All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
1
vote
0
answers
151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
11
votes
1
answer
566
views
Integral representation of product of two Whittaker functions
Does anyone know anything about the following formula involving special functions:
$$\begin{multline*}
W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
- 373
4
votes
0
answers
84
views
Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms
Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
- 128k
1
vote
1
answer
285
views
Why are the two ODE solutions linearly independent?
I notice that some second-order ODEs can be related to the triconfluent Heun's equation
$$u''(z)-(3z^2+\gamma)u'(z)+(\alpha-(3-\beta) z)u(z)=0.$$
And people usually say the general solution of the ...
- 155
1
vote
0
answers
218
views
Asymptotic inverses and de Bruijn conjugates (etc.) for complex-valued functions
I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT"), and it's been an absolute revelation for my research. The thing is, my current work centers ...
- 1,284
0
votes
0
answers
230
views
Ramanujan's infinite sum for pi
Ramanujan's famous pi formula states that
\begin{equation}
\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum_{k=0}^{\infty}\frac{(4k)!}{k!^4}\frac{26390k+1103}{396^{4k}}
\end{equation}
How can one prove this?...
- 45
2
votes
1
answer
317
views
Recommendation for books on boundary-value problems that include perturbed boundaries and many solved problems
I am looking for a book or resource that contains applied math analytical methods and a lot of solved problems in Boundary-Value Problems for second-order PDEs, and if it could be related to wave-...
- 183
1
vote
1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
4
votes
1
answer
2k
views
How to mathematically characterize a feedback loop in ODEs?
I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of ODEs. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there ...
- 513
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
5
votes
0
answers
91
views
Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line
Let $\newcommand{\ptp}{\widehat{\otimes}}\ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, ...
- 25.8k
1
vote
2
answers
859
views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
- 128k
5
votes
0
answers
163
views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
- 20.2k
3
votes
1
answer
488
views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
- 751
5
votes
1
answer
1k
views
Continuous dependence on initial parameters of an ODE for non-Lipschitz functions?
For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If ...
- 215
2
votes
1
answer
809
views
Square-integrable unbounded function
In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity:
$$f(x)=x^2\exp{(−...
- 16.5k
0
votes
1
answer
117
views
Lorenz ODEs with negative parameters
Consider the Lorenz system
$$\dot{x}(t) = \sigma(y-x) \, ,$$
$$\dot{y}(t) = x(\rho-z) - y \, ,$$
$$\dot{z}(t) = xy-\beta z \, .$$
Usually one considers the parameters $\sigma, \rho,$ and $\beta$ to be ...
- 3,574
11
votes
1
answer
2k
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Reference request: proof of Ramanujan's Cos/Cosh Identity
The Ramanujan Cos/Cosh Identity, as stated here, is
$$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+
\left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=...
3
votes
1
answer
142
views
Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
- 839
4
votes
0
answers
126
views
Relationship between three different definitions of solutions for ODE with irregular coefficient
What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
- 839
1
vote
2
answers
424
views
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the ...
- 839
1
vote
1
answer
331
views
Can divergence free vector fields be approximated by smooth ones?
If $M$ is a compact Riemannian manifold, is the space of $C^{\infty}$ divergence-free vector fields dense in the space of $C^r$ divergence-free vector fields, in the $C^r$ topology ($r\geq 1$)? How ...
- 19
3
votes
0
answers
159
views
Upper bound on the geodesic distance in a Lipschitz domain
I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
- 895
3
votes
1
answer
172
views
Translation to English of Brillouin's analysis of Airy's integral
I am trying to read the following paper by Leon Brillouin (the part on page 16 onwards):
Léon Brillouin, Sur une méthode de calcul approchée de certaines intégrales dite méthode du col, Annales ...
- 1,594
4
votes
1
answer
118
views
almost linear ODE
Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$
$$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$
where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $...
- 315
3
votes
1
answer
301
views
Reference request: Oldest books on series with unsolved exercises?
Per the title, what are some of the oldest books on series out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
3
votes
2
answers
322
views
Hausdorff dimension of the graph of the sum of two continuous functions
How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:
Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user139844
2
votes
2
answers
317
views
Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative
What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?
More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of
a function
$$...
- 839
2
votes
1
answer
391
views
Entropy solution for linear transport equation
Consider the transport equations
$$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$
and
$$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$
Can we define a notion of entropy solutions for (1) ...
- 839
3
votes
0
answers
141
views
Partially BV vector fields and renormalization
Why does the approach used to prove Theorem 4.1 in the paper by Le Bris and Lions on Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications not work ...
user123672
1
vote
1
answer
178
views
Growth assumption and example of finite (arbitrarily small) time blow up for ODE
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 839
3
votes
1
answer
224
views
Flow of ODE with monotone source
Let $\Phi$ be the flow (defined as in page 6 of this paper) of the ODE
$$\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}$$
Is ...
user124345
1
vote
1
answer
169
views
Difference quotient for solutions of ODE and Liouville equation
Suppose that $\Phi$ is the solution of
$$\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}$$
How does one prove that
$$\...
- 839
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
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
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{...
user124345
1
vote
0
answers
107
views
Level sets of a BV function and its derivative
Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$?
More specifically, does Alberti ...
- 839
2
votes
0
answers
187
views
Role of absolute continuity of divergence of BV function in proof of renormalization property
In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
Heuristically, ...
- 839
7
votes
2
answers
593
views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
- 91.8k
13
votes
1
answer
661
views
Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
- 91.8k
3
votes
0
answers
55
views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
- 1,385
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
27
votes
2
answers
1k
views
Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
- 28k
1
vote
1
answer
247
views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
user60665
7
votes
1
answer
771
views
Famous but unavailable paper of Jan Boman
The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...
- 28k
14
votes
3
answers
664
views
(Sharp) inequality for Beta function
I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...
- 315
13
votes
5
answers
3k
views
Reference request: Oldest calculus, real analysis books with exercises?
Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there.
Edit. Unsolved exercises ...
7
votes
0
answers
356
views
Is this proof of Basel identity known?
Today, to divert myself, I tried to find a new proof of Basel identity $\boxed{\sum_{j=1}^\infty\frac{1}{j^2}=\frac{\pi^2}{6}}$. I came up with the following, which essentially interprets the identity ...
- 3,146
18
votes
0
answers
439
views
An integral in Gradshteyn and Ryzhik
Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
- 189
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 1,692
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 ...
- 1,759