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.
134 questions from the last 365 days
5
votes
1
answer
282
views
An intriguingly simple integral functional for star-shaped, planar, simple, closed, smooth curves
The following is a question that popped up in my research in geometric analysis some time ago and that I dropped and kept coming back to multiple times. I will first state the problem, or rather my ...
- 159
0
votes
0
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47
views
Is the composition of a logarithmic multiplier and a singular integral operator bounded at $L^\infty$?
I need to consider an operator of the form
$$
W=T\circ (\log|D|)^{-1}\chi(|D|) \quad \text{in } \mathbb{R}^2,
$$
where $\chi$ is a smooth cut-off function supported near zero and $T$ is an operator ...
2
votes
1
answer
112
views
Example: the equilibrium point is attractive but unstable
It is well known that asymptotic stability of ODEs consists of two aspects: attractivity and stability. So, are there any examples where an equilibrium point is attractive but not stable?
- 105
4
votes
0
answers
119
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Is the one-sided concentration function of a non-negative integrable function absolutely continuous?
For $g\in L^1_+\!(0,1)$ define the one-sided concentration function
$$
Q_g(\delta)\;:=\;\sup_{0\le x\le 1-\delta}\int_x^{x+\delta} g(t)\,dt,
\qquad 0\le \delta\le 1.
$$
It is easy to check that $Q_g$ ...
- 12.4k
4
votes
0
answers
232
views
Geometric lemma in Mockenhaupt-Seeger-Sogge
I've been trying to read the paper of G. Mockenhaupt, A. Seeger & C. Sogge, Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem, 136 Ann. Math. 207 (1992).
In estimating a ...
- 343
5
votes
2
answers
454
views
Pointwise limit of cadlag functions
Suppose I have a sequence of cadlag functions $f_{n}$ defined on $\mathbb{R}$ and a function $f$ such that $f_{n} \to f$ pointwise. Can the limit $f$ have uncountably many points of discontinuity? ...
- 145
1
vote
1
answer
165
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Third moment of the Riemann zeta function
We know asymptotically
$$\int_1^T|\zeta^{(m)}(1/2+it)|^2dt$$
and
$$\int_1^T|\zeta^{(m)}(1/2+it)|^4dt,$$
but is anything known for the third moment
$$\int_1^T|\zeta^{(m)}(1/2+it)|^3dt?$$
Since the ...
- 161
4
votes
1
answer
233
views
Asymptotic for fourth moment of derivatives of Riemann zeta function
Due to work of Ingham, it is known that
$$\int_1^T|\zeta^{(m)}(1/2+it)|^2\;dt\sim\frac{1}{2m+1}T(\log T)^{2m+1}.$$
Is there a similar result for the fourth moment? That is, is there an explicit result ...
- 161
0
votes
0
answers
58
views
Sharper concentration bounds for Gram matrices of Müntz–Szász systems?
Let $\mathcal{D}_K = \{x^\alpha\} \cup \{p_k\}_{k=0}^K$ be a dictionary on $[0,1]$, where $p_k$ are the orthonormal Jacobi polynomials with respect to the measure $d\mu = x^\beta dx$ (with $\beta > ...
2
votes
0
answers
207
views
Edge-vertex inequality EV ≥ 4^(n-1)
Show that for any set $A \subset \{0,1\}^{n}$ of half size $2^{n-1}$ we have $EV\geq 4^{n-1}$
Here $E$-denotes edge-boundary of $A$, i.e., the number of edges between $A$ and its complement; and $V$ ...
- 4,219
19
votes
5
answers
1k
views
For which $n$ is the minimum max entry of an $n$ by $n$ orthogonal matrix known? Is $n=3$ already an open problem?
Crossposted on Mathematics SE, where the question Orthogonal matrices with small entries was brought to my attention, though it is about bounds rather than exact values.
Let $\| A \|_{\max} := \max\...
- 4,887
1
vote
0
answers
66
views
More on the theme of differentiation of asymptotic equivalences
CONTEXT
This question follows another question of mine about a class of functions for which differentiation of asymptotic equivalence is always legit. I briefly summarize the situation here, to make ...
- 203
3
votes
1
answer
156
views
Global asymptotic stability of ordinary differential equation systems
Consider the stability issues related to systems of ordinary differential equations.
$$
\begin{split}
\frac{{\rm d} x}{{\rm d} t} & = w (x) - \lambda \int_{0}^{x} w(s) \, {\rm d} s \equiv G_1 (x); ...
- 105
6
votes
1
answer
306
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Van der Corput's B-process and partial summation
I've been reading through Montgomery's "Ten lectures on the interface between analytic number theory and harmonic analysis", and am currently reading through his description of van der ...
- 63
-7
votes
1
answer
227
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How many iterations to solve this 'ultimate challenge' equation?
I've been benchmarking root-finding methods on this equation :
$$f(x) = e^{\sin(10x)} \cdot \arctan(100x) + \ln(x+1) \cdot \cos\left(\frac{1}{x+0.1}\right) - 5$$
Properties :
Contains exponentials, ...
