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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 ...
user40021's user avatar
3 votes
0 answers
170 views

Is there such a matrix in $SO(n)$?

Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and $$ \frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = \frac{B_{ij}}{\sqrt{B_{ii}B_{jj}}},...
user25607's user avatar
  • 131
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
0 votes
1 answer
606 views

Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]

Is there a difference between $L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ? Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue $\sigma$-algebra $\...
user avatar
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 ...
driss-alamilouati's user avatar
3 votes
1 answer
352 views

Integral Equation with "convolution"

I've got the following problem I'm working on which is related to some of my research: Solve: $f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$ for f, given $G$ which has whatever smoothness ...
Chris's user avatar
  • 31
4 votes
1 answer
261 views

Minimizing action squared versus action

I have a very basic question in the calculus of variations: Suppose I want to minimize the functional $$A[r, r'] = \int_\Omega L(r, r') dx $$ When is it possible to say that extremals of $A$ agree ...
user avatar
2 votes
0 answers
227 views

Is there an absolutely continuous function $f$

Is there an absolutely continuous function $f$ satisfying $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1, $$ which is not $C^{1}$?
Ravi's user avatar
  • 111
1 vote
0 answers
341 views

spaces of smooth functions with bounds on partial derivatives

EDIT: As there were no takers at all... I have added below a possible approach I came up with... I would like to ask the following elementary but tricky question about the density of spaces of smooth ...
santker heboln's user avatar
1 vote
0 answers
267 views

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:...
Robert M.'s user avatar
2 votes
1 answer
239 views

Volumes of families of semialgebraic sets

Let $f_0(T),f_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f_1(T)X+f_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is $\sqrt{\max\{0,f_1(T)...
user31600's user avatar
3 votes
0 answers
176 views

Tauberian theorem wanted

At least, I think it might deserve to be called a Tauberian theorem, inasmuch as it would generalize the Tauberian theorem mentioned by Liviu Nicolaescu in his reply to my question Using a quadratic ...
James Propp's user avatar
  • 19.7k
-1 votes
1 answer
349 views

A question about approximation of Real analytic functions

Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$ for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in ...
Umberto's user avatar
  • 105
3 votes
1 answer
2k views

What is the pure intuition for topological continuity and topology? [closed]

I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity. The ...
Nick's user avatar
  • 191
0 votes
0 answers
63 views

The union of weighted compact supported continuous function

Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
JumpJump's user avatar
  • 679
5 votes
1 answer
878 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
Tom LaGatta's user avatar
  • 8,512
1 vote
0 answers
85 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
user6818's user avatar
  • 1,893
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
2 votes
0 answers
917 views

Guessing game with guess cost

This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
Alex R.'s user avatar
  • 4,952
5 votes
1 answer
781 views

Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
Tom LaGatta's user avatar
  • 8,512
1 vote
0 answers
145 views

convergence of supergradient

Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set $$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$ Assume there exists a concave function ...
CodeGolf's user avatar
  • 1,835
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}+\...
Pi314's user avatar
  • 11
5 votes
0 answers
760 views

two versions of the nested interval property

There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...
James Propp's user avatar
  • 19.7k
1 vote
0 answers
70 views

The jump set of $SBV$ function over a hyper surface

Assume $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. Also assume $S\subset \Omega$ is a smooth hyper surface such that $0<\mathcal H^{N-1}(S)<+\infty$. Now, given a positive ...
JumpJump's user avatar
  • 679
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
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
0 votes
0 answers
94 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ \frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
Uchiha's user avatar
  • 87
3 votes
1 answer
464 views

smooth families of analytic functions

My question is essentially whether taking partial derivatives of a smooth family of analytic functions yields again a smooth family of analytic functions. The precise question is the following: Let $...
Florian's user avatar
  • 31
1 vote
1 answer
685 views

This limit converges to the partial derivative?

Let a function $f:X \times \mathbb{R} \rightarrow \mathbb{R}$ continuous, with $X \subset \mathbb{R}$ compact, and supose that $\partial_2 f(x,t)$ exists for all $x \in X$ and is continuous. (here $\...
Ferraiol's user avatar
  • 121
5 votes
1 answer
543 views

Acceleration via smoothing

Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature? Let $f(t)=(\sum_{1 \leq n \leq ...
James Propp's user avatar
  • 19.7k
1 vote
0 answers
93 views

Schoenberg correspondence on $L^p$

Schoenberg correspondence states that $\psi: \mathbb R^d\longrightarrow \mathbb C$ is conditionally positive definite and hermitian if and only if $e^{t\psi}$ is positive definite for each $t>0$. ...
Thomas's user avatar
  • 630
2 votes
0 answers
263 views

A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$

Hi to everyone, The ingredients of my problem are the following: I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
Jerry's user avatar
  • 21
0 votes
1 answer
169 views

Are all discrete-analytic funtions as defined here also natural?

Let's define a discrete-analytic function as a function that is equal to its Newton expansion: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=...
Anixx's user avatar
  • 10.1k
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\}, \...
Tom LaGatta's user avatar
  • 8,512
1 vote
1 answer
879 views

Countable discrete abelian group amenable

For me the definition of amenability of an at most countable discrete group (with counting measure) is existence of a Folner sequence. Assuming this, why is every countable discrete abelian group ...
Kestutis Cesnavicius's user avatar
1 vote
0 answers
146 views

Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...
Helsi's user avatar
  • 11
2 votes
1 answer
897 views

Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings, I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
-1 votes
1 answer
187 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} f(x)=...
Sue's user avatar
  • 25
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 $x_1=\dots=x_n$? (...
user36162's user avatar
  • 259
1 vote
0 answers
305 views

Adjoint operator in sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta u\|...
reseacher's user avatar
1 vote
0 answers
115 views

Inequality for an integral [closed]

How to prove that the function $$f(r)=(1 - r^2) \int_0^{2\pi}|Re[\frac{e^{i a} (2 - e^{-i s} r)}{(-e^{i s} + r)^2}]|ds$$ for real $a$ and $r\in[0,1]$ attains its maximum for $r=0$ with $f(0)=8$.
Marjo's user avatar
  • 11
1 vote
0 answers
324 views

Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
Felix Goldberg's user avatar
3 votes
1 answer
500 views

Hausdorff measure on product spaces of p-adic integers

This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
Alan Haynes's user avatar
  • 1,723
2 votes
1 answer
137 views

Young transform reference

The Young transform of nonnegative function $f(x)$, $x \in \mathbb R^n_+$ is defined to be $$ (\mathscr Yf)(y) = \inf \left[ \left. \frac{x_1 y_1 + \ldots + x_n y_n}{f(x)} \; \right|\; x \colon f(...
Appliqué's user avatar
  • 1,329
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 ...
Chris Gerig's user avatar
  • 17.5k
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 ...
dave's user avatar
  • 13
1 vote
0 answers
92 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 ...
Hans's user avatar
  • 3,031
0 votes
1 answer
302 views

An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and $\phi(s)$ such ...
Felice's user avatar
  • 45
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 ...
Ritwik's user avatar
  • 3,245
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 ...
nick's user avatar
  • 61