All Questions
5,848 questions
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An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
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0
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618
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The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
- 305
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answers
126
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A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
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122
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Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
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53
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Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 393
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0
answers
36
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Convergence of numerical scheme for HJB equation
Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...
- 393
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180
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Proof that the zeroes of certain polynomials are increasing with respect to degree
Choose $k+1$ positive integers $d_j\in\{0,1,2,3,\ldots\}$ and let $d=(d_1,\ldots,d_k)$.
Consider the following polynomial equation over the positive reals:
$$
\sum_{j=1}^{k}\frac1{x^{d_j}} = x^{d_{k+1}...
- 21
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0
answers
52
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Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
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76
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Implicit function theorem for non $C^1$ mappings
I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
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272
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How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
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answers
97
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Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
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34
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It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
- 677
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119
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About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
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120
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Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
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0
answers
66
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Lower bound of the derivative $(f*g_\sigma)'$ at the zero-crossing point
I am stuck with the following problem. Let consider $f$ a smooth real function such that:
$f$ is negative before 0,
$f$ is positive after 0,
we have $|f'(0)|>0$.
Let $\sigma>0$ and $g_\sigma$ ...
- 393
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votes
2
answers
125
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Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
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145
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Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
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149
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Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477
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101
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Does the tensor product of mollifiers work for $L^{p,q}$ spaces?
Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively.
For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \...
- 3,477
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0
answers
94
views
Oscillation of a polynomial
Recently I came across a statement in a paper that I am unable to verify. Namely, it roughly says that the oscillation of a polynomial on a cube can be controlled by the oscillation of the polynomial ...
- 1,045
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0
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138
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Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
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votes
1
answer
175
views
Asymptotic of ratio between l1 / l2 norm of a structured vector
As suggested in this discussion, I would like to inquire about the following question:
Consider a matrix B of size $n\times n$ defined as:
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\...
- 405
0
votes
1
answer
127
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asymptotic of ratio between two summations (l1 / l2 norm)
Let $B$ as a $n\times n$ matrix where
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\theta_j), 1\leq i<j\leq n$$ and other entries equals to $0$, and $$\theta=[\theta_1,\cdots,\theta_n]\...
- 405
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102
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Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \Delta u=0, $$
the quantity $|\nabla u|^2$ is subharmonic, i.e.
$$\Delta (|\nabla u|^2) \geq 0. $$
Reason:
$$\Delta (|\nabla u|^2)= 2 \nabla (...
- 371
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answers
122
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A bound for the Bessel function of the first kind J_0
I have proved the following bound for the Bessel function of the first kind:
$$
J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2}
$$
which is
$$
|J_0(x)|\le \frac1{\sqrt[4]{1+x^2}}
$$
but I ...
- 258
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48
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On $[0,1]$ with the Lesbegue measure, is it possible to have $\lVert \cdot \rVert^{1/n}_p \leq \lVert \cdot \rVert_q$ for $p>q$ and $n$ large?
The question is as in the above.
In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$.
(I deleted the last question ...
- 3,477
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75
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Morse functions on subset $\bar \Omega$ of $\mathbb {R}^d$ and its level sets
Let $f$ be a $C^{2}(\bar{\Omega})$ Morse function, where $\Omega$ is a bounded open set of $\mathbb{R}^d$: this means that
$$
\begin{cases}
f(x) = 0 \\
\nabla f (x) \neq 0
\end{cases}\text{ on }\...
- 61
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0
answers
111
views
Sum power series not continuous unit circle
This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
- 7,300
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0
answers
101
views
Sobolev estimates on domain with boundary
Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...
- 61
0
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0
answers
107
views
$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 48.9k
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0
answers
108
views
linear span of linear in C_0(R)
We consider the set $C_0(\mathbb R)$ of real continous functions $f:\mathbb R\rightarrow \mathbb R$ with $\lim_{|x|\rightarrow \infty}f(x)=0$ endowed with the supremum norm.
Is there $f\in C_0(\mathbb ...
- 1
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0
answers
317
views
What is the "best" good kernel?
A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied:
$\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$,
$...
- 3,062
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0
answers
120
views
The sufficient condition for a function equals to zero when it vanishes in an open set
For the class od analytic functions in one variable, there is an identity theorem: "If two holomorphic functions $f$ and $g$ on a domain $D$ agree on a set $S$ which has an accumulation point $c$ ...
- 101
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0
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175
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Does l2 projection of sequences preserve l1 norm convergence?
Let $\ell^2$ denote the set of square summable sequences with inner product $\langle x,y\rangle=\sum_{i=1}^{n}x(i)y(i)$ and $\ell^2$ norm $\|x\|_2=\sqrt{\langle x,x\rangle}$. Let $\|x\|_1=\sum_{i=1}^{\...
- 45
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0
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85
views
Lagrange's interpolating polynomial
Let $f:[a,b]\rightarrow R$ be a function that is not $C^{(n+1)}$ on $[a,b]$ but its $n$-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does ...
- 1
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0
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124
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Calculation of first correction to Selberg type integral
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix.
$\Tr U$ will denote the character ...
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0
answers
342
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Main ideas behind the proof of the Carleson theorem
I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a ...
- 587
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142
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Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
0
votes
0
answers
155
views
Implicit function theorem on curves
I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
0
votes
1
answer
161
views
Verifying the proof of a bilinear estimate in $L^2$
$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
- 159
0
votes
0
answers
131
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
- 307
0
votes
0
answers
168
views
How does one make sense of singular solutions to constant mean curvature equation?
Background:
Consider the following ODE:
$$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$
where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
- 537
0
votes
1
answer
51
views
Is the number of uncrossing invariant under time-change?
Let $X=(X_t:t\ge 0)$ be a stochastic process (martingale in general) starting at $X_0=0$. For $T>0$ and $a<b$, let $U^T_{a,b}(X)$ be the number of upcrossings of $X$ across the interval $[a,b]$ ...
- 1,399
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votes
0
answers
130
views
Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$
I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral:
\begin{equation}
\int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
0
votes
0
answers
251
views
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?
Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$.
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...
- 521
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votes
0
answers
53
views
$\displaystyle\dfrac{a(x)}{x^{2}}\int_{0}^{x}b(\tau)d\tau \sim \dfrac{2}{1-\alpha}\dfrac{\sqrt{a(x)}}{x^{3/2}}, \ \text{when} \ x \to 0.$
Knowing that $b,a \in C^{0}((0,L]) \cap C^{1}((0,L))$, are positive and $b(x) = \dfrac{1}{\sqrt{xa(x)}}$. Assume that $0 < \alpha < 1$ and
$$
\int_{0}^{x}b(\tau)d\tau \sim \dfrac{2}{1-\alpha}\...
- 241
0
votes
0
answers
120
views
How to prove an equality involving Laguerre polynomials
Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$.
How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
- 521
0
votes
0
answers
174
views
Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
- 91
0
votes
0
answers
80
views
An interpolation of $n!$ such that its derivatives have few zeros
The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties:
$\Gamma(n)=(n-1)!$ for $n=1,2,3,...$.
The $k$'th derivative $\Gamma^{(k)}$ has no zeros on $(0,+\infty)$ when $k$ is ...
- 700
0
votes
0
answers
134
views
From convergence pointwise to convergence of the supremum for semicontinuous functions
Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
- 449