All Questions
5,856 questions
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94
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Estimating convolutions of powers
I would like an asymptotic estimate of
$$
\sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}}
$$
that does not involve any infinite summation. In order to lighten the notation, I ...
- 562
0
votes
0
answers
241
views
Continuity of a function
Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$:
$$ F(z)=\bigg(\alpha-i\...
- 71
3
votes
0
answers
205
views
convolution of surface measures
Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\...
- 171
1
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1
answer
279
views
Conjecture that two nested convex curves have a point with the same slope
I'm trying to prove a conjecture and need some help.
Consider a continuous, twice differentiable function $p(a)$ such that $p(0) = 0$ and $\forall a$, $p'(a) > 0$ and $p''(a) < 0$ and $p$ is ...
- 317
2
votes
1
answer
90
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Expressions in "continued" monotone functions
Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction
Now take a look at this question: https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-...
- 2,205
1
vote
0
answers
102
views
Differentiable Path of Operators and their Inverses
Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
- 11
0
votes
1
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316
views
Modulo dynamics on [0,1)
For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\...
- 2,619
1
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0
answers
52
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Condition for maximizer of convex combination to be expansion mapping
I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...
- 43
0
votes
1
answer
298
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Asymptotic behavior of convex functions
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^2$ convex function which is strictly
positive. If $x_n$ is a sequence of points such that $f(x_n)\rightarrow 0$, show that (or
give a counterexample)...
- 3
2
votes
1
answer
1k
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How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?
I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "...
- 466
1
vote
0
answers
94
views
Determining the exact form of a projection in a Hilbert space
Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue square-...
- 108
1
vote
1
answer
158
views
variational characterization of the average of an $L^p$ function
Let $\Omega$ be a measurable set having finite Lebesgue measure. Let $p\geq 1$ and $u\in
L^p(\Omega)$. Is it true that the minimum value of the real function
$$
c\in
\mathbb{R}^n\mapsto\int_\Omega |u-...
- 13
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 $\epsilon >0$...
2
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0
answers
76
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question about a genralized Skorokhod topology
Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
$$\rho(f,g):=\inf_{\lambda\in\Lambda}\Big\{\max\...
- 1,835
1
vote
1
answer
420
views
density of a set
let $S=\{\sin (n)|n \in N\}$. We can prove $S$ is dense in $[-1,1]$. So is the set $\{\sin( n^2)|n \in N\}$; but the set $\{\sin (n^3)| n \in N\}$ is not dense in $[-1,1]$. How to prove this?
- 11
2
votes
1
answer
276
views
Conformal Extension from a closed set to open
Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
- 541
0
votes
1
answer
659
views
Under what condition will this set contain a limit point of [0,1)?
Let $T_1,T_2,....T_n$ be numbers such that
$T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. ...
- 230
2
votes
0
answers
495
views
Characterization of weak Lebesgue spaces [closed]
I would be interested to know whether the following is true:
Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a ...
- 373
1
vote
2
answers
641
views
Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$
Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the ...
- 595
0
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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
0
answers
131
views
Representing quasianalytic functions in several variables
For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
- 807
3
votes
0
answers
47
views
Limit Behavior of Iterated Curvature-Function
What can happen, if one defines an infinite sequence of functions as follows
$f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$
$f_{n+1}: \int_0^x \sqrt{1+(f_n'(t))^2}dt\mapsto\frac{f_n''(x)}{\...
- 13.2k
3
votes
0
answers
409
views
Continuous function sort
If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
- 529
1
vote
0
answers
42
views
Error bounds for approximation with dyadic sums of polynomials
Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?
In the 2-dimensional case the ...
- 13.2k
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 ...
- 11
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1
answer
238
views
A property of a quasiperiodic function
Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b.
Does the limit of
$\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$
Exist ...
- 213
6
votes
1
answer
778
views
Inverse function theorem for DC-functions
I would like to have an inverse (or/and) implicite function theorem for DC-functions.
It seems that I have right definitions, but I fail to prove it...
Definitions:
Let $h:\mathbb R^n\to\mathbb R$ ...
- 45k
1
vote
2
answers
382
views
A question about zeros of Tate type integral
Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $...
- 2,709
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votes
0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91
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{...
- 281
4
votes
0
answers
939
views
Proofs of Baire category theorem
I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere).
My motivation is the ...
- 315
2
votes
1
answer
466
views
What is the regularity of the argument of a complex function?
Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
- 305
1
vote
1
answer
56
views
Can we implicitly fit a system of linear ODEs by reduced information?
Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...
- 749
1
vote
1
answer
409
views
Encoding information about submatrix determinants
$M$ is an $n\times n$ matrix. Consider the submatrices $M(P;Q)$ formed from $P$ rows and $Q$ columns of $M$ where $P$ and $Q$ are disjoint indices.
Is there some way to encode the various ...
- 1,533
3
votes
0
answers
69
views
Almost everywhere in a rectangle [duplicate]
I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
1
vote
0
answers
91
views
Tubular neighbourhood which is nowhere piecewise linear
I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
- 11
3
votes
0
answers
289
views
How well do continuously differentiable functions behave from R^2 to R^2 ?
The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question.
In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ...
- 529
1
vote
1
answer
193
views
If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative?
Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the ...
- 3,245
5
votes
0
answers
270
views
Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
- 156k
0
votes
0
answers
115
views
Quasi-simmetric function and bi-Lipschitz functions
Assume that $f$ is a homeomorphism of the unit circle onto itself. If $$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
1
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0
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122
views
Does the difference quotient of an absolut cont. funct. converge in L^1?
Assume that $\mu$ is a finite Radon measure on the real line and $f$ is integrable wrt. $\mu$. Define
$F(x)=\int_{]\infty;t]}f(y)d\mu(y) $
Is the following statement true?
The functions $d_h:x\...
- 11
0
votes
0
answers
345
views
Jacobian of the inversion map
Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\...
- 657
3
votes
1
answer
367
views
A differential inclusion relating to the slope of a convex function
This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the ...
- 6,206
10
votes
0
answers
315
views
Does antidifferentiability of continuous functions imply Dedekind completeness?
Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
- 19.7k
0
votes
0
answers
127
views
A question of the weights $A_\infty$' equvalent condition in Real &Harmonic analysis
I have a question. The question is to prove:
The weight $w \in A_\infty $if and only if
$\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all
...
0
votes
1
answer
116
views
to find a function with a property
We need to Find a non constant map $f:\mathbb{C}^3\to \mathbb{C}$ such that for any three distinct complex numbers $z_1,z_2,z_3$ and any automorphism $\phi$ of $\mathbb{C}$, we have
$f(z_1,z_2,z_3)= ...
- 3
2
votes
0
answers
160
views
Radius of convergence to be proved more precisely (differential equation)
There is a differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
It is possible to get a solution which is a power series (see below). However, I am looking for an ...
3
votes
0
answers
227
views
Mesh for 3d dungeons game. [closed]
Hallo, I look for some F: R^2->R height function which would generate the Speleothem ceiling http://en.wikipedia.org/wiki/Speleothem for 3d game taking place in dungeons/caves.
The function might be ...
- 139
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
4
votes
0
answers
273
views
Real Analytic Function and nth Prime
It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...
- 153