All Questions
5,909 questions
4
votes
1
answer
554
views
Smoothing operator raising the smoothness exactly by one
Is there a continuous map $S: C^k(M)\to C^{k+1}(M)$ with the following properties?
(1) if $S(f)$ is $C^{k+2}$, then $f$ is $C^{k+1}$,
(2) if $f$ is $C^\infty$, then so is $S(f)$,
(3) $f$ and $S(f)$ ...
- 45k
5
votes
1
answer
327
views
Convergence in energy of bounded (semi)subharmonic functions
Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...
- 660
1
vote
1
answer
657
views
Local Uniform Convergence
Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as $x\...
- 109k
26
votes
5
answers
8k
views
Proof that no differentiable space-filling curve exists
Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere ...
CommunityBot
- 1
5
votes
1
answer
501
views
Hausdorff measure of the graph
Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure?
Of course if such ...
- 14k
4
votes
1
answer
1k
views
What is the value of the infinite product: $(1+ \frac{1}{1^1}) (1+ \frac{1}{2^2}) (1+ \frac{1}{3^3}) \cdots $? [closed]
What is the value of the following infinite product?
$$\left(1+ \frac{1}{1^1}\right) \left(1+ \frac{1}{2^2}\right) \left(1+ \frac{1}{3^3}\right) \cdots $$
Is the value known?
- 179
9
votes
2
answers
2k
views
Stokes theorem with corners
I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19:
Notation:
for $1 \le n \le m$
$\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ \...
CommunityBot
- 1
-2
votes
1
answer
193
views
Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 902
3
votes
1
answer
176
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 $a_0=...
- 4,588
12
votes
1
answer
1k
views
Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are "sort of increasing" or "sort of decreasing" (as defined below)?
Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...
CommunityBot
- 1
2
votes
1
answer
152
views
Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets
Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over $\...
- 14k
5
votes
1
answer
254
views
Blow-Up for Semi-Linear Wave Equations
I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
CommunityBot
- 1
3
votes
0
answers
314
views
Is a particular set of polynomials dense in a set of functions?
Let us consider the set $\mathcal{F}$ of strictly increasing continuous functions from $[0;1]$ on $[0,1]$ that cancel in $0$ and are equal to $1$ in $1$. So, if $f\in \mathcal{F}$ one has $f(0)=0$ and ...
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1
vote
2
answers
3k
views
Are there dense sets of positive but not full measure? [closed]
This topic came out today during a discussion with a colleague. I realized that a counter-example to his claim could be constructed if there exits a subset $A \subset [0,1]$ such that $0 < \mu(A) &...
CommunityBot
- 1
5
votes
1
answer
136
views
Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'
Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
- 1,771
5
votes
0
answers
254
views
A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
8
votes
2
answers
2k
views
Is it true that for each bounded continuous function we can find a set of analytic functions to uniformly converge it?
Is it true that for each bounded continuous function $f:\mathbb R \to \mathbb R$, we can find a set of analytic functions $g_i:\mathbb R \to \mathbb R, i=1,2,...$ such that $g_i$ uniformly converges ...
- 25.3k
7
votes
2
answers
2k
views
Baire Category Theorem Application
In Antoine Henrot Michel Pierre -
Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
- 3,993
-1
votes
1
answer
148
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 ...
CommunityBot
- 1
5
votes
0
answers
620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
2
votes
2
answers
421
views
Bi-Lipschitz constant of arc-length parametrisation of convex curve
Assume that $f:[0,2\pi]\to [0,2\pi]$ is a an increasing diffeomorphism, and let $\underline f = \min f'$ and $\overline f = \max f'$ and define $$g(s) = \int_0^s e^{if(t)} dt.$$ Assume that $g(0)=g(2\...
- 89
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 ...
- 1,893
7
votes
1
answer
234
views
When is this sum of perfect powers bounded
For any positive integers $n,d$, let
$$
A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)}
$$
It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is
a polynomial of degree $2d-2$. Then we ...
CommunityBot
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4
votes
1
answer
332
views
Limit of a hypergeometric integral
Let $n,N,T$ be positive integers, with $N=\binom{n}{2}$, and $3\leq n\leq T\leq N$. Define:
$$P(z):=z^{N+1-T}\int_0^1\frac{(1-t^2)^{n-2}}{(1-(1-z)t)^{N+1}}{}_2F_1\left[-T,N+1,N+2-T; \frac{tz}{1-(1-z)...
- 1,273
5
votes
2
answers
2k
views
Constructive Proof to Show that Algebraic Numbers are Algebraically Closed
EDIT2: After reading some papers, I think the question can best be rephrased as "How can the minimal polynomial for a polynomial with algebraic coefficients be calculated. I have seen papers and ...
CommunityBot
- 1
0
votes
1
answer
137
views
Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]
By playing around with assoc. Legendre polynomials, I arrived at
$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$
Now, I want to show that we don't have equality ...
- 3
2
votes
0
answers
98
views
What does integrability of a strictly monotonic function imply about the tails of that function?
In particular, if $f:\mathbb{R}_{+}\rightarrow[0,1]$ is a strictly monotonic decreasing function and $f$ is integrable then does it necessarily hold that $f^{-1}(1/t)=o(t)$?
3
votes
0
answers
262
views
About the small set expansion conjecture
Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - \...
- 1,893
3
votes
1
answer
283
views
Extension of Sobolev Functions
Let $\,D\subseteq\mathbb{R}^{n-1}$ be a convex bounded domain. Let$A:D\to(0,\infty)$
be a Lipschitz continuous function. Let $\,\Omega\,$ be bounded domain in $\,\mathbb{R}^{n}\,$
of the form
\begin{...
- 2,494
1
vote
1
answer
215
views
Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$
My singular integral operator is defined by
\begin{align}
Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t},
\end{align}
that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast f....
- 16.2k
0
votes
0
answers
121
views
A special approximation of BV functions
Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., $\...
CommunityBot
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5
votes
1
answer
239
views
Function and its Gradient with Prescribed Norms
I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case.
Question:
Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
- 902
0
votes
1
answer
80
views
Graph of bounded continous functions with distance 1
Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$
We set $E = \{\{f,g\}: f,g \in V\...
- 109k
9
votes
3
answers
4k
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Is there a reference for compact imbedding theory of Hölder space?
This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(...
7
votes
3
answers
393
views
Non-smooth function with all differences of translates smooth?
Suppose $f:\mathbb{R} \to \mathbb{R}$ has the property that for every fixed $t\in\mathbb{R}$ the function
$$
g_t : x \mapsto f(x) - f(x-t)
$$
is $C^\infty(\mathbb{R})$. Does it follow that $f$ is ...
- 853
0
votes
1
answer
237
views
Hadwiger-Nelson problem in higher dimensions
Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by
$V(\text{HN}_n) = \mathbb{R}^n$;
$E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ and ...
- 48.9k
1
vote
1
answer
188
views
"Schwarz symmetrization" on annulus
If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...
- 2,834
7
votes
1
answer
497
views
Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?
Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
- 24.2k
2
votes
1
answer
238
views
Asymptotic behaviour of eigenvalues
If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
- 1,164
0
votes
1
answer
247
views
Proving a complicated inequality with powers of logarithms
I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...
- 379
4
votes
2
answers
3k
views
Power series with non-integer exponents
Motivation:
For the sake of concreteness, I'll state a very particular context, but my question is a little more general. I'm trying to find a function $\gamma\colon [0,\delta) \to [0,\delta')$ that ...
- 151
0
votes
1
answer
200
views
Number of critical points
Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and $f(t)=(x(t)...
- 343
1
vote
0
answers
100
views
Summing a function at integer points
For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum
$$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$
If $F_f(y)$ is defined for all $y$, it is periodic of period 1.
...
- 37.7k
2
votes
1
answer
186
views
Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?
I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that
$$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to \...
CommunityBot
- 1
3
votes
2
answers
940
views
Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \...
- 692
1
vote
2
answers
931
views
A question on the Lebesgue differentiation theorem
In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit
$$
\lim_{\...
- 109k
17
votes
1
answer
3k
views
Integrals of pullbacks and the Inverse function theorem(s?)
The usual story goes like this:
Smooth picture (?):
For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following
is true:
$\phi^{-1}$ is a local diffeomorphism a.e.
...
- 7,817
6
votes
1
answer
843
views
Orlicz Norm and A result on expectation
I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined:
Consider arbitrary, non-negative, convex function $\psi:\...
- 3,993
1
vote
1
answer
191
views
What is the fractional derivative smoothness of functions from the Zygmund class?
Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional ...
- 902
1
vote
0
answers
99
views
Set nor its compliment contain an uncountable closed set [closed]
Does there exist a set $X$ subset of the real numbers such that no uncountable closed set is contained in $X$ or $X^c$?
- 121