Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
Guo's user avatar
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Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping $$ f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n} $$ surjective? If not, what is its image? If yes, what can be said about ...
Stefan Kohl's user avatar
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A counterexample showing $BV_p \neq AC_p$

I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity. Let $p > 1$. ...
maxematician's user avatar
7 votes
1 answer
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Asymptotics of truncated logarithm on a cricle

Consider $f_n(x) = \min_{|z|=x} \Re \sum_{j=1}^{n} \frac{z^j}{j}$, a real function of positive variable $x>0$. I am interested in lower bounds on $f_n(x)$. Specifically, I ask: what lower bounds ...
Ofir Gorodetsky's user avatar
7 votes
1 answer
353 views

Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...
Sascha's user avatar
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An equivalent condition for differentiability almost everywhere?

Given a function $f \in L^1 (\mathbb R)$, define the roughness $R_f$ of $f$ at $x \in \mathbb R$ by $$\DeclareMathOperator{\esssup}{\operatorname{esssup}} R_f (x) := \limsup_{r \to 0+}\dfrac{r \...
Nate River's user avatar
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Kolmogorov superposition on the Hilbert Cube

A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form $$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...
James Hanson's user avatar
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7 votes
1 answer
587 views

Expectation involving maximum of Gaussian variables

Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
Steve's user avatar
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When the value of a function in a point is equal to its integral average over the point's neighborhood?

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
Grove's user avatar
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Avoiding equal distances

Is the following consistent? There exists $X \subseteq [0, 1]$, such that $X$ does not have measure zero and for every $Y \subseteq X$, if $Y$ does not have measure zero, then there are $y_1 < y_2 ...
Ashutosh's user avatar
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1 answer
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An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
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Is this inequality in two variables true?

It it true that for all $p\in(0,1/3]$ and all real $t$ we have $$4 \ln(1-p +p\cosh t) \ln\frac{1+\sqrt{1-2p}}{1-\sqrt{1-2p}} \le t^2 (1+c p) \sqrt{1-2p} ,$$ where $c:=2\sqrt{3}\, \ln(2+\sqrt{3})-3$? ...
Iosif Pinelis's user avatar
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1 answer
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Normal distribution by successive approximation?

$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
Iosif Pinelis's user avatar
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1 answer
361 views

Function of two sets

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
pi66's user avatar
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A criterion on a vector field for its flow to extend continuously at $t=\infty$

In my work in algebraic topology I need to build a special homotopy and I came up with a construction based on some ordinary differential equation in which I am not an expert. I miss some argument to ...
Pascal Lambrechts's user avatar
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1 answer
1k views

Signed variant of the Flint Hills series

I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one: $$ \sum_{n = 1}^\infty \...
Mateusz Kwaśnicki's user avatar
7 votes
1 answer
731 views

Conserved positive charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
Maurizio Barbato's user avatar
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1 answer
301 views

Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
Twi's user avatar
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1 answer
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When is this sum of perfect powers bounded

For any positive integers $n,d$, let $$ A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)} $$ It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is a polynomial of degree $2d-2$. Then we ...
Ewan Delanoy's user avatar
7 votes
2 answers
2k views

Tails of sums of Weibull random variables

Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
ilyaraz's user avatar
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1 answer
762 views

Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all $x\...
Ali Reza's user avatar
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0 answers
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The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
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0 answers
238 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\...
Tian LAN's user avatar
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Permutations which change the value of a convergent series

I'm interested in the following combinatorial problem: What is a necessary and sufficent condition on a permutation $\sigma : \mathbb{N} \rightarrow \mathbb{N}$, so that there exist a summable ...
Et-'s user avatar
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Can you identify this irrational number?

There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
Gerald Edgar's user avatar
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Between real analysis and mathematical logic

This question lies in the intersection of real analysis and logic, so I try to keep things rather basic. First of all, logicians care about the following kind of formula: Let $\varphi(n, x)$ be a ...
Sam Sanders's user avatar
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475 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 $\epsilon>0$ ...
No-one's user avatar
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Sard's theorem for superharmonic functions: less regularity required?

A function $f:\mathbb{R}^d \to \mathbb{R}$ must be at least $C^d$ in order to guarantee in general that $$\{\phi\in \mathbb{R}|\,\exists x\in \mathbb{R}^d:\,f(x)=\phi,\,(\nabla f)(x)=0\}$$ is a zero-...
5th decile's user avatar
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7 votes
0 answers
257 views

On the "Collected Works" of Charles Bradfield Morrey, Jr

Why Charles Bradfield Morrey, Jr.'s "Collected works" haven't been published yet? I've been thinking of this question for a while, at least from the first time I started to improve the ...
Daniele Tampieri's user avatar
7 votes
0 answers
289 views

How does Conway's proposed compromise for constructing the real numbers in ONAG actually work?

I have also asked this question on Math Stack Exchange (link). In On Numbers and Games, after discussing the inclusion of the real numbers in the surreal numbers, No, Conway discusses the merits of ...
Mike Earnest's user avatar
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0 answers
420 views

A discontinuous construction

Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $...
James Baxter's user avatar
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7 votes
0 answers
247 views

When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?

Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
Mozhgan Mirzaei's user avatar
7 votes
0 answers
106 views

The first homotopic Baire class

Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ belongs to the first Baire class (to the first homotopic Baire class), if there exists a continuous map $H:X\times \omega\to Y$ (a continuous ...
MasleniZZa's user avatar
7 votes
0 answers
567 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 Lip([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])...
Carlo Mantegazza's user avatar
7 votes
0 answers
199 views

Results that are easier in a metric space

Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces? In particular, I'm ...
Alec Rhea's user avatar
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498 views

Counter-example to the completeness of the Wasserstein metric

$\newcommand{\P}{\mathcal{P}}$ Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
Oleg's user avatar
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7 votes
0 answers
181 views

distance distributions on a hypersphere?

Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define $$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$ where ...
T. Amdeberhan's user avatar
7 votes
0 answers
219 views

integrality of a Riccati-type equation

The following is a problem we were unable to prove and left stated in the paper "Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846. Define ...
T. Amdeberhan's user avatar
7 votes
0 answers
356 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)...
Snoop Catt's user avatar
7 votes
0 answers
211 views

Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$. These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
TomH's user avatar
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0 answers
603 views

Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
user avatar
7 votes
0 answers
225 views

Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
Taras Banakh's user avatar
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7 votes
0 answers
325 views

About the first decimal of $\sqrt {n!}$

Do we have : $$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$ Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
Med's user avatar
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0 answers
111 views

A monoid-structure on pairs of interlacing polynomials

Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace ...
Roland Bacher's user avatar
7 votes
0 answers
172 views

On derivatives of polynomials majorized by $\max(1,|x|^d)$

In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question. Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How ...
Yuval Filmus's user avatar
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7 votes
0 answers
337 views

Polynomials and divided differences

I would greatly appreciate any hint for proving the following. Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then $...
George's user avatar
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7 votes
1 answer
1k views

Summary of sufficient conditions for convergence of Fourier series

I would like to summarize various sufficient conditions for various modes of convergence of Fourier series. The followings are what I have gathered so far: $L^p$ convergence: if $f \in L^p(\mathbb{T}...
user141240's user avatar
7 votes
1 answer
225 views

Hausdorff dimension and sigma finiteness

If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$. I would like to see an example of such a ...
Longyearbyen's user avatar
6 votes
4 answers
777 views

roots of higher derivatives of exponential

Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$. Question. Is it true that $D^nf(z)=0$ has only real roots ...
T. Amdeberhan's user avatar
6 votes
3 answers
464 views

Evaluating the infinite product $\prod_{k\geq 2}(1-\frac{1}{k^3})$

Does anyone know how to evaluate the infinite product $$ \prod_{k = 2}^{\infty} \left( 1 - \frac{1}{k^3} \right)? $$ I know that a generalized quadratic version has a nice closed form $$ \frac{\sin(\...
kodlu's user avatar
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