All Questions
1,485 questions with no upvoted or accepted answers
1
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0
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43
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Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball
Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [...
- 12.5k
1
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0
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64
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Defining boundary conditions for spline interpolation via the Euler–Maclaurin formula
The Euler–Maclaurin formula states an interdependency between
\begin{align}
I\quad:=&\quad\int_m^nf(x) \, dx,\ \ m,n\in\mathbb{Z},\\[6pt]
S\quad:=&\quad\sum_{k=m}^n f(k), \\[6pt]
D\quad:=&\...
- 13.2k
1
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0
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49
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On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
1
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0
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922
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A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
1
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0
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74
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Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
user127450
1
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0
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113
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Higher Order Partial Derivatives Test
For a nonconstant analytic function $ℝ→ℝ$, a point is a local minimum iff at that point, the order of the first nonzero derivative is even and that derivative is positive. Is there an analogous test ...
- 7,545
1
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0
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117
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Estimation of the integral $\int_a^b e^{2\pi i f(x)} dx $
Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that
$$
\min_{x\in [a,b]} |f'(x)|>\lambda
$$
It is ...
- 675
1
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0
answers
93
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Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
- 33
1
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0
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211
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Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
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0
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156
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Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
- 105
1
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0
answers
85
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Conditional expectation with respect to a topological factor map
Let $\pi: (X,T) \to (Y,S)$ be a factor map of minimal topological dynamical systems, and let $U \subseteq X$ be open, non-empty.
Question: Does there exist a $T$-invariant, Borel probability measure $...
- 859
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0
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136
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Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x↦x^{-x}$
The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...
1
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0
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114
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density of fractal measures
Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
- 11
1
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0
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45
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Shifting Sobolev norms in a hyperbolic estimate
Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
- 4,143
1
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0
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136
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Reference for Existence and uniqueness of an Integro-Differential Equation
I have an Integro-Differential Equation (IDE) of the following form:
$$
x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds,
$$
I have found this classical reference, but the IDEs considered therein ...
- 121
1
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0
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143
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The average order of Mobius function in different intervals
Let $\mu$ be the Mobius function. Davenport proved that for any $\alpha\in \mathbb{R}$, for any $N\in \mathbb{N}$ and for any $A>0$, there exists a constant $C_A$ such that
$$
\left|\sum_{n=1}^{N} \...
- 675
1
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0
answers
126
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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
1
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0
answers
116
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Class of Borel mapping of multivalued map
At the students scientific conference I seen paper where were this propositions:
Let $X$, $Y$ $-$ compact metric spaces, $2^X$ $-$ the set of all closed subsets of $X$.
Proposition 1. Let $f:X→Y$ ...
- 21
1
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0
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237
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On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
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0
answers
50
views
Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
1
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0
answers
101
views
Non standard Lipschitz extension
Consider a ball B and let $f(x) \in L^1(B)$ such that $\int_B f(x) dx = 0$. Furtheremore, there exists a closed set $E \subset B$ such that $f|_E$ is Lipschitz. The standard Lipschitz extension ...
- 455
1
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0
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138
views
A Gagliardo--Nirenberg inequality in $H^2(\mathbb R^4)$
Does the following inequality hold in $H^2(\mathbb R^4)$
$$
\sup_{u \in H^2(\mathbb R^4), u\not\equiv 0} \frac{\|u\|_4^4}{\|\Delta u\|_2^2 \|u\|_2^2} > \frac1{16 \pi^2}?
$$
- 379
1
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0
answers
96
views
System of Poisson equations
Let $(M,g)$ be a closed (compact and without boundary) and oriented Riemannian manifold and let us consider the Poisson equation for a smooth function $\varphi$:
$\Delta \phi = f$,
where $f$ is a ...
- 2,818
1
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0
answers
137
views
Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
- 3,804
1
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0
answers
112
views
Question regarding the image of a polynomial map containing a small box
I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously.
Let $\delta, \varepsilon > 0$.
Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
- 3,625
1
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0
answers
213
views
Restriction of a Sobolev function to a straight line
I have been asked the following question, and I have to admit that I have no idea about the answer.
Assume that $f \colon (a,b) \to \mathbb{R}$ is a function. Assume also that there exists a ...
- 459
1
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0
answers
94
views
Measure of the boundary of the support of a certain function defined by an expectation
Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : ...
- 111
1
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0
answers
326
views
Approximation of Borel sets
Let $\nu$ be a finite Radon measure on $\mathbb{R}^2$ and denote the Lebesgue measure on $\mathbb{R}^2$ by $\mathcal{L}^2$. Assume that $\nu<<\mathcal{L}^2$.
We denote the boundary of $A\subset\...
- 347
1
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0
answers
100
views
Higher order derivative of negative power of cosine function
This is a question I encountered in my own research on Generalized
Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the
basis measure for this family is
$$L\left( \...
- 984
1
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0
answers
315
views
Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?
In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given:
The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
1
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0
answers
134
views
Convolution integral of series involving the non-trivial zeros of $\zeta(s)$
Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
- 219
1
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0
answers
585
views
A Lemma on convex domain which is a Lipschitz domain
I am reading the following paper:
https://docs.wixstatic.com/ugd/1de1d9_cd82cb002eaa4eefa9af574eb5efdff2.pdf
I am stuck on the proof of lemma 2.3 on page 6.
I don't see why does the property (i) of ...
- 1,594
1
vote
0
answers
150
views
Proving the existence of a sequence with recursive growth constraints
Fix $0 < c < 1/3$. Show that there exists a strictly increasing sequence $a_k > 0$, a sequence $b_k \geq 0$, and an infinite set $K \subseteq \mathbb{Z}_{>0}$ such that
\begin{align}\...
- 161
1
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0
answers
124
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Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
1
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0
answers
74
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Is the vanishing on boundary condition for the eigenvalue problem of the $p$-Laplacian important?
Consider the eigenvalue problem of the $p$-Laplacian, $$-\Delta _p u=\lambda |u|^{p-2}u,\ u\in W_0^{1,p}(\Omega)$$
In most of the literature I saw, an extra condition is mentioned that $u$ vanish on ...
- 698
1
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0
answers
100
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singular integral operators
Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...
- 4,143
1
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0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
1
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0
answers
206
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About filters on real numbers
While trying to solve an open problem which I formulated earlier (currently "conjecture 1857" in this file), I met the following problem:
Let $\Delta_{\geq}+\alpha$ be the filter on $\mathbb{R}$ ...
- 765
1
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0
answers
116
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Eigenvalues of elliptic operator analytic with respect to a parameter
I am interested when one can say the eigenvalues of an elliptic operator
are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
- 1,385
1
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0
answers
186
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Using continuity + commutativity to define "limit"
Let $S$ be the set of real sequences, and $S_0\subset S$ be the set of convergent real sequences. For each $f:\mathbf R\to\mathbf R$, we define $\bar{f}:S\to S$ by $$\bar{f}(\{x_n\})=\{f(x_n)\}.$$
It ...
- 1,630
1
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0
answers
41
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Necessary additive and multiplicative properties to characterize a mildly growing function
Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...
- 13.9k
1
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0
answers
105
views
Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
- 623
1
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0
answers
71
views
Proving an Algorithm that generates minimal $\|x\|_0$ for the underdetermined system $Ax=b$
Let $A \in \mathbb {F}^{m \times n}$ with $m< n,$ $b \in \mathbb{F}^m$ and let $x$ be unknown in $\mathbb{F}^n.$ Assume $0<p<1.$ Then $$\arg \min\limits_{x: Ax=b} \|x\|_0 = \lim\limits_{p \to ...
- 259
1
vote
0
answers
180
views
Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
- 115
1
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0
answers
91
views
Modulus of continuity of the Dirichlet Laplacian problem
I remember the following statement is correct but I cannot find a reference for that, can anybody help me to give one?
Let $\Omega\subseteq\mathbb{R}^{n}$ be an open, bounded, smooth domain,
$\varphi\...
- 31
1
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0
answers
63
views
Direct proof of fact $u \in C(U)$ satisfies $|Du| \ge 1$ in sense of viscosity if and only if property holds
Is it possible anybody could sketch me a direct proof of the fact that $u \in C(U)$ satisfies $|Du| \ge 1$ in the sense of viscosity if and only if the following property holds?
If $V \subseteq U$ is ...
- 349
1
vote
0
answers
105
views
Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
- 1,385
1
vote
0
answers
99
views
simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
- 43.2k
1
vote
0
answers
105
views
compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
1
vote
0
answers
194
views
Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171