**0**

votes

**0**answers

46 views

### Orthogonality relation for associated Legendre functions

The associated Legendre Polynomials $P_l^m(x)$ obey orthogonality relations for fixed $m$ and fixed $l$:
$$
\begin{align}
\int_0^\pi P_k^m(\cos\theta)P_l^m(\cos\theta)\sin\theta d\theta ...

**1**

vote

**1**answer

83 views

### Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$

Let $\Omega
\subset
\mathbb{R}^{N}$
be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Caratheodory function such that $g(x,t)=0$
for $t\leq0$
. Suppose that ...

**0**

votes

**1**answer

76 views

### Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...

**3**

votes

**4**answers

250 views

### Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1.
Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite ...

**4**

votes

**1**answer

126 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 $S(f)$ ...

**4**

votes

**1**answer

94 views

### Existence of doubling non-Polish metric measure spaces

Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is doubling if there exists a constant $C \geq 1$ such that $0 < ...

**1**

vote

**1**answer

83 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 ...

**15**

votes

**5**answers

2k 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 ...

**4**

votes

**1**answer

188 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 ...

**24**

votes

**2**answers

1k views

### Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?

**0**

votes

**0**answers

50 views

### What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$
\begin{equation}
V_{ij}^{kl}=V_{ji}^{lk}
\end{equation}
(So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...

**5**

votes

**1**answer

520 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?

**-1**

votes

**1**answer

146 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 $H^1$ ...

**4**

votes

**1**answer

156 views

### Generalisation of Lebesgue differentiation theorem to Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow ...

**2**

votes

**1**answer

72 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 ...

**3**

votes

**1**answer

90 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 ...

**0**

votes

**0**answers

82 views

### if $u_\epsilon \rightarrow u$ weakly in $L^2$ then also $\partial_t u_\epsilon \rightarrow \partial_t u $ weakly in $L^2$

I am looking for a citation of the above claim, I am not sure under what conditions on the sequence $u_\epsilon$ exactly this applies. $u_\epsilon$ is a sequence of functions that depend on $x,t$ ...

**38**

votes

**1**answer

411 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longmapsto \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in ...

**0**

votes

**2**answers

168 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) ...

**4**

votes

**1**answer

82 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 ...

**7**

votes

**2**answers

666 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\} \ | \ ...

**7**

votes

**2**answers

580 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 ...

**1**

vote

**0**answers

58 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 ...

**0**

votes

**1**answer

116 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 ...

**0**

votes

**0**answers

63 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

**1**answer

172 views

### Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla ...

**6**

votes

**1**answer

170 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 ...

**2**

votes

**2**answers

305 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 ...

**2**

votes

**0**answers

110 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 - ...

**0**

votes

**0**answers

40 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., ...

**4**

votes

**1**answer

162 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 $\alpha,\beta>0$. ...

**0**

votes

**1**answer

99 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 ...

**2**

votes

**1**answer

56 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 ...

**34**

votes

**6**answers

2k views

### On an example of an eventually oscillating function

For $x\in(0,1)$, put
$$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$
This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...

**0**

votes

**0**answers

27 views

### Deriving inequalities from a polynomially-bounded derivative

In this paper (p. 2, definition/remark) the following notion of ‘polynomial growth’ is defined for a non-negative real function $g(x)$ and a real constant $b\in(0;1)$:
There exist positive ...

**2**

votes

**1**answer

73 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{ ...

**1**

vote

**1**answer

81 views

### About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...

**6**

votes

**1**answer

111 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 $f$ ...

**1**

vote

**1**answer

75 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**

votes

**1**answer

169 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 ...

**0**

votes

**1**answer

149 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 $\varepsilon>0$ ...

**1**

vote

**0**answers

87 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.
...

**3**

votes

**1**answer

106 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
...

**2**

votes

**1**answer

120 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 ...

**0**

votes

**1**answer

91 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 ...

**1**

vote

**0**answers

83 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 $X^c$?

**1**

vote

**1**answer

93 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 ...

**3**

votes

**0**answers

83 views

### On Rényi entropy/divergence

The Rényi entropy for a probability density function $f$ with dominating measure $\mu$ of order $\alpha>0$ is defined as
$$H_\alpha(f)={1 \over {\alpha-1}}\log\int f^\alpha d\mu.$$
If $f$ is ...

**0**

votes

**0**answers

45 views

### minimal entropy approximation of a truncated discrete measure

Consider a measure $\mu$ on $\mathbb{N}$ given by the sequence $(\mu(n))_{n \geq 0}$ with $\mu(0)>0$. For example $\mu(n)=n^2+1$ on the figure below.
For each $n$, let $X_n \sim \mu(\cdot \mid ...

**0**

votes

**0**answers

73 views

### Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...