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-2
votes
1answer
23 views

Systems of ODEs that fulfill a matrix relationship at steady state

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$ with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...
-2
votes
0answers
49 views

An analytic characterization of eigenvalues of a Hermitian matrix [on hold]

[..the following is trying to understand a certain argument of Terence Tao in a lecture notes of his..] If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. ...
-2
votes
0answers
22 views

why is $\frac {dy}{dx} dx = dy$? [migrated]

if $y$ is a function of $x$, why is $\frac {dy}{dx} dx = dy$? This has been bugging me. Why is it you can treat that as a fraction? I would like the traditional calculus view first if possible, then ...
2
votes
1answer
76 views

Do the sequences with divergent associated $\zeta$-function form a vector space?

Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication. ...
0
votes
1answer
123 views

“Almost” zeta function

Given a sequence $(a_n)_{n\in\mathbb{N}}$ with $a_n > 0$ for all $n\in \mathbb{N}$ and $\lim_{n\to\infty}a_n = 0$ the series \begin{eqnarray} \zeta((a_n)_{n\in\mathbb{N}}) := \sum_{n=1}^\infty ...
4
votes
1answer
47 views

Existence of an equivariant Morse function

Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse ...
-1
votes
0answers
52 views

Real number and axiom of continuity [on hold]

I have just read Courant's Introduction to Calculus and Analysis. What makes me confusion is the section "Real Number and Nested Intervals". In the Postulate of Nested Intervals or the axiom of ...
0
votes
0answers
94 views

A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...
1
vote
1answer
154 views

On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
-2
votes
0answers
167 views

Is this statement which sounds like Jordan's test generalized for two dimensions and Limit of Dirichlet integral in 2 dimensions, True?

(If this question is inappropriate to MO, please move it to math.stachexchange) First a few definitions and then the question. This is related to this question. Definition-1 A function ...
0
votes
0answers
47 views

A question related to the Nikolskii fractional spaces

Consider a Nikolskii space, that is $$ N^{s,p}=\{f\in L^{p}(I, d\ell): \|f\|_{\overline{N}^{s,p}}=\underset{h>0}{\sup}h^{-s}\|\tau_{h}f-f\|_{L^{p}(I_{h})}<\infty \}, $$ where ...
1
vote
0answers
80 views

continuity with respect to weak-${\ast}$ topology

Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
0
votes
0answers
54 views

Domains of weyl's law

I found this nice generalization of Weyl's law on wikipedia see here. Unfortunately, it is not explicitely stated over which set we are supposed to integrate there. I would definitely guess (not ...
2
votes
1answer
95 views

A question about Skorokhod metric

I have a question related to the Skorokhod distance. Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
5
votes
1answer
101 views

Jackson's theorem for partial sum of Fourier series

There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies $$ |f(x) - ...
3
votes
1answer
69 views

Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$

Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g. http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g Now define ...
1
vote
0answers
30 views

What is the purpose of the definition of “metric regularity”/“regularity modulus”?

A set mapping $F:X \rightrightarrows Y$ is said to be metrically regular for $\overline{x}\in X$ and $\overline{y} \in Y$ if there exists a $\kappa\in(0,\infty)$ for which $$ d(x,F^{-1}(y))\leq ...
1
vote
1answer
82 views

Another type of derivative, and the associated primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that: $ ...
3
votes
1answer
158 views

Constructive Proof to Show that Algebraic Numbers are Algebraically Closed

EDIT2: After reading some papers, I think the question can best be rephrased as "How can the minimal polynomial for a polynomial with algebraic coefficients be calculated. I have seen papers and ...
1
vote
0answers
57 views

vector space of ternary forms with real rooted property

Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that ...
2
votes
3answers
109 views

Determining Roots of a Polynomial with Interval Estimates of Coefficients

Let $f$ be a monic univariate polynomial with real coefficients: $$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$ The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as ...
1
vote
1answer
129 views

meromorphic extention of a function

Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in ...
4
votes
1answer
190 views

The closed form of $\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$

The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$ where $\psi(n)$ is digamma function arose in the evaluation of an integral I posted on MSE, ...
3
votes
1answer
168 views

A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm

Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...
0
votes
0answers
26 views

Approximation of a degenerate continuous quadratic form

Let $a(\cdot)$ be a real positive definite function and let $b(\cdot)$ be a real function in $L^2$. Consider: $$ P_T(x)=\frac{1}{\sqrt{T}}\int_0^T\int_0^T a(u-v)b(u-x)b(v-x)dudv. $$ The goal is to ...
-3
votes
1answer
368 views

A question about pointwise convergence of Fourier transform in $N$-dimensions

I am retreating back on this statement, after some explorations and calculation Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
5
votes
2answers
258 views

Exotic Lebesgue Measurable Function

Measurable functions whose graphs are dense in the plane are well known. Examples include, the Conway 13 function, as given in the answer in this link: When is the graph of a function a dense set ? ...
3
votes
0answers
144 views

An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n ...
2
votes
4answers
157 views

Simple bound for generalized geometric series

Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound $$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$ where $c>0$ is independent from $m$. Is there a simple way of proving this ...
0
votes
0answers
167 views

A question about Assaf Naor's review in Bourbaki about the Batson-Spielman-Srivastava result

I am referring to this article - http://www.cims.nyu.edu/~naor/homepage%20files/Exp.1033.pdf If I understand right, the author states that his equations (8) and (9) are equivalent to the equations ...
0
votes
3answers
81 views

On the existence of compactly supported functions whose its Fourier transform satisfies a given condition

My question is concerned with the existence of compactly supported functions whose its Fourier transform satisfies a given condition: For $\gamma\ge 1$, one can prove that there is no compactly ...
2
votes
0answers
91 views

Uniqueness of the maximum derivative of a rational function

This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function $$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$ with all $a_i\geq 0$ and ...
3
votes
1answer
174 views

Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
2
votes
1answer
125 views

Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
1
vote
1answer
194 views

Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1]. Germs of analytic functions can be distinguished by derivatives at a point. So in both cases we see ...
0
votes
0answers
49 views

Existence and Uniqueness of Volterra integral equations of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
3
votes
1answer
210 views

“Nice” functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point. Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
-2
votes
1answer
172 views

A calculus question [closed]

Fix $q>1$. Define the function $$ f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r. $$ The problem is whether the following is true, $$ \lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
4
votes
1answer
281 views

Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...
-1
votes
2answers
73 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
0
votes
0answers
102 views

Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$ $$f(\alpha x+(1-\alpha)y)\leq ...
0
votes
1answer
61 views

Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain. Definition of Newtonian ...
1
vote
0answers
58 views

How is this transformation related to the Legendre transform?

I stumbled over the following transform in a statistical mechanics paper: Unfortunately, no mathematical details were given there, which is why I wanted to ask here about this transform. Let $s : ...
0
votes
0answers
153 views

Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...
28
votes
1answer
2k views

What did Rolle prove when he proved Rolle's theorem?

Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...
0
votes
0answers
34 views

Stability of simple conditions on functions under convolution and/or mixture

We consider families of smooth probability densities defined on $\mathbb{R}^+$, $p=(x\in \mathbb{R}^+ \mapsto p_n(x))_{n\in\mathbb{N}}=(p_n)_{n\in\mathbb{N}}$ satisfying (i) $\int_{\mathbb{R}^+} ...
1
vote
1answer
135 views

Estimates on evolution operator

Let's consider the following evolution operator in $\mathbb{R}^3$ $$S(t)=e^{(i+\delta)t\Delta }$$ How to get the following estimate $$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert ...
0
votes
1answer
125 views

Relationship between LlogL and Hardy spaces

I think that for positive, one-dimensional, periodic functions, the following statement is true: $$ f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}), $$ where $$ LlogL=\{f\in ...
0
votes
0answers
63 views

Equivalence of ensembles and $\sum_{N=0}^{\infty} \int_{-Bn}^{\infty} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $

I am thinking about a step in the proof of the equivalence of ensembles in StatMech on page 25. step from 3.19 to 3.20. It seems to be argued there that for $A$ and $K$ large enough, the term ...
7
votes
2answers
253 views

Non-smooth function with all differences of translates smooth?

Suppose $f:\mathbb{R} \to \mathbb{R}$ has the property that for every fixed $t\in\mathbb{R}$ the function $$ g_t : x \mapsto f(x) - f(x-t) $$ is $C^\infty(\mathbb{R})$. Does it follow that $f$ is ...