All Questions
5,877 questions
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117
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Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
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votes
0
answers
58
views
Quadrics over the univariate function field with discriminant of minimal degree
Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...
- 1,833
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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
0
votes
0
answers
115
views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
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votes
1
answer
212
views
Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$
Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...
user124345
0
votes
0
answers
113
views
Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user123457
0
votes
1
answer
275
views
An extension for lower semi continuous lower bounded real valued functions class
Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
- 179
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votes
1
answer
51
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Strict positive type function on hypersurface also of positive type in neighborhood?
Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
- 13
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0
answers
170
views
What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?
I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...
- 723
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votes
0
answers
75
views
Dense Egoroff theorem
Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...
- 5,405
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votes
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124
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Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]
Recall a very famous theorem due to Hartogs for complex analytic functions of several variables.
Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...
- 671
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126
views
$p$-volume of $n$-dimensional hyper-ellipsoids
I read that the unit hypersphere has maximum volume for dimension five and would like to generalize this result. (If you think that integrating over an $n$-dimensional $p$-hyper-ellipsoid area ($x_1^...
- 4,829
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1
answer
113
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Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
- 1,903
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112
views
On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
- 505
0
votes
0
answers
60
views
Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
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323
views
Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
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0
answers
58
views
$N-$Green function in $\mathbb R^N$
Let $N \geq 3$. Does there exist solution of the following equation
$$-\Delta_N G + G^{N-1} = \delta_0,$$
where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
- 379
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0
answers
299
views
When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
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votes
0
answers
100
views
Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
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votes
0
answers
123
views
Reference for the Hardy maximal function on the torus
I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
- 3,425
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votes
0
answers
311
views
Approximations of Polylogarithm and Lerch transcendent?
For the Gamma function $\Gamma(x+1)$, we have beautiful approximations of the function in terms of elementary function, such as the Stirling approximation and its refinements, that give sharp ...
- 1,324
0
votes
1
answer
347
views
Asymptotic behaviour of fixed points in permutations
For any $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
- 48.9k
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71
views
Existence of local minimizer
For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$
$$\| \nabla F(x) \| < \epsilon$$
and a sufficiently large $\alpha$ where
$$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$
Can ...
- 719
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0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
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votes
0
answers
140
views
Lipschitz extensions preserving the convex hull of the range
We assume that $X$ is a metric space and that $A \subseteq X$ is a subset. Let $f : A \rightarrow \mathbb R$ be a Lipschitz-continuous function with Lipschitz constant $L$.
By the Kirszbraun theorem, ...
- 5,327
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votes
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308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 617
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votes
0
answers
58
views
in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
- 43.2k
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0
answers
84
views
Number of solutions to non-linear equations
As part of our project, we are required to determine the total number of distinct solutions to the following equations.
There are $n$ variables of one type, say $\{p_i\}_{i=1}^n$, $m$ variables of ...
- 109
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votes
0
answers
93
views
What is the class of real sequences satisfying these conditions?
I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions:
$\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
0
votes
0
answers
80
views
Comparison of two functions
Given a function $f$ from $R^2$ to $R$ satisfying tha following:
$1)$ $f$ is a convex function which vanishes on $(0,1)$ and on $(1,0).$
$2)$ $f$ is a decreasing function on $x$ and on $y$ and $f$...
- 1,257
0
votes
2
answers
144
views
Optimization function of two variables
Let $A, B, C, D \in \mathbb{R^*_+}$.
Is it possible to solve
$$
\max_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)}
$$
The KKT conditions give for an extrema $(x^*,y^*)$
...
user83947
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votes
0
answers
42
views
What (analytical or numerical) method can I use to solve scalar optimal problem?
I got the following optimization problem in mind and I am looking for some (analytic or numerical) methods to solve it. Can anyone have any ideas? Here is problem
\begin{aligned}
& {\text{...
- 221
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votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
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votes
0
answers
271
views
Convolution Integral involving an unknown function
I've got the following problem I'm working on which is related to some of my research.
I am trying to solve the following equation for the function $f$.
$$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \...
- 281
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votes
0
answers
131
views
Measurable sets of probability measures $\{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}$
Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the ...
- 119
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votes
0
answers
55
views
Representation of multi-variable functions as a composition of 1- or 2-variable functions [duplicate]
This is a re-post of my question from M.SE that remains unanswered for several months.
I'm familiar with Kolmogorov–Arnold representation theorem, but AFAIK their construction makes essential use of ...
- 6,819
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votes
0
answers
85
views
Some problems about symmetric convolution semigroup on the unit circle
These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes".
Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
- 369
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votes
0
answers
64
views
Approx the jump point of a $BV$ function from both hand side
Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as
$$
u(x)=
\begin{cases}
0,&\text{ if }x\in(-1,0)\\
1,&\text{ if }x\in(0,1)
\end{cases}
$$
Clearly, we have $u\in ...
- 679
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votes
0
answers
470
views
Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
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votes
0
answers
344
views
Beurling density $D(X)$ of $X=\{x_j\in\mathbb R, \ |x_j-x_{i}|>\gamma>0: \ i,j\in\mathbb Z\}$
Beurling density of set $X$ is defined (see, for example "Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces" by Aldroubi and Grochenig) as:
$$D(X)=\lim_{r\rightarrow \infty} \inf_{y\in\...
- 297
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votes
1
answer
186
views
Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...
user76167
0
votes
0
answers
116
views
Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane
We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...
- 5,053
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votes
0
answers
63
views
The union of weighted compact supported continuous function
Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
- 679
0
votes
0
answers
173
views
Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
- 679
0
votes
0
answers
82
views
Construction of a path of quadratic variation
This question has been posted to Stack Exchange earlier, and no answer is available yet.
Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval
is defined by
$$V_{p}(x, [a, b]) =...
- 1,399
0
votes
0
answers
808
views
Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$
Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
0
votes
0
answers
152
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 $\hat{...
- 141
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votes
0
answers
89
views
Degree of permutation of hypercube
Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
- 13.9k
0
votes
1
answer
111
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 ...
user70876
0
votes
0
answers
121
views
A special approximation of BV functions
Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., $\...
- 679