All Questions
5,857 questions
1
vote
0
answers
170
views
non-analytic functions with arbitrary large derivatives [closed]
This may be a trivial question but I can't see it immediately.
Suppose $\{a_k\}$ is an increasing sequence of positive reals. Does there exist a smooth function $f \in C^{\infty}([0,1])$ such that $\...
- 4,115
2
votes
1
answer
289
views
Laplacian dissipative?
is it true that the Laplacian $\Delta:=\frac{d^2}{dx^2}$ on $(0,1)$ with Neumann boundary conditions is dissipative on $C[0,1]?$
For this we have to show that there is for any $x \in D(\Delta)$a $x' \...
- 21
4
votes
1
answer
293
views
Points of differentiability of $f(x) = \sum\limits_{n : q_n < x} c_n$
Let, $\{q_n\}_{n \in \mathbb{N}}$ be an enumeration of rational numbers. Consider the function $f : \mathbb{R} \to \mathbb{R}$ given by, $$\displaystyle f(x) = \sum\limits_{n : q_n < x} c_n$$
...
- 810
1
vote
0
answers
211
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
vote
1
answer
166
views
Image of a Jordan compact set under a degenerate map
This is crossposted from MSE, I hope this is suitable here, since there is no reaction there. I need this lemma for teaching, and I would appreciate any help.
Briefly:
Is the image of a Jordan ...
- 7,426
1
vote
1
answer
335
views
Orthonormal basis and decay
Edit: I added smoothness, hoping to simplify the problem with this additional assumption.
Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
- 501
5
votes
0
answers
166
views
global estimate for biharmonic function
My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions
Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,...
- 914
2
votes
1
answer
800
views
Interpolation in Sobolev spaces
Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...
- 843
0
votes
1
answer
2k
views
Collection of graduate research projects in Real Analysis [closed]
While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate ...
6
votes
1
answer
1k
views
About the generating structure of Borel field
This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
- 8,071
3
votes
0
answers
169
views
Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?
Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$?
Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
-6
votes
1
answer
141
views
Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]
Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
...
- 105
5
votes
1
answer
461
views
Integrals involving the Lambert function W
I'm actually struggling on a calculation of an integral involving the Lambert function W.
Let $\tilde{w}$>0 a parameter that I will tune to $0^+$ at the end of my calculation.
I'm interested in the ...
0
votes
1
answer
139
views
Change of variables for double integral [closed]
Thank you for your time.
My basic question is whether the following change of variables allowed
$$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$
I fail to ...
- 11
4
votes
2
answers
5k
views
Ratio of Sequences Sum Inequality
I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
- 1,182
0
votes
1
answer
166
views
Can this result be proven by using only two (or maybe three) results listed below? [closed]
I wan to show that there is no continuous real-valued function of a real variable that sends rationals to irrationals and irrationals to rationals by using only $1)$ and $2)$:
$1)$ We can use the ...
user114642
1
vote
1
answer
183
views
Where find proof of such theorem about uniform convergence of differences
Where to find a proof of theorem which says that:
if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesgue measure or on the set of second category with Baire ...
- 13
1
vote
1
answer
153
views
Mild solution of 2D surface quasi-geostrophic (SQG) equation
I was reading one of Kato's papers on Navier-stokes equations. A mild solution can be denoted as $u= e^{t\Delta}u_0 + \int_{0}^{t} e^{(t-s)\Delta} \mathbb P\nabla \cdot(u \otimes u)ds$, where $\mathbb ...
- 11
2
votes
2
answers
328
views
Minimum of an apparently harmless function of two variables
DISCLAIMER: I already posted this question on Mathematics a month ago, here. However, since it has not been solved yet on that platform, I decided to ask it also here on mathoverflow. At a first ...
- 837
4
votes
1
answer
875
views
comparing norms of tensor product of two Hilbert spaces
Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...
- 41
4
votes
1
answer
861
views
Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 21.8k
3
votes
4
answers
1k
views
Simple bound for generalized geometric series
Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.
Is there a simple way of proving this ...
- 85
1
vote
0
answers
143
views
The average order of Mobius function in different intervals
Let $\mu$ be the Mobius function. Davenport proved that for any $\alpha\in \mathbb{R}$, for any $N\in \mathbb{N}$ and for any $A>0$, there exists a constant $C_A$ such that
$$
\left|\sum_{n=1}^{N} \...
- 675
0
votes
0
answers
72
views
Looking for example of integral transformations that preserve number of zeros
Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros.
I am looking for non-trivial examples of integral transformation
\begin{align}
g(x)= \int f(t) h(t,x) dt
\end{align}
such that $f$ and $g$...
- 671
3
votes
1
answer
139
views
Bilipschitzian maps and densities
Let $ A \subseteq \mathbf{R}^{m} $ and suppose that $ \mathbf{R}^{m} \setminus A $ has $ m $ dimensional density equals $ 0 $ at a point $ a \in A $. Let $ B \subseteq \mathbf{R}^{m} $ and let $ f : A ...
- 541
4
votes
0
answers
212
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
1
vote
0
answers
51
views
Convergence acceleration of a series by using optimal parameters
One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
- 5,470
5
votes
0
answers
315
views
Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.9k
3
votes
0
answers
106
views
Dependency of the Wasserstein distance on the parameter: a differential perspective
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below:
$$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
- 345
2
votes
0
answers
116
views
Growth estimates for polynomials with natural coefficients
Suppose $p(x)$ is a degree $m$ polynomial whose coefficients are natural numbers. Suppose further that we have $p(1)=n$, $p(2)\leq nm$ and $p(3)=n^2$, and assume that $m\leq \log n$. So $p$ only grows ...
- 21
5
votes
2
answers
1k
views
Equation between the two branches of the lambert w function
My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r_1(y)$ and $r_2(y)$ of ...
- 497
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
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 ...
user56512
10
votes
1
answer
700
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 ...
- 632
31
votes
1
answer
2k
views
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
I ...
user46896
3
votes
1
answer
2k
views
Minkowski inequality
In the Wikipedia proof of the Minkowski inequality (http://en.wikipedia.org/wiki/Minkowski_inequality), the following inequality is used:
$$|f+g|^p\leq2^{p-1}(|f|^p+|g|^p).$$
I was just wondering if ...
- 31
4
votes
1
answer
559
views
How to construct i.i.d. standard normal random variables on $\Omega = [0, 1]$ with the Lebesgue measure
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Then we can find independent random variables $X_1, X_2, X_3, \dots$ defined on $(\Omega, \...
user64615
3
votes
0
answers
53
views
Controlling a Schwartz kernel near the diagonal
Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
- 1,913
13
votes
1
answer
1k
views
Is there an algebra for divergent series summation operators?
Let $D$ denote a divergent series and let $C$ denote a convergent series.
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
- 4,829
2
votes
0
answers
73
views
Proof of a technical fact in the book of Schapire and Freund on boosting
Disclaimer: I asked this question on math.stackexchange.com two weeks ago but it has not been answered yet so I figured that I might as well try to also post it here.
I am currently looking at ...
- 121
-1
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
1
vote
0
answers
127
views
Name for class of functions satisfying $\lim_{x\to 0^+}\lambda g(x)/g(\lambda x)>1$
I would like to ask whether is used some name for functions $g:A\to\mathbb{R}$, $A\subset \mathbb{R}$, for which $$\exists \lambda>1:\;\; \lim_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}>1.$$
- 51
9
votes
2
answers
2k
views
Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
user37929
3
votes
2
answers
129
views
ODEs whose finite-time solutions are not L^2 on their interval of definition
Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be analytic and consider the ODE
$$x'(t)=f(x(t)).$$
It is well-known that if $(t_{min},t_{max})$ is the maximal domain of a solution $x$ and $t_{max}<\infty$, ...
- 31
3
votes
0
answers
240
views
About optimizing decay rate of Fourier transforms?
Suppose we have a density function $f(t)$ of a random variable and $f \in C^1(R)$. If characteristic function of $f$ is $\phi_f(x) \asymp O(x^{-\beta})$ and $f$ satisfies some restrictive conditions ...
- 31
13
votes
3
answers
2k
views
Set of real numbers with positive measure containing no midpoints
Does there exists a subset E of R with positive measure and without containing any midpoints (i.e. x,y distinct in E, (x+y)/2 not in E)?
- 133
3
votes
1
answer
935
views
Beurling density and interpolation
Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density (...
- 859
1
vote
1
answer
245
views
Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?
Let $X$ be a metric space.
In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
- 623
1
vote
0
answers
126
views
identity involving spectral functions
Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true :
$$ A^* f(AA^*) = f(A^* A) A^*$$
- 519
7
votes
2
answers
788
views
Riemannian distance functions on the real line
A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties:
$d$ is a length metric (...
- 13.5k