All Questions
1,533 questions with no upvoted or accepted answers
2
votes
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answers
227
views
Is there an absolutely continuous function $f$
Is there an absolutely continuous function $f$ satisfying
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1,
$$
which is not $C^{1}$?
- 111
2
votes
0
answers
131
views
Representing quasianalytic functions in several variables
For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
- 807
2
votes
0
answers
428
views
Weak relative compactness in $L^1_{loc}$.
In my work I stumbled upon a proposition (without proof, alas), which I can't really prove.
Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...
- 173
2
votes
0
answers
104
views
Fourier multiplier with a singularity on a convex curve
Let $h$ be a strictly convex function such that $h(0) = h'(0)=0$. Let $\Phi: \mathbb{R}^2 \to \mathbb{R}$ be a $C^{\infty}$-function with compact support (say, $\Phi$ is supported on $[-1,1]\times[-1,...
2
votes
0
answers
890
views
Obtaining a pointwise bound on the convolution of two singular measures
I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids.
We are in Section 7, near equation (34) (pag.16 of the arxiv).
Notations and ...
- 692
2
votes
0
answers
263
views
A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$
Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
- 21
2
votes
0
answers
563
views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
- 19.6k
2
votes
0
answers
160
views
Radius of convergence to be proved more precisely (differential equation)
There is a differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
It is possible to get a solution which is a power series (see below). However, I am looking for an ...
2
votes
0
answers
184
views
Checking for error in conjugate gradient algorithm
What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
- 489
2
votes
0
answers
131
views
Bounding an integral with a small parameter by log
I have been working through Erdos & Yau's `Linear Boltzmann equation as the weak coupling limit of a random Schrodinger Equation,'
(arXiv link: http://arxiv.org/abs/math-ph/9901020), and for an ...
- 21
2
votes
0
answers
114
views
Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.
Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.)
...
- 88
2
votes
0
answers
564
views
Young inequality in weighted spaces
Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...
- 1,205
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
2
votes
0
answers
187
views
Recovering a linear map from a non-linear approximation
The problem described here is algorithmic. We are given "black box access" to a map $f:R^d\to R^d$. By this we mean that one may query the value of $f(v)$ for an arbitrary $v\in R^d$.
We assume that ...
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 8,512
2
votes
0
answers
470
views
Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?
If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
- 10.1k
2
votes
0
answers
241
views
subspace separation and M-matrices
The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$
\operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\...
- 20.2k
2
votes
0
answers
354
views
What is this effect in Fourier/additive synthesis called?
Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...
- 165
2
votes
0
answers
517
views
When deRham curve is bijection?
Motivation: Suppose we have deRham curve. From wikipedia:
Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M:
$d_0:\ M \...
- 1,626
1
vote
0
answers
146
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 521
1
vote
0
answers
59
views
Asymptotic behavior of the Hermite functions
I would like to understand the asymptotic behavior of the Hermite function :
$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$
where $H_k(x)$ is the $k-$th Hermite polynomial. For ...
- 21
1
vote
0
answers
100
views
Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
1
vote
0
answers
39
views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
1
vote
0
answers
45
views
Existence of optimal entropic weights for empirical modeling
Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
- 111
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
1
vote
0
answers
125
views
Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$
Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
user538396
1
vote
0
answers
46
views
Optimal transport and the geometry of singular measures on fractal Sets
Let $K$ be a self-similar fractal set in $\mathbb{R}^n$ with Hausdorff dimension $d < n$, equipped with a self-similar measure $\mu$ supported on $K$. Let $\mathcal{P}(K)$ denote the space of ...
- 11
1
vote
0
answers
42
views
Approximation of the function $f(z)=z^2/|z|$ by $C^1$ immersions
Let $D$ denote the unit disk in $\mathbb C=\mathbb R^2$. We consider the function $f:D\rightarrow\mathbb C $ defined by $$f(z):=\frac{z^2}{|z|}.$$ Then as proved in Global invertibility (p324 Remark 4)...
- 435
1
vote
0
answers
100
views
Difference of two completely monotonic functions
We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \...
1
vote
0
answers
58
views
Asymptotics of Jacobi form
What are the large $x\in\mathbb R$ asymptotics of $f(x)=\theta_3(c_1+c_2 x^3,e^{-x^2})$ where $c_1,c_2$ are a pair of complex numbers (say, $\Re(c_2)>0$ and $\Im(c_2)<0$), and $\theta_3(a,b)=\...
- 11
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
1
vote
0
answers
56
views
Extension of this maximisation problem : finite or not?
$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
- 1,399
1
vote
0
answers
215
views
Computing a closed form representation for a Fourier series summation
I want to compute a closed form representation for the below given summation expession.
$$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
- 698
1
vote
0
answers
82
views
Counting the number of local minima of a function that is the sum of square roots of cosines
Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows
$$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$
where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
- 21
1
vote
0
answers
67
views
Distribution of zeros for arbitrary Bessel functions
Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
1
vote
0
answers
54
views
Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
1
vote
1
answer
127
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
- 171
1
vote
0
answers
95
views
Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
1
vote
0
answers
96
views
Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
- 4,359
1
vote
0
answers
92
views
Modulus of Continuity, Heat Flow, and Derivative Estimates
Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by
\begin{align}
(P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right],
\end{align}
where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
- 801
1
vote
0
answers
96
views
Function whose derivatives eventually vanish almost everywhere
As a takeaway of this post we have the following result.
P. Let $f:[0,1]\to\mathbb{R}$ be infinitely differentiable such that for all $x\in[0,1]$ the sequence $\{f^{(n)}(x)\}$ is eventually $0$. Then ...
- 637
1
vote
0
answers
95
views
Vandermonde-type factorization of moment matrix?
Consider $n,d \in \mathbb{N}_{>0}$, there are many functions $y:\mathbb{N}^{n} \to \mathbb{R}$. Now for simplicity, we denote $y(\alpha)$ to be $y_{\alpha}$. Let $|\alpha| = \sum_{i=1}^{n}\alpha_{i}...
- 275
1
vote
0
answers
128
views
Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
- 1,043
1
vote
0
answers
43
views
If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?
Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be:
$m(x) \cdot \text{div} ( s(x) \nabla f(x))$.
What ...
- 93
1
vote
0
answers
141
views
Can this integral be solved analytically
I have an integral of the form
$$\int_{t_1}^{t_2} \frac{\sum_{i=1}^n a_i e^{b_i t}}{\sum_{i=1}^n c_i e^{d_i t}} dt$$
Where $a_i,b_i,c_i,d_i$ are $4n$ real constants, and $t_1,t_2$ are positives. Is ...
- 686
1
vote
0
answers
79
views
How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
- 111
1
vote
0
answers
86
views
The intersection of $ n $ cylinders in $ 3D$ space
I posted the question on here, but received no answer
I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
- 387
1
vote
0
answers
48
views
How to derive a lower bound of a MinMax inequality?
Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$.
The goal
For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
- 405
1
vote
0
answers
148
views
Rational solutions to $\cos(\lambda \pi) = \cos^2(a\pi) - \cos(b\pi) \sin^2(a\pi) $, with $a,b \in \mathbb{Q}$
In a similar vein to this question, I am trying to understand the occurrence of rational solutions $\lambda$ to the following equation $$\cos(\lambda \pi) = \cos^2 (a\pi) - \cos ( b\pi ) \sin^2 \left(...
- 51