All Questions
1,487 questions with no upvoted or accepted answers
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59
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Asymptotic behavior of the Hermite functions
I would like to understand the asymptotic behavior of the Hermite function :
$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$
where $H_k(x)$ is the $k-$th Hermite polynomial. For ...
- 21
1
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0
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101
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
1
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0
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39
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Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
1
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0
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45
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Existence of optimal entropic weights for empirical modeling
Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
- 111
1
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60
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Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
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0
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125
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Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$
Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
user538396
1
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0
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46
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Optimal transport and the geometry of singular measures on fractal Sets
Let $K$ be a self-similar fractal set in $\mathbb{R}^n$ with Hausdorff dimension $d < n$, equipped with a self-similar measure $\mu$ supported on $K$. Let $\mathcal{P}(K)$ denote the space of ...
- 11
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42
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Approximation of the function $f(z)=z^2/|z|$ by $C^1$ immersions
Let $D$ denote the unit disk in $\mathbb C=\mathbb R^2$. We consider the function $f:D\rightarrow\mathbb C $ defined by $$f(z):=\frac{z^2}{|z|}.$$ Then as proved in Global invertibility (p324 Remark 4)...
- 435
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100
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Difference of two completely monotonic functions
We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \...
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0
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58
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Asymptotics of Jacobi form
What are the large $x\in\mathbb R$ asymptotics of $f(x)=\theta_3(c_1+c_2 x^3,e^{-x^2})$ where $c_1,c_2$ are a pair of complex numbers (say, $\Re(c_2)>0$ and $\Im(c_2)<0$), and $\theta_3(a,b)=\...
- 11
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175
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Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
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56
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Extension of this maximisation problem : finite or not?
$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
- 1,399
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215
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Computing a closed form representation for a Fourier series summation
I want to compute a closed form representation for the below given summation expession.
$$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
- 698
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82
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Counting the number of local minima of a function that is the sum of square roots of cosines
Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows
$$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$
where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
- 21
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67
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Distribution of zeros for arbitrary Bessel functions
Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
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0
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54
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Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
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0
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78
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Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
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1
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127
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approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
- 171
1
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0
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95
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Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
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96
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Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
- 4,359
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0
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92
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Modulus of Continuity, Heat Flow, and Derivative Estimates
Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by
\begin{align}
(P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right],
\end{align}
where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
- 801
1
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0
answers
96
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Function whose derivatives eventually vanish almost everywhere
As a takeaway of this post we have the following result.
P. Let $f:[0,1]\to\mathbb{R}$ be infinitely differentiable such that for all $x\in[0,1]$ the sequence $\{f^{(n)}(x)\}$ is eventually $0$. Then ...
- 637
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128
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Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
- 1,043
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43
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If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?
Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be:
$m(x) \cdot \text{div} ( s(x) \nabla f(x))$.
What ...
- 93
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141
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Can this integral be solved analytically
I have an integral of the form
$$\int_{t_1}^{t_2} \frac{\sum_{i=1}^n a_i e^{b_i t}}{\sum_{i=1}^n c_i e^{d_i t}} dt$$
Where $a_i,b_i,c_i,d_i$ are $4n$ real constants, and $t_1,t_2$ are positives. Is ...
- 686
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79
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How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
- 111
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0
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86
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The intersection of $ n $ cylinders in $ 3D$ space
I posted the question on here, but received no answer
I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
- 387
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48
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How to derive a lower bound of a MinMax inequality?
Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$.
The goal
For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
- 405
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0
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148
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Rational solutions to $\cos(\lambda \pi) = \cos^2(a\pi) - \cos(b\pi) \sin^2(a\pi) $, with $a,b \in \mathbb{Q}$
In a similar vein to this question, I am trying to understand the occurrence of rational solutions $\lambda$ to the following equation $$\cos(\lambda \pi) = \cos^2 (a\pi) - \cos ( b\pi ) \sin^2 \left(...
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162
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Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
- 43.2k
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96
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Regularity of Feynman-Kac formula for a simple diffusion
Let consider the diffusion process given by:
$$dX_t = \alpha(X_t) dW_t$$
where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
- 393
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70
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
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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
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
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98
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Periodicity in one Fourier variable
Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument.
We can thus write $f$ as $$ f(x) = \sum_{n \in \...
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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:
$$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
- 393
1
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0
answers
59
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Study of the properties of a non-local ODE
I am studying the following non-local ODE
$$\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].$$
The number $x_0$ can ...
- 452
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82
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How to calculate the Integral with confluent hypergeometric function
How to prove this.Thank you in advance
Let $\delta,\beta>0$ How to prove this
\begin{align}
& \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
- 531
1
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0
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79
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Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$
The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
- 1,295
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0
answers
66
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Parameter estimation of a Taylor expansion
Let $a,b$ two real numbers, $\theta$ a real parameter and suppose that you have an analytic function of the form:
$$
f_\theta(x)\triangleq \sum_{k\in\mathbb{N}}a_k(\theta)x^k \quad\forall x\in[a,b],
$$...
- 393
1
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0
answers
122
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Implicit function theorem / Implicit selections when Jacobian not invertible
I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
- 393
1
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0
answers
113
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Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,153
1
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0
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166
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Monotone likelihood ratio of convolved power function kernel, $p\ge 3$
It was shown in a previous answer that for: $f(x)=|x|^p$, $\;x\in \mathbb{R}$, $\;p>2$, defining the density:
$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(
\hspace{-1pt}...
- 391
1
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0
answers
144
views
Analyticity of a function in two complex variables
Let $f$ be a function defined on $\mathbb{C}^2$ given by
$$
f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
- 11
1
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0
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269
views
Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$
Let $f(x) = \log(\cosh(x))$, and define the kernel density:
$$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
answers
57
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Monotonicity of root of hyperbolic function
For $\kappa \geq \alpha>0$ and $y \geq 0$, consider the following equation:
$$\sqrt{1-\frac{\alpha }{\kappa }} \tanh \left(y \sqrt{1-\frac{\alpha }{\kappa }}\right)=\tanh \left(y-\frac{\alpha }{2}\...
- 11
1
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0
answers
67
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Estimating commutator of Fourier integral
Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by
$$ 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
answers
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
answers
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
answer
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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60
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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