Nonnegative Coefficients Research Articles

Strategic behavior, social optimization, and revenue maximization in queues with infinite servers

Motivated by the impact of emerging technologies on (toll) parks, this paper studies a problem of customers’ strategic behavior, social optimization, and revenue maximization for infinite-server queues. More specifically, we assume that a customer’s utility consists of a positive reward for receiving service minus a cost caused by the other customers in the system. In the observable setting, we show the existence, uniqueness, and expressions of the individual equilibrium threshold, the socially optimal threshold, and the optimal revenue threshold, respectively. Then, we prove that the optimal revenue threshold is smaller than the socially optimal threshold, which is smaller than the individual one. Furthermore, we also extend the cost functions to any finite polynomial function with nonnegative coefficients. In the unobservable setting, we derive the joining probabilities of individual equilibrium and optimal revenue. Finally, using numerical experiments, we complement our results and compare the social welfare and the revenue under these two information levels.

The volume polynomial of lattice polygons

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

Longitudinal Relationship Between Brain Atrophy Patterns, Cognitive Decline, and Cerebrospinal Fluid Biomarkers in Alzheimer's Disease Explored by Orthonormal Projective Non-Negative Matrix Factorization.

Longitudinal magnetic resonance imaging (MRI) has been proposed for tracking the progression of Alzheimer's disease (AD) through the assessment of brain atrophy. Detection of brain atrophy patterns in patients with AD as the longitudinal disease tracker. We used a refined version of orthonormal projective non-negative matrix factorization (OPNMF) to identify six distinct spatial components of voxel-wise volume loss in the brains of 83 subjects with AD from the ADNI3 cohort relative to healthy young controls from the ABIDE study. We extracted non-negative coefficients representing subject-specific quantitative measures of regional atrophy. Coefficients of brain atrophy were compared to subjects with mild cognitive impairment and controls, to investigate the cross-sectional and longitudinal associations between AD biomarkers and regional atrophy severity in different groups. We further validated our results in an independent dataset from ADNI2. The six non-overlapping atrophy components represent symmetric gray matter volume loss primarily in frontal, temporal, parietal and cerebellar regions. Atrophy in these regions was highly correlated with cognition both cross-sectionally and longitudinally, with medial temporal atrophy showing the strongest correlations. Subjects with elevated CSF levels of TAU and PTAU and lower baseline CSF Aβ42 values, demonstrated a tendency toward a more rapid increase of atrophy. The present study has applied a transferable method to characterize the imaging changes associated with AD through six spatially distinct atrophy components and correlated these atrophy patterns with cognitive changes and CSF biomarkers cross-sectionally and longitudinally, which may help us better understand the underlying pathology of AD.

Multigraded algebras and multigraded linear series

AbstractThis paper is devoted to the study of multigraded algebras and multigraded linear series. For an ‐graded algebra , we define and study its volume function , which computes the asymptotics of the Hilbert function of . We relate the volume function to the volume of the fibers of the global Newton–Okounkov body of . Unlike the classical case of standard multigraded algebras, the volume function is not a polynomial in general. However, in the case when the algebra has a decomposable grading, we show that the volume function is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

Khinchin Families, Set Constructions, Partitions and Exponentials

In this paper, we give a simple criterion to verify that functions of the form eg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^g$$\\end{document} are in the Hayman class when g is a power series with nonnegative coefficients. Thus, using the Hayman and Báez-Duarte formulas, we obtain asymptotics for the coefficients of generating functions that arise in many examples of set construction in analytic combinatorics. This new criterion greatly simplifies the one obtained previously by the authors.

Multivariable versions of a lemma of Kaluza's

AbstractLet and be a convergent multivariable power series in . In this paper, we present two conditions on the positive coefficients that imply that for nonnegative coefficients . If , then both of our results reduce to a lemma of Kaluza's. For , we present examples to show that our two conditions are independent of one another. It turns out that functions of the type satisfy one of our conditions, whenever is a product of probability measures on . Our results have applications to the theory of Nevanlinna–Pick kernels.

Positive rigs

Abstract A positive rig is a commutative and unitary semi-ring A such that 1 + x {1+x} is invertible for every x ∈ A {x\in A} . We show that the category of positive rigs shares many properties with that of K-algebras for a (non-algebraically closed) field K. In particular, it is coextensive and, although we do not have an analogue of Hilbert’s basis theorem for positive rigs, we show that every finitely presentable positive rig is a finite direct product of directly indecomposable ones. We also describe free positive rigs as rigs of rational functions with non-negative rational coefficients, and we give a characterization of the positive rigs with a unique maximal ideal.

Polynomiality of the faithful dimension for nilpotent groups over finite truncated valuation rings

Given a finite group G \mathrm {G} , the faithful dimension of G \mathrm {G} over C \mathbb {C} , denoted by m f a i t h f u l ( G ) m_\mathrm {faithful}(\mathrm {G}) , is the smallest integer n n such that G \mathrm {G} can be embedded in G L n ( C ) \mathrm {GL}_n(\mathbb {C}) . Continuing the work initiated by Bardestani et al. [Compos. Math. 155 (2019), pp. 1618–1654], we address the problem of determining the faithful dimension of a finite p p -group of the form G R ≔ exp ⁡ ( g R ) \mathscr {G}_R≔\exp (\mathfrak {g}_R) associated to g R ≔ g ⊗ Z R \mathfrak {g}_R≔\mathfrak {g}\otimes _\mathbb {Z}R in the Lazard correspondence, where g \mathfrak {g} is a nilpotent Z \mathbb {Z} -Lie algebra and R R ranges over finite truncated valuation rings. Our first main result is that if R R is a finite field with p f p^f elements and p p is sufficiently large, then m f a i t h f u l ( G R ) = f g ( p f ) m_\mathrm {faithful}(\mathscr {G}_R)=fg(p^f) where g ( T ) g(T) belongs to a finite list of polynomials g 1 , … , g k g_1,\ldots ,g_k , with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra g \mathfrak {g} . Furthermore, for each 1 ≤ i ≤ k 1\le i\leq k the set of pairs ( p , f ) (p,f) for which g = g i g=g_i is a finite union of Cartesian products P × F \mathscr P\times \mathscr F , where P \mathscr P is a Frobenius set of prime numbers and F \mathscr F is a subset of N \mathbb N that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) was only established under the assumption that either f = 1 f=1 or p p is fixed. Next we formulate a conjectural polynomiality property for the value of m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) in the more general setting where R R is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras g \mathfrak {g} that are defined by partial orders, m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) is given by a single polynomial-type formula. Finally, we compute m f a i t h f u l ( G R ) m_\mathrm {faithful}(\mathscr {G}_R) precisely in the case where g \mathfrak {g} is the free metabelian nilpotent Lie algebra of class c c on n n generators and R R is a finite truncated valuation ring.

An Extension of the Perplexing Polynomial Puzzle

Summary Originally published as a puzzle in 2005 [3], the Perplexing Polynomial Puzzle indeed is perplexing: any given polynomial p ( x ) with nonnegative integer coefficients can be completely determined by just two evaluations. In this article, an extension is made to polynomials with arbitrary integer coefficients, by considering a simple translation x ↦ x + k with k ∈ N such that the result is a new polynomial with only nonnegative integer coefficients on which the original solution can be used. A proof is given that this is indeed always possible, and a method is constructed to determine a suitable k to do so.

Weighted Pseudo-θ-Almost Periodic Sequence and Finite-Time Guaranteed Cost Control for Discrete-Space and Discrete-Time Stochastic Genetic Regulatory Networks with Time Delays

This paper considers the dual hybrid effects of discrete-time stochastic genetic regulatory networks and discrete-space stochastic genetic regulatory networks in difference formats of exponential Euler difference and second-order central finite difference. The existence of a unique-weight pseudo-θ-almost periodic sequence solution for discrete-time and discrete-space stochastic genetic regulatory networks on the basis of discrete constant variation formulation is discussed, as well as the theory of semi-flow and metric dynamical systems. Furthermore, a finite-time guaranteed cost controller is constructed to reach global exponential stability of these discrete networks via establishing a framework of drive, response, and error networks. The results indicate that spatial diffusions of non-negative dense coefficients have no influence on the global existence of the unique weighted pseudo-θ-almost periodic sequence solution of the networks. The present study is a basic work in the consideration of discrete spatial diffusion in stochastic genetic regulatory networks and serves as a foundation for further study.

Bounds for invariants of numerical semigroups and Wilf’s conjecture

Given coprime positive integers g_1< ldots < g_e, the Frobenius number F=F(g_1,ldots ,g_e) is the largest integer not representable as a linear combination of g_1,ldots ,g_e with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F+1 le e n. We provide bounds for g_1 and for the type of the numerical semigroup S=langle g_1,ldots ,g_e rangle in function of e and n, and use these bounds to prove that F+1 le q e n, where q= Bigg lceil frac{F+1}{g_1} Bigg rceil , and F+1 le e n^2. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup S=langle g_1,ldots ,g_e rangle is almost-symmetric.

Positivity preservers forbidden to operate on diagonal blocks

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at “opposite ends”, and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified “diagonal/principal blocks”, in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.

Volterra-type operators mapping weighted Dirichlet space into H^infty

The problem of describing the analytic functions g on the unit disc such that the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H^infty is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of T_g acting from a weighted Dirichlet space D^p_omega , induced by an upper doubling weight omega , to H^infty . We also characterize, in terms of neat conditions on omega , the upper doubling weights for which T_g: D^p_omega rightarrow H^infty is bounded (or compact) only if g is constant.

ADCoC: Adaptive Distribution Modeling Based Collaborative Clustering for Disentangling Disease Heterogeneity from Neuroimaging Data.

Conventional clustering techniques for neuroimaging applications usually focus on capturing differences between given subjects, while neglecting arising differences between features and the potential bias caused by degraded data quality. In practice, collected neuroimaging data are often inevitably contaminated by noise, which may lead to errors in clustering and clinical interpretation. Additionally, most methods ignore the importance of feature grouping towards optimal clustering. In this paper, we exploit the underlying heterogeneous clusters of features to serve as weak supervision for improved clustering of subjects, which is achieved by simultaneously clustering subjects and features via nonnegative matrix tri-factorization. In order to suppress noise, we further introduce adaptive regularization based on coefficient distribution modeling. Particularly, unlike conventional sparsity regularization techniques that assume zero mean of the coefficients, we form the distributions using the data of interest so that they could better fit the non-negative coefficients. In this manner, the proposed approach is expected to be more effective and robust against noise. We compared the proposed method with standard techniques and recently published methods demonstrating superior clustering performance on synthetic data with known ground truth labels. Furthermore, when applying our proposed technique to magnetic resonance imaging (MRI) data from a cohort of patients with Parkinson's disease, we identified two stable and highly reproducible patient clusters characterized by frontal and posterior cortical/medial temporal atrophy patterns, respectively, which also showed corresponding differences in cognitive characteristics.

Conjectural Positivity of Chern–Schwartz–MacPherson Classes for Richardson Cells

Abstract Following some work of Aluffi–Mihalcea–Schürmann–Su for the CSM classes of Schubert cells and some elaborate computer calculations by Rimanyi and Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square $ coming from the Segre-SM classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture that the structure constants of this new product $\square $ in the standard Schubert basis have alternating sign behavior. I prove that this conjecture on the sign of the structure constants of $\square $ would follow from my above positivity conjecture about the CSM classes of Richardson cells.

Nonmodular infinite products and a conjecture of Seo and Yee

We will tackle a conjecture of S. Seo and A. J. Yee, which says that the series expansion of 1/(q,−q3;q4)∞ has nonnegative coefficients. Our approach relies on an approximation of the generally nonmodular infinite product 1/(qa;qM)∞, where M is a positive integer and a is any of 1,2,…,M.

Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws

The numerical solution of hyperbolic conservation laws requires algorithms with upwind characteristics. Conventional methods such as the numerical difference method can realize this characteristic by constructing special distributions of nodes. However, there are still no reports on how to construct algorithms with upwind characteristics through wavelet theory. To solve this problem, a system of high-order and stable wavelet collocation upwind schemes was successfully proposed by constructing interpolation wavelets with specific symmetry and smoothness. The effects of the characteristics of the scaling functions on the schemes were explored based on numerical tests and Fourier analysis. The numerical results revealed that the stability of the constructed scheme is affected by the smoothness order, N, and the asymmetry of the scaling function. The dissipation analysis suggested that schemes with N ∈ even have negative dissipation coefficients, leading to unstable behaviors. Only scaling functions with N ∈ odd and a bias magnitude of 1 can be used to construct stable upwind schemes due to the non-negative dissipation coefficients. Typical numerical examples verified the effectiveness of the proposed method, which is proved to have high accuracy and efficiency in solving high-speed flow problems with multi-scale smooth structures and discontinuities.

Formula omitted]-partition power sums

We develop the theory of weighted P-partitions, which generalises the theory of P-partitions from labelled posets to weighted labelled posets. We define the related generating functions in the natural way and compute their product, coproduct and other properties. As an application we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions and the reverse basis, both of which refine the power sum symmetric functions. These bases share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and moreover expand into the monomial basis of quasisymmetric functions with nonnegative integer coefficients. We prove formulas for products, coproducts and classical quasisymmetric involutions via the combinatorics of P-partitions, and give combinatorial interpretations for the coefficients when expanded into the monomial and fundamental bases.

The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for p=0 and by Komatsu and Ying for the general non-negative integer p.

THE GENERAL KOVALEVSKAYA THEOREM FOR A CAUCHY PROBLEM

The Kovalevskaya theorem is a well-known mathematical result that provides a powerful tool for solving systems of ordinary differential equations (ODEs). However, there has been a longstanding question regarding whether the theorem can be extended to more general systems of partial differential equations (PDEs). In this paper, we address this question by presenting a discussion of the only general Kovalevskaya theorem that can be extended from the theory of ODEs to systems of PDEs. Our discussion begins by reviewing the classical Kovalevskaya theorem for ODEs and its various applications in mathematics and physics. We then introduce the concept of Frobenius manifold, which provides a framework for extending the Kovalevskaya theorem to PDEs. We show how the Frobenius manifold structure can be used to transform a system of PDEs into a system of ODEs, which can then be solved using the classical Kovalevskaya theorem. Overall, our discussion provides a novel perspective on the Kovalevskaya theorem and its potential applications in PDE theory. By extending the theorem to this broader class of equations, we open up new possibilities for solving complex problems in mathematics, physics, and engineering.