All Questions
742 questions
14
votes
6
answers
3k
views
What's a natural candidate for an analytic function that interpolates the tower function?
I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
- 4,466
12
votes
4
answers
2k
views
Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(...
- 7,831
12
votes
1
answer
5k
views
Points of continuity of Baire class one functions
This is an idle question motivated by two comments I made to a previous MO question (which I just searched for, unsuccessfully). That question asked if the characteristic function of the rationals is ...
- 65.4k
12
votes
1
answer
1k
views
Kolmogorov-Arnold theorem for (just-)functions
There is famous Kolmogorov-Arnold theorem for continuous functions composition - continuous function of several variables can be composed of continuous functions of two variables.
Specialization of ...
- 1,626
11
votes
0
answers
374
views
A game of harmonic series(s)
Given a set $A\subseteq\mathbb{R}_{>0}$, consider the following (two-player, perfect-information, length-$\omega$) game $H_A$:
Players $1$ and $2$ alternately play strictly increasing natural ...
- 21.2k
11
votes
2
answers
1k
views
Is sigma-additivity of Lebesgue measure deducible from ZF?
Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. ...
- 16.6k
11
votes
1
answer
1k
views
Has anyone seen this series?
I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...
- 1,649
11
votes
1
answer
2k
views
Transcendentality of all irrationals in the Cantor set
Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic ...
- 113
11
votes
1
answer
1k
views
Smallest positive zero of Weierstrass nowhere differentiable function
Consider the Weierstrass nowhere differentiable function $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos(4^n \pi x)$. It seems that the smallest positive zero of $f(x)$ occurs at $x=\frac{1}{5}$, but I ...
- 413
11
votes
1
answer
704
views
Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?
A function $f:\omega\to\omega$ is called
$\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$;
$\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite.
Let us ...
- 41.8k
11
votes
2
answers
528
views
Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$
I'm interested in the asymptotics of
$$\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$$
as $x\to\infty$. I expect the results to behave similarly to $e^{x^2}=\sum_{k\ge 0}\frac{x^{2k}}{k!}$. However, I'...
- 840
10
votes
2
answers
2k
views
A result attributed to Whitney
One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...
- 727
10
votes
3
answers
913
views
Inequality for functions on [0,1]
Let $a\in (0,1), \;\;\psi_a(x):=\prod_{j=0}^\infty (1-a^{2j+1}x).$
Question. Is it true that, for all $x\in [0,1]$ and all $k\in\mathbb{N},$ the following inequality holds:
$$\frac{x^k}{(1-a)(1-a^3)\...
- 783
10
votes
2
answers
3k
views
Gluing two diffeomorphisms together
A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
$\psi(...
- 8,903
10
votes
2
answers
834
views
Functions that are approximately differentiable a.e
The classical definition of an approximately differentiable function is as follows:
Definition.
Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
- 28k
9
votes
5
answers
2k
views
Convexity of distance-to-boundary function
Let $\Omega\subset\mathbb{R}^{n}$ be an open,
bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$
the distance-to-boundary function, that is,
$$
d_{\Omega}\left(x\right):=\inf\left\...
- 91
9
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
8
votes
1
answer
388
views
A dichotomy for the quadratic variation of differentiable functions?
For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula
$$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$
where the $\limsup$ is taken over all "partitions" ...
- 128k
8
votes
3
answers
519
views
Invertibility of specific function
This is my first post. I'm not a mathematician, just an electronics engineer who loves mathematics. In one of my projects, I arrived at the following function:
$$V\left(\varphi\right)=\frac{A\sqrt{\pi-...
8
votes
2
answers
2k
views
Expression for the sum of square roots of zeros of a polynomial
Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$.
General question. Does there exist a simple expression for the ...
- 34.3k
8
votes
1
answer
609
views
Hausdorff distance and Cauchy sequences
This is a generalization of an older question.
Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of non-empty closed subsets such that for all $\varepsilon > ...
- 48.8k
7
votes
3
answers
986
views
Mixtures of log-convex functions are log-convex: a reference
A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
- 128k
7
votes
2
answers
455
views
On a monotonicity property of Fourier coefficients of truncated power functions
Is it true that
$$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$
is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$?
This question is related to this previous one.
Twice integrating by parts, one ...
- 128k
7
votes
3
answers
2k
views
A question on fractional derivatives
I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.
I just wanted to ask if there is a notion of ...
user168611
7
votes
1
answer
753
views
Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
- 71
7
votes
2
answers
267
views
Meeting a set of lines in $\mathbb{R}^n$
Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?
$M$ intersects all the elements of ${\...
- 48.8k
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 1,881
6
votes
1
answer
340
views
Inequality for functions on [0,1], continued
Let $0<a<1,\; \psi_a(x)=\displaystyle \prod_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set
$$f_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi_a(x).$$
Question. Is it true that, ...
- 783
6
votes
2
answers
632
views
Interpolation space between $L^1\cap L^2$ and $L^1$
In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
- 193
6
votes
1
answer
1k
views
Level sets of a Weierstrass nowhere-differentiable function
Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known ...
- 413
6
votes
1
answer
181
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence of ...
- 6,741
5
votes
1
answer
345
views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
- 692
5
votes
1
answer
958
views
Does a nonlinear additive function on R imply a Hamel basis of R?
A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
- 4,587
5
votes
1
answer
288
views
Is there any continuous ternary function which can not be represented by composition of continuous binary functions?
Let $f : X^3 \rightarrow X$.
If $X$ is $\mathbb Z$, then there will be a couple of functions $g,h$ from $\mathbb Z^2$ to $\mathbb Z$ that satisfies $f(x,y,z) = g(h(x,y),z)$ since there is a bijection ...
- 151
5
votes
3
answers
1k
views
Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
5
votes
1
answer
534
views
Minimiser of a certain functional
Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \...
- 6,155
5
votes
2
answers
458
views
Backward heat equation and forward perturbed heat equation well posed?
I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
- 536
4
votes
1
answer
538
views
Derivatives of Riemann $\xi$ and traces of zeros
Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
- 10.5k
4
votes
1
answer
410
views
Using a quadratic kernel instead of a linear kernel in the Laplace transform
Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
- 19.7k
4
votes
1
answer
785
views
What is the dual space of $L^p$(conservative vector fields on a bounded set)?
First, some background: I wanted to prove that, if $f$ is a measurable function such that $\nabla f\in L^p_\text{loc}(\mathbb R^n)$, then $f\in L^p_\text{loc}(\mathbb R^n)$, $p\in(1,\infty)$. This is ...
- 391
4
votes
1
answer
1k
views
Fourier coefficients of real analytic functions on an n-dimension torus
Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $...
- 7,609
4
votes
2
answers
479
views
Smooth functions with zeros of infinite order on a closed set
It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(...
- 541
4
votes
1
answer
374
views
An open mapping theorem for homogeneous functions?
I am researching different generalizations of the familiar open mapping theorem from functional analysis. Every "proof" I attempt while simply assuming positive-homogeneity, even in the finite-dim ...
- 123
4
votes
1
answer
213
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
- 536
4
votes
1
answer
3k
views
optimization of inverse matrix with constraint on matrix elements
everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...
- 49
4
votes
1
answer
8k
views
Detection of Redundant Constraints
Suppose I pose the following query to a constraint logic programming
system:
?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3.
Are there systems that would recognize the last inequality as
...
Countably Infinite
4
votes
2
answers
245
views
On the monotonicity of the ratio of two logarithmic expressions
According to this comment and this comment, a positive answer to this recent question (about Bernoulli numbers) would be sufficient to prove the following:
$r:=f/g$ is increasing on $(0,\pi/2)$ from $...
- 128k
4
votes
1
answer
977
views
Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
- 41
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
4
votes
3
answers
499
views
Are $\pm f\sqrt{1+g^2}$ and $\pm fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
This is a follow-up on the previous question.
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb ...
- 128k