Nonnegative Coefficients Research Articles

Palindromic Riordan arrays, classical orthogonal polynomials and Catalan triangles

We characterize the palindromic generalized Riordan arrays and their Sheffer sequences showing that, apart from a trivial case, these arrays all arise from a given array P(q,t,u) by suitably choosing the parameters q,t,u. Remarkably, well-known polynomial and numerical sequences arise as special cases of such Sheffer sequences, including classical orthogonal polynomials. After a suitable normalization, both the palindromic Sheffer sequences and their gamma polynomials have non-negative integer coefficients. We prove that these coefficients count a family of directed graphs, then through this combinatorial setting we obtain a bijective proof of the palindromic property of P(q,t,u), and we recover the combinatorial expansion of orthogonal polynomials due to F. Bergeron. Finally, we state explicit connections between the inverse Q(q,t,u) of P(q,t,u) and an array C(q,t) which generalizes Aigner's array of ballot numbers, the Pascal triangle and the Catalan triangle of Shapiro.

A combinatorial identity for the p-binomial coefficient based on abelian groups

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $\mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underline{\lambda}$, which is a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian $p$-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in $p$ with non-negative integer coefficients.

Special cases of critical linear difference equations

In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.

Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn equation has been an open problem in recent years. This work provides a positive answer by designing and analyzing a class of up to fourth order maximum principle preserving integrators for the Allen-Cahn equation. First, the second order finite difference discretization is applied to the Allen-Cahn equation in the space direction. The obtained semi-discrete system also preserves the maximum principle and the energy dissipation law. Then the fully discrete numerical scheme is obtained by applying the Lawson transformation and the Runge-Kutta integration in the time direction. We define sufficient conditions for explicit integration factor Runge-Kutta scheme to preserve the maximum principle, namely, the Shu-Osher form of the underlying Runge-Kutta scheme has non-negative coefficients αi,j, nondecreasing abscissas ci and the time step size τ>0 satisfies τ{βi,jαi,j}∈[−4,12]. We prove that the proposed method is convergent with order O(τp+h2) in the discrete L∞ norm. A fast solver is then applied to the discrete system to accelerate numerical computations. Various experiments for 1D, 2D and 3D problems are provided to illustrate the high-order convergence and maximum principle preserving of the proposed algorithms over a long time and verify the theoretical analysis.

Piecewise polynomial curves with normalized derivatives

In this paper, a new method for constructing piecewise polynomial curves is presented. The first derivative and the second derivative of the given curve are non-negative normalized combination of the first order divided differences and non-negative combination of the second order divided differences of the control points respectively. Under the derivative constraint conditions, the given curves contain shape parameters that can locally be used to adjust a curve. The proper ranges for the adjustable shape parameters are given. The optimal value of the shape parameter is obtained for minimizing the non-negative combination coefficients of the second derivatives of the curves. Some geometric examples are provided to show the feasibility of such curve representations as well as the approximation effect to the control polygon.

Characters of 2-layered Heisenberg groups

We give a classification of irreducible representations of generalized Heisenberg groups which is the pattern group associated to the closed set In light of the conjectures of Higman, Lehrer and Isaacs for unitriangular groups, this result shows that the number of irreducible characters of with a fixed degree is a polynomial in q−1 with non-negative integer coefficients.

Iterative Oscillation Criteria in Deviating Difference Equations

The oscillation of the first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients is investigated, using an iterative procedure. The conditions obtained by this method achieve a marked improvement on all known conditions in the literature. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the results.

OSCILLATORY BEHAVIOR OF HIGHER ORDER NONLINEAR DIFFERENCE EQUATIONS

The authors present some new oscillation criteria for higher order nonlinear difference equations with nonnegative real coefficients of the form ... Both of the cases n even and n odd are considered. They give examples to illustrate their results.

Subdivision schemes on the dyadic half-line

We consider subdivision schemes, which are used for the approximation of functions and generation of curves on the dyadic half-line. In the classical case of functions on the real line, the theory of subdivision schemes is widely known because of its applications in constructive approximation theory and signal processing as well as for generating fractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-line endowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role of exponentials is played by Walsh functions. We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectral properties of matrices and in terms of the smoothness of solutions of the corresponding refinement equation. We also investigate the problem of convergence of subdivision schemes with non-negative coefficients. We obtain an explicit criterion for the convergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-line and prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.

Strongly Exponential Separation between Monotone VP and Monotone VNP

We show that there is a sequence of explicit multilinear polynomials P n (x 1 , … ,x n ) ϵ R [x 1 , … ,x n ] with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for P n must have size exp (Ω ( n )) This builds on (and strengthens) a result of Yehudayoff (STOC 2019) who showed a lower bound of exp (Ω(√n)).

The [formula omitted]-positivity of bivariate Eulerian polynomials via the Hetyei–Reiner action

The bivariate Eulerian polynomials are defined by An(p,q)=∑π∈Snpodes(π)qedes(π),where odes(π) and edes(π) are the number of descents of permutation π in odd and even positions, respectively. In this paper, by the Hetyei–Reiner action, we show that for k≥1, the bivariate Eulerian polynomials A2k+1(p,q) and (1+p)−1A2k(p,q) are γ-positive, namely, they can be expressed in terms of the basis Bn≔{(pq)i(p+q)j(1+pq)n−2i−j|i,j≥0,2i+j≤n}with nonnegative coefficients.

ИССЛЕДОВАНИЕ СОГЛАСОВАННОСТИ ЭКСПЕРТНЫХ ОЦЕНОК ПРИ ЗАВИСИМЫХ ВЫБОРКАХ

The problem of the agreement in experts’ opinions is one of the main ones in making collective decisions. The quality of the collective decision (accuracy, reliability, reliability, number of experts, time and laboriousness of decision making) depends on the degree of consistency of expert decisions. The article discusses the issues of reliability of estimates given by a group of experts for dependent samples. The dependence of random variables in samples is manifested, first of all, in testing systems. The dependence can be between the test questions, as well as between the student’s answers to the test questions on different topics. A connection can also be found between the answers of different trainees with a similar assimilation of the educational material. In the article, to determine the consistency of expert assessments, the indicator is the sum of centered values of random variables corresponding to the assigned estimates. This indicator characterizes the total scatter of expert assessments relative to the arithmetic mean of the collective assessment. In case of complete agreement of opinions, the value of the indicator is zero. The characteristics of the scatter of estimates are derived from the Chebyshev theorem. The lower and upper bounds of the estimation error are obtained. The confidence interval for the correlation coefficient is built. A formula for finding the proportion of possible deviations in the score is derived. The behavior of the boundaries of the confidence interval was investigated for different values of the point estimate. The case is considered when all correlation coefficients are non-negative. A comparison is made of the behavior of the boundaries of the confidence intervals for non-negative correlation coefficients and the general case. The results obtained can be used, for example, in testing systems for personnel training, in product quality control systems, to select promising technologies, and medical diagnostics.

The distribution of sums and products of additive functions

The celebrated Erdős--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to hold for certain non-additive functions and for amenable arithmetic functions over certain subsets of the natural numbers. Continuing in this vein, we show that if $g_1(n), \ldots, g_k(n)$ is a collection of functions satisfying certain mild hypotheses for which an Erdős--Kac-type normal distribution law holds, and if $Q(x_1, \ldots, x_k)$ is a polynomial with nonnegative real coefficients, then $Q(g_1(n), \ldots, g_k(n))$ also obeys a normal distribution law. We also show that a similar result can be obtained if the set of inputs $n$ is restricted to certain subsets of the natural numbers, such as shifted primes. Our proof uses the method of moments. We conclude by providing examples of our theorem in action.

Generalized $q,t$-Catalan numbers

Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers.

Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws

In [4], we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favorably to the inverse Lax-Wendroff method in [5,6] where analysis and numerical experiments have previously verified the presence of order reduction [5,6]. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all nonnegative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws.

Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers

The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with {n}. It is then natural to ask if the resulting rational function is actually a polynomial in s and t with nonnegative integer coefficients and, if so, what it counts. The first simple combinatorial interpretation for this polynomial analogue of the binomial coefficients was given by Sagan and Savage, although their model resisted being used to prove identities for these Lucasnomials or extending their ideas to other combinatorial sequences. The purpose of this paper is to give a new, even more natural model for these Lucasnomials using lattice paths which can be used to prove various equalities as well as extending to Catalan numbers and their relatives, such as those for finite Coxeter groups.

The Karcher mean of three variables and quadric surfaces

The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean is problematic even for 2×2 triples. In this paper we study (1) the linear formula for the Karcher mean of 2×2 positive definite Hermitian matrices: Λ(A,B,C)=xA+yB+zC with nonnegative coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem, and (2) the quadric surface induced by the determinantal formula: det⁡(ABC)13=det⁡(xA+yB+zC). We show that the solution set forms a simplex of dimension less than equal 2 and settle the first problem for linearly dependent case. A classification of the quadric surfaces from the linear form of Karcher means is presented in terms of linear (in)dependence of A,B,C: hyperboloid of two sheets, hyperbolic cylinder, and parallel planes.

Estimates of Dirichlet eigenvalues for degenerate $$\triangle _{\mu }$$-Laplace operator

Let $$\varOmega $$ be a bounded open domain in $$\mathbb {R}^n$$ and $$\triangle _{\mu }=\sum _{j=1}^{n}\partial _{x_{j}}(\mu _{j}^2(x)\partial _{x_{j}})$$ be a class of degenerate elliptic operator with continuous nonnegative coefficients $$\mu _{1},\mu _{2},\ldots ,\mu _{n}$$ . Denote by $$\lambda _{k}$$ the kth Dirichlet eigenvalue of the self-adjoint degenerate elliptic operator $$-\triangle _{\mu }$$ on $$\varOmega $$ . If the coefficients $$\mu _{1},\mu _{2},\ldots ,\mu _{n}$$ satisfy some general assumptions, we give an explicit lower bound estimate of $$\lambda _{k}$$ . Moreover, if the coefficients $$\mu _{1}\ldots ,\mu _{n}$$ are homogeneous functions with respect to a group of dilation, then we obtain an explicit sharp lower bound estimate for $$\lambda _{k}$$ , which has a polynomially growth in k of the order related to the homogeneous dimension. Finally, we also establish an upper bound estimate of $$\lambda _{k}$$ for general self-adjoint degenerate elliptic operator $$\triangle _{\mu }$$ .

Zero-one Schubert polynomials

We prove that if sigma in S_m is a pattern of w in S_n, then we can express the Schubert polynomial mathfrak {S}_w as a monomial times mathfrak {S}_sigma (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on mathfrak {S}_w being zero-one. In this case, the Schubert polynomial mathfrak {S}_w is equal to the integer point transform of a generalized permutahedron.

On the partitions into distinct parts and odd parts

In this paper, we show that the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n satisfies Euler’s recurrence relation for the partition function p(n) when n is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of n is also derived. In this context, we conjecture that for k > 0, the series has non-negative coefficients.