All Questions
5,630 questions
6
votes
1
answer
440
views
Are there superexponential Pfaffian functions?
This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
- 1,979
9
votes
0
answers
979
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
8
votes
1
answer
526
views
Interpolation between L^1 and Sobolev Space
Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that
$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ||f||_{L^...
- 101
0
votes
2
answers
190
views
Is real analytic function good enough (see problem)? [closed]
Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then ...
- 67
0
votes
0
answers
145
views
Does there exist this special kind of homeomorphism?
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
- 183
2
votes
0
answers
374
views
How to solve $f(f(x))=x^2+x$ [duplicate]
Now I just have the equation $f(f(x))=x^2+x$. How can I find $f(x)$?
I have already tried many times, but I cannot solve it by any way I know. Is a solution possible?
8
votes
4
answers
4k
views
Non-zero smooth functions vanishing on a Cantor set
It is easy to give examples of continuous functions $f:[0,1]\to \mathbb R_+\cup\{0\}$ non-zero but vanishing on a Cantor set (ex: Can Cantor set be the zero set of a continuous function?). It is ...
- 1,805
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
12
votes
1
answer
742
views
If the generating function summation and zeta regularized sum of a divergent series exist, do they always coincide?
One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
- 4,829
3
votes
1
answer
496
views
Prove that these two definitions of "natural" integration constant coincide when both converge
These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...
- 10.1k
-5
votes
1
answer
754
views
Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]
The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
- 10.1k
38
votes
13
answers
5k
views
Continuous relations?
What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?
I ...
- 862
15
votes
4
answers
1k
views
Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...
- 41.1k
3
votes
2
answers
487
views
Integrating over the Intersection of Convex Regions
Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?
The ...
- 13.2k
2
votes
2
answers
277
views
Set of distinct real numbers such that all combination of sums are distinct
Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers.
For a given $p \in \mathbb N$, consider further the set
$$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, \...
user45183
4
votes
1
answer
1k
views
Can't figure out "standard application" of the Garsia-Rodemich-Rumsey Lemma
I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
- 617
5
votes
2
answers
922
views
What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?
In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
Troelstra,...
- 16.6k
1
vote
0
answers
94
views
Estimating convolutions of powers
I would like an asymptotic estimate of
$$
\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
8
votes
2
answers
555
views
Finiteness as a motivation for compactness
Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...
- 339
2
votes
1
answer
383
views
Continuous real function on germs
Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
- 21.5k
5
votes
1
answer
493
views
On a.e. approximate differentiability of certain continuous real functions
I have the following question:
If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on $[...
- 1,881
2
votes
2
answers
152
views
Name of a generalized version of semi-continuity
I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.
We will say a ...
- 799
3
votes
1
answer
284
views
Measure of the same set in different models of ZF
Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.
Is $\mu^\mathcal{M}(A^\mathcal{...
- 409
2
votes
1
answer
275
views
The class of uniformly accelerated curves and surfaces
Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...
2
votes
2
answers
219
views
Boundedness of ratio of linear functions
Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \...
- 23
5
votes
2
answers
3k
views
Product of Lebesgue and counting measures
Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
- 405
5
votes
1
answer
271
views
Is this graph of reciprocal power means always convex?
Let
$$
p = (p_1, \ldots, p_n)
$$
be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$.
Is the function
$$
q \...
- 27.7k
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 $f(0)=8$.
- 11
1
vote
0
answers
153
views
On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$
Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property
$$
\left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert
...
- 1,697
4
votes
0
answers
451
views
Why does it seem that $rca=rba$? [closed]
The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-...
- 459
3
votes
0
answers
69
views
Almost everywhere in a rectangle [duplicate]
I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
2
votes
1
answer
416
views
What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]
I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$?
Only I've calculated the following identity:
$$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln \...
- 21
2
votes
0
answers
112
views
Asymptotic analysis involving a circular multiple integral
Let $t_1,\ldots,t_m>0$, and $m\ge 4$ be an even integer. Consider the function:
$$
f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m} |x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots |x_{m-1}-...
- 87
11
votes
2
answers
802
views
Functions that Calculate their $L_p$ Norm
are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\...
- 13.2k
-1
votes
1
answer
1k
views
derivatives and uniformly convergence [closed]
Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has
$$f(x)=\sum_{n\...
- 3,998
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 113
2
votes
1
answer
151
views
A question on existence of solutions of a linear ODE system
I am working on a problem of harmonic functions on surfaces, and in one step I got the following system of ODEs with prescribed asymptotes. I was wondering what methods could give us the existence or ...
1
vote
0
answers
248
views
Uniform bound for an alternating series of functions
I have mainly two questions, the first one being motivated by the second one.
1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
- 266
4
votes
1
answer
185
views
Reference: Hardy space regularity of the Jacobian determinant
I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes.
Theorem:
For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, $\...
- 97
21
votes
0
answers
658
views
A multiple integral
Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos {(s_{2n-1}...
- 16.5k
3
votes
1
answer
1k
views
For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1
This is related to a question asked on mathstackexchange https://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...
- 516
1
vote
2
answers
259
views
Markov-Bernstein like inequalities for monotone polynomials
Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...
- 311
0
votes
0
answers
206
views
About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
3
votes
1
answer
967
views
Continuity of minimizers to distance function from point to convex set
Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...
- 81
4
votes
2
answers
655
views
Differentiability: Partially Defined Functions
These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds.
(Cf. discussion on p. 45.)
Definition
Let $E$ and $F$ be two Banach spaces together with a plain subset $A\subseteq ...
- 598
0
votes
1
answer
394
views
Legendre differential equation with additional term
In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( 8\,...
user37929
7
votes
3
answers
670
views
Cofinality of a $\sigma$-ideal of $\mathbb{R}$
The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...
- 73
1
vote
0
answers
94
views
Determining the exact form of a projection in a Hilbert space
Let $$\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
6
votes
1
answer
216
views
Estimates of Hausdorff dimension (and its derivatives)
For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
- 165
4
votes
0
answers
896
views
A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
- 2,232