All Questions
5,858 questions
3
votes
1
answer
435
views
Is the countable intersection of residual sets in [0,1] with Hausdorff dimension 1 of full Hausdorff dimension?
Let $E_k\subset [0,1]$ be residual subsets (i.e. containing dense $G_\delta $ set) with $E_{k+1}\subset E_k$ and $\dim_HE_k=1, \forall k.$ My question is : $\dim_H\bigcap_k E_k=1?$ Thanks.
- 31
2
votes
1
answer
152
views
Existence of non-negative extensions of smooth functions on axes
I am struggling to solve an extension problem of smooth functions, and I would like someone to help me.
The setting is as follows:
Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
- 393
2
votes
1
answer
129
views
A singular integral of several functions
While playing with some PDE I came across a singular integral that looks something like
$$T(f_1,f_2,\ldots,f_n)(x)=p.v.\int_{-\infty}^\infty\frac{(f_1(x)-f_1(y))(f_2(x)-f_2(y))\cdots(f_n(x)-f_n(y))}{(...
- 247
1
vote
1
answer
859
views
Continuous and dense embeddings and the density of sets in Hilbert space
Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose $B$...
- 113
2
votes
0
answers
46
views
increasing inter-class distances results in decreasing linear regression error
Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set.
Define $\mathbf{...
- 183
1
vote
2
answers
270
views
Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]
In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...
- 19
1
vote
0
answers
192
views
The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
2
votes
0
answers
374
views
How to solve $f(f(x))=x^2+x$ [duplicate]
Now I just have the equation $f(f(x))=x^2+x$. How can I find $f(x)$?
I have already tried many times, but I cannot solve it by any way I know. Is a solution possible?
6
votes
2
answers
720
views
Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 293
1
vote
3
answers
845
views
$H^{-1}$ conservative gradient flow and $L^2$ projection
Consider Cahn-Hilliard (see this) equation hich is known as the $H^{-1}$ gradient flow of Cahn-Hilliard energy functional, also it is easy to verify that this equation is mass preserving i.e. measure ...
- 53
6
votes
2
answers
2k
views
How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
- 71
1
vote
0
answers
71
views
Proving an Algorithm that generates minimal $\|x\|_0$ for the underdetermined system $Ax=b$
Let $A \in \mathbb {F}^{m \times n}$ with $m< n,$ $b \in \mathbb{F}^m$ and let $x$ be unknown in $\mathbb{F}^n.$ Assume $0<p<1.$ Then $$\arg \min\limits_{x: Ax=b} \|x\|_0 = \lim\limits_{p \to ...
- 259
2
votes
1
answer
191
views
Sobolev inequality involving summing from $j = 0$ to $m - 2$, exists constant
Let $I = (0, 1)$ and $1 \le q < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|D^{(m - 1)}u\|_{L^q(I)} + \sum_{j = 0}^{m - 2} \|D^ju\|_{L^\infty(I)} \le \...
user80013
3
votes
1
answer
147
views
Number and asymptotic for cyclic sequences
Cyclic sequence is equivalence class of cyclic shift action.
If $a = (a_1, ... , a_i)_c$ is cyclic sequence then $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, ...
- 123
6
votes
3
answers
1k
views
Dependence of error on mesh for Riemann sums
Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$,
and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest
$\delta > 0$ such that every Riemann sum arising from a ...
- 19.7k
2
votes
1
answer
223
views
Infinite sum of asymptotic expansions
I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^...
- 21
3
votes
0
answers
63
views
Is the collection of Schur convex functions sequentially compact?
We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
- 185
1
vote
0
answers
117
views
The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,835
5
votes
1
answer
133
views
If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result
Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have that$...
user75246
5
votes
1
answer
550
views
Weakest assumption for pointwise convergence of Fourier series
This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose $f(x)=\sum_{...
- 53
3
votes
1
answer
442
views
Error of midpoint method for functions that are not twice-differentiable
All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...
- 19.7k
3
votes
1
answer
379
views
Lipschitz map of the circle onto a triangle
Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane.
- 303
7
votes
1
answer
234
views
When is this sum of perfect powers bounded
For any positive integers $n,d$, let
$$
A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)}
$$
It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is
a polynomial of degree $2d-2$. Then we ...
- 621
2
votes
3
answers
365
views
Construct a fixed-point set operator
How to find an uncountable set $S$, and construct an function $f : 2^S
\longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in
T$?
for example, let $S =\mathbb{R}$, how can I ...
- 73
5
votes
1
answer
254
views
Blow-Up for Semi-Linear Wave Equations
I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
- 51
1
vote
4
answers
620
views
Do there exist nonconstant functions such that...
Do there exist nonconstant real valued functions $f$ and $g$ such that the expression:
$$f(x) -v/g(x)$$
is maximized at $x = v$ for all positive real $v$?
- 13
8
votes
3
answers
813
views
Strange real functions
I know there are a lot of strange functions $f~:~\mathbb R \to \mathbb R$.
I'm looking for an "elementary but complete" exposition of a result discovered by W. Sierpi\'nski and A. Zygmund in "Sur une ...
- 2,829
-1
votes
1
answer
230
views
Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]
Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
...
- 31
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 19.6k
1
vote
2
answers
220
views
reference needed for sobolev type estimates
I'm reading a paper and the authors applied the following sobolev type estimates
$$
||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}
$$
for $\alpha>\frac{1}{4}$,
where $v$ ...
- 113
5
votes
0
answers
116
views
For $f$ a polynomial, does strict convexity of $\log f(e^s)$ imply that the second derviative of $\log f(e^s)$ has no zeros?
Let $f(t)$ be a monic real polynomial such that $f(t) > 0$ for all $t \ge 0$. Suppose that $\log f(e^x)$ is strictly convex on $\mathbb{R}$, i.e.
$f(s^2) \cdot f(t^2) > f(st)^2$
for all $s, t \...
user94803
1
vote
0
answers
106
views
Identifying a notion of integration
Let $f$: $I\longrightarrow\mathbb{R}$ be a (not necessarily bounded) function on an interval $I\subseteq\mathbb{R}$.
Suppose $f$ admits a function $F$: $I\longrightarrow\mathbb{R}$ such that
(1) $F$ ...
- 277
1
vote
1
answer
90
views
Inverting two paraboloid relations
Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations:
$$
k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right|
$$
$$
...
3
votes
1
answer
155
views
Smoothening a probability measure
Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define
$$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\},
\ z\in{\mathbb ...
- 23k
4
votes
1
answer
388
views
Dependence of the constant in Korn's inequality on the domain
Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and
$$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i
j} ( v) \varepsilon_{i j} (...
- 2,222
1
vote
2
answers
1k
views
Is there a periodic function without minimum period such that all the possible periods are irrationals? [closed]
Let $f:\mathbb R\to\mathbb R$ be a periodic function. We say $f$ is without minimum period if, $\forall t$ such that $f(x+t)=f(x)\forall x$, there is a $t'$ such that $0<t'<t$ and $f(x+t')=f(x)\...
- 188
2
votes
1
answer
140
views
interpret of Picone inequality for non-regular functions
Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
- 371
2
votes
1
answer
65
views
interchange of infinite intersection and taking convex hull of a set
Let $B(x,\delta)$ be an open ball centered at $x\in R^n$ with radius $\delta>0$. Let $F:R^n\rightarrow R^m$ be a vector-valued function. Then $F(B(x,\delta))$ would be a subset of $R^m$. Let $\...
- 35
4
votes
0
answers
136
views
Classifying countable sets of weighted dots on a real line
Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of ...
- 10.1k
1
vote
1
answer
168
views
Does the Abel transform preserve analyticity?
Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$.
If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$,
$$
A_wf(y)=\int_y^1\frac{f(x)w(x,y)}{\sqrt{x^2-...
- 8,086
4
votes
0
answers
896
views
A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
- 2,232
3
votes
1
answer
250
views
Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
- 1,835
2
votes
0
answers
184
views
Modify the jump set of $BV$ function
Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
- 679
2
votes
1
answer
297
views
A raceway problem
Let $f(x)=\sin x$, and $g(x)=\sin x + 1$. Consider a set
$S=\{(x,y)| f(x)\leq y \leq g(x), x\in [0,2\pi]\}$. This set $S$ can be considered as "Raceway"
My question is finding the shortest path in $S$...
- 3,320
4
votes
1
answer
465
views
Julia sets without Montel's theorem
Let $J(c)$ be the Julia set of $f(z)=z^2 +c$ defined as the closure of repelling periodic orbits. Is there a way to prove that $J(c)$ is the boundary of the basin of attraction of attractive fix ...
- 1,648
4
votes
1
answer
414
views
Convergence of the Double Integral of a Polynomial Reciprocal
Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...
- 3,142
1
vote
1
answer
237
views
Poisson kernel, expectation, an absolute value comes in
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
-1
votes
1
answer
173
views
For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$
Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
- 11
3
votes
0
answers
237
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k