All Questions
5,630 questions
1
vote
1
answer
918
views
Pros and cons of probability model for permutations
I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by L....
2
votes
1
answer
578
views
When is the bound in Riesz-Thorin Interpolation Theorem attained?
Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
- 717
3
votes
1
answer
390
views
An elementary inequality: reference request
Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$.
Now ...
- 6,962
23
votes
2
answers
2k
views
Which smooth compactly supported functions are convolutions?
If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...
- 1,017
1
vote
1
answer
220
views
There is a horseshoe with positive measure
Here is a theorem by Bowen :
My question is about the highlighted part in the picture. why there such a function $g$ exist?
- 279
3
votes
1
answer
693
views
Equivalence of negative Sobolev norm of derivative to $L^2$-norm
Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the $L^2(S)...
- 2,270
2
votes
1
answer
152
views
Is there a dense rational sequence of positive separation?
Let us consider the set $\ell_\neq$ of bounded sequences of unequal real terms. We use the following descriptions. A sequence $x=(x_0,x_1,...)\in\ell_\neq$ is dense if, for all $\varepsilon>0$, ...
- 2,437
1
vote
1
answer
370
views
A question which belongs to a class of Zygmund functions
Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, \,\,\,\...
- 197
2
votes
1
answer
158
views
Positive kernel property
Let $k:[0,1]^2\rightarrow (0,+\infty)$ be a continuous function and let
$f,g:[0,1]\rightarrow (0,+\infty)$ be measurable functions. We assume that
$$\forall x\in [0,1],\quad
f(x)=\int_0^1 k(x,y) g(y) ...
- 16.2k
2
votes
0
answers
229
views
Analytic varieties for the primes and the twin primes
I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...
- 25.4k
1
vote
1
answer
112
views
Looking for methods/results for explicitly bounding iterations of rational functions
In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series.
That is, suppose that
$$
f(z)=az+b_{1}z^{r+1}+\...
- 11
1
vote
1
answer
94
views
On weak linear continuous functions
This is what I have first asked in SE but I think it is more suitable for here. I am interested in the set of all continuous functions $f: (0, \infty) \longrightarrow \Bbb{R}$ with the following ...
- 49
3
votes
1
answer
685
views
Is the countably infinite product of locally convex topological vector spaces locally convex?
Let $(X,\tau)$ be a locally convex topological vector space and denote the product space
$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$
If we endow $X^{\infty}$ ...
- 1,835
3
votes
1
answer
495
views
Inequality in the Sobolev space $H^1$
I've found the following inequality
$$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ^2\bigg)^{\frac{q}{2}-a}+\frac{c}{r^{2a}}\bigg(\int_{B_r}...
user45822
4
votes
1
answer
670
views
A generalization of a theorem of Grothendieck
In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$.
Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.
Assume that $S$ is a subvector space of ...
- 356
1
vote
1
answer
2k
views
Constructing a continuous matrix valued function
Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
- 133
1
vote
3
answers
267
views
Formalism for moving from a metric space into a vector space for mathematical/statistical modeling given a data
I have a metric space $(X,d)$. I have a physical situation (data) where each physical entity corresponds to an $x \in X$.
I want to do some mathematical/statistical modeling of this data, but the ...
- 698
2
votes
0
answers
161
views
Improving a bound from Taylor's Theorem
For this problem, suppose $g:\mathbb{R}\rightarrow\mathbb{R}$ is such that $g\in\mathcal{C}^{k}(\mathbb{R})$, and there exists $\epsilon>0$ such that
\begin{align*} \epsilon<|g^{(k)}(x)|<\...
- 21
3
votes
2
answers
135
views
series representation of bivariate functions
Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
- 53
3
votes
1
answer
97
views
Number of small projections
Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of $\...
- 133
3
votes
1
answer
2k
views
Is the space of test functions separable? [closed]
Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ...
- 33
2
votes
3
answers
3k
views
dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
- 1,835
10
votes
3
answers
2k
views
The intersection of $n$ cylinders in $3$-dimensional space
A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
- 11.4k
9
votes
1
answer
224
views
Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
- 19.6k
1
vote
1
answer
163
views
Pohozaev result for equations with weights
I am interested in nonnegative solutions of
$-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$.
Or instead the equation $ -\Delta u + \...
- 539
4
votes
1
answer
465
views
Julia sets without Montel's theorem
Let $J(c)$ be the Julia set of $f(z)=z^2 +c$ defined as the closure of repelling periodic orbits. Is there a way to prove that $J(c)$ is the boundary of the basin of attraction of attractive fix ...
- 1,648
5
votes
1
answer
279
views
A problem on the boundedness of maximal operator by using linearization method
We know that the maximal operator is bounded on $L^{p}(\mathbb{R}^{n})$ where $n\geq 1$ and $1<p<\infty$ and the proof would be contained in many classical harmonic analysis books. Here I find a ...
- 333
19
votes
5
answers
1k
views
Floors of powers of reals, how much do the first few determine the next?
Call an integer sequence $\mathbf{x}=\left( x_1,x_2,\cdots \right)$ feasible if it is $f(r)=\left(\lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\ldots, \lfloor r^n \rfloor, \ldots \...
- 30.1k
6
votes
1
answer
218
views
Approximating an iteratively defined function
Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows:
$$f_0(x) =1+2x$$
$$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } f_{n-1}\left(\frac{x}{t}...
- 13k
12
votes
1
answer
991
views
The geometric-mean factorial
Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$,
where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing
$\odot$ with other ...
- 151k
12
votes
4
answers
831
views
Relating the roots of polynomials to the solution sets of certain functional equations
Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in \mathbb{R},\;\text{and}\;f^0=\text{...
- 453
49
votes
2
answers
3k
views
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 ...
- 2,933
18
votes
2
answers
2k
views
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
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 1,881
1
vote
1
answer
100
views
Estimating a quantity from an estimate in its integral
I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int_0^x \int_0^y f(z) \ dz \ dy \leq g(x).$$
It is then ...
- 13
13
votes
2
answers
2k
views
An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
- 1,727
33
votes
5
answers
12k
views
Differentiable functions with discontinuous derivatives
For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...
- 19.7k
1
vote
1
answer
289
views
Compactly supported smooth function with Laplace transform bounded on a cone
My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
- 11
1
vote
0
answers
184
views
A bound for a product in BMO
The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true
$$
\|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}.
$$
...
- 843
1
vote
1
answer
715
views
Quantitative version of of Riemann Lebesgue Lemma
I'm wondering if there exists a "Quantitative version of of Riemann Lebesgue Lemma" at least for the following case
$
\int_{1}^{\infty}F(t)e^{-2\pi i wt}dt
$
where $F(t)$ is a Piecewise cont. ...
- 167
2
votes
1
answer
90
views
Expressions in "continued" monotone functions
Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction
Now take a look at this question: https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-...
- 2,205
3
votes
0
answers
314
views
Is a particular set of polynomials dense in a set of functions?
Let us consider the set $\mathcal{F}$ of strictly increasing continuous functions from $[0;1]$ on $[0,1]$ that cancel in $0$ and are equal to $1$ in $1$. So, if $f\in \mathcal{F}$ one has $f(0)=0$ and ...
- 31
9
votes
2
answers
1k
views
Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?
From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
- 2,049
2
votes
1
answer
951
views
A special case of the Divergence theorem
I am interested in the following statement:
Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
domain $U$, continuous up to the boundary $\partial U$, and vanishing on $\...
0
votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
6
votes
0
answers
2k
views
Are planar Lipschitz curves countable unions of graphs?
More precisely:
Question:
Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be Lipschitz. Do there exist Borel (or Suslin) sets $A_i \subset \mathbb{R}^2$ and directions $v_i \in \mathbb{R}^2$, for ...
- 3,270
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
5
votes
1
answer
893
views
Isolated critical points
Is the following statement true or false?
Let $f:U\subset{\bf R}^n\to{\bf R}$ be a $C^2$-function (or $C^k$, with $k>2$; or real analytic) defined in a neighborhood $U$ of $0$. Assume that $0$ is ...
5
votes
1
answer
857
views
Hausdorff metric on C[0,1]
Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in $...
- 283
27
votes
2
answers
1k
views
Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent
Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
- 447