All Questions
5,848 questions
1
vote
0
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267
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Extension of solutions of PDE
Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed.
Let $F:...
- 11
4
votes
1
answer
860
views
Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 21.8k
1
vote
1
answer
603
views
The smallest altitude amongst the triangles formed by points in the unit circle
Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this ...
- 19
0
votes
2
answers
319
views
Fixed point theorem that does not require the hemi-continuity of the set valued map?
All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...
- 383
1
vote
0
answers
295
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Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?
I first specify the setting and then formulate the question precisely. (A very long post follows.)
Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
- 3,584
6
votes
1
answer
791
views
Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable.
Suppose that a continuous function $f$ on the line and satisfies
$$
|f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1]
$$
...
- 111
17
votes
2
answers
905
views
Intersection of compact sets in the unit interval
Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...
- 3,584
1
vote
2
answers
292
views
specific improper integral involving erf
I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:
$$
\int_{1}^{\infty} \left[\mathrm{erf}\...
- 21
-1
votes
1
answer
369
views
Would this go to 0 [closed]
Let $t_{m}$ be the sup of the sum of the pairwise distances
between any $2m$ points in the unit disk. Does $t_{m}/m^{2}$ go to
$0$ as $m\rightarrow\infty$?
- 1
2
votes
2
answers
509
views
Banach algebra of BV functions
I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.
- 16.2k
0
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0
answers
115
views
Quasi-simmetric function and bi-Lipschitz functions
Assume that $f$ is a homeomorphism of the unit circle onto itself. If $$1/M \le \frac{|f(e^{i(t+s)})-f(e^{i(t)})|}{|f(e^{i(t)})-f(e^{i(t-s)})|}\le M,$$ then we say that $f$ is $M-$quasi-symmetric ...
- 259
2
votes
1
answer
447
views
Original source for a well-known result of convergence in measure and almost everywhere
A well-known result in measure theory states that given a sequence $(f_n)_{n=1}^\infty$ of measurable functions from a $\sigma$-finite measure space $(X,\mathcal{A},\mu)$ to $\mathbb{R}$ then the ...
- 729
5
votes
1
answer
330
views
Compactness of a semi algebraic set
Suppose I have a polynomial $p\in R[x_1,\ldots,x_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?
- 65
3
votes
1
answer
247
views
If $f(x)+f(2x)$ is quasianalytic, is $f(x)$ necessarily quasianalytic?
Assume that $f\in C^{\infty}$ and that $M_n$ is a sequence such that $$\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty$$
and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, ...
- 807
13
votes
3
answers
2k
views
"Values" of divergent integrals
Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
- 19.7k
12
votes
2
answers
607
views
Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set
It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary set?...
user36763
1
vote
0
answers
196
views
Extension of diffeomorphisms preserving bilateral bounds of the derivatives
Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
- 31
8
votes
3
answers
800
views
Continuous functions as uniformly continuous function
Three question concerninng metrics on the real line:
Is there a metric $d$ on $\Bbb{R}$ such that a function $f : (\Bbb{R},d) \longrightarrow (\Bbb{R},d)$ ( or $f : \Bbb{R} \longrightarrow (\Bbb{R},...
user38122
1
vote
0
answers
1k
views
Convergence of the integral of step functions
This is a question about the proof of Lemma A in §16 of the book Functional Analysis by F. Riesz and B. Sz.-Nagy.
Lemma A: For every sequence of step functions $\{\varphi_n\}$ which decreases to ...
- 236
9
votes
1
answer
947
views
Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...
- 3,245
17
votes
2
answers
1k
views
Kolmogorov superposition for smooth functions
Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as
$$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$
for ...
- 807
1
vote
1
answer
859
views
Continuous and dense embeddings and the density of sets in Hilbert space
Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose $B$...
- 113
0
votes
1
answer
250
views
Equation in integers of irrational degree
Are there any algebraic irrational numbers in $\{log_xy|x,y\in\mathbb{N},x,y\geq2\}$?
- 7,377
4
votes
1
answer
260
views
Weak continuity of Lebesgue decomposition
Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function $f:X\to\...
- 1,205
3
votes
0
answers
205
views
convolution of surface measures
Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\...
- 171
4
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1
answer
388
views
Dependence of the constant in Korn's inequality on the domain
Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and
$$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i
j} ( v) \varepsilon_{i j} (...
- 2,222
4
votes
1
answer
164
views
An algebraically independent set of real as a range of an increasing function
Is there an strictly increasing function $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that its image is algebraically independent (over $\Bbb{Q}$) ?
user39070
-2
votes
1
answer
395
views
non-trivial convergent sequence [duplicate]
I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...
- 147
2
votes
0
answers
104
views
Fourier multiplier with a singularity on a convex curve
Let $h$ be a strictly convex function such that $h(0) = h'(0)=0$. Let $\Phi: \mathbb{R}^2 \to \mathbb{R}$ be a $C^{\infty}$-function with compact support (say, $\Phi$ is supported on $[-1,1]\times[-1,...
4
votes
1
answer
280
views
Approximation of an integral over the unit ball of L_1
For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-...
- 4,705
3
votes
1
answer
784
views
Expected number of random binary vectors so that the form a basis
I would like to compute the expected number of vectors in $\mathbb{F}_2^n$ we need to draw (following a uniform distribution) so that they form a basis of $\mathbb{F}_2^n$, i.e., that we have $n$ ...
- 43
2
votes
1
answer
208
views
Does a particular iteration produce a weak solution to a non linear pde?
Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...
- 3,245
5
votes
0
answers
428
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
2
votes
1
answer
546
views
Limit involving regularized gamma function and its inverse
Let
$$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$
where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function $\Gamma(...
- 917
7
votes
4
answers
6k
views
Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
In the book, "Pi and the AGM" by Borwein and Borwein, it is mentioned that Gauss computed the following integral to the eleventh decimal palce.
$\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How did he do it? ...
- 263
1
vote
1
answer
133
views
Special finite subcover of a compact
Let $(a,b)\in \mathbb R^n$. We consider the following open cover of the compact line segment $[a,b]$: $$[a,b]\subset\underset{x\in [a,b]}{\bigcup}B(x,\rho_x),$$
where for $x\in K,B(x,\rho_x)$ is a ...
0
votes
0
answers
100
views
Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
0
votes
2
answers
720
views
Is there a probability density function satisfying the following conditions?
I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here.
...
- 5,470
6
votes
0
answers
223
views
Sum of product maximum
For which pairs of integers $(n,m)$ is the maximum of the following function $$f(x)=\sum_{i_1+\dots +i_n=m}\prod_{k=1}^n x^{i_k}_{k},\ \ x=(x_1,\dots,x_n), \|x\|=1$$ attained when $x_1=\dots=x_n$?
(...
- 259
4
votes
1
answer
214
views
The d-dimensional matrix with columns (1,0,0…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d)
During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I ...
- 41
12
votes
2
answers
732
views
Is it possible to have the set $f^{-1}(\lbrace x \rbrace)$ perfect for every $x$?
There are examples of functions $f \colon [0,1] \longrightarrow [0,1]$ such that
for any $\alpha $, $f^{-1}(\lbrace \alpha \rbrace)$ is uncountable. My favorite example is $$f(r) = \limsup_n \frac{...
user38090
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
- 16.6k
3
votes
1
answer
403
views
Is there a probability density function providing the least expected value?
Fix constant reals $A>1$ and $D>0$. Let $f:\mathbb{R}\to[0,\infty)$ be a probability density function on $\mathbb{R}$, i.e. $\int_{-\infty}^\infty f(x)\, dx=1$, that is continuous almost ...
- 31
18
votes
2
answers
1k
views
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 ...
- 351
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
4
votes
1
answer
461
views
How to get an expression for this integral (Numerically/Analytically)
I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^1 P_l(t)P_{l'}(t) \, dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the $l$-th ...
user37929
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
0
answers
416
views
When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 61
2
votes
0
answers
890
views
Obtaining a pointwise bound on the convolution of two singular measures
I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids.
We are in Section 7, near equation (34) (pag.16 of the arxiv).
Notations and ...
- 692
-1
votes
2
answers
418
views
An inequality involving multi-index [closed]
I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:
For $x \in \mathbb{R}^{n}$ and $\alpha = (\alpha_{...
- 339