Questions tagged [ca.classical-analysis-and-odes]
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
3,575 questions
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28
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Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
1
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0
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31
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Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
-1
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0
answers
115
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Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
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1
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104
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If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
- 481
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0
answers
34
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Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
- 395
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70
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+50
When is a first-order delay differential equation equivalent to a higher-order ordinary differential equation?
The proportional delay differential equation
$$
xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0
$$
with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
- 3,065
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0
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71
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Fourier decay implies what kind of regularity
We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that
$$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$
...
0
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1
answer
74
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Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem
Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$.
Q: Suppose for some reason (e.g. numerical) we know that there is ...
- 593
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1
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115
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Fourier transform of exponential over torus
I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
7
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1
answer
160
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When is a non-linear first-order ODE equivalent to a linear second-order ODE?
The Riccati equation $y'(x)+y(x)^2=f(x)$ is non-linear, but can be transformed into the linear equation $-u''(x)+f(x)u(x)=0$ via $y(x)=\frac{u'(x)}{u(x)}$.
Is there a general statement known about ...
- 3,065
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When is Laplace transform of a function power-law and relation to the behavior of the function near zero?
I want to see when the Laplace transform of a non-negative function $f$ defined on $[0, +\infty)$ is a power function in the loose sense, i.e.,
$$g(s) = \mathcal L\{f\}(x) = \int_0^\infty f(x) e^{-sx} ...
- 103
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1
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437
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Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
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1
answer
554
views
Example of continuous function which is not differentiable everywhere in a strong sense
Is there a continuous function
$$u\colon (0,1)\to \mathbb{R}$$
such that at every point $x\in (0,1)$ one has
$$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$
In particular $u$ is not ...
- 21.8k
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96
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Derivative bounds for self convolution of the spherical measure in $R^d$
While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate
$$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
- 1
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71
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Upper bound on higher order derivatives of $\frac{1}{v(t)}$
Suppose that $ v(t) >l>0$ and
$$
\vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}.
$$
Can we give an upper bound for
$$
(\frac{1}{v(t)})^{(k)}
$$
?
Attempt:
We first compute the first fourth order ...
- 269
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votes
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answers
204
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A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
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95
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Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 312
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votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
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votes
4
answers
474
views
A certain inequality involving square roots of polynomials
I want to prove the inequality
$$\begin{aligned}
&\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\
&- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + ...
- 1,109
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votes
1
answer
53
views
Exponentially weighted norms are not equivalent
Let $\|u\|^2_{L^2_\eta}$ be the exponentially weighted norm of the space of functions $u(x)$ for which $u(x)\mathrm{e}^{\eta\cdot x}$ with $\eta\in \mathbb{R}$ is in $L^2(\mathbb{R})$. How can I show ...
1
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0
answers
59
views
Asymptotic behavior of positive solution to nonlinear scalar field equation
It is well-known that the radial positive solution
$u=u(r)$ to nonlinear scalar field equation $$-\Delta u+u=u^p\text{ in } ~\mathbb{R}^d, 1<p<\frac{d+2}{d-2}$$ has the following asymptotic ...
- 705
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1
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72
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The asymptotic at infinity of ODE
It may be a simple problem in ODE.
Let $u$ be the positiv solution to
$$u''(t)-f(u,t)u(t)=0, u(0)=1, \lim_{t\to+\infty}=0,$$
with $f>0$ and $\lim_{t\to+\infty}f(u(t),t)=1$.
Can we prove that there ...
- 705
2
votes
0
answers
43
views
A distribution defined via an ODE for its Laplace trnsform
Fix a parameter $0 < c < \infty$.
As the solution to a certain problem,
there is a probability density function $f_c(t)$ on $0 < t < \infty$ with mean $1$ and
whose Laplace transform $L(\...
- 236
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votes
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answers
85
views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 6,195
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vote
1
answer
192
views
How to evaluate the following integral?
How to (analytically) calculate the following integral,
$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$
where $\langle z, \zeta \...
- 521
1
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1
answer
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views
Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?
This question is related to This question.
When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
- 178
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votes
1
answer
195
views
Asymptotic spectrum of a complex Sturm-Liouville differential operator
Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by
$$
\mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x),
$$
with Neumann ...
- 333
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votes
2
answers
364
views
Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
- 178
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votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
3
votes
1
answer
80
views
Solution of $d Y_t/dt = A(t) Y_t, Y_0 = I_d$ is positive definite?
Let $\{A(t)\}_{t \in [0,1]}$ be time-varying symmetric matrices in $\mathbb{R}^{d\times d}$. We consider the following ODE for $Y_t \in \mathbb{R}^{d \times d}$
$$
\tag{1}
\frac{d Y_t}{dt} = A(t) Y_t, ...
- 399
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votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
2
votes
1
answer
104
views
Looking for review of delay differential equations involving $f(x)$ and $f(x/k)$
A research problem unexpectedly leads me to a delay differential equation of the form
$$
f(x)=\alpha(f(x),f(x/2))\,f'(x)+\beta(f(x),f(x/2))\,f'(x/2)+\gamma(f(x),f(x/2))
$$
For special cases of $\alpha,...
- 3,065
2
votes
1
answer
208
views
Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors
Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
- 349
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votes
0
answers
33
views
Non-positive definite solution for differential Riccati equation
Consider the matrix-valued differential Riccati equation (DRE):
$$
\dot P_t +PA+A^\top P+Q-(B^\top P+S)^\top (B^\top P+S)=0, \quad t\in [0,T];\quad P(T)=G,
$$
where all coefficients are continuous.
...
- 503
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0
answers
57
views
Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
16
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0
answers
351
views
The convergence domain of the function $\sum \{n!x\}$
This is a problem from a mathematics competition: Does there exist an irrational number $x$ such that the series
$$\sum_{n=1}^{\infty}\{n!x\}<+\infty$$
where $\{ \}$ means the fractional part of a ...
- 505
2
votes
2
answers
175
views
Great literature on discrete dynamical systems and/or qualitative theory of difference equations
I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
1
vote
0
answers
52
views
Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
- 562
0
votes
0
answers
79
views
Is there an asymptotic expansion for the reciprocal of the classical Euler beta function?
The classical Euler beta function can be defined by
$$
B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t
$$
for $\Re(p),\Re(q)>0$.
The beta function and the classical Euler gamma function $\...
- 1,091
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
9
votes
1
answer
429
views
A curious norm related to the L¹ norm
If $f \in C^0([0,1])$, one can define
$$\Vert f \Vert_? = \sup_{J \subset [0,1]} \left\lvert \int_J f \right\rvert,$$
where $J$ runs among all subintervals of $[0,1]$.
This is a norm on $C^0([0,1])$ (...
- 575
0
votes
1
answer
119
views
Detecting singular points from a parametrization
Suppose $r(t)$ parametrizes some, say algebraic, curve in the plane. It can certainly be that $r$ is smooth but the curve is not, since $r$ resolves double points by passing through them at different ...
- 13
0
votes
0
answers
61
views
Bounding the coefficients of a polynomial written as the sum of powers of linear forms
Crosspost: Just a heads up, I've posted this question on math.stackexchange as well. I have made new attempts at solving it though, and figured I'd ask here since it's more of a research-level ...
- 121
1
vote
2
answers
120
views
Expansion of the associated Legendre polynomials $P^m_l(\cos\vartheta)$ for $\vartheta \rightarrow 0$
A simple test with Mathematica indicates that for the associated Legendre polynomials $P^m_l$ the following relation should hold:
$$ \lim_{\vartheta\rightarrow 0} P^m_l(\cos\vartheta) =a_{lm} \...
- 369
3
votes
1
answer
111
views
Sobolev inequalities and Wiener algebra
It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$)
such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and
$$
\...
- 16.2k
2
votes
1
answer
160
views
Is the function $(z-1)(2^{z}-1)\zeta(z)$ logarithmically concave and convex in $z\in(0,\infty)$?
For proving that the sequence
\begin{equation}\label{first-proof-decreas-seq}
\frac{1}{(2k-1)(k+1)} \frac{2^{2k+2}-1}{2^{2k}-1} \biggl|\frac{B_{2k+2}}{B_{2k}}\biggr|
\end{equation}
is decreasing in $k\...
- 1,091
2
votes
1
answer
315
views
Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$
I_k=\int_{0}^{\...
- 1,091
3
votes
1
answer
240
views
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and
$$
[f]_{\frac{2}{\...
- 1,219