Nonnegative Integer Parameter Research Articles

On Stancu operators depending on a non-negative integer

In this paper, we deal with Stancu operators which depend on a non-negative integer parameter. Firstly, we define Kantorovich extension of the operators. For functions belonging to the space Lp [0, 1] , 1 ? p < ?, we obtain convergence in the norm of Lp by the sequence of Stancu-Kantorovich operators, and we give an estimate for the rate of the convergence via first order averaged modulus of smoothness. Moreover, for the Stancu operators; we search variation detracting property and convergence in the space of functions of bounded variation in the variation seminorm.

Гафниан некоторых трехпараметрических теплицевых матриц и совершенные паросочетания дуговых и хордовых диаграмм

We obtain an explicit formula for exact calculatingthe hafnian of 3-parameter Toeplitz matrices of aspecial type in polynomial time. We also give anasymptotic estimate for the hafnian of this type ofmatrices. Separately, we consider a case of non-negativeinteger parameters, when calculating the hafnianis equivalent to enumerating perfect matchings of arc and chord diagrams.

Generalizations of the Feline and Texas Chainsaw Josephus Problems

We define and study the Extended Feline Josephus Game, a game in which n players, each with ℓ lives, stand in a circle. The game proceeds by alternating between hitting k consecutive players—each of whom will consequently lose a life—and skipping s consecutive players. This cycle continues until every player except one loses all of their lives. Given the nonnegative integer parameters n, k, s and ℓ, the goal of the game is to identify the surviving player. In this paper, we show how the defining parameters n, k, s, and ℓ affect the survivor of games with specific constraints on those parameters and our main results provide new closed formulas to determine the survivor of these Extended Feline Josephus Games. Moreover, for cases where these formulas do not apply, we provide recursive formulas for reducing the initial game to other games with smaller parameter values. For the interested reader, we present a variety of directions for future work in this area, including an extension which considers players lying on a general graph, rather than on a circle.

Counting all self-avoiding walks on a finite lattice strip of width one and two

In this paper, a closed-form expression for counting all SAWs, irrespective of length, but restricted to the finite lattice strip $\{ -a,\ldots ,0,\ldots ,b\}\times \{0,1\}$, shall be obtained in terms of the non-negative integer parameters $a$ and $b$. In addition, the argument used to prove this result will be extended to establish an enumerating formula for counting all SAWs, irrespective of length, but restricted to the half-finite lattice strip of width two $\{ 0,1,\ldots ,n\}\times \{ 0,1,2\}$, in terms of $n$.

Pluriassociative algebras II: The polydendriform operad and related operads

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of dendriform algebras, called γ-polydendriform algebras, so that 1-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the γ-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, γ-polydendriform algebras seem adapted structures to split associative operations into 2γ operation so that some partial sums of these operations are associative. We provide a complete study of the γ-polydendriform operads, the underlying operads of the category of γ-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.

Lowest-dimensional example on non-universality of generalized Inönü–Wigner contractions

We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found.

Trees and Meta-Fibonacci Sequences

For $k>1$ and nonnegative integer parameters $a_p, b_p$, $p = 1..k$, we analyze the solutions to the meta-Fibonacci recursion $C(n)=\sum_{p=1}^k C(n-a_p-C(n-b_p))$, where the parameters $a_p, b_p$, $p = 1..k$ satisfy a specific constraint. For $k=2$ we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include the Conolly recursion and all of its generalizations that have been studied to date. We show that the solutions for all the recursions defined by the parameters in these families have a natural combinatorial interpretation: they count the number of labels on the leaves of certain infinite labeled trees, where the number of labels on each node in the tree is determined by the parameters. This combinatorial interpretation enables us to determine various new results concerning these sequences, including a closed form, and to derive asymptotic estimates. Our results broadly generalize and unify recent findings of this type relating to certain of these meta-Fibonacci sequences. At the same time they indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.

Generalized cluster complexes and Coxeter combinatorics

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

Evaluation of the remainder in an approximation formula of D. D. Stancu

Abstract:In a paper published in 1983 in this journal [6] D. D. Stancu introduced and investigated a positive linear operator , of Bernstein type, depending on a non-negative integer parameter r and two real parameters α and β, such that 0 ≤α≤β. Special attention was given to the case of the operator . This operator enjoys the variation diminishing property in the sense of I. J. Schoenberg [3]. He evaluated the orders of approximation, the remainder and the point spectrum of this operator. In the present paper we evaluate the remainder of the approximation formula of a function by means of the general operator of D.–D. Stancu, by making use of first and second order divided differences.

The remainder in the approximation by a generalized Bernstein operator: a representation by a convex combination of second-order divided differences

We investigate the remainder of the approximation formula of a function f∈C[0,1] by means of a generalized Bernstein operator (1.3), depending on two nonnegative integer parameters, introduced by the author in 1984 in the paper [11]. The remainder is expressed in (2.1)–(2.2) by a formula generalizing the author's earlier representation (1.2) of the remainder in Bernstein's classical approximation formula. In (2.13)–(2.14) an expression is given for the remainder involving a linear functional (2.14) which is a convex combination of second-order divided differences.