All Questions
5,856 questions
4
votes
0
answers
340
views
Viscosity solution of the PDE
Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and
$$|Du|-f(x,u)=0$$
where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
- 61
2
votes
1
answer
351
views
Operation on measurable sets in lines, containing an interval?
Question 1: In $\mathbb{R}^2$, let $l_1$,$l_2$ be two parallel lines and $l_3$ another line which is not parallel to $l_1$. Given two measurable sets $E_1$ and $E_2$ in $l_1$ and $l_2$ respectively, ...
- 435
13
votes
1
answer
1k
views
Is it necessary to use AC to solve this problem ?
Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, $A\...
CommunityBot
- 1
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 8,246
3
votes
1
answer
399
views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\...
1
vote
1
answer
383
views
Solution of a PDE and its uniqueness
Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
$\...
- 108k
6
votes
2
answers
3k
views
Approximating erf by tanh
It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...
- 41.1k
26
votes
2
answers
12k
views
About the definition of Borel and Radon measures
I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...
CommunityBot
- 1
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 435
12
votes
4
answers
2k
views
Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(...
- 7,831
6
votes
2
answers
425
views
orderings of the field R((x, y))
I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the ...
- 328
3
votes
1
answer
6k
views
Classical Derivative, Weak Derivative and Integration by Parts
Hello,
While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.
QUESTION
Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $...
- 882
1
vote
1
answer
168
views
Mollification with prescribed boundary values
Suppose you are given a $C^1$-function $f:\mathbb R^n\to\mathbb R$ which restricts to a smooth function $f|_{\partial B}:\partial B\to \mathbb R$, where $B$ is the unit ball in $\mathbb R^n$. Can one ...
- 13k
8
votes
3
answers
2k
views
Does a weaker condition than vanishing derivative imply a function being constant?
I learned this question from math.stackexchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does
$$g(x):=\lim_{\epsilon\to 0}\frac{f(x+...
CommunityBot
- 1
1
vote
1
answer
1k
views
Calculating the Lebesgue decomposition of a measure [closed]
How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
- 60.5k
2
votes
3
answers
365
views
Construct a fixed-point set operator
How to find an uncountable set $S$, and construct an function $f : 2^S
\longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in
T$?
for example, let $S =\mathbb{R}$, how can I ...
- 73.2k
1
vote
1
answer
620
views
Smallest Lipschitz constant on non-convex domains
It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...
- 54.2k
4
votes
1
answer
319
views
Concerning strata in $C^\infty(M)$
The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf ...
- 17.5k
9
votes
1
answer
1k
views
Determinantal formula for the nullspace of a singular matrix
In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...
- 39.3k
11
votes
1
answer
2k
views
Transcendentality of all irrationals in the Cantor set
Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic ...
- 6,548
5
votes
1
answer
550
views
Weakest assumption for pointwise convergence of Fourier series
This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose $f(x)=\sum_{...
- 31.5k
0
votes
1
answer
341
views
Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
0
votes
1
answer
155
views
Ratio of eventually close sequences
Let $a_n$,$b_n$ with $b_n>0$ be two bounded sequences which are eventually close to, respectively, two other sequences $\bar a_n$,$\bar b_n$ with $\bar b_n>0$, that is, for every $\epsilon >0$...
- 12.6k
9
votes
1
answer
1k
views
Why is this generality in Vitali's Lemma useful?
In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?
Vitali's Lemma:
...
CommunityBot
- 1
3
votes
1
answer
975
views
Generalized Cesàro means of a bounded sequence
While studying the convergence of a certain iterative algorithm, I have come across the following generalization of the Cesàro mean: given a sequence $\{a_k\}$ and an integer $m\geq 0$, define
$c_k^{(...
CommunityBot
- 1
0
votes
1
answer
337
views
Integral inequality
Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
- 882
3
votes
2
answers
1k
views
Function with all but mixed second partial derivatives twice differentiable?
Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
- 555
3
votes
1
answer
258
views
Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
- 2,270
3
votes
1
answer
2k
views
A question about a formal power series manipulation
I want to find a function $f(x,y)$ which can satisfy the following equation,
$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum _{n=1} ^\infty \frac{f(x^n,y^n)}{...
- 3,541
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
2
votes
2
answers
2k
views
Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?
Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
CommunityBot
- 1
2
votes
1
answer
190
views
Completeness for spaces of eventually bounded nets
Let $A$ be a directed set, and $\ell^\infty_A$ the (complex vector) space of all
eventually bounded nets $A\to \mathbb{C}$. We can define the limit superior seminorm on $\ell^\infty_A$:
$$
\vert\vert{...
- 42.8k
23
votes
4
answers
2k
views
Which is the correct ring of functions for a topological space?
There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...
CommunityBot
- 1
0
votes
0
answers
382
views
Lambert W-function
I asked this question MSE, but didn't get any answers. Maybe here someone can help.
I need to solve
$$
\theta \rho^{\theta}+r \theta>v
$$
where $\theta \in \mathbb{R}^{+}, -1 < r,v<1, \ 0&...
CommunityBot
- 1
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 586
6
votes
3
answers
1k
views
functional subrings
I should recall the notion of maximal subring of a commutative unitary ring $R$.
Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
CommunityBot
- 1
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
17
votes
12
answers
5k
views
Looking for an interesting problem/riddle involving triple integrals.
Does anyone know some good problem in real analysis, the solution of which involves triple integrals, and which is suitable for second semester Analysis students?
Thanks!
- 5,759
1
vote
0
answers
115
views
A question about smoothness
$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold :
$\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all ...
- 1,441
4
votes
0
answers
462
views
System of Equations Upper Bound
I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:
For $i=1,2,\...
CommunityBot
- 1
6
votes
1
answer
634
views
Arbitrary small positive lower semi continuous functions
This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...
CommunityBot
- 1
0
votes
1
answer
116
views
Root and sign of a complicated bivariate function
Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let
$$
\Phi(p,i) := \frac{1}{2^p+1}
+ \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right),
$$
where $\lg x$ is ...
CommunityBot
- 1
5
votes
1
answer
400
views
Estimating the volume of a semialgebraic set from above
Suppose $S$ is a subset of $\mathbb{R}^n$ of finite volume defined by a system of finitely many polynomial inequalities with integer coefficients. Can anyone describe an algorithm that, given such a ...
CommunityBot
- 1
1
vote
1
answer
199
views
On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis
Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...
- 91.8k
-8
votes
2
answers
1k
views
why do we need algorithms, and why is non-convex optimization difficult? [closed]
A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...
- 28.6k
1
vote
1
answer
393
views
On methods for dealing with recursively defined sequences
Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...
- 376
5
votes
1
answer
225
views
Extending Jordan loops
I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.
Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\...
- 44.8k
-1
votes
1
answer
4k
views
Lipschitz condition on the first derivative of a function? [closed]
If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?
- 2,135
0
votes
1
answer
372
views
Does this sequence converge to zero?
Description
Let $\{e_n\}$, $e_n\in \mathbb{R}^p$ be a sequence of vectors, $\{U_n\}$, $U_n\in\mathbb{C}^{p\times p}$ be a sequence of unitary matrices (that is $U_i^*=U_i^{-1}$, $^*$denonts conjugate ...
CommunityBot
- 1
0
votes
1
answer
298
views
Asymptotic behavior of convex functions
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly
positive. If $x_n$ is a sequence of points such that $f(x_n)\rightarrow 0$, show that (or
give a counterexample)...
- 91.8k