shown by the counterexample provided in the answers by the ...
- 6,223
9
votes
2
answers
700
views
Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?
The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...
- 3,477
9
votes
0
answers
287
views
The approximate mean value theorem / Rolle's theorem in pure constructive mathematics
In the replies of this very similar question, there is a fascinating answer that is beautiful in its simplicity. In particular, it seems to use perhaps the most minimal assumptions one can possibly ...
- 191
1
vote
0
answers
48
views
How to derive a lower bound of a MinMax inequality?
Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$.
The goal
For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
- 405
1
vote
0
answers
96
views
Regularity of Feynman-Kac formula for a simple diffusion
Let consider the diffusion process given by:
$$dX_t = \alpha(X_t) dW_t$$
where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
- 393
12
votes
2
answers
1k
views
Counterexamples to differentiation under integral sign, revisited
Let $f\colon\mathbb R^2\to\mathbb R$ be a measurable function such that
\begin{equation*}
F(t):=\int_{\mathbb R}dx\,f(t,x)
\end{equation*}
exists and is finite for all real $t$. Suppose that
\...
- 128k
6
votes
1
answer
379
views
An inequality for a concave function $f(x)=x^{p/2}$
Assume that $p\in(1,2]$, $a,b\ge 1$, $b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
- 333
5
votes
2
answers
564
views
Stone-Weierstrass without the "subalgebra" condition
Suppose I consider $C_0(\mathbb{N})$ consisting of function on the natural numbers vanishing at $\infty$. For an irrational $1<\alpha<2$, let $p_{m\alpha}(\cdot)$ be the function $p_{m\alpha}(n)=...
- 161
1
vote
1
answer
184
views
Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
- 6,223
3
votes
1
answer
401
views
What does the Jacobian of a vector field at an equilibrium tell you about local behavior of integral curves when the Jacobian is not a stable?
I have a soft question regarding the Jacobian of vector fields and isolated equilibria, and what they imply about local behavior of nearby integral curves near.
Let $V:U \subset_{open} \mathbb{R}^n \...
- 518
4
votes
1
answer
334
views
Is this approximation for $\pi$ enough to make this value converge? And how to find an upper bound for it
Update:
\begin{align*}
|I_n-J_n| = (\pi-S_n)\sum_{k=0}^n |\frac{a_kp_k(\ln\pi)}{\ln^{k+1}\pi}|
\end{align*}
and
\begin{align*}
|I_n| = \sum_{k=0}^n | \frac{a_k\pi p_k(\ln\pi)}{\ln^{k+1}\pi}
-\sum_{k=...
- 521
23
votes
2
answers
3k
views
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
- 493
0
votes
1
answer
106
views
The sequence has a stationary accumulation point
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that
(a) $\{x^k\}$ has an ...
- 21
5
votes
2
answers
708
views
Approximation of Hölder continuous functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
11
votes
1
answer
1k
views
New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
- 53
1
vote
1
answer
150
views
Is the Boltzmann entropy continuous in the supremum norm?
We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\...
- 825
3
votes
4
answers
2k
views
Representation of the Dirac delta function
The Dirac delta function appears in the Sokhotsky formula,
$$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$
to be understood in the integral sense
$$\text{Im}\lim_{\epsilon\to ...
- 188k
10
votes
1
answer
936
views
Derivative without extrema is monotone
This is a cross-post from Math.SE.
The question was asked there 3 months ago but didn't receive much attention aside from one comment asking for clarification. I feel like it might be non-trivial and ...
- 103
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 434
0
votes
0
answers
28
views
Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
0
votes
1
answer
68
views
Box dimension and graph of Hölder function
In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that,
if a function $f:I^{d}\...
- 73
3
votes
1
answer
140
views
Oscillation functions and similar constructs
For given $f$ from reals to reals, the associated oscillation function is defined as follows:
$$\textstyle
osc_f(x):= \lim_{n\rightarrow \infty} [\sup_{y \in B(x, \frac{1}{2^n}) } f(y)-\inf_{z \in B(x,...
- 4,359
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
0
votes
1
answer
117
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
- 4,058
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
- 833
3
votes
1
answer
211
views
Blowup of Sobolev norms in approximating a non-absolutely continuous function
Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that if $f$ fails to be absolutely ...
- 6,223
4
votes
1
answer
253
views
The number of roots of the sum of radicals
Let $n\in \mathbb{N}$ and $$-\infty < a_1 < b_1 < a_2 < b_2 < a_3 < b_3<\cdots<a_n<b_n<+\infty$$ and $k_i\in \mathbb{R}, i=1,2,\ldots,n$. Is there any information about ...
- 89
8
votes
3
answers
429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,153
8
votes
1
answer
688
views
Measure without measurable sets
This question is a little on the softer and speculative side, so bear with me.
Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
- 3,574
3
votes
2
answers
248
views
Exceptional set for Marstrand's projection theorem
If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $...
- 265
4
votes
3
answers
369
views
Non-negativity of a complicated function
Show that $f(x)\ge 0$ for $0\le x \le 1$, where:
$$f(x) = \arccos(x)^2 -8x(5x^2-2) \sqrt{1-x^2}\arccos(x)+36 x^8-112 x^6+93 x^4-17 x^2$$
The endpoints are $f(0)=\pi^2/4$ and $f(1)=0$. Plotting ...
- 391
5
votes
1
answer
508
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 219
4
votes
1
answer
837
views
Can a function that is continuous on a dense set be almost extended to a continuous function?
Note: All sets and functions defined below are assumed measurable. $\mu$ denotes the Lebesgue measure.
Let $D$ be a dense subset of $[0, 1]$, and $f: D \to \mathbb R$ a function. Given $\varepsilon &...
- 6,223
5
votes
1
answer
366
views
Quantitative Lebesgue density theorem
Let $A \subset [0, 1]$ be a measurable set, and $\mathbf 1_A$ its indicator function, viewed as a function on $\mathbb R$. Define for each $\delta > 0$, the function $f_{A, \varepsilon}: \mathbb R \...
- 6,223
7
votes
0
answers
254
views
$C^0$-limit of volume-preserving maps on $\mathbb R^n$
Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
- 435
2
votes
1
answer
433
views
Stone-Weierstrass theorem: coefficients of approximating sequence bounded?
Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$.
The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points ...
- 463
6
votes
2
answers
319
views
Does control on the “magnitude” of the rearrangement give control of the rearranged Cesaro sums?
Let $a_n$ be a nonnegative sequence that Cesaro converges to $K > 0$. We recall this means
$$\frac{1}{N} \sum_{n = 1}^N a_n \to K$$
as $N \to \infty$.
Suppose $a_{\phi_n}$ with $\phi: \mathbb N \to ...
- 6,223
1
vote
1
answer
126
views
Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?
Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that
$$
\int_0^1 |y-x| f(x) \, dx = 0
$$
for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
- 136
4
votes
2
answers
415
views
The set of all possible values of subseries of a convergent positive term series
Inspired by The set of all limits of sub-series of an absolute convergent series is the following true?:
Let $a_n$ be a strictly decreasing sequence and $\sum_1^\infty a_n=\ell<\infty$ is a ...
- 356
6
votes
3
answers
1k
views
Discontinuous functions without removable discontinuities
A function $f:\mathbb{R}\rightarrow \mathbb{R}$ has a removable discontinuity at a given real $x$ in case the left and right limits are equal but not to the function value, i.e. $f(x+)=f(x-)$ but $f(x)...
- 4,359