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.
3,455
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Regarding a Feature of Multivariate Real Function
Any real function can be expressed as a function of the sum of two monotonic real functions?
More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such ...
- 103
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Bounding near the boundary for a Sobolev function.
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
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Cardinality of $\eta$-bush; on a Lemma from Wolff's paper.
The Question
This question is about Lemma 1.2 on the fifth page of Thomas Wolff's paper, "A sharp bilinear cone restriction estimate", Annals of Mathematics, 153 (2001), 661--698. The Lemma states (...
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Functions which form continuous curve with its own iterations
The following function
$$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+1$$
has interesting property to form a continuous curve with its own integer iterations. The following ...
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Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
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Why not evaluate integrals using ODE-solvers?
Hello!
I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral
$I(x) = \int_{0}^{x} f(...
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Polynomial upper approximation with respect to the Gaussian measure
Let $f = 1_{[a,+\infty)}$ be the indicator function of a half-line. Does there exist a sequence $(P_n)$ of polynomials such that $f(x) \leq P_n(x)$ for every real $x$ and
$$ \lim_{n\to \infty} \int_{\...
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How to "globalize" the inverse function theorem?
Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function ...
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How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial?
Suppose that $\sum^\infty_{n=1} u_n = s,$ where the series converges conditionally, and $s'>s$. How to prove the existence of such a permutation $\sigma,$ such that
1) $u_n\geq 0 \rightarrow \sigma(...
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Regarding Kolmogorov's Superposition Theorem
Hi Experts,
I have question regarding Kolmogorov's Superposition Theorem:
It is known that:
Let ${f(x_1,x_2,...,x_m): \Re^m :=[0,1]^m \to \Re}$ be an arbitrary multivariate continuous function. From ...
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Uncountable preimage of every point
Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable?
Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the ...
- 4,761
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1
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Bounding a smooth function near the endpoint
Let $\Omega=(a,b)$ a finite interval, $g\in \mathcal{H}^k(\Omega)$ some integer $k$, with $g(a)=0$ and let $\epsilon>0$. Is there an $\alpha\geq 1+k$ such that:
$
\left\|g\right\|_{L_2(a,a+\...
- 3,177
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A tricky integral
Let $\alpha>0$ and $\beta\in\mathbb{R}$. I am looking for an explicit formula for the integral
$$\int_{-\infty}^{\infty} (1+x^2)^{-1/2}e^{-\alpha x^2}e^{-i \beta x}dx.$$
I tried several changes ...
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Series of squared Fourier coefficients
Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is
$$
g(t)=\sum_{-\infty}^{+\infty}a_n e^{in\omega t}
$$
does the series
$$
\sum_{-\infty}^{+\infty}\frac{a_n^2}{n^2}?
$$
...
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Uses of Divergent Series and their summation-values in mathematics?
This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see.
Dear MO-Community,
When I was trying to find closed-form representations for odd zeta-...
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2
answers
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Justifying the definition of arclength
Background
One of my friends told me the following story: A child must walk from his home at point A = (1,0) to his school at point B = (0,1). The laws in his country state that you can only walk ...
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General Sobolev Inequalities
In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
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Continuity in terms of lines
Let $f: \mathbb R^n \rightarrow \mathbb R^n$, where $n> 1$ be a bijective map such that the image of every line is a line.
Is $f$ continuous?
I think it is, but the proof isn't immediately ...
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What is the difference between hard and soft analysis?
I have heard references to "hard" vs. "soft" analysis. What is the difference? It seems to do with generality versus nitty-gritty estimates, but I haven't gotten any responses more clear than that.
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Bounding a smooth function near the boundary
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>0$ and that $f$ ...
- 3,177
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1
answer
403
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Shrinking a Lipschitz smooth domain.
Let $\Omega\subset \mathbb{R}^d$ be an open and bounded domain with Lipschitz smooth boundary. Let $\delta>0$ and
$
\Omega_\delta = \{ x\in\Omega : \inf_{y\in\partial\Omega} \left\|x-y\right\|_{...
- 3,177
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Functions whose antiderivative behaves like xf(x)
I'm wondering if a classification of analytic functions, $f\,$ (it may be that $C^1$ is enough, but I'm not taking any chances, if you have a reason why I only need to consider a larger class of ...
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5
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Beginners text on calculus of variations
I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options.
I work on Machine Learning, and that where I intend to apply this.
...
2
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1
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668
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Transfinite sums related to a sequence
Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products of all $...
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Finite interpolation by nondecreasing indefinitely differentiable functions in a finite-dimensional space
Some time ago, I asked about inite interpolation by
a nondecreasing polynomial here at Finite interpolation by a nondecreasing polynomial. This turned out to be an already solved problem; it also ...
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644
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Complete extensions of valuations from Q to R.
This is somewhat related to the question and the answers here:
Is completeness of a field an algebraic property?
My question is (to which I believe the answer must have been known), does every ...
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Examples and importance of Embedding (and Non-Embedding) Theorems
An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
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A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
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Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
I have spent some time using gp-pari. There is, of course, a formal power series solution to
$ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
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How small can the set of $p$ such that the $L^p$ norms are different for two fixed functions?
What does it tell you about two functions if their $L^p$ norms are the same for all $p\in[1,\infty]$? Certainly they could be related by composition with a diffeomorphism with Jacobian of norm 1, or ...
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Approach to solving a differential-functional equation
What could be an approach to solving such equations?
$$f'(x)=C \prod_{k=0}^x f(k)$$
$$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$
Here the product and the sum are understood as indefinite sum and ...
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Big O notation and the maximal set of comparable functions
One can easily find a set of functions that are comparable with respect to the big O notation that is,
$$f \leq g \Leftrightarrow \exists c \exists x_0 \forall x\geq x_0: |f(x)| \leq c|g(x)|,$$
for $f,...
- 1,592
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maximal coordinate on a sphere
What is the easiest (preferably without calculations) way to see that the mean value of $\max(x_1,x_2,\dots,x_n)$ on the sphere $\mathbb{S}^{d-1}= \{ (x_1,\dots,x_n):\ x_1^2+\dots+x_n^2=1 \}$ behaves ...
- 103k
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Another special property of the exponential function?
For $x>0$, define
$\tilde f(x) = \sum\limits_{k = 0}^\infty {\frac{{(x - k) ^k }}{{k!}} {\bf 1}(x>k)}$,
where ${\bf 1}$ is the indicator function.
I know (actually, proved) that $\tilde f(x)$ ...
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1
answer
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Laplace equation over concentric spheres
Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over ...
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Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
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Non-Hölder continuous devil's staircases
Let $f:[0,1]\to[0,1]$ be a devil's staircase in the usual sense. (That is, $f$ is continuous, non-decreasing, $f'=0$ on a set of full Lebesgue measure.) We also require the complement to the set where ...
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Everywhere totally discontinuous function [duplicate]
Possible Duplicate:
Function with range equal to whole reals on every open set
I was told that it is possible to define a function $f:\mathbb R\to \mathbb R$ such that it takes every value on ...
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Evil Fourier Coefficients
Let $f:[0,1]\to[0,1]$ be the classical devil's staircase.
Has anybody ever computed (or studied) the fourier coefficient of $f(x)$?
Related question: is the fourier series of $f(x)-x$ normally ...
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3
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Do these properties characterize differentiation?
Let $L: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ be a linear operator which satisfies:
$L(1) = 0$
$L(x) = 1$
$L(f \cdot g) = f \cdot L(g) + g \cdot L(f)$
Is $L$ necessarily the derivative? ...
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Is evaluating limits with dual numbers sound?
Let $D$ be the ring $\mathbb{C}[\epsilon]/\langle \epsilon^2\rangle$. Define the functions $dual : \mathbb{C} \to D$ and $stdPart : D \to \mathbb{C}$ by $dual(x) := x+0\cdot \epsilon$ and $stdPart(x+...
user5810
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When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
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Riemann-Roch as an index theorem [closed]
I am sorry to make this a new question. I would have liked to leave a comment, but I suppose I don't have enough rep for that....
So, in the accepted answer to this question I don't understand why in ...
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Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection?
In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is ...
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Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
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votes
1
answer
625
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Upper bound for the quality of an $abc$-triple
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$...
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Decay of the Fourier transform
Suppose $f(z)$ is a function analytic in the strip $|Re(z)|\leq a$. Is the fourier transform $\hat{f}(w)=o(e^{-a|w|})$?
It seems plausible but I can't seem to prove it either.
There is similar ...
- 3,177
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3
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Product of sine
For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} \...
- 2,752
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3
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Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
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4
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Is the analysis as taught in universities in fact the analysis of definable numbers?
Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...
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