All Questions
1,549 questions with no upvoted or accepted answers
8
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103
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Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
- 1,429
8
votes
0
answers
414
views
For $f$ Lipschitz with $|\nabla f| = 1$ a.e., what is the supremal Hausdorff dimension of the set on which $\varepsilon< |\nabla f| < 1-\varepsilon$?
Let $f$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere.
Let $\varepsilon \geq 0$. What is the supremal Hausdorff dimension of the set on which $f$ is differentiable with $\varepsilon &...
- 6,155
8
votes
1
answer
258
views
Sequential colimit of iterated quotients of Cauchy sequences
We work in constructive mathematics.
The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
- 2,624
8
votes
0
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296
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Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
- 1,459
8
votes
0
answers
422
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
- 399
8
votes
0
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518
views
Concave and other bounded functions: Series representation and converging polynomials
Main Question
Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
- 697
8
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0
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314
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How to prove that $ \sum_{m=0}^{\infty} { \Gamma\{(1+2m)/\alpha\}\over \Gamma(1/2+m)} { (-t^2/4)^{m}\over m !} \ge (\alpha/2)^{3}\exp(-t^{2}/4) $
I would love to prove the following inequality
$$
{1\over \sqrt{\pi} } \sum_{m=0}^{\infty}
\Gamma\{(1+2m)/\alpha\}
{ (-t^2)^{m}\over (2m) !}=$$
$$
\sum_{m=0}^{\infty}
{ \Gamma\{(1+2m)/\alpha\}\over \...
- 178
8
votes
0
answers
327
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How does Conway's proposed compromise for constructing the real numbers in ONAG actually work?
I have also asked this question on Math Stack Exchange (link).
In On Numbers and Games, after discussing the inclusion of the real numbers in the surreal numbers, No, Conway discusses the merits of ...
- 455
8
votes
0
answers
256
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Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
- 333
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
8
votes
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492
views
Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$
Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 99
8
votes
0
answers
210
views
Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
- 379
8
votes
0
answers
342
views
Conjecture on matrix with reciprocal principal minors
Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the ...
- 909
8
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433
views
Heisenberg group: function without vertical derivative
Let $\mathbb H$ be Heisenberg group with vector fields
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
and $U\subset\mathbb H$ is an open set.
I am ...
- 399
8
votes
1
answer
2k
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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}...
- 315
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
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0
answers
249
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
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0
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150
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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 ...
- 21.2k
7
votes
0
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254
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$C^0$-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
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0
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203
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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 ...
- 71
7
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0
answers
220
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Why are these two determinants equal?
This question is a follow up on Mark Wildon's comment from an earlier MO question.
As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by
$$\binom{n}...
- 43.2k
7
votes
0
answers
270
views
Can you identify this irrational number?
There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
- 41.1k
7
votes
0
answers
270
views
Between real analysis and mathematical logic
This question lies in the intersection of real analysis and logic, so I try to keep things rather basic.
First of all, logicians care about the following kind of formula:
Let $\varphi(n, x)$ be a ...
- 4,359
7
votes
0
answers
481
views
A seemingly trivial property of continuous functions differentiable at the origin (PART 2)
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous function such that $F(0)=0$, $F$ is differentiable at $0$ and $DF(0)$ is invertible. Is there an elementary way to show that for all $\epsilon>0$ ...
- 1,149
7
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0
answers
240
views
Sard's theorem for superharmonic functions: less regularity required?
A function $f:\mathbb{R}^d \to \mathbb{R}$ must be at least $C^d$ in order to guarantee in general that
$$\{\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
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
$$
f(x)=\frac{1}{|B(x,r)...
- 527
7
votes
0
answers
211
views
Increasing derivatives of recursively defined polynomials
Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$.
These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
- 225
7
votes
0
answers
628
views
Proving Richardson's theorem for constants
(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
user5810
7
votes
0
answers
227
views
Uniform approximation of separately continuous functions on zero-dimensional spaces
For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
- 41.8k
7
votes
0
answers
327
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About the first decimal of $\sqrt {n!}$
Do we have :
$$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$
Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
- 79
7
votes
0
answers
111
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A monoid-structure on pairs of interlacing polynomials
Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace ...
- 17.9k
7
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174
views
On derivatives of polynomials majorized by $\max(1,|x|^d)$
In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question.
Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How ...
- 1,906
7
votes
0
answers
340
views
Polynomials and divided differences
I would greatly appreciate any hint for proving the following.
Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then $...
- 71
7
votes
1
answer
233
views
Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
- 541
7
votes
1
answer
548
views
Does this Linear Algebra Construction have a Name?
Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordinates of $w$ vanish (...
- 5,191
6
votes
0
answers
130
views
Do there exist strictly contracting eikonal functions on $\mathbb R^n$?
A function $f: \mathbb R^n \to \mathbb R$ is said to be a strict contraction if
$$|f(x) - f(y)| < |x - y|$$
for all $x \neq y$.
A function $f$ is said to be eikonal if it is differentiable ...
- 6,155
6
votes
0
answers
156
views
Generalized Rademacher theorem for fractional derivatives
It is known that if $f$ is $\alpha$ Holder and $\gamma<\alpha$ then $f$ is $\gamma$ fractional differentiable. See Theorem 14 in the paper by G. H. Hardy and J. E. Littlewood, "Some properties ...
- 1,904
6
votes
0
answers
431
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
- 2,836
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
votes
0
answers
220
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
6
votes
0
answers
108
views
Archimedean ordered field in which every function is smooth
In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \...
- 2,624