Nonnegative Reals Research Articles

Multiplicative arithmetic functions and the generalized Ewens measure

Random integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from Sn. These similarities include the Erdős–Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley’s theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions.Given a multiplicative function α: ℕ → ℝ≥0, one may associate with it a measure on the integers in [1, x], where n is sampled with probability proportional to the value α(n). Analogously, given a sequence {θi}i≥1 of non-negative reals, one may associate with it a measure on Sn that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure.We study the case where the mean value of α over primes tends to some positive θ, as well as the weights α(p) ≈ (log p)γ. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.

Maximal chain-based Choquet-like integrals

The original idea of the discrete Choquet integral on nonnegative reals is based on a distinguished permutation making the inputs reordering nonincreasingly or nondecreasingly. This permutation generates the weights from the given capacity, and then a weighted arithmetic mean yields the value of the corresponding Choquet integral. Since the permutation relates to a maximal chain, we introduce a concept of MCC-integrals using the setting of maximal chains on the power set of a finite set. This approach provides a unified framework for several integrals of nonnegative vectors with respect to games. We study basic properties and various representations of these MCC-integrals subsuming known results for the Choquet integral as a special case. For supermodular games it is shown that the Choquet integral is the minimum of all product-based MCC-integrals. Further, we discuss symmetric and asymmetric extensions for real-valued inputs providing new insight into the construction of fuzzy integral quadruplets. During the way, we point out certain inaccuracies in the literature.

BAR-CODES OF SIERPIŃSKI RELATIVES WITH TRIANGLE CONVEX HULLS

This paper presents results about the triangle Sierpiński relatives. These are Sierpiński relatives that have the same convex hull (whose boundary is a right isosceles triangle) as the Sierpiński gasket. In general, the Sierpiński relatives all have the same fractal dimension but different topologies. The special subset of triangle relatives includes are all both path-connected and multiply-connected. We investigate the epsilon hulls of the relatives (sets of all points within a distance of [Formula: see text] to the relative) to characterize and compare. This analysis includes the topological bar-codes which convey information about the connectivity of the [Formula: see text]-hulls of the relatives as [Formula: see text] ranges over the non-negative reals. We also prove that the growth rate of holes in the [Formula: see text]-hulls is equal to the fractal dimension.

The Walsh-Fourier transform on the real line

The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.

Universal points in the asymptotic spectrum of tensors

Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiringX\mathcal {X}of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements ofX\mathcal {X}in terms ofthe asymptotic spectrum ofX\mathcal {X}, which is defined as the collection of semiring homomorphisms fromX\mathcal {X}to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information.Universal spectral pointsare elements in the asymptotic spectrum of the semiring ofalltensors. Finding all universal spectral points suffices to find the asymptotic spectrum ofanysub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, calledquantum functionals. We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes.

Two-weight L^p \to L^q bounds for positive dyadic operators in the case 0

Let $\sigma$, $\omega$ be measures on $\mathbb{R}^d$, and let $\{\lambda_Q\}_{Q\in\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\mathcal{D}$ of dyadic cubes in $\mathbb{R}^d$. We characterize the two-weight norm inequality, \begin{equation*} \lVert T_\lambda(f\sigma)\rVert_{L^q(\omega)}\le C \, \lVert f \rVert_{L^p(\sigma)}\quad \text{for every $f\in L^p(\sigma)$,} \end{equation*} for the positive dyadic operator \begin{equation*} T_\lambda(f\sigma):= \sum_{Q\in \mathcal{D}} \lambda_Q \, \Big(\frac{1}{\sigma(Q)} \int_Q f\mathrm{d}\sigma\Big) \, 1_Q \end{equation*} in the difficult range $0<q<1 \le p<\infty$ of integrability exponents. This range of the exponents $p, q$ appeared recently in applications to nonlinear PDE, which was one of the motivations for our study. Furthermore, we introduce a scale of discrete Wolff potential conditions that depends monotonically on an integrability parameter, and prove that such conditions are necessary (but not sufficient) for small parameters, and sufficient (but not necessary) for large parameters. Our characterization applies to Riesz potentials $I_\alpha (f \sigma) = (-\Delta)^{-\frac{\alpha}{2}} (f\sigma) $ ($0<\alpha<d$), since it is known that they can be controlled by model dyadic operators. The weighted norm inequality for Riesz potentials in this range of $p, q$ has been characterized previously only in the special case where $\sigma$ is Lebesgue measure.

Some reduction operations to pairwise compatibility graphs

A graph G=(V,E) with a vertex set V and an edge set E is called a pairwise compatibility graph (PCG, for short) if there are a tree T whose leaf set is V, a non-negative edge weight w in T, and two non-negative reals dmin≤dmax such that G has an edge uv∈E if and only if the sum of the edge weights in the unique path between u and v in the weighted tree (T,w) is in the interval [dmin,dmax]. PCG is a relatively new graph class motivated from bioinformatics. In this paper, we give some necessary conditions and sufficient conditions for a graph to be in PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete k-partite graphs, cacti, and some other graph classes are subsets of PCG.

On the Duffin-Schaeffer conjecture

Let $\psi: \mathbb{N}\to \mathbb{R}_{\ge 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q| \le \psi(q)/q$. If $\sum_{q=1}^\infty \psi(q)\varphi(q)/q = \infty$, we show that $\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha-a/q| \le \psi(q)/q)$, giving a refinement of Khinchin's Theorem.

Log-epsilon-skew normal: A generalization of the log-normal distribution

The log-normal distribution is widely used to model non-negative data in many areas of applied research. In this paper, we introduce and study a family of distributions with non-negative reals as support and termed the log-epsilon-skew normal (LESN) which includes the log-normal distributions as a special case. It is related to the epsilon-skew normal developed in Mudholkar and Hutson (2000) the way the log-normal is related to the normal distribution. We study its main properties, hazard function, moments, skewness and kurtosis coefficients, and discuss maximum likelihood estimation of model parameters. We summarize the results of a simulation study to examine the behavior of the maximum likelihood estimates, and we illustrate the maximum likelihood estimation of the LESN distribution parameters to two real world data sets.

Category Theory for Autonomous and Networked Dynamical Systems.

In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin–Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szüsz’s inhomogeneous (1958) version of Khintchine’s Theorem (1924). For example, given any real sequence [Formula: see text], we build a divergent series of non-negative reals [Formula: see text] such that for any [Formula: see text], almost no real number is inhomogeneously [Formula: see text]-approximable with inhomogeneous parameter [Formula: see text]. Furthermore, given any second sequence [Formula: see text] not intersecting the rational span of [Formula: see text], and assuming a dynamical version of Erdős’ Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously [Formula: see text]-approximable with any inhomogeneous parameter [Formula: see text]. Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin–Schaeffer Conjecture and Duffin–Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher’s Zero-One Law (1961) for inhomogeneous approximation by reduced fractions.

The rise and fall of V-functors

In this article we study function spaces (rise) and descent (fall) in quantale-enriched categories, paying particular attention to enrichment in the non-negative reals, the quantale of distribution functions and the unit interval equipped with a continuous t-norm.

A note on square roots of nonnegative matrices

The max-times algebra is the set R+ of nonnegative reals with operations ⊕:(a,b)→max⁡{a,b} and ⊙:(a,b)→ab. We discuss the property of matrices to be squares of max-times or conventional nonnegative matrices. We prove that there exists a matrix having a conventional nonnegative square root but no max-times square root. Also, we present a set S of cardinality three for which there is a nonnegative matrix M with spectrum S, and every such M has both conventional and max-times square roots. These results answer two questions from the recent paper by Tam and Huang.

First-Order Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints

In this paper we consider a mathematical program with second-order cone complementarity constraints (SOCMPCC). The SOCMPCC generalizes the mathematical program with complementarity constraints (MPCC) in replacing the set of nonnegative reals by second-order cones. There are difficulties in applying the classical Karush--Kuhn--Tucker (KKT) condition to the SOCMPCC directly since the usual constraint qualification such as Robinson's constraint qualification never holds if it is considered as an optimization problem with a convex cone constraint. Using various reformulations and recent results on the exact formula for the proximal/regular and limiting normal cone, we derive necessary optimality conditions in the forms of the strong, Mordukhovich, and Clarke (resp., S, M, and C) stationary conditions under certain constraint qualifications. We also show that, unlike the MPCC, the classical KKT condition of the SOCMPCC is in general not equivalent to the S-stationary condition unless the dimension of each seco...

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $$c \left( {\begin{array}{c}|V(H)| r-1\end{array}}\right) $$c|V(H)|r-1 has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd?s and Simonovits. One interesting problem, first proposed by ?uczak and Thomass? and then solved by Allen, B?ttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turan density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobas---Erd?s graphs defined earlier by the authors.

A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.

Superadditive and subadditive transformations of integrals and aggregation functions

We propose the concepts of superadditive and of subadditive transformations of aggregation functions acting on non-negative reals, in particular of integrals with respect to monotone measures. We discuss special properties of the proposed transforms and links between some distinguished integrals. Superadditive transformation of the Choquet integral, as well as of the Shilkret integral, is shown to coincide with the corresponding concave integral recently introduced by Lehrer. Similarly the transformation of the Sugeno integral is studied. Moreover, subadditive transformation of distinguished integrals is also discussed.

An intersection theorem for systems of finite sets

For nonnegative reals ω, ψ and natural t≤k≤(n+t−1)/2, the maximum of ω|A∩([n]k)|+ψ|A∩([n]n+t−1−k)| among all t-intersecting set systems A⊆2[n] is determined.

On the Chromatic Thresholds of Hypergraphs

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

A short proof of the Schröder–Simpson Theorem

We give a short and elementary proof of the Schröder–Simpson Theorem, which states that the space of all continuous maps from a given space X to the non-negative reals with their Scott topology is the cone-theoretic dual of the probabilistic powerdomain on X.