All Questions
757 questions
6
votes
1
answer
289
views
Archimedean ordered fields without maxima and minima in constructive mathematics
In constructive mathematics, let us define an ordered (Heyting) field $F$ to be a commutative ring with a binary relation $<$ which is
irreflexive, where for all $x$, $\neg (x < x)$
asymmetric, ...
- 2,624
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
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
309
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
6
votes
1
answer
808
views
Must the Minkowski sum of a Borel set and a *closed* ball be Borel?
Let A be a Borel set in R^n. Must then A + B(0,1) be Borel?
Here B(0,1) is the closed ball centered at 0 of radius 1.
I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a ...
- 63
6
votes
2
answers
231
views
Subsets $X$ such that their Hausdorff outer measure is not finite
Let $H^d:\mathcal{P}(\mathbf{R}^n) \to \mathbf{R}\cup \{\infty\}$ be the $d$-dimensional Hausdorff outer measure on $\mathbf{R}^n$, for some $0<d<n$ with $n$ integer, which is constructed in the ...
- 1,511
5
votes
2
answers
483
views
Are there any known approaches of generalizing functions that do not have a limit at infinity to values at infinity?
Let's consider the affinely extended real line. The functions that have a limit on positive or negative infinity $\lim_{x\to+\infty} f(x)$ or $\lim_{x\to-\infty} f(x)$ can be generalized to the values ...
- 10.1k
5
votes
0
answers
195
views
What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
- 13.8k
5
votes
1
answer
410
views
Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$. Is there always a real number $x$ such that
$$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$
- 1,053
5
votes
2
answers
297
views
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with
$f_n \to f$ uniformly for some continuous $f$.
$f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,
where we ...
- 6,155
5
votes
1
answer
712
views
Does this condition imply absolute continuity?
Let $f: [0, 1] \to \mathbb R$ be a measurable function. Define the (possibly infinite valued) upper and lower Dini derivative $D^+ f, D^- f: [0, 1] \to [-\infty, \infty]$ by
$$D^+ f (x) := \limsup_{y \...
- 6,155
5
votes
1
answer
213
views
Matrix-valued periodic Fibonacci polynomials
Consider the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=xf_{n-1}(x)-f_{n-2}(x)$ with initial values $f_0(x)=0$ and $f_1(x)=1$. It is well known that the values of these ...
- 5,622
5
votes
1
answer
199
views
Find the inverse of a more general matrix that is similar to the Hilbert matrix
In the last MO question , the following matrix is given:
$$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$
and its inverse has been discussed.
Now the problem is further extended to a more general ...
- 153
5
votes
0
answers
143
views
Error of midpoint method for differentiable functions
Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$?
...
- 19.7k
5
votes
2
answers
301
views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
- 446
5
votes
1
answer
243
views
How much time does a function spend above or below its average value around a point?
Given a locally integrable function $f: \mathbb R \to \mathbb R$, define $
K: \mathbb R \times \mathbb R+ \to \mathbb R$ by
$$
K(x, r) :=
\begin{cases}
1, & \text{if }f(x) > \dfrac{1}{2r}\...
- 2,069
5
votes
2
answers
565
views
Geometry of Level sets of elliptic polynomials in two real variables
Updated:
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
- 356
5
votes
0
answers
150
views
monomer-dimer tiling of a Young diagram
As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.
Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
- 43.2k
5
votes
0
answers
1k
views
Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
5
votes
3
answers
1k
views
Constant rank theorem for Banach spaces
Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
- 2,099
5
votes
2
answers
922
views
What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?
In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
Troelstra,...
- 16.6k
5
votes
0
answers
221
views
Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?
Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
- 698
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
5
votes
2
answers
503
views
Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Look at the expression
$$
f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.
$$
The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
- 2,207
5
votes
1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 536
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
5
votes
0
answers
270
views
Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
- 156k
5
votes
1
answer
425
views
"Interlacing property" of certain polynomials
I posted this question on MO which was quickly and decidedly answered by Noam D. Elkies.
Once more referring to the same set of polynomials
$$u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\...
- 43.2k
5
votes
2
answers
594
views
Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
- 41.8k
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 124
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
5
votes
2
answers
647
views
Dominated convergence 2.1?
After this question : Dominated convergence 2.0?
I want to know, what about the case when $h\in L^1([0,1])$.
The completed question :
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging ...
- 4,074
5
votes
2
answers
1k
views
Equation between the two branches of the lambert w function
My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r_1(y)$ and $r_2(y)$ of ...
- 497
5
votes
3
answers
1k
views
Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.
Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
- 698
5
votes
0
answers
247
views
Involutions on $[0,1]$ given by power series (related to probability generating functions)
Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$.
Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and $a_k\...
- 3,937
5
votes
0
answers
140
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
- 1,101
5
votes
1
answer
2k
views
Commuting with self-adjoint operator
Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...
- 501
5
votes
2
answers
248
views
Hausdorff dimension of the zero set of $\nabla f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ ...
- 6,155
5
votes
3
answers
1k
views
Non-continuous higher differentiability
The standard definition is that a function $f:\mathbb{R}^n\to \mathbb{R}$ is differentiable at a point $x$ if there exists a linear map $\mathrm{d}f_x: \mathbb{R}^n \to \mathbb{R}$ such that
$$f(x+h) ...
- 66.8k
5
votes
1
answer
542
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
- 1,075
5
votes
0
answers
190
views
Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
5
votes
4
answers
589
views
Looking for a reference on conformal mapping on $\Bbb R^n$
A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e.,
if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then
$$\cos (Tx(t_0),Ty(t_0))= \...
- 1,101
5
votes
1
answer
474
views
An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 128k
5
votes
1
answer
598
views
An inequality related to Lagrange's identity and $L_p$ norm
Let $a_1, a_2, \cdots , a_n$, $b_1, b_2, \cdots, b_n$ be real numbers, $p \in [1, +\infty)$, prove that
$$\sum_{1\leq i < j \leq n} |a_ib_j - a_jb_i|^p \leq c_p \sum_{i=1}^n |a_i|^p \sum_{i=1}^n |...
- 563
5
votes
1
answer
618
views
Is the harmonic series worse than any summable series?
It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values.
We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...
- 536
5
votes
1
answer
350
views
Set of translations of a real function having a dense linear span
Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{x\rightarrow \pm \infty} f(x)=0$, and consider the sup-norm topology on $W$.
Problem. does there ...
- 537
4
votes
1
answer
321
views
Can planar set contain even many vertices of every unit equilateral triangle?
Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle?
I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
- 18.8k
4
votes
1
answer
313
views
A determinant of perfect square polynomials
Usually, I like working with determinants related to the Vandermonde matrix, i.e.
$$\det(x_j^{i-1})=\prod_{i<j}(x_j-x_i).$$
However, I run into some unusual matrix and its determinant. Define the $(...
- 43.2k
4
votes
0
answers
281
views
Dual space of ${\rm Lip}_0(\mathbb R^d)$
This question comes to me when I read this paper : https://arxiv.org/pdf/1702.06049.pdf
Let ${\rm Lip}_0(\mathbb R^d)$ be the space of Lipschitz functions $F$ on $\mathbb R^d$ with $F(0)=0$. Then is $...
user128095
4
votes
1
answer
642
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 697