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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 ...
CantorSet's user avatar
  • 113
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, ...
Jeff's user avatar
  • 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 $\epsilon >0$...
Roberto López-Valcarce's user avatar
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^{(...
Roberto López-Valcarce's user avatar
1 vote
2 answers
938 views

Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
Analysis Now's user avatar
  • 1,471
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 ...
QuantumLogarithm's user avatar
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_{...
icurays1's user avatar
5 votes
3 answers
1k views

Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.

Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
Rajesh D's user avatar
  • 698
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....
Hammerhead's user avatar
  • 1,211
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 ...
Gro-Tsen's user avatar
  • 32.5k
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 ...
io0's user avatar
  • 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{...
Jeremy's user avatar
  • 281
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&...
Alex's user avatar
  • 151
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 ...
Ritwik's user avatar
  • 3,245
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 ...
A Blumenthal's user avatar
3 votes
1 answer
325 views

Measuring almost-critical values of smooth functions.

Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $x \in X$ is a critical point of $f$ if the gradient $\...
Vidit Nanda's user avatar
  • 15.5k
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(...
Benjamin Dickman's user avatar
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 ...
Jeremy's user avatar
  • 281
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 ...
Adterram's user avatar
  • 1,441
18 votes
6 answers
3k views

What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
user23078's user avatar
  • 1,644
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 ...
Ali Reza's user avatar
  • 1,788
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)}{...
Anirbit's user avatar
  • 3,541
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 ...
Christian Rinderknecht's user avatar
1 vote
3 answers
188 views

sequences of plane measures converging to a singular one: terminology, etc

We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let $a$, $b$, and $c$ be vertices of a triangle in $\mathbb{R}^2$, and $...
Dima Pasechnik's user avatar
23 votes
4 answers
5k views

Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$. I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
Victor's user avatar
  • 1,437
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 : ...
Analysis Now's user avatar
  • 1,471
1 vote
2 answers
692 views

Can we extend an a.e. Lipschitz map defined on a closed subset of R^N to the whole space so that it is still a.e. Lipschitz?

I have the following question. Let $A$ be a metrically oriented $n$-dimensional subset of $\mathbb{R}^N$ and $f$ a continuous map from $A$ to $\mathbb{R}^M$. We know that $\operatorname{Lip} f < +\...
Changyu Guo's user avatar
  • 1,881
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,\...
Alex R.'s user avatar
  • 4,952
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 ...
Federico's user avatar
  • 133
-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 ...
Casella's user avatar
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 ...
Tiffy's user avatar
  • 107
10 votes
2 answers
9k views

When do maximum and expectation commute?

Hi, I'm looking for conditions on $G(t,x)$ such that $$ \sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)] $$ where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [...
martin's user avatar
  • 111
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\...
Andrés E. Caicedo's user avatar
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 ...
Zhang Changhe's user avatar
-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?
Hafiz ul Asad's user avatar
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)...
Henrique's user avatar
10 votes
1 answer
539 views

Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are ...
juan's user avatar
  • 7,024
5 votes
3 answers
349 views

minimum of two probability densities

Consider a smooth probability density $\pi(x)$ on $\mathbb{R}^d$. I am looking for natural for the integral $\iint_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv$ to be finite. If $\pi$ is a radially ...
Alekk's user avatar
  • 2,133
20 votes
2 answers
1k views

a determinantal identity

Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity $$ \det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB) $$ ...
Joe Fu's user avatar
  • 340
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 ...
Sidney Raffer's user avatar
1 vote
1 answer
273 views

Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies $$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$ where $|\cdot|$ denotes the ...
Shiyu's user avatar
  • 61
3 votes
2 answers
175 views

Decay rate of nonlocal differential operator?

Hi, Moers. Let $m(\xi) \in S^0$, that is, $$ |D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n. $$ It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$. ...
Wang Ming's user avatar
  • 425
2 votes
1 answer
403 views

The set of Upper semi-continuous functions as a ring.

I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set {$[a , b): a,b \in \mathbb{R} $} as it's base. If $X$ is a topological space, an upper ...
Ali Reza's user avatar
  • 1,788
0 votes
1 answer
138 views

question about the closed form of a function

Hi everyone! I have a question about how to find the closed form of a function defined by $$\phi(\theta)=\inf_{x\geq 2}f(x;\theta)\equiv\inf_{x\geq 2}\frac{(x+2)^2}{\frac{1}{\theta}\left(\frac{x-1}{2}...
Higgs88's user avatar
  • 69
7 votes
5 answers
6k views

Advantages of the sequence definition of limits

I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
31 votes
4 answers
8k views

Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...
bort's user avatar
  • 313
9 votes
5 answers
2k views

Homeomorphism of the rationals

In working with the classification of stable vector bundles on $\mathbb{P}^2$, I've found that I need to answer a fairly basic question from analysis/point set topology. Here it is. Suppose $f:\...
Jack Huizenga's user avatar
2 votes
1 answer
413 views

Technique: Compactness => (Finite -> Reals)

Context I'm studying a classical results of Erdos and Lovasz, on colorings of the real line. The theorem to be proved is as follows: Let $m, k$ be two positive integers satisfying: $$e(m(m-1)+1)k\...
anon's user avatar
  • 23
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 ...
Ali Reza's user avatar
  • 1,788
0 votes
0 answers
176 views

search for a function satisfying some conditions

Hi everyone, I would like to find a function $$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R_+}$$ satisfying the following conditions: $$1-\frac{z\Psi'(z)}{\Psi(z)}+8s\Psi''(z)...
Higgs88's user avatar
  • 69

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