We outline here a proof that a certain rational function C(n)(q, t), which has come to be known as the "q, t-Catalan," is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because C(n)(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection (n) of Dyck paths Pi of le...

in this paper, we present some inequalities for the co-pi index involving the some topological indices, the number of vertices and edges, and the maximum degree. after that, we give a result for trees. in addition, we give some inequalities for the largest eigenvalue of the co-pi matrix of g.

For i = 1,2, ••• ,n, pi shall denote the length of the mesh interval [xi-i,Xi\. Let p = m.sxi<i<npi and p' = mini<i<n p{. P is said to be a uniform mesh if pi is a constant for all i. Throughout, h will represent a given positive real number. Consider a real function s(x, h) defined over [a,6] which is such that its restriction Si on [XÌ-I,XÌ] is a polynomial of degree 3 or less for i = 1,2, • ...

Let S ⊂ Rk+m be a compact semi-algebraic set defined by P1 ≥ 0, . . . , P` ≥ 0, where Pi ∈ R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ `. Let π denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of π(S) is bounded by (k +m). We also present an algorithm for computing the the first q Betti numbers of π(S), whose com...

We consider the question of finding deep cuts from a model with two rows of the type PI = {(x, s) ∈ Z×R+ : x = f + Rs}. To do that, we show how to reduce the complexity of setting up the polar of conv(PI) from a quadratic number of integer hull computations to a linear number of integer hull computations. Furthermore, we present an algorithm that avoids computing all integer hulls. A polynomial...

In 1996 Goulden and Jackson introduced a family of coefficients $( c_{\pi, \sigma}^{\lambda} ) $ indexed by triples partitions which arise in the power sum expansion some Cauchy for Jack symmetric functions $(J^{(\alpha )}_\pi )$. The can be viewed as an interpolation between structure constants class algebra double coset algebra. suggested that are polynomials variable $\beta := \alpha-1$ with...

Let P := {x∈Rn: Ax6 b} be a polyhedron and PI its integral hull. A Chv atal–Gomory (CG) cut is a valid inequality for PI of the form ( A)x6 b , with ∈R+; TA∈ Z and b ∈ Z . We give a polynomial-time algorithm which, given some x∗ ∈P, detects whether a totally tight CG cut exists, i.e., whether there is a CG cut such that ( TA)x∗= b. Such a CG cut is violated by as much as possible under the assu...

The development of a Self-Tuning Neuro-Fuzzy Generalized Minimum Variance (GMV) controller is described. It uses fuzzy expert knowledge of the dynamic weightings to meet desired closed-loop stability and performance requirements. The controller is formulated in a polynomial system approach mixed with a Neuro-Fuzzy model and Fuzzy Self-Tuning mechanism. The proposed method is applied to a model ...

For 1 i k, let Ri denote pi(y)Fi + Gi , where pi(y) is a polynomial in y with integer coefficients, and Fi,Gi are linear polynomials in x1, . . . , xn with integer coefficients. Let P(z1, . . . , zk) be a Presburger relation over the nonnegative integers. We show that the following problem is decidable: Given: R1, . . . , Rk and a Presburger relation P. Question:Are there nonnegative integer va...