All Questions
5,910 questions
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
0
votes
0
answers
68
views
Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.
If ...
2
votes
0
answers
116
views
Existence of a sequence of real numbers
Let
$$g_{c;k}(z):=\frac{2 (c-z-1)^{k+2}}{(k+1) (k+2)}+\frac{1}{2} (-c+z+2)^2 z^k+\frac{-2 c (k+2)+4 k+6}{(k+1) (k+2)}+\frac{2z}{k+1}.$$
Do there exist $c\in(1,3/2)$ and a sequence $(a_k)_{k=0}^\infty$ ...
CommunityBot
- 1
3
votes
1
answer
220
views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 498
2
votes
0
answers
189
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
0
votes
1
answer
97
views
Numerically bounding a Exponential-Trigonometric Integral [closed]
I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer.
I have tried decomposing into Riemann sum and ...
- 188k
0
votes
1
answer
57
views
Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 128k
3
votes
1
answer
227
views
"Essential values" of a function at a point?
Recall that the essential range $\operatorname{ess.im} f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, ...
- 6,155
2
votes
2
answers
154
views
Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 60.5k
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,155
4
votes
1
answer
297
views
Oscillation of monotone real-analytic function
Let $f:(a,\infty)\rightarrow \mathbb{R}$ be a real-analytic and strictly monotone function. I have been wondering how much this function can "oscillate". Namely, can we always find a ...
- 128k
2
votes
1
answer
547
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
CommunityBot
- 1
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
1
vote
1
answer
124
views
$d(x,y) = \min\{|x_1−y_1|+|x_2−y_2|, 1−|x_1−y_1|+|x_2−(1−y_2)|\}$ defines a metric on $[0,1)\times[0,1]$? [closed]
For $x,y \in [0,1)\times[0,1]$, let $d(x,y)$ be the minimum of $|x_1−y_1|+|x_2−y_2|$ and $1−|x_1−y_1|+|x_2−(1−y_2)|$. Prove or disprove that $d$ is a metric.
I was unable to find a counterexample to ...
- 109k
7
votes
0
answers
619
views
Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 6,367
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
5
votes
2
answers
372
views
Weak Archimedean property instead of Archimedean property
We say that a sequence $(z_n)$ of real numbers is a modulated Cauchy sequence, whenever there exists a function $\alpha:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
|z_i-z_j| \le \frac{1}{k} \quad ...
- 6,367
3
votes
0
answers
100
views
How to compute the partial derivatives of this function?
For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
- 1,389
0
votes
1
answer
231
views
Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 60.5k
6
votes
1
answer
193
views
The most even partition of $\mathbb R$ into measure dense sets
Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every ...
- 6,155
3
votes
1
answer
343
views
Asymptotic behavior of a recursion
Let $x_n(0)=1$,
$$
x_n(N+1) = \frac{1}{N+1}\sum_{k=0}^N \sum_{j=1}^n x_j(k)x_{n+1-j}(N-k) + \frac{10}{N+1} x_{n+1}(N) , \quad\quad N\ge 0 .
$$
So the recursion is on $N$, and at each level, we compute ...
- 128k
2
votes
1
answer
149
views
Proof that superlinearly convergent sequence converges faster than linearly convergent sequence
Given real sequences $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$, both converging to the same limit $A$ and such that $|a_n-A|\neq 0$ and $|b_n-A|\neq 0$ for every $n$ sufficiently large, we ...
- 6,155
3
votes
2
answers
118
views
Does the derivative of the antiderivative of a BV function $f$ agree with $f$ at all but countably many points of differentiability?
Let $f: (a, b) \to \mathbb R$ be a function of bounded variation, and write
$$F(x) := \int_a^x f(t) \, dt$$
for the antiderivative. Is it true that at all but countably points of differentiability of $...
- 60.5k
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 105k
9
votes
1
answer
553
views
Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
- 166
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 ...
- 60.5k
5
votes
0
answers
156
views
What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,155
3
votes
1
answer
490
views
Space derivative of flow of ODE with monotone source
Consider the ODE
$$
\begin{cases}
\partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\
\Phi(0,x) = x, & x \in \mathbb R
\end{cases}
$$
where $f$ is function which is a non-...
CommunityBot
- 1
1
vote
1
answer
179
views
For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$
Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that
\begin{align*}
0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
CommunityBot
- 1
4
votes
1
answer
204
views
Stationary phase formula for a complex valued phase
I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form
$$
I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx}
$$
where $\varphi : \mathbb{R} \...
- 6,367
0
votes
0
answers
121
views
Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?
Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
- 7,127
12
votes
2
answers
866
views
Sets that project to zero measure on all lines except one
It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
- 613
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
-1
votes
1
answer
167
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 6,155
8
votes
1
answer
342
views
How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
- 3,312
4
votes
1
answer
374
views
An open mapping theorem for homogeneous functions?
I am researching different generalizations of the familiar open mapping theorem from functional analysis. Every "proof" I attempt while simply assuming positive-homogeneity, even in the finite-dim ...
- 60.5k
4
votes
1
answer
217
views
$2$ continuous, commuting functions doesn't always have a common fixed point
The question is as such:
If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point.
-- Zorich ...
1
vote
1
answer
50
views
Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula
I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$)
\begin{equation}
f(\theta)= \frac{h(t,\...
- 128k
9
votes
1
answer
492
views
Dispersion points of Lipschitz functions
For a function $f: \mathbb R^n \to \mathbb R^m$ with $m < n$, we say that $x \in \mathbb R^n$ is a dispersion point of $f$ if
$$\liminf_{y \to x} \frac{|f(y) - f(x)|}{|y - x|} > 0.$$
Question: ...
- 6,155
21
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
- 35.5k
1
vote
1
answer
330
views
Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?
I am trying to study the converge of the series
$$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$
But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
- 348
0
votes
1
answer
139
views
Proving negativeness of function involving $-\log t$
I have been trying to solve the following function is non-increasing with respect $\theta$
\begin{equation}
h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
- 128k
7
votes
1
answer
270
views
Sequential continuity and the Axiom of Choice
It is well-known that ZF cannot prove the following:
"for a function $f$ from reals to reals and any real $x$, $f$ is continuous at $x$ if and only if $f$ is sequentially continuous at $x$."...
- 4,316
1
vote
2
answers
102
views
About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$
Suppose we have a non-decreasing sequence of positive real numbers that tend to infinity: $0<w_1\leq w_2\leq w_3\leq...$ It is known that:
For every $n$ and $p\geq 2n$, we have $w_p \geq (1-\frac {...
- 6,476
3
votes
1
answer
966
views
Continuity of minimizers to distance function from point to convex set
Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...
CommunityBot
- 1
2
votes
0
answers
99
views
Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$
Do you know if there exists a closed form for the integral :
$$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$
where $p$, $q$, $r$ are natural integers such as this integral ...
- 12.9k
0
votes
0
answers
73
views
An example of a groupoid that satisfy the following hypothesis
In the paper titled, 'Tannaka–Krein duality for compact groupoids I, Representation theory', the author proves the Peter Weyl theorem on compact groupoids. In the statement, he gives the hypothesis ...
0
votes
0
answers
60
views
Criteria for log-absolute-monotonicity
Consider a function $f: [0,1] \rightarrow \mathbb R$ defined by a power series $f(x) = a_0 + a_1 x + a_2 x^2 + \dots$, where all $a_i$ are positive.
Is there are any criterion in terms of the ...
- 3,475
2
votes
1
answer
133
views
Points of differentiability of convex functions
Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
- 128k