Nonnegative Coefficients Research Articles

A pseudo restricted MLE under multivariate order restrictions and its algorithm

It is shown in this paper that a quasi order for the vectors in Rp is a cone induced if and only if the order is preservable under limits and under linear combinations with non-negative coefficients. For the mean vectors in MANOVA subject to the restriction of simple ordering, a pseudo restricted MLE is proposed. This estimator is a matrix projection onto a closed convex set inside the restricted domain. An algorithm for the pseudo restricted MLE is developed, that computes the matrix projections using only vector projections.

Improved iterative oscillation tests in difference equations with several arguments

ABSTRACTAn iterative procedure is used to establish the oscillatory sufficient conditions of first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients. The conditions obtained by this method, achieve a marked improvement on all known conditions in the literature.

Local limit theorems for one class of distributions in probabilistic combinatorics

AbstractLet a functionf(z) be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radiusR. Let the distribution of the random variableξn,n∈ {1, 2, …}, be defined by the formula$$\begin{array}{} \displaystyle P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots \end{array} $$for some ∣z∣ <R(if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions ofξnare proved in two cases: a) iff(z) = (1 −z)−λ,λ= const ∈ (0, 1] for ∣z∣ < 1, and b) if all positive coefficients of expansion f (z) in a power series are equal to 1 and the setAof their numbers has the form$$\begin{array}{} \displaystyle A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}. \end{array} $$A hypothetical general local limit normal theorem for random variablesξnis stated. Some examples of validity of the statement of this theorem are given.

An asymptotically optimal Bernoulli factory for certain functions that can be expressed as power series

Given a sequence of independent Bernoulli variables with unknown parameter p, and a function f expressed as a power series with non-negative coefficients that sum to at most 1, an algorithm is presented that produces a Bernoulli variable with parameter f(p). In particular, the algorithm can simulate f(p)=pa, a∈(0,1). For functions with a derivative growing at least as f(p)∕p for p→0, the average number of inputs required by the algorithm is asymptotically optimal among all simulations that are fast in the sense of Nacu and Peres. A non-randomized version of the algorithm is also given. Some extensions are discussed.

Factorizations of polynomials with integral non-negative coefficients

We study the structure of the commutative multiplicative monoid $$\mathbb {N}_0[x]^*$$ of all the non-zero polynomials in $$\mathbb {Z}[x]$$ with non-negative coefficients. The monoid $$\mathbb {N}_0[x]^*$$ is not half-factorial and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $$\mathbb {N}_0$$ with derivations into $$\mathbb {N}_0$$ . We study ideals, chain of ideals, prime ideals and prime elements of $$\mathbb {N}_0[x]^*$$ . Our monoid $$\mathbb {N}_0[x]^*$$ is a submonoid of the multiplicative monoid of the ring $$\mathbb {Z}[x]$$ , which is a left module over the Weyl algebra $$A_1(\mathbb {Z})$$ .

Oscillation tests for difference equations with several non-monotone deviating arguments

Abstract The purpose of this paper is to derive sufficient conditions for the oscillation of all solutions of a difference equation with several non-monotone deviating arguments and nonnegative coefficients. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the significance of the results are also given.

Root system chip-firing I: interval-firing

Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle = 0$, where $\Phi^{+}$ is the set of positive roots of a root system of Type A and $\lambda$ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle \in [-k-1,k-1]$ or $\langle\lambda,\alpha^\vee\rangle \in [-k,k-1]$, for any $k \geq 0$. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of $k$ so that the number of weights with given stabilization is a polynomial in $k$. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.

Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

Let (W,S) be a finite Coxeter system with root system R and with set of positive roots R+. For α∈R, v,w∈W, we denote by ∂α, ∂w and ∂w/v the divided difference operators and skew divided difference operators acting on the coinvariant algebra of W. Generalizing the work of Liu [15], we prove that ∂w/v can be written as a polynomial with nonnegative coefficients in ∂α where α∈R+. In fact, we prove the stronger and analogous statement in the Nichols–Woronowicz algebra model for Schubert calculus on W after Bazlov [4]. We draw consequences of this theorem on saturated chains in the Bruhat order, and partially treat the question when ∂w/v can be written as a monomial in ∂α where α∈R+. In an appendix, we study related combinatorics on shuffle elements and Bruhat intervals of length two.

Hecke relations in rational conformal field theory

We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their q expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.

On the Minimizers of Energy Forms with Completely Monotone Kernel

Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with constant free term. Our main result states that minimizers are analytic and have a power series development in terms of even powers of the distance to the midpoint of the domain of definition and with nonnegative coefficients. We show moreover that our minimization problem is equivalent to the minimization of the energy functional under a nonnegativity constraint.

Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers

Let n and k be integers such that 1≤k≤n and f(x) be a nonzero polynomial of integer coefficients such that f(m)≠0 for any positive integer m. For any k-tuple s→=(s1,...,sk) of positive integers, we defineHk,f(s→,n):=∑1≤i1<⋯<ik≤n∏j=1k1f(ij)sj andHk,f⁎(s→,n):=∑1≤i1≤⋯≤ik≤n∏j=1k1f(ij)sj. If all sj are 1, then let Hk,f(s→,n):=Hk,f(n) and Hk,f⁎(s→,n):=Hk,f⁎(n). Hong and Wang refined the results of Erdös and Niven, and of Chen and Tang by showing that Hk,f(n) is not an integer if n≥4 and f(x)=ax+b with a and b being positive integers. Meanwhile, Luo, Hong, Qian and Wang established the similar result when f(x) is of nonnegative integer coefficients and of degree no less than two. For any k-tuple s→=(s1,...,sk) of positive integers, Pilehrood, Pilehrood and Tauraso proved that Hk,f(s→,n) and Hk,f⁎(s→,n) are nearly never integers if f(x)=x. In this paper, we show that if f(x) is a nonzero polynomial of nonnegative integer coefficients such that either deg⁡f(x)≥2 or f(x) is linear and sj≥2 for all integers j with 1≤j≤k, then Hk,f(s→,n) and Hk,f⁎(s→,n) are not integers except for the case f(x)=xm with m≥1 being an integer and n=k=1, in which case, both of Hk,f(s→,n) and Hk,f⁎(s→,n) are integers. Furthermore, we prove that if f(x)=2x−1, then both Hk,f(s→,n) and Hk,f⁎(s→,n) are not integers except when n=1, in which case Hk,f(s→,n) and Hk,f⁎(s→,n) are integers. The method of the proofs is analytic and p-adic.

On Two-Sided Gamma-Positivity for Simple Permutations

Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin.We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.

On a transport problem and monoids of non-negative integers

A problem about how to transport profitably a group of cars leads us to study the set $T$ formed by the integers $n$ such that the system of inequalities, with non-negative integer coefficients, $$a_1x_1 +\cdots+ a_px_p + \alpha \leq n \leq b_1x_1 +\cdots+ b_px_p - \beta$$ has at least one solution in ${\mathbb N}^p$. We will see that $T\cup\{0\}$ is a submonoid of $({\mathbb N},+)$. Moreover, we show algorithmic processes to compute $T$.

Nonnegative linearization coefficients of the generalized Bessel polynomials

In this work, we solve the general linearization problem for the generalized Bessel polynomials using their inversion formula. For some particular values, we get a recurrence relation satisfied by the linearization coefficients from which we deduce their nonnegativity. We also recover a result given by Berg and Vignat (Constr Approx 27:15–32, 2008) and derived an explicit formula that generalizes a result by Atia and Zeng (Ramanujan J 28:211–221, 2012).

Hyperplane-Approximation-Based Method for Many-Objective Optimization Problems with Redundant Objectives.

For a many-objective optimization problem with redundant objectives, we propose two novel objective reduction algorithms for linearly and, nonlinearly degenerate Pareto fronts. They are called LHA and NLHA respectively. The main idea of the proposed algorithms is to use a hyperplane with non-negative sparse coefficients to roughly approximate the structure of the PF. This approach is quite different from the previous objective reduction algorithms that are based on correlation or dominance structure. Especially in NLHA, in order to reduce the approximation error, we transform a nonlinearly degenerate Pareto front into a nearly linearly degenerate Pareto front via a power transformation. In addition, an objective reduction framework integrating a magnitude adjustment mechanism and a performance metric are also proposed here. Finally, to demonstrate the performance of the proposed algorithms, comparative experiments are done with two correlation-based algorithms, LPCA and NLMVUPCA, and with two dominance-structure-based algorithms, PCSEA and greedy MOSS, on three benchmark problems: DTLZ5(I,M), MAOP(I,M), and WFG3(I,M). Experimental results show that the proposed algorithms are more effective.

On Bergeron's Positivity Problem for $q$-Binomial Coefficients

F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$ has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for $a\le 3$ and any $b,c\ge 4$. The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem.

Long-time dynamics of the strongly damped semilinear plate equation in [formula omitted

We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in Rn, then the semigroup generated by the considered problem possesses a global attractor in H2(Rn)×L2(Rn). We also establish the boundedness of this attractor in H3(Rn)×H2(Rn).

Doubly Nonlinear Equations of Porous Medium Type

In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by $$\partial_t u^m- {\rm div}\, D_{\xi}\, f(x,Du) =0,$$ with $${m\in (0,\infty )}$$ , or more generally with an increasing and piecewise C1 nonlinearity b and a function f depending on u $$\partial_{t}b(u) - {\rm div}\, D_{\xi}\, f(x,u,Du)= -D_u f(x,u,Du).$$ For the function f we merely assume convexity and coercivity, so that, for instance, $${f(x,u,\xi)=\alpha(x)|\xi|^p + \beta(x)|\xi|^q}$$ with 1 < p < q and non-negative coefficients α, β with $${\alpha(x)+\beta(x)\geqq \nu > 0}$$ , and $${f(\xi)=\exp(\tfrac12|\xi|^2)}$$ are covered. Thus, for functions $${f(x,u,\xi )}$$ satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if $${f(x,u,\xi )}$$ grows naturally when $${\left\vert \xi \right\vert \rightarrow +\infty }$$ as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations.

On the convergence criterion for branched continued fractions with independent variables

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

Oscillation Criteria for Delay and Advanced Differential Equations with Nonmonotone Arguments

We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficients. New oscillation criteria, involving lim sup and lim inf, are obtained based on an iterative method. Examples, numerically solved in MATLAB, are given to illustrate the applicability and strength of the obtained conditions over known ones.