All Questions
5,657 questions
7
votes
1
answer
772
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\...
- 1,788
7
votes
0
answers
313
views
Did Lebesgue like non-measurable set or not?
I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:
Vitali's nonmeasurable set, ...
- 877
7
votes
0
answers
250
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
7
votes
0
answers
150
views
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 ...
- 20.5k
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
204
views
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 ...
- 71
7
votes
0
answers
270
views
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. ...
- 41.1k
7
votes
0
answers
270
views
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 ...
- 4,359
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
241
views
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-...
- 1,461
7
votes
0
answers
265
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 ...
- 6,367
7
votes
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 $...
- 2,069
7
votes
0
answers
264
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 $\mu_X, \mu_Y$...
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 ...
- 367
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
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 1,309
7
votes
0
answers
219
views
Results that are easier in a metric space
Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces?
In particular, I'm ...
- 10.1k
7
votes
0
answers
549
views
Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
7
votes
0
answers
221
views
integrality of a Riccati-type equation
The following is a problem we were unable to prove and left stated in the paper
"Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846.
Define ...
- 43.2k
7
votes
0
answers
394
views
Fixed radius mean value property implies harmonicity?
Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent:
$f$ is harmonic.
$f$ satisfies the ball mean value property
$$
f(x)=\frac{1}{|B(x,r)...
- 527
7
votes
0
answers
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 ...
- 225
7
votes
0
answers
628
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 ...
user5810
7
votes
0
answers
227
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 ...
- 41.9k
7
votes
0
answers
327
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 $n!=1\times 2...\times n$.
- 79
7
votes
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 ...
- 17.9k
7
votes
0
answers
174
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 ...
- 1,906
7
votes
0
answers
340
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 $...
- 71
7
votes
1
answer
233
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 ...
- 541
6
votes
4
answers
780
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 ...
- 43.2k
6
votes
3
answers
747
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(\...
- 10.4k
6
votes
7
answers
5k
views
Best way to teach concept of real numbers using a hands-on activity?
I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
- 163
6
votes
3
answers
852
views
Almost everywhere-periodic functions with many periods
Let $f : \mathbb{R} \to \mathbb{R}$ be a Lebesgue measurable function and $D$ be a countable dense subset of $\mathbb{R}$.
Suppose that for a.e. $x \in \mathbb{R}$ we have
\begin{equation*}
f(x + d) = ...
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
- 63
6
votes
3
answers
11k
views
Sums of uncountably many real numbers [closed]
Suppose $S$ is an uncountable set, and $f$ is a function from $S$ to the positive real numbers. Define the sum of $f$ over $S$ to be the supremum of $\sum_{x \in N} f(x)$ as $N$ ranges over all ...
- 15.4k
6
votes
3
answers
1k
views
Discontinuous functions without removable discontinuities
A function $f:\mathbb{R}\rightarrow \mathbb{R}$ has a removable discontinuity at a given real $x$ in case the left and right limits are equal but not to the function value, i.e. $f(x+)=f(x-)$ but $f(x)...
- 4,359
6
votes
2
answers
755
views
Prove positivity of a binomial sum
Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
- 43.2k
6
votes
2
answers
3k
views
Approximating erf by tanh
It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...
- 6,541
6
votes
3
answers
1k
views
Is there an entropy proof for bounding a weighted sum of binomial coefficients?
Given a probability $p \in (0,1)$ and parameter $\alpha \in (0,1)$, is there an entropy-based proof which yields a good upper bound for the sum
$$\sum_{\ell = 0}^{\alpha n} \binom{n}{\ell}p^\ell(1-p)^{...
- 557
6
votes
2
answers
303
views
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances?
Formal version of question. If $X$ is a set, let $[X]...
- 48.9k
6
votes
2
answers
509
views
Summing Bernoulli numbers
Consider the Bernoulli numbers denoted by $B_n$, which are rational numbers.
It is known that the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$ are not integers once $n>1$.
I am curious about the ...
- 43.2k
6
votes
2
answers
437
views
Matrix-valued ordinary differential equation with symmetry
I am considering the following equation
$$\begin{pmatrix} -\frac{d}{dx} + \lambda \sin(2\pi x) & \lambda - \lambda \cos(2\pi x) \\ -\lambda-\lambda \cos(2\pi x) & -\frac{d}{dx} - \lambda \sin(...
- 192
6
votes
2
answers
1k
views
Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable
This could be a soft question. I am trying to show that the $n$-th Taylor series coefficient of a function is $O(n^{-5/2})$. However, because the function is a function composition of another function ...
- 212
6
votes
1
answer
234
views
What about of periodic points of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function?
Let $\mu(n)$ the Möbius function, we define $F:[0,1]\to[0,1]$ as $$F(x)=\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n.\tag{1}$$
For a function of this kind (I presume that this continuous function has image $[...
user75829
6
votes
2
answers
773
views
Finiteness of an integral
In a paper I am reading, the following seems to be claimed:
Let $f:[0,\infty)\to [2,\infty)$ be a continuous, monotonically increasing function with $\lim_{x\to\infty}f(x)=\infty$ and let $\alpha>3/...
- 460
6
votes
2
answers
622
views
Forcing the uniqueness of a solution of an ODE
For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and
$$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$
Let $y_n$ be the ...
- 449
6
votes
2
answers
953
views
If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?
Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function.
Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous almost ...
- 6,213
6
votes
2
answers
1k
views
Zero points of a smooth function on $\mathbb{R}$
Assume $f(x)$ is a smooth function on $\mathbb{R}$ and $f$ does not vanish on any interval. In other words, $f$ can have zero points but we cannot find any interval $(a, b)$ such that $f(x)=0$ for all ...
- 903
6
votes
3
answers
555
views
Uniformly approximating a function and its derivative using polynomials
I'm struggling either proving or disproving the following statement:
Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
- 263
6
votes
1
answer
2k
views
Analysis of solutions to a nonlinear ODE
Consider the following ODEs:
$\phi^2=\phi''\sqrt{1-\phi'^2}$, or $\phi^2=-\phi''\sqrt{1-\phi'^2}$.
Is there any theory (e.g. comparison theorems) which analyzes solutions of the above ODEs? I am only ...
- 591
6
votes
3
answers
749
views
Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski
I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that
$$
|S_f(N)|^2 \leq N + \frac{2N^2}{q} ...
- 111