Nonnegative Coefficients Research Articles

Random periodic sequence of globally mean-square exponentially stable discrete-time stochastic genetic regulatory networks with discrete spatial diffusions

<abstract><p>This paper regards the dual effects of discrete-space and discrete-time in stochastic genetic regulatory networks via exponential Euler difference and central finite difference. Firstly, the global exponential stability of such discrete networks is investigated by using discrete constant variation formulation. In particular, the optimal exponential convergence rate is explored by solving a nonlinear optimization problem under nonlinear constraints, and an implementable computer algorithm for computing the optimal exponential convergence rate is given. Secondly, random periodic sequence for such discrete networks is investigated based on the theory of semi-flow and metric dynamical systems. The researching findings show that the spatial diffusions with nonnegative intensive coefficients have no influence on global mean square boundedness and stability, random periodicity of the networks. This paper is pioneering in considering discrete spatial diffusions, which provides a research basis for future research on genetic regulatory networks.</p></abstract>

Matrix Factorization With Framelet and Saliency Priors for Hyperspectral Anomaly Detection

Hyperspectral anomaly detection aims to separate sparse anomalies from low-rank background components. A variety of detectors have been proposed to identify anomalies, but most of them tend to emphasize characterizing backgrounds with multiple types of prior knowledge and limited information on anomaly components. To tackle these issues, this paper simultaneously focuses on two components and proposes a matrix factorization method with framelet and saliency priors to handle the anomaly detection problem. We first employ a framelet to characterize nonnegative background representation coefficients, as they can jointly maintain sparsity and piecewise smoothness after framelet decomposition. We then exploit saliency prior knowledge to measure each pixel’s potential to be an anomaly. Finally, we incorporate the pure pixel index with Reed-Xiaoli’s method to possess representative dictionary atoms. We solve the optimization problem by using a block successive upper-bound minimization framework with guaranteed convergence. Experiments conducted on benchmark hyperspectral datasets demonstrate that the proposed method outperforms some state-of-the-art anomaly detection methods.

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

We study the homogeneous Dirichlet problem for the equation ut-divF(z,∇u)∇u=f,z=(x,t)∈QT=Ω×(0,T),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t-{\ ext {div}}\\left( \\mathcal {F}(z,\ abla u)\ abla u\\right) =f, \\quad z=(x,t)\\in Q_T=\\Omega \ imes (0,T), \\end{aligned}$$\\end{document}where Omega subset mathbb {R}^N, is a bounded domain with partial Omega in C^2, and mathcal {F}(z,xi )=a(z)vert xi vert ^{p(z)-2}+b(z)vert xi vert ^{q(z)-2}. The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that frac{2N}{N+2}<p(z), q(z), and that the modulating coefficients and growth exponents satisfy the balance conditions a(z)+b(z)≥α>0,|p(z)-q(z)|<2N+2inQ¯T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} a(z)+b(z)\\ge \\alpha >0,\\quad \\vert p(z)-q(z)\\vert <\\frac{2}{N+2}\\hbox { in }\\overline{Q}_T \\end{aligned}$$\\end{document}with alpha =const. We find conditions on the source f and the initial data u(cdot ,0) that guarantee the existence of a unique strong solution u with u_tin L^2(Q_T) and avert nabla uvert ^{p}+bvert nabla uvert ^qin L^infty (0,T;L^1(Omega )). The solution possesses the property of global higher integrability of the gradient, |∇u|min{p(z),q(z)}+r∈L1(QT)with anyr∈0,4N+2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\vert \ abla u\\vert ^{\\min \\{p(z),q(z)\\}+r}\\in L^1(Q_T)\\quad \ ext {with any }r\\in \\left( 0,\\frac{4}{N+2}\\right) , \\end{aligned}$$\\end{document}which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The global second-order differentiability of the strong solution is proven: DiF(z,∇u)Dju∈L2(QT),i=1,2,…,N.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} D_i\\left( \\sqrt{\\mathcal {F}(z,\ abla u)}D_j u\\right) \\in L^{2}(Q_T),\\quad i=1,2,\\ldots ,N. \\end{aligned}$$\\end{document}The same results are obtained for the equation with the regularized flux mathcal {F}(z,sqrt{epsilon ^2+(xi ,xi )})xi .

A heuristic technique for decomposing multisets of non-negative integers according to the Minkowski sum

We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.

Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter b with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate b-weighting that “measures” in some sense their non-orientability.Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the “b-conjecture” of Goulden and Jackson from 1996, expansions of the β-ensemble matrix model, deformations of the HCIZ integral, and b-Hurwitz numbers that we introduce here and that are b-deformations of classical (single or double) Hurwitz numbers obtained for b=0.A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable b-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.

An Exclusion Sector for Zeros of Polynomials with Nonnegative Coefficients

Summary It is demonstrated both algebraically and geometrically that the zeros of a polynomial with nonnegative coefficients cannot lie in a sector around the positive real axis. An example shows how the restriction of nonnegativity may sometimes be circumvented. An extensive generalization of the ideas in this note can be found in Melman, Aaron (2021). Zero exclusion sectors for some polynomials with structured coefficients. Amer. Math. Monthly 128: 322–336.

Upper tails via high moments and entropic stability

Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph Gn,p. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in Gn,p. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in Gn,p for all p=p(n) satisfying n−1logn≪p≪1.

The anti-spherical category

We study a diagrammatic categorification (the “anti-spherical category”) of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localization procedure for the anti-spherical category, from which we construct a “light leaves” basis of morphisms. Our techniques may be used to calculate many new elements of the p-canonical basis in the anti-spherical module.

Infinite families of crank functions, Stanton-type conjectures, and unimodality

Dyson’s rank function and the Andrews–Garvan crank function famously give combinatorial witnesses for Ramanujan’s partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton’s conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.

ALLOCATION OF CAPITAL INVESTMENTS OF AN OIL COMPANY BASED ON INDEPENDENT OPPORTUNITY INFORMATION

To solve the problem of investment allocation with optimization of the total level of return on investment for projects under conditions of risk associated with uncertainty of project income, a linear combination with non-negative coefficients of two loss-making models was used: the model of the usual minimax loss-making and the model of the minimax loss-making level. The problem of probabilistic programming obtained as a result of this combination, provided that the probabilistic variables characterizing the profitability of investments in projects are independent (unrelated), is reduced to a linear programming problem. The solution of this problem by the simplex method gives a vector of the equity distribution of investments that determine the total return of investments by the criterion of the minimum of the loss function. On the basis of the proposed approach to the optimal solution of the problem of investment allocation, an algorithm for project investment planning has been developed using a database of invested and returned funds for projects in previous planning periods, according to which distributions of possible investment returns are modeled. The numerical implementation of the algorithm is illustrated by the example of investment planning in an oil company for oil and gas production projects. Keywords: probability programming, investments, investment portfolio, minimax criterion, loss-making function, return on investment, income, concentrated distribution, probability distribution, measure of opportunity, measure of necessity, fractile model, probability variable, fuzzy variable membership function.

Some Combinatorial Properties of Skew Jack Symmetric Functions

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew diagram. It follows that, in some special cases, the coefficients of the skew Jack symmetric functions with respect to the basis of the monomial symmetric functions are polynomials with nonnegative integer coefficients.

M-dissipative boundary conditions and boundary tuples for Maxwell operators

For Maxwell operators ▪ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted to the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a translation of the problem to operator-theoretic settings of abstract Maxwell operators. In particular, we show that Calkin reduction operators are naturally connected with Leontovich boundary conditions and provide an abstract version of impedance boundary condition applicable to other types of wave equations. Taking Friedrichs and Krein-von Neumann extensions of related boundary operators, it is possible to associate m-dissipative Maxwell operators to arbitrary non-negative measurable impedance coefficients.

Integral Operators Induced by Symbols with Non-negative Maclaurin Coefficients Mapping into H^infty

For analytic functions g on the unit disk with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta from a space X of analytic functions in the unit disk to H^infty , in terms of neat and useful conditions on the Maclaurin coefficients of g. The choices of X that will be considered contain the Hardy and the Hardy–Littlewood spaces, the Dirichlet-type spaces D^p_{p-1}, as well as the classical Bloch and mathord {mathrm{BMOA}} spaces.

Linear recurrent relations, power series distributions, and generalized allocation scheme

Abstract We consider distributions of the power series type determined by the generating functions of sequences satisfying linear recurrence relations with nonnegative coefficients. These functions are represented by power series with positive radius of convergence. An integral limit theorem is proved on the convergence of such distributions to the exponential distribution. For the generalized allocation scheme generated by these linear relations a local normal theorem for the total number of components is proved. As a consequence of more general results of the author, a limit theorem is stated containing sufficient conditions under which the distributions of the number of components having a given volume converge to the Poisson distribution.

On the Hopf algebra of multi-complexes

We introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of multi-complexes which is defined as the algebra which has a formal basis $${\mathscr {C}}$$ of all isomorphism types of multi-complexes, and multiplication is to take the disjoint union. This is a Hopf algebra with an operation encoding the disassembly information for such objects and extends the Hopf algebra of graphs. In our main result, we explicitly describe here the structure of this Hopf algebra of multi-complexes H. We find an explicit basis $${\mathscr {B}}$$ of the space of primitives, which is of combinatorial relevance: it is such that each multi-complex is a polynomial with non-negative integer coefficients of the elements of $${\mathscr {B}}$$ , and each $$b\in {\mathscr {B}}$$ is a polynomial with integer coefficients in $${\mathscr {C}}$$ . Using this, we find the grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of sub-multi-complexes in a multi-complex. We also explicitly illustrate how our results specialize to the graph Hopf algebra, and observe how they specialize to results in all of the above-mentioned particular cases. We also investigate applications of these results to the graph reconstruction conjectures and rederive some results in the literature on these questions.

Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments

In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving \(\lim \sup\), which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions.

The p-Frobenius and p-Sylvester numbers for Fibonacci and Lucas triplets.

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p $-Frobenius number by $ g_p (a_1, a_2, \dots, a_l) $, which is the largest integer that can be represented at most $ p $ ways by a linear combination with nonnegative integer coefficients of $ a_1, a_2, \dots, a_l $. When $ p = 0 $, the $ 0 $-Frobenius number is the classical Frobenius number. When $ l = 2 $, the $ p $-Frobenius number is explicitly given. However, when $ l = 3 $ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when $ p > 0 $, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers [1] or of repunits [2] for the case where $ l = 3 $. In this paper, we show the explicit formula for the Fibonacci triple when $ p > 0 $. In addition, we give an explicit formula for the $ p $-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $ p $ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

Self-products of rationally elliptic spaces and inequalities between the ranks of homotopy and homology groups

We survey recent results on inequalities between the ranks of homotopy and cohomology groups (resp., graded components of mixed Hodge structures on these groups) for rationally elliptic spaces (resp., quasi-projective varieties which are rationally elliptic). We also discuss a refinement of these results describing a new invariant of rationally elliptic spaces allowing to compare the ranks of homotopy and homology groups. Its definition is based of the study of the range of parameter r such that rP(t)<Q(t)r for all t≥ε, where (P(t),Q(t)) is a pair of arbitrary polynomials with non-negative integer coefficients. We show that this range is related to the classical Lambert W-function W(z).

Polynomials that preserve nonnegative matrices

In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London ((1978/1979) [8]) posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If Pn denotes the set of all polynomials that preserve all n-by-n nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to Pn. However, it is known that Pn contains polynomials with negative entries. In this work, novel results for Pn with respect to the coefficients of the polynomials belonging to Pn. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.