All Questions
1,533 questions with no upvoted or accepted answers
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162
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Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
- 43.2k
1
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0
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96
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Regularity of Feynman-Kac formula for a simple diffusion
Let consider the diffusion process given by:
$$dX_t = \alpha(X_t) dW_t$$
where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
- 393
1
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0
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61
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Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues
I am looking for a reference justifying the following statement.
Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$
$$
...
- 5,380
1
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0
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70
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
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0
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123
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Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
1
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0
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98
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Periodicity in one Fourier variable
Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument.
We can thus write $f$ as $$ f(x) = \sum_{n \in \...
1
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0
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102
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Proving that a quantity is positive (Gaussian density and Gaussian CFD)
$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{1/2}}$
Hi everyone,
I am interested in the following problem:
Let consider the heat equation problem:
$$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
- 393
1
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0
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59
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Study of the properties of a non-local ODE
I am studying the following non-local ODE
$$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$
The number $x_0$ can ...
- 452
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0
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82
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How to calculate the Integral with confluent hypergeometric function
How to prove this.Thank you in advance
Let $\delta,\beta>0$ How to prove this
\begin{align}
& \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
- 521
1
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0
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79
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Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$
The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
- 1,295
1
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66
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Parameter estimation of a Taylor expansion
Let $a,b$ two real numbers, $\theta$ a real parameter and suppose that you have an analytic function of the form:
$$
f_\theta(x)\triangleq \sum_{k\in\mathbb{N}}a_k(\theta)x^k \quad\forall x\in[a,b],
$$...
- 393
1
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0
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122
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Implicit function theorem / Implicit selections when Jacobian not invertible
I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
- 393
1
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0
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113
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Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,143
1
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166
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Monotone likelihood ratio of convolved power function kernel, $p\ge 3$
It was shown in a previous answer that for: $f(x)=|x|^p$, $\;x\in \mathbb{R}$, $\;p>2$, defining the density:
$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(
\hspace{-1pt}...
- 391
1
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0
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143
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Analyticity of a function in two complex variables
Let $f$ be a function defined on $\mathbb{C}^2$ given by
$$
f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
- 11
1
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0
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269
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Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$
Let $f(x) = \log(\cosh(x))$, and define the kernel density:
$$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}...
- 391
1
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0
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57
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Monotonicity of root of hyperbolic function
For $\kappa \geq \alpha>0$ and $y \geq 0$, consider the following equation:
$$\sqrt{1-\frac{\alpha }{\kappa }} \tanh \left(y \sqrt{1-\frac{\alpha }{\kappa }}\right)=\tanh \left(y-\frac{\alpha }{2}\...
- 11
1
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0
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67
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Estimating commutator of Fourier integral
Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by
$$ s_n(g)(x) = \frac{nf(x)}{4\pi i} \int_{\mathbb R^2} e^{\frac{in\vert x-y\...
1
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0
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167
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Sobolev embedding into the space of bounded, uniformly continuous function?
Let $\Omega$ be a sufficiently smooth open domain of $\mathbb R^d$. Is any embedding of the Sobolev spaces $W^{s,p}(\Omega)$, $s>0$, into the target space $C^0(\overline{\Omega})$ (the space of ...
- 3,388
1
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0
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89
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Pre-images of the critical point of $3.83 x(1-x)$
This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
- 333
1
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1
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76
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Upper bounds for the spatial differential of the inverse of a flux
It is well known that given a regular velocity field $b: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ (say, continuous in time and uniformly Lipshitz in space), the flux $X$ associated to $b$ is a ...
- 697
1
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0
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59
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Factoring a smooth map as a function to a linear map
I am searching for a reference to the following fact about smooth functions.
If $f \in C^k(\mathbb{R}^n, \mathbb{R}^m)$ such that $f(0) = 0$, then there exists $g \in C^{k - 1}(\mathbb{R}^n, \...
- 2,834
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58
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Cardinality of maximal elements in terms of set-theoretic inclusion in the space $c_0(\mathbb{N})$
Let $c_0(\mathbb{N})$ be the space of real-valued sequences converging to zero.
Here, each element $\{\alpha_n\} \in c_0(\mathbb{N})$ itself is a "set", so that we can think about set-...
- 3,477
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68
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Hilbert like transform on the circle
Suppose that $u $ is a smooth function real valued function defined on an open neighborhood of the unit disc $ \mathbb{D} $, which satisfies a second order elliptic partial differential equation;
\...
1
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0
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64
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Variation of the fractional derivatives
$\DeclareMathOperator\AC{AC}\DeclareMathOperator\Lip{Lip}$Suppose we have $f\in L^1(\mathbb{R})\cap \AC(\mathbb{R})\cap \Lip(\mathbb{R})$ and $f$ piecewise linear function, bounded and $|f|\leqslant \...
- 89
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108
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Existence of a smooth extension
In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface
$$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$
Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...
- 4,143
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165
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Question about stationary phase with Hessian close to $0$
Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define
$$
I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
- 3,625
1
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115
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Value of $\pi$ and algorithm for Bernoulli numbers
Chowla and Hartung provide an "algorithm" for computing Bernoulli numbers in this paper.
In particular, if the Bernoulli numbers are defined by
$$\frac{x}{e^x-1}=1-\frac{x}{2}+\sum_{n=1}^\...
- 41
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0
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91
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limsup of sequence
Let $\mathbb{Z}_{\geq 0}[|t|]$ be the ring of power series with non-negative integer coefficients and consider the power series
$$P(t) = \sum_{i=0}^ \infty a_i t^i \in \mathbb{Z}_{\geq 0}[|t|]$$
$$P^2(...
- 81
1
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0
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50
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Bound an integral with parameter
Let us define, for $x_0 > 0$ and $x_0 \ll 1$,
$$K(x) = \int_{-x_0}^{x - g(x)^2} \frac{f(y)}{x - y}dy, \quad \text{for } x \in [-x_0/4, x_0],$$
and
$$g(x) = \frac{(x_0 - x)|\log(x_0)|}{|\log(x_0 - x)...
- 452
1
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0
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135
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Continuity of derivative
Consider a homeomorphism of the real line $F : \mathbb{R} \to \mathbb{R}$ such that it is differentiable everywhere and the derivative is bounded. Does it follow that the derivative is continuous, ...
1
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0
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77
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Find a vector in the null space of a large dense matrix, where elements in the matrix are not directly accessible
I am working with Conjugate Gradient method to solve for 𝐴𝑥=𝑏, where 𝐴 is an extremely large PSD and Singular matrix. I cannot directly access the elements of 𝐴. The only thing I can do is ...
1
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0
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131
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Integral flow that can commute to Laplacian operator
Firstly, considering the vector field in $ \mathbb{R}^3 $, $ X=x_2e_1-x_1e_2 $, we can see that
$$
\phi(t,x)=\phi(t,x_1,x_2,x_3)=(t,x_1\cos t+x_2\sin t,-x_1\sin t+x_2\cos t,x_3)
$$
is the ...
- 1,219
1
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64
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Sequential Hölder-norm for functions in $H_{\alpha}([0,1]^{d})$?
I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even ...
- 149
1
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106
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Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 286
1
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0
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179
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Getting rid of complex zeros of function with zeros the primes?
From our Note: simple real function with zeros greater than one the primes
simple real function with zeros greater than one the primes:
$j_1(x)=(\sin(\pi x))^2+(\sin(\pi \frac{\Gamma(x)+1}{x}))^2$.
...
- 25.4k
1
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0
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214
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Bound for the laplacian of a strictly convex function from above by the gradient of it
Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$
I am trying to show that
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
- 185
1
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217
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Inequality on matrix trace
Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems :
$$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma ...
- 1,399
1
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96
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On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
- 61
1
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0
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127
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Functions $f: \mathbb R \to \mathbb R$ such that $\det [f(a_j-b_k)]_{j,k} \neq 0$ for all $a_1,b_1, \dots, a_N,b_N$ and all $N \in \mathbb N$
A function $f: \mathbb R \to \mathbb R$ is called totally positive if for every $N \in \mathbb N$, every $a_1< a_2< \dotsb < a_N \in \mathbb R$ and every $b_1 < b_2 < \dotsb < b_N \...
- 437
1
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0
answers
57
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Necessary and sufficient conditions so that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?
First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
- 870
1
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0
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165
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Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
- 11
1
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0
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216
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Is there an approximate formula for this summation function?
Consider the function $$\sum_{n=1}^\infty \frac{\cos(nx)}{n^r},$$ where $r\in\mathbb{N}$. Is there any approximate formula (closed form possibly avoiding this type of summation) for this function? I ...
- 59
1
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0
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106
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When are the level curves of a polynomial bounded?
Let $f \in \mathbb{R}[x,y]$. I want to understand when $f$ has the following property: for all sufficiently large (positive) $k$, the level curves defined by
$$\displaystyle f(x,y) = k$$
consist of a ...
- 26.9k
1
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0
answers
125
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Second derivative of the logarithm of the modified Bessel function of the first kind
This question makes sense entirely without the probabilistic perspective, but let us quickly describe how it arises in our setting. Let $X,X’$ denote two i.i.d. random variables having the ...
- 183
1
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1
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191
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Comparison of solutions of Hamilton-Jacobi equations with different initial conditions
Consider a Hamilton-Jacobi equation:
$$u_{t} + f(u_{x}) = 0 \quad (x,t) \in \mathbb{R}\times [0,+\infty)$$
with two possible initial conditions $u(x,0) = g_{i}(x)$ for $x \in \mathbb{R}$ and $i=1,2$. ...
- 3,515
1
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0
answers
65
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Prescribed class of measurable sets
Let $X\neq\emptyset$ and let $\mu:P(X)\to[0,\infty]$ be an outer measure. Recall that, a set $A\subseteq X$ is $\mu$-measurable if
$$
\mu(B)=\mu(A\cap B)+\mu(B\setminus A), \text{ for all }B\subseteq ...
- 895
1
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1
answer
179
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For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$
Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that
\begin{align*}
0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
- 19
1
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0
answers
195
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The image of zero-measure set under normal mapping is Lebesgue measurable
Let $u$ be a convex function defined on a bounded open set $\Omega$ in $\mathbb{R}^n$. Then $u$ is twice differentiable a.e. Let $E_u$ be the set on which $u$ is not twice differentiable. Then $E_u$ ...
- 11
1
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0
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119
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On some integral equation
Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows:
$$
1-t= \int\limits_0^\infty p(x)\Phi\left(\frac{...
- 1,399