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.
875 questions with no upvoted or accepted answers
32
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$f\circ f=g$ revisited
This may be related to solving $f(f(x))=g(x)$. Let
$C(\mathbb{R})$ be the linear space of all continuous functions from
$\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \...
- 4,060
29
votes
0
answers
3k
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Why do polytopes pop up in Lagrange inversion?
I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
21
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0
answers
416
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
- 281
21
votes
0
answers
658
views
A multiple integral
Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos {(s_{2n-1}...
- 16.5k
21
votes
0
answers
1k
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Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$
A while ago, I came across the following problem, which I was not able to resolve one way or the other.
Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
- 17.1k
18
votes
0
answers
718
views
Are these continued fractions of integrals known?
Simplified repost of Are these continued fractions of integrals known? on MSE
EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
- 1,459
18
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0
answers
571
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Fundamental Theorem of Algebra via multiple integrals
Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
- 60.5k
18
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0
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759
views
An intriguing calculus question
Let $f:{\bf R}^n\to {\bf R}$ ($n\geq 2$) be a $C^1$ function. Is it true that
$$\sup_{x\in {\bf R}^n}f(x)=\sup_{x\in {\bf R}^n}f(x+\nabla f(x))\hskip 3pt ?$$
- 283
18
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0
answers
439
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An integral in Gradshteyn and Ryzhik
Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
- 189
18
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0
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310
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Profiles of very high dimensional functions
This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...
- 96.4k
16
votes
0
answers
351
views
The convergence domain of the function $\sum \{n!x\}$
This is a problem from a mathematics competition: Does there exist an irrational number $x$ such that the series
$$\sum_{n=1}^{\infty}\{n!x\}<+\infty$$
where $\{ \}$ means the fractional part of a ...
- 505
16
votes
0
answers
539
views
Identification of a curious function
The following question was asked on MSE but there were no replies.
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\...
- 1,906
16
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0
answers
910
views
Polynomials with presumably positive coefficients
After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...
- 13.5k
15
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0
answers
284
views
Solution spaces of algebraic differential equations and derived geometry
We consider potentially non-linear differential equations on the formal one dimensional disc $\Delta$. Such equations are given by expressions $$P(z,f,f',f'',...)=0,$$ where $P$ is an element of the ...
user108998
15
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0
answers
303
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Surprising approximate identity
While answering this MO question Connection between Bernoulli numbers and Riemann-Siegel theta function? Dan Romik
found the following surprising approximate identity:
$$\ln{8\pi}\approx \pi\left[ 2\...
- 16.5k
13
votes
0
answers
497
views
Is it possible that the following integral is $0$?
Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.
It is not difficult to see that
$$\int_{1<|z|&...
- 1,997
13
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0
answers
1k
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Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
- 1,839
12
votes
0
answers
828
views
Multiple Integral (American Mathematical Monthly problem 11621 and its generalization)
AMM problem 11621 asks to calculate the integral
$$I_2=\int\limits_{-\infty}^{\infty}ds_1\int\limits_{-\infty}^{s_1}ds_2
\int\limits_{-\infty}^{s_2}ds_3\int\limits_{-\infty}^{s_3}ds_4
\;\cos{(s_1^2-...
- 16.5k
11
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0
answers
3k
views
Eric T. Sawyer's proof of Fourier restriction conjecture
Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
- 111
11
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0
answers
726
views
Are Kähler differentials the same as one-forms for compact manifolds?
Let $M$ be a manifold and let $A = \mathcal{C}^\infty(M)$ be the ring of smooth real-valued functions.
An old posting asks about the relationship of Kähler differentials and ordinary differential ...
user13113
11
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0
answers
361
views
Positivity of polynomial sequences via generating series
In this question I address
the problem of proving the nonnegativity of a numerical sequence
$a_0,a_1,a_2,\dots$ via generating series technique. In the notation
$A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...
- 13.5k
10
votes
0
answers
263
views
Bi-Lipschitz mappings
Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...
- 28k
10
votes
0
answers
345
views
Is this elliptic integral identity known?
Thinking about some physical problem, I came across the following identity:
$$\phi^2\Pi\left(-\phi,\frac{1}{\sqrt{2}}\right)+\phi^{-2}\Pi\left(\phi^{-1},\frac{1}{\sqrt{2}}\right)=\frac{\pi}{\sqrt{2}}+...
- 16.5k
10
votes
0
answers
2k
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Questions on de Branges' work on the Riemann hypothesis
According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...
- 163
10
votes
0
answers
245
views
A slightly generalized existence and uniqueness theorem for integral equations (reference request)
I want to use the following statement without including the proof, which is completely straightforward but rather tedious:
Let $G_0:\mathbb R\times\mathbb R^m\to\mathbb R^m$, $\Theta_0:\mathbb R\...
- 61.9k
9
votes
0
answers
347
views
Can one prove Rademacher’s theorem via the rising sun lemma?
The classical Rademacher’s theorem states that Lipschitz continuous functions on $\mathbb R^n$ are differentiable almost everywhere.
In dimension one, a stronger result holds - it can be shown that ...
- 6,195
9
votes
0
answers
546
views
Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
- 21.3k
9
votes
0
answers
512
views
On Riesz criteria for Riemann hypothesis:
Marcel Riesz defined a function :
$R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$
The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$)
For any $\varepsilon$
We have ...
- 790
9
votes
0
answers
264
views
Regularity class of certain diffeomorphisms of the real line
I care about the following class of homeomorphisms of $\mathbb R$, which I'll call $\mathcal C^?$.
For simplicity, let us restrict attention to compactly supported homeomorphisms
(a homeomorphism $\...
- 43.2k
9
votes
0
answers
262
views
Semi-norms for Schwartz-Bruhat space over Q_p
I'm an analyst beginning to do some work over the p-adics, in particular work with spaces of functions from $\mathbb{Q}_p$ to $\mathbb{C}$. The Schwartz-Bruhat space in this case is given by the space ...
- 91
9
votes
0
answers
412
views
min/max of degenerate critical points and Newton diagrams
Given a smooth function of several variables, whose first derivatives vanish at the origin. Suppose the matrix of second derivatives is degenerate at the origin. For example all the second derivatives ...
- 2,232
8
votes
0
answers
191
views
Lifting a determinant map
This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^...
- 96.4k
8
votes
0
answers
277
views
a question on the paper of Łaba and Wolff
I'm reading the paper A local smoothing estimate in higher dimensions by Izabella Łaba and Thomas Wolff. The paper can be found at J. Anal. Math. 88 (2002), 149–171, doi: 10.1007/BF02786576, arxiv: ...
- 463
8
votes
0
answers
221
views
Inertia group vs. differential equations
The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...
- 2,040
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
8
votes
0
answers
525
views
Phase perturbations in oscillatory integrals
I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in ...
- 852
8
votes
0
answers
1k
views
G-delta of measure 0 containig the rationals.
It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...
- 1,981
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
7
votes
0
answers
254
views
$C^0$-limit of volume-preserving maps on $\mathbb R^n$
Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
- 435
7
votes
0
answers
481
views
A seemingly trivial property of continuous functions differentiable at the origin (PART 2)
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous function such that $F(0)=0$, $F$ is differentiable at $0$ and $DF(0)$ is invertible. Is there an elementary way to show that for all $\epsilon>0$ ...
- 1,149
7
votes
0
answers
215
views
Helmholtz decomposition of compactly supported fields
Let $\mathbf F$ be a smooth vector field in $\mathbb R^3$, which is null outside a finite compact domain $V$. By the Helmholtz decomposition theorem, there exist a scalar field $\Phi$ and a vector ...
- 687
7
votes
0
answers
2k
views
Algebraizing topology and analysis via condensed mathematics
I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow.
I've just come across a Twitter thread by Laurent Fargues explaining a work ...
- 79
7
votes
0
answers
369
views
On the solvability of a nonlinear differential system
A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
- 79
7
votes
0
answers
418
views
How would have Bezout proved Bezout's theorem?
How would Bezout have proved Bezout's theorem bounding the number of points in the intersection of two plane (polynomial) curves in $\mathbb{R}^2$?
I have looked at a couple of modern algebraic ...
- 679
7
votes
2
answers
824
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
- 401
7
votes
0
answers
461
views
On a paper of Alain Connes entitled 'Around Wilson's Theorem '
A relatively recent paper Alain Connes - Around Wilson's theorem
introduced the function
$$
S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr).
$$
In the same paper, he proved that the ...
user145059
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
7
votes
0
answers
619
views
Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 1,309
7
votes
0
answers
393
views
Fixed radius mean value property implies harmonicity?
Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent:
$f$ is harmonic.
$f$ satisfies the ball mean value property
$$
f(x)=\frac{1}{|B(x,r)...
- 527
7
votes
0
answers
317
views
An inequality which involves a sum of integrals
Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
- 71