**1**

vote

**0**answers

11 views

### L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that
$$
\|q\|_1 \leq O(n) \|p\|_1
$$
where we define $\|f\|_p := (\int_{-1}^1 |f(x)|^pdx)^{1/p}$?
...

**2**

votes

**0**answers

121 views

### Parametric normalized Fermat curves (Fermat functions)

The normalized Fermat curve is $X^n+Y^n=1$. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here.
E.g. $\cos^2x+\sin^2x=1$ has ...

**-3**

votes

**0**answers

75 views

### Is a surjective mapping of R2 to itself with full rank derivative everywhere injective? [on hold]

Let $f:\mathbb R^2 \rightarrow \mathbb R^2$, and $rank(\frac{df}{dx}) = 2$ everywhere. If $f$ is surjective $f$ necessarily injective?
Also, what if $f$ maps $\mathbb R^{2+}$ (i.e. $\{x_1>0, x_2&...

**1**

vote

**4**answers

351 views

### Sum and interpolation of hurwitz zeta functions

$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$
Can I apply Euler-Maclauren formula to this sum?
where $a\in(0,0.5)$, p is a natural number, and $x$ is a ...

**0**

votes

**0**answers

67 views

### some strange sums ramanujan type

i found sum as
$$\sum _{k=0}^{\infty } \frac{e^{k x} \left(-\frac{1}{y}\right)^k}{p-e^{k x}}=\sum _{n=1}^{\infty } \left(\frac{-1+\, _2F_1\left(1,\frac{2 i n \pi -\log \left(-\frac{1}{y}\right)}{x};\...

**2**

votes

**0**answers

90 views

### Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...

**11**

votes

**6**answers

694 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...

**1**

vote

**0**answers

71 views

### Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...

**10**

votes

**7**answers

4k views

### a^b = b^a when a is not equal to b.

I am attempting to find a closed form solution or a nice series for $x^{x+1}=(x+1)^{x}$. First of all I looked at $a^b=b^a$. Fixing a, this means finding out when $a^{1/a} = x^{1/x}$. $f(x)=x^{1/x}$ ...

**9**

votes

**3**answers

2k views

### erfc lower bound

I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following?
$$erfc(x) > \frac{ x \exp(-x^...

**0**

votes

**0**answers

48 views

### Min-max theorem - Subspaces over $H_0^1$ [closed]

I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem
$$\left\{ \begin{aligned}
-u''(x)&=\lambda u(x)\quad 0<x<\...

**0**

votes

**1**answer

201 views

### The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...

**5**

votes

**1**answer

62 views

### Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that
$$
\lvert u\rvert_s = \...

**1**

vote

**0**answers

45 views

### How much can the integrability at zero tell about the decay rate around zero? [migrated]

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...

**1**

vote

**1**answer

130 views

### Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...

**13**

votes

**1**answer

416 views

### Why sum of three squares of real polynomials is a sum of two squares?

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...

**0**

votes

**1**answer

39 views

### characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...

**3**

votes

**0**answers

49 views

### Riesz basis of Paley-Wiener space

Let us consider the Paley-Wiener space:
$$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$
Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real ...

**6**

votes

**1**answer

122 views

### The “Peano phenomenon” for differential equations

Consider the following statement:
If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$
the "Peano phenomenon" can arise only at those values ...

**3**

votes

**1**answer

172 views

### On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...

**7**

votes

**1**answer

335 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...

**7**

votes

**1**answer

287 views

### Rotation invariance of an integral

Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...

**5**

votes

**3**answers

275 views

### Nonlinear ODE: $y'=(1+axy)/(1+bxy)$

Consider the first order nonlinear ODE problem:
$$
y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0
$$
where $a, b>0$ are some constants. I would like to know if these kind of equations were ...

**3**

votes

**0**answers

81 views

### Optimal Kakeya Maximal Bound for Bushes

Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In ...

**3**

votes

**2**answers

352 views

### An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity
$$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$
...

**0**

votes

**1**answer

76 views

### Behavior of a Solution of a Nonlinear ODE

In my work, I encountered the following equation:
$$
(a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x).
$$
I would like to know as much as possible about the solution. More particularly, I would ...

**14**

votes

**2**answers

307 views

### Needing proof of convergence for a sequence

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. ...

**5**

votes

**1**answer

150 views

### Domain of Laplacian

Let $L$ be an operator on $C^2(\mathbb R)$, defined by
$$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$
for a measure $\nu(dy) = |y|^{-2} ...

**3**

votes

**1**answer

496 views

### Mandelbrot and “log-derivative”

I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent")
$$\lim_{\varepsilon\to0}\frac{\log(f(x+\varepsilon) - f(x))}{\log(\varepsilon)}.$$
To simplify, lets ...

**9**

votes

**2**answers

771 views

### On the average of continuous functions $f:\mathbb{R}^2\rightarrow[0,1]$

Is it true that if the average of a continuous function $f:\mathbb{R}^2\rightarrow[0,1]$ over a unit circle centered around $(x,y)$ is $f(x,y)$ for all $(x,y)\in\mathbb{R}^2$, then $f$ is necessarily ...

**2**

votes

**1**answer

223 views

### Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely
Generalizations to broader ...

**0**

votes

**1**answer

247 views

### Uniformly continuous functions and Borel hierarchy in the compact-open topology

Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}_u(\...

**1**

vote

**1**answer

96 views

### Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem.
For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...

**3**

votes

**0**answers

172 views

### Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...

**17**

votes

**3**answers

2k views

### Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...

**5**

votes

**0**answers

66 views

### Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only):
Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then
$$
\left| \int_{\mathbb{R}^3} f - \...

**2**

votes

**1**answer

82 views

### Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$.
I assume that my ODE ...

**0**

votes

**0**answers

24 views

### diffusion and potentials in several dimensions

In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ...

**29**

votes

**3**answers

3k views

### What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...

**2**

votes

**1**answer

177 views

### Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com
http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe
Now I ...

**6**

votes

**1**answer

214 views

### Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that
$$
\|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|...

**2**

votes

**1**answer

150 views

### Ramanujan-type sum

Could you show that
$$\sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi }}^{(...

**7**

votes

**1**answer

150 views

### $C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...

**9**

votes

**1**answer

484 views

### Find functions such that f(f(x))=f(x)e^x

Are there monotonically increasing functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = e^x f(x)$?

**1**

vote

**0**answers

103 views

### Ramanujan sum type

I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...

**1**

vote

**0**answers

45 views

### Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694
Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...

**7**

votes

**4**answers

4k views

### Applications of Euler-Cauchy ODEs

The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...

**1**

vote

**0**answers

56 views

### History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -
$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...

**3**

votes

**1**answer

76 views

### Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...

**19**

votes

**12**answers

3k views

### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...