# 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.

2,158 questions

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votes

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63 views

### Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which ...

**0**

votes

**1**answer

43 views

### Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...

**2**

votes

**2**answers

109 views

### Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...

**1**

vote

**1**answer

54 views

### Uniqueness of minimizers in the Calculus of Variations

Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...

**2**

votes

**1**answer

66 views

### The regularity of ODE with Zygmund coefficients

A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.
According to Markus' paper A uniqueness theorem for ...

**1**

vote

**0**answers

44 views

### How to prove a set of delay differential equations never converge (the delay is not constant)

Two functions $x(t)$ and $y(t)$ are coupled via:
$$\dot x(t) = a y(t)-b,y(t+x(t))=x(t)$$
where $a<0$, $b\neq 0$ is some constant.
I am mostly confused with the second equation. What is the ...

**7**

votes

**2**answers

803 views

### Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...

**-1**

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**0**answers

45 views

### Fractional differential equation and inverse Laplace transform

I am writing my bachelor thesis to prove that a certain linear fractional differential equation of order $(n,q)$ has $N$ linearly independent solutions, where $N$ is the smallest possible integer ...

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vote

**1**answer

63 views

### Interpolation of a trilinear functional

Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...

**3**

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**1**answer

187 views

### A certain generalisation of the golden ratio

Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...

**8**

votes

**1**answer

2k views

### Is There an Induction-Free Proof of the 'Be The Leader' Lemma?

This lemma is used in the context of online convex optimisation. It is sometimes called the 'Be the Leader Lemma'.
Lemma:
Suppose $f_1,f_2,\ldots,f_N$ are real valued functions with the same ...

**0**

votes

**1**answer

60 views

### Bringing a Heun equation into canonical form

It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...

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**0**answers

46 views

+50

### Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...

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**0**answers

33 views

### Anisotropic elliptic problems

Where can I find a reference on the following anisotropic problem?
$$(\bullet) \qquad \begin{align*} \nabla_x (A(x) \nabla_x u(x,y)) + \Delta_y(K(y) \ast u(x,y))\end{align*}=0, $$
where $(x,y) \in \...

**3**

votes

**1**answer

105 views

### Lower estimate of the minimal eigenvalue of a Hamiltonian

Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...

**1**

vote

**1**answer

57 views

### Elliptic interface problem without conditions on the interface

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...

**6**

votes

**1**answer

440 views

### Famous but unavailable paper of Jan Boman

The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...

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**0**answers

28 views

### Geodesic connectedness in static Lorentz manifold vs connectedness by trajectories with potential in Riemann manifold

What is the relationship between the study of geodesic connectedness in a standard static Lorentz manifold and the connectedness of two points by trajectories with potential (i.e. solutions to $x''(t) ...

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votes

**0**answers

54 views

### Prove that an iterative estimate implies Holder continuity

Let $u$, $w$ be nonnegative continuous functions such that $\frac{u}{w}$ is bounded on $B_{2^{-1}}$. Why the inequality
$$a_k \le \frac{u}{w} \le b_k \quad \text{ on $B_{2^{-k}}$} , \qquad b_k - a_k ...

**12**

votes

**3**answers

307 views

### (Sharp) inequality for Beta function

I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...

**2**

votes

**1**answer

85 views

### Approximate sequence of numbers

Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers
$$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$
It is easy to see that these numbers satisfy
$$x_{n,0} = \frac{1}{n+1} ...

**0**

votes

**0**answers

41 views

### Limiting a sequence of moment generating functions [migrated]

I was trying to solve the following problem:
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...

**0**

votes

**1**answer

64 views

### Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$

Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...

**1**

vote

**1**answer

125 views

### How to recognize if a continuous vector field in the Euclidean space is a gradient

How to recognize, by "analytic" methods, if a $C^0$ vector field $v:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is the gradient of a function $h:\mathbb{R}^n \rightarrow \mathbb{R}$, given that the ...

**1**

vote

**0**answers

24 views

### Fredholm integral equation of third kind

Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...

**1**

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**0**answers

56 views

### On the convergence of an integral of Hardy's maximal function

Let $f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$ be an $L^1$ function.
Assume that
$$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0 $$ as $\bar r \to 0$ ...

**1**

vote

**0**answers

54 views

### Time-varying perturbations of continuous-time hyperbolic orbits

My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...

**1**

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**0**answers

61 views

### Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...

**2**

votes

**1**answer

94 views

### The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$
...

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41 views

### symmetry of solutions

Consider the problem $$-u''+u= u^p \text{ on }\mathbb I; \quad u=0 \text{ on } \partial \mathbb I$$ where $\mathbb I $ is symmetric bounded interval. If $u$ is a least energy solution, then it is ...

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**0**answers

123 views

### $L^1$-continuity estimate for ODE solutions in terms of $L^1$ distance of vector fields (only one of them being Lipschitz)

Consider the following ODE initial value problems
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in ...

**2**

votes

**1**answer

130 views

### Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...

**2**

votes

**1**answer

130 views

### Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...

**6**

votes

**1**answer

179 views

### Asymptotic Expansion of Bessel Function Integral

I have an integral:
$$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...

**6**

votes

**1**answer

336 views

### Interesting behaviour of binomial coefficients

Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...

**3**

votes

**1**answer

216 views

### Prescribing a gradient direction

Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...

**4**

votes

**1**answer

103 views

### Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...

**0**

votes

**1**answer

100 views

### Delay equations

In an effort to solve a delay partial differential equation
$$\partial_t f(t,x)= \Phi(x) f(t,x)+\Psi(x) f(t,x-\alpha),$$
with
$$f(0,x)=1,\hspace{0.3cm} f(t,0)=1$$
Where $\alpha$ is the delay ( a real ...

**1**

vote

**0**answers

59 views

### Best constant for Hölder inequality in Lorentz spaces

It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely
$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...

**33**

votes

**2**answers

2k views

### When can a function be made positive by averaging?

Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\...

**-2**

votes

**2**answers

112 views

### What is the limit of this integral as $n$ approaches infinity for integer $k\geq 0$ and real $m\geq 1$? [closed]

$\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$

**-5**

votes

**1**answer

147 views

### How do you prove the validity of this formula for $H(n)$? [closed]

I'm looking for a proof of the identity
$$\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{2n}+\pi\int_{0}^{1} (1-u)\cot{\pi u}\left(1-\cos{2\pi n u}\right)\,du.$$
There is a generalization of this formula for $...

**12**

votes

**5**answers

2k views

### Reference request: Oldest calculus, real analysis books with exercises?

Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there.
Edit. Unsolved exercises ...

**0**

votes

**0**answers

47 views

### Convex functions and infinite convex combinations

Let $f: D\rightarrow \mathbb R$ be a convex function on a convex subset in $\mathbb R^n$. Let $t_i>0$ with $\sum_{i=1}^\infty t_i=1$ and $x_\in D$ be such that the series $\sum_{i=1}^\infty t_i x_i ...

**1**

vote

**1**answer

96 views

### Are bilinear sparse bounds for local operators trivial?

I'm thinking about a recent result by M. Lacey which says that a dyadic spherical maximal function satisfies a sparse bilinear bound. To be precise, define the unit scale dyadic spherical maximal ...

**0**

votes

**0**answers

23 views

### Convergence of the asymptotic expansion solution of homogeneous linear ODE of order 2

Consider ODE $w''+pw'+qw=0$, $p$ and $q$ are functions of $z$. Denote $w_1$ and $w_2$ the ODE's two linear independent solutions. As $z\to0$, in which situation:
Do both $w_1$ and $w_2$'s ...

**4**

votes

**2**answers

181 views

### Integral equality of 1st intrinsic volume of spheroid

Computations suggest that
$$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$
The question ...

**5**

votes

**1**answer

209 views

### Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...

**3**

votes

**2**answers

284 views

### how to calculate the following integral related to Chebyshev polynomials

Chebyshev polynomials of the second kind $V_n(x)$ can be defined as
$$V_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$
or through the recurrence relation
$$V_{n+1}=xV_n-V_{n-1}, V_0=1, V_1=...

**-2**

votes

**3**answers

172 views

### A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

Is there an analytical solution satisfying these 3 equations with non-constant z?
$$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$
$$\frac{dy}{dt}=z\cdot\sin(\omega t)$$
$$\frac{dz}{dt}=x\cdot\cos(\omega t) - ...