All Questions
5,630 questions
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extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 10.7k
2
votes
1
answer
128
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Characterization of a subset of [0,1] $III$
I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to $...
CommunityBot
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votes
1
answer
222
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Does the divergence of the sum of reciprocals of a set of integers imply this density statement about the set?
Suppose $A \subseteq \mathbb{N}$ is such that $\displaystyle{\sum_{n \in A} n^{-1}} = \infty$. Suppose $B \subseteq \mathbb{N}$ is infinite.
Is there a set $X \subseteq [1,\infty)$ and a increasing ...
- 23.2k
3
votes
1
answer
166
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Does directional limits along any given direction, always exist for a function of bounded variation?
If a function $f:\mathbb{R}^N\to\mathbb{R}$ is of bounded variation (in modern sense or in Tonelli sense or according to any of the existing definitions), then can we say that, given any point $x\in \...
- 24.8k
3
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295
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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
10
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1
answer
2k
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Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
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3
votes
1
answer
250
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Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
CommunityBot
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2
votes
1
answer
247
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is $x_{n}\ll \overline{x}_{n}^{2}$?
This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\...
- 105k
0
votes
1
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195
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Existence of bounded $n-$th derivative of the solution of differential equation
This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)...
CommunityBot
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5
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0
answers
310
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Reference for Hodge decomposition
Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
- 1,253
1
vote
1
answer
192
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Characterization of a subset of $[0,1]$
Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property:
For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...
- 24.2k
15
votes
1
answer
595
views
Solving a non linear equation
I've been trying to prove that the following equation has a unique solution in interval 0 < x < 1 :
$$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$
Where K is a number (integer, if it ...
- 60.6k
0
votes
1
answer
321
views
Is the span of those vectors dense in $\ell_2$?
For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} \...
- 3,031
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1
answer
432
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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 ...
CommunityBot
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4
votes
3
answers
711
views
Inequality of arithmetic, geometric and harmonic means
Let $a_1,\dots,a_n$ be positive numbers, does the following inequality holds?
$$\frac{a_1+a_2+\cdots+a_n}{n}-\sqrt[n]{a_1a_2\cdots a_n}\geq\sqrt[n]{a_1a_2\cdots a_n}-\frac{n}{\frac{1}{a_1}+\frac{1}{...
- 30.1k
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2
answers
348
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Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians?
This question originates an engineering application.
There is a certain process that is presumed to be a sequence of diffusions and is usually modelled as a sum of Gaussians:
$$\Sigma_n w_ne^{-\...
- 34.7k
2
votes
1
answer
81
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Monotonicity of Trapezoid Approximations
Here's a numerical analysis question which may not be very important, especially in practice, but has been bugging me.
Suppose $f$ is a continuous function on an interval $[a,b]$. Let $T_n(f)$ be ...
- 54.2k
2
votes
1
answer
3k
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Inequality for the tail of normal distribution function
Let $ Ф(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt $ be the cumulative distribution function of the standard normal distribution.
Numerical calculations suggest the following ...
- 54.2k
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 ...
- 54.2k
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votes
1
answer
101
views
Extending point-wise bound to uniform bound
Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$
$$ C(t):= \sup_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$
Does it follow that $f$ is bounded?
Note ...
- 6,210
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4
answers
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Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...
CommunityBot
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6
votes
1
answer
440
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Are there superexponential Pfaffian functions?
This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
CommunityBot
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5
votes
0
answers
596
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Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
28
votes
4
answers
3k
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"Converse" of Taylor's theorem
Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a_0,a_1,\ldots,a_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum_{i=0}^k a_i(c)t^i+o(t^k)$ (for ...
- 25.3k
0
votes
2
answers
190
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Is real analytic function good enough (see problem)? [closed]
Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then ...
- 105k
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0
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145
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Does there exist this special kind of homeomorphism?
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
- 183
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1
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754
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Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]
The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
- 50.6k
2
votes
2
answers
277
views
Set of distinct real numbers such that all combination of sums are distinct
Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers.
For a given $p \in \mathbb N$, consider further the set
$$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, \...
CommunityBot
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4
votes
1
answer
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Can't figure out "standard application" of the Garsia-Rodemich-Rumsey Lemma
I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
- 2,703
1
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0
answers
94
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Estimating convolutions of powers
I would like an asymptotic estimate of
$$
\sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}}
$$
that does not involve any infinite summation. In order to lighten the notation, I ...
- 562
2
votes
1
answer
382
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Continuous real function on germs
Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
- 42.8k
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1
answer
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Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?
I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{...
CommunityBot
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2
votes
2
answers
152
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Name of a generalized version of semi-continuity
I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.
We will say a ...
- 3,312
3
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0
answers
69
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Almost everywhere in a rectangle [duplicate]
I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
2
votes
1
answer
275
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The class of uniformly accelerated curves and surfaces
Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...
3
votes
1
answer
284
views
Measure of the same set in different models of ZF
Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.
Is $\mu^\mathcal{M}(A^\mathcal{...
- 32.5k
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2
answers
1k
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An Entropy Inequality (generalized)
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability measure ...
CommunityBot
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votes
4
answers
3k
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Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 9,366
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2
answers
219
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Boundedness of ratio of linear functions
Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \...
- 1,991
18
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2
answers
2k
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Generalization of Darboux's Theorem
Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let $U\...
- 2,933
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2
answers
2k
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Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
CommunityBot
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1
vote
0
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153
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On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$
Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property
$$
\left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert
...
- 1,697
1
vote
0
answers
115
views
Inequality for an integral [closed]
How to prove that the function $$f(r)=(1 - r^2) \int_0^{2\pi}|Re[\frac{e^{i a} (2 - e^{-i s} r)}{(-e^{i s} + r)^2}]|ds$$ for real $a$ and $r\in[0,1]$ attains its maximum for $r=0$ with $f(0)=8$.
- 11
4
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0
answers
451
views
Why does it seem that $rca=rba$? [closed]
The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-...
- 459
10
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1
answer
772
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Nondifferentiability set of an arbitrary real function
A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...
CommunityBot
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11
votes
2
answers
802
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Functions that Calculate their $L_p$ Norm
are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\...
- 149k
49
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2
answers
3k
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Is a function with nowhere vanishing derivatives analytic?
My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to a ...
CommunityBot
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2
votes
1
answer
416
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What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]
I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$?
Only I've calculated the following identity:
$$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln \...
- 655
1
vote
0
answers
248
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Uniform bound for an alternating series of functions
I have mainly two questions, the first one being motivated by the second one.
1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
CommunityBot
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2
votes
0
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112
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Asymptotic analysis involving a circular multiple integral
Let $t_1,\ldots,t_m>0$, and $m\ge 4$ be an even integer. Consider the function:
$$
f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m} |x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots |x_{m-1}-...
CommunityBot
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