Search results for "FOS: Mathematics"
showing 10 items of 1448 documents
Resonance between Cantor sets
2007
Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is irrational, then \[ \dim(C_a+C_b) = \min(\dim(C_a) + \dim(C_b),1), \] where $\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\dim(K)+\dim(K') \le 1$ (``geometric resonance''), then there exists $r<1$ such that all contraction ratios of the similitudes defining $K$ and $K'$ are powers of $r$ (``algebraic resonance…
Finite 2-groups with odd number of conjugacy classes
2016
In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-$p$ techniques and, in particular, we use the Kneser classification of semi-simple $p$-adic algebraic groups.
$(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces
2019
In this paper we study the $(BV,L^p)$-decomposition, $p=1,2$, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case $p=1$ we also study the associated geometric problem and the thresholding parameters.
On the cardinality of almost discretely Lindelof spaces
2016
A space is said to be almost discretely Lindelof if every discrete subset can be covered by a Lindelof subspace. Juhasz et al. (Weakly linearly Lindelof monotonically normal spaces are Lindelof, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelof first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhasz et al. (First-countable and almost discretely Lindelof $$T_3$$ spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few rel…
Efficient computation of the branching structure of an algebraic curve
2012
An efficient algorithm for computing the branching structure of a compact Riemann surface defined via an algebraic curve is presented. Generators of the fundamental group of the base of the ramified covering punctured at the discriminant points of the curve are constructed via a minimal spanning tree of the discriminant points. This leads to paths of minimal length between the points, which is important for a later stage where these paths are used as integration contours to compute periods of the surface. The branching structure of the surface is obtained by analytically continuing the roots of the equation defining the algebraic curve along the constructed generators of the fundamental gro…
New spaces of matrices with operator entries
2019
In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in $\mathcal B(H)$. We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as $H^\infty(\mathbb{D})$ and $A(\mathbb{D})$ when dealing with upper t…
Lehmer code transforms and Mahonian statistics on permutations
2012
Abstract In 2000 Babson and Steingrimsson introduced the notion of vincular patterns in permutations. They show that essentially all well-known Mahonian permutation statistics can be written as combinations of such patterns. Also, they proved and conjectured that other combinations of vincular patterns are still Mahonian. These conjectures were proved later: by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In this paper we give an alternative proof of some of these results. Our approach is based on permutation codes which, like the Lehmer code, map bijectively permutations onto subexcedant sequences. More precisely, we give several code transforms (i.e., bijections…
Dimensions of random affine code tree fractals
2014
We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.
Unconditionally convergent multipliers and Bessel sequences
2016
Abstract We prove that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained, thus providing positive answers to a conjecture by Balazs and Stoeva in some particular cases.
Generalized Lebesgue points for Sobolev functions
2017
In this article, we show that a function $f\in M^{s,p}(X),$ $0<s\leq 1,$ $0<p<1,$ where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h.$