All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
3
votes
2
answers
322
views
Hausdorff dimension of the graph of the sum of two continuous functions
How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:
Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
CommunityBot
- 1
2
votes
1
answer
391
views
Entropy solution for linear transport equation
Consider the transport equations
$$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$
and
$$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$
Can we define a notion of entropy solutions for (1) ...
- 555
3
votes
0
answers
141
views
Partially BV vector fields and renormalization
Why does the approach used to prove Theorem 4.1 in the paper by Le Bris and Lions on Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications not work ...
CommunityBot
- 1
1
vote
1
answer
178
views
Growth assumption and example of finite (arbitrarily small) time blow up for ODE
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 1,277
1
vote
0
answers
107
views
Level sets of a BV function and its derivative
Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$?
More specifically, does Alberti ...
- 839
3
votes
1
answer
224
views
Flow of ODE with monotone source
Let $\Phi$ be the flow (defined as in page 6 of this paper) of the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}.
\end{cases}$$
Is ...
- 34.7k
2
votes
0
answers
165
views
Jacobian and Jacobian matrix of solutions of ODE with Sobolev vector field
Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{...
CommunityBot
- 1
1
vote
1
answer
169
views
Difference quotient for solutions of ODE and Liouville equation
Suppose that $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
How does one prove that
$$\...
- 60.5k
2
votes
0
answers
199
views
Convergence of the difference quotient of a BV function
Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
What can be said about the difference quotient
$$
\frac{u(x+\epsilon y)-u(x)}{\epsilon}
$$
regarding its convergence as $\epsilon \to 0$...
- 839
2
votes
0
answers
187
views
Role of absolute continuity of divergence of BV function in proof of renormalization property
In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
Heuristically, ...
- 839
13
votes
1
answer
661
views
Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
- 356
7
votes
4
answers
4k
views
Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 2,821
3
votes
0
answers
55
views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
- 1,385
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
CommunityBot
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7
votes
1
answer
771
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 ...
- 28k
1
vote
1
answer
247
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 ...
CommunityBot
- 1
14
votes
3
answers
664
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) = \...
- 3,255
13
votes
5
answers
3k
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 ...
- 4,713
13
votes
2
answers
2k
views
An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
CommunityBot
- 1
7
votes
0
answers
356
views
Is this proof of Basel identity known?
Today, to divert myself, I tried to find a new proof of Basel identity $\boxed{\sum_{j=1}^\infty\frac{1}{j^2}=\frac{\pi^2}{6}}$. I came up with the following, which essentially interprets the identity ...
CommunityBot
- 1
18
votes
0
answers
439
views
An integral in Gradshteyn and Ryzhik
Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
- 189
12
votes
4
answers
2k
views
History of ODE and PDE reference request
Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
26
votes
9
answers
14k
views
Graduate ODE textbook
Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
- 4,713
12
votes
1
answer
1k
views
Proof of Green's formula for rectifiable Jordan curves
$\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
- 8,992
0
votes
1
answer
242
views
Harnack inequality for fractional laplacian
Let u be a positive solution of $s\in (0, 1) $
\begin{equation}
\left\{\begin{aligned}
(-\Delta )^{s} u &= 0 \text{ in } (-2T, 2T)\\
u &=g\quad\text{in}\quad \mathbb R\setminus(-2T, 2T).
\...
- 17.2k
2
votes
1
answer
239
views
Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
- 17.2k
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
CommunityBot
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1
vote
0
answers
324
views
Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
1
vote
2
answers
215
views
vector valued BVP for ODE's
I am dealing with a vector valued second order homogeneous BVP:
$\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$
where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and $...
- 52.3k
3
votes
0
answers
203
views
Identity for the product of two different associated Legendre polynomials
In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated:
$$
\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...
- 16.5k
8
votes
4
answers
4k
views
Approximation by exponential polynomials
Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of order $n$.
Define $E_n$ to be the collection of all exponential ...
- 12.4k
2
votes
1
answer
662
views
Clausen’s identity for associated Legendre polynomials
Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493)
$$P_n(\cos{\...
- 5,624
6
votes
0
answers
227
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
- 10.5k
1
vote
1
answer
3k
views
Is there a standard proof that the L^1 norm > constant * sup norm for functions with derivative bounded above by K on the unit disk in R^n?
Suppose that you have a bounded function $f(x)$ on a compact domain in $\mathbb{R}^n$. It's easy to see from Holder's inequality that
$$
||f||_1 \leq \operatorname{Volume}(D) ||f||_\infty.
$$
There ...
- 42.8k
9
votes
1
answer
749
views
property of convex functions
I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
- 1,211
2
votes
2
answers
406
views
Solution to semilinear heat equation at $t=0$: $u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$
Consider the following Cauchy problem
$$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Suppose that $u \...
- 52.3k
24
votes
2
answers
2k
views
Reference for exponential Vandermonde determinant identity
I recently stumbled upon the following identity, valid for any real numbers $\alpha_1,\dots,\alpha_n$ and $\lambda_{n1} \leq \dots \leq \lambda_{nn}$:
$$ \mathrm{det}( e^{\alpha_i \lambda_{nj}} )_{1 \...
- 114k
1
vote
1
answer
262
views
Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)
Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
- 16.4k
5
votes
0
answers
329
views
Could there be something like a Grzegorczyk hierarchy in Analysis?
My most prevalent interest in mathematics has always been hyper-operators. I first learned about them when I was in highschool, and quite frankly, they amazed and dazzled me. For those who've yet to ...
CommunityBot
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4
votes
2
answers
371
views
Literature on ZS-AKNS systems with independent potentials
For those with some familiarity with integrable systems, I'll summarize my question as such:
Where can I find literature on ZS-AKNS systems, and their solution via the inverse scattering transform, ...
- 1,826
5
votes
2
answers
840
views
Decompostition of a Lipschitz domain
We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \...
- 4,713
1
vote
0
answers
176
views
Coefficient perturbations of polynomials with real roots only
Let
\begin{align}
P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\
Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\
p_i, q_i& \in \mathbb{R},\ 0<...
- 19.6k
2
votes
2
answers
261
views
Viscosity solutions for $u'(x) + \alpha u(x) - f(x) = 0$: supersolutions dominate subsolutions
Let $$u'(x) + \alpha u(x) - f(x) = 0,$$ with $x \in [0,\infty)$ and $\alpha \in \mathbb{R}$. Suppose $f \in C(\mathbb{R})$.
If
$u_1$ is a viscosity supersolution (or a viscosity solution, or a $C^...
- 671
3
votes
2
answers
579
views
More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya
Is there a comprehensive reference book on inequalities in the
spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...
- 2,600
5
votes
2
answers
1k
views
real analyticity, Fourier coefficients [duplicate]
Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
- 105k
3
votes
0
answers
111
views
When does the constant term in the following expansion is nonzero?
Dyson's Theorem
The constant term in the expansion of
$$\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_i}$$
is the multinomial coefficient
$$\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!},$$
...
3
votes
2
answers
306
views
Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
- 23.2k
1
vote
1
answer
380
views
Infinite compositions of holomorphic functions, is there literature on the subject?
I've recently become very intrigued by infinite compositions. To get at what I mean by the term, I'll be as explanatory as possible.
Consider a sequence of holomorphic functions $\{\phi_j\}_{j=0}^\...
- 63.9k
12
votes
2
answers
552
views
On the independence of lower and upper asymptotic and Banach densities
Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \...
CommunityBot
- 1
11
votes
2
answers
1k
views
Two divergent series conspiring?
Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus
$$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad
\sum_{n\geq0}\frac{a_n}{2n+...
- 105