**6**

votes

**1**answer

282 views

### An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...

**3**

votes

**0**answers

95 views

### Regularity in PDE theory

I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not?
Let $U$ be connected,open and bounded in $\mathbb{R}^n$ ...

**12**

votes

**1**answer

742 views

### Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$

The following question has a 500 points bounty on MSE that soon comes to an end, and no answer
as expected was given yet. How would a professional solve the problem? Wish you succcess.
...

**0**

votes

**0**answers

87 views

### The Lebesgue measure of the low level sets of the two-dimmension Fourier transform of a compactly supported function

Let $f\in {{L}^{1}}\left( {{\mathbb{R}}^{2}} \right)$ be a density function with the support $\operatorname{supp}\left( f \right)\subset \left[ a,b \right]\times \left[ c,d \right]$. Denoted by ...

**-1**

votes

**1**answer

53 views

### Glueing smooth functions give a smooth function if reparametrized [closed]

Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and
$$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 ...

**1**

vote

**1**answer

116 views

### Interesting property of analytic functions

Let $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$, be an analytic application, such that: $f(t)=0\Longleftrightarrow\ t=t_0$.
Is it true that there is an analytic function $g:(t_0-\varepsilon, ...

**5**

votes

**0**answers

118 views

### Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential
$$
I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.
$$
The classical Hardy-Littlewood-Sobolev ...

**2**

votes

**1**answer

41 views

### Convergence in an infinite matrix

Let $\omega$ be the first infinite ordinal, and let $A$ be a real $(\omega+1)\times(\omega+1)$-matrix, that is $A$ is a map $A:(\omega+1)\times(\omega+1) \to \mathbb{R}$.
Suppose that $A$ has the ...

**3**

votes

**1**answer

93 views

### Horizontal Sobolev space on Carnot group

This question is connected with my previous: Heisenberg group: function without vertical derivative.
Here I am trying to look from another side: what is a difference between Sobolev space and ...

**5**

votes

**0**answers

168 views

### Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...

**2**

votes

**0**answers

55 views

### Weak Morrey Spaces

As is well known, Morrey spaces are widely used to
investigate the local behavior of solutions to second order elliptic partial differential
equations. Recall that the classical Morrey spaces ...

**8**

votes

**3**answers

481 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**0**

votes

**0**answers

131 views

### Orthogonal complements of intersections of closed subspaces

Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$.
$\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...

**0**

votes

**1**answer

98 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**13**

votes

**1**answer

400 views

### Positive roots of a polynomial

Let $a_i>0$, $i=1,\dots,n$, and put $\overline{a}:=\frac{1}{n}\sum_{i=1}^n a_i$. Assuming not all $a_i$'s are equal, take
$$
p(x):=\sum_{i=1}^n a_i (a_i-\overline{a})\prod_{k=1,\dots,n\;k\neq i} ...

**6**

votes

**0**answers

242 views

### Heisenberg group: function without vertical derivative

Let $\mathbb H$ be Heisenberg group with vector fields
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
and $U\subset\mathbb H$ is an open set.
I am ...

**3**

votes

**0**answers

68 views

### Maximizing the discrepancy in Jensen's inequality for a certain function

Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter.
Define
$$
D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}}
...

**6**

votes

**1**answer

265 views

### how wiggly is a generic level set?

Typical level sets of smooth real-valued functions are manifolds, so they cannot be fractals. If we coarse grain a bit though, sometimes we get space-filling behavior, eg. every point could be within ...

**3**

votes

**1**answer

75 views

### random odes adapted solution

Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode
$$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$
Where $b$ is a bounded continuous function (not ...

**9**

votes

**1**answer

167 views

### Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem ...

**8**

votes

**1**answer

151 views

### Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...

**5**

votes

**3**answers

431 views

### Introductory texts to mathematics [closed]

I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would ...

**2**

votes

**1**answer

51 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**4**

votes

**1**answer

86 views

### Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.
Let $a_k >0$ be an increasing sequence ...

**2**

votes

**1**answer

335 views

### Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature.
Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...

**7**

votes

**2**answers

226 views

### A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...

**1**

vote

**1**answer

165 views

### Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of ...

**48**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**0**

votes

**0**answers

48 views

### Characterization of the maximizer of a function based on a parameter's value

Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter.
I have two optimization problems. ...

**1**

vote

**0**answers

176 views

### What does the Riemann–Stieltjes integral measure? [closed]

The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum:
$$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$
where ...

**6**

votes

**0**answers

172 views

### Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define
$$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$
...

**3**

votes

**1**answer

88 views

### Gradient estimate of convex functions

Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its ...

**2**

votes

**1**answer

87 views

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

This is a follow up question of the question Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$
Let $\Omega
\subset
\mathbb{R}^{N}$
...

**0**

votes

**0**answers

53 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

84 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

77 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

254 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

133 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

95 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

89 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

189 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 ...

**25**

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

51 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

524 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

151 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

160 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

73 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

91 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 $x,t$ ...