PERFORMANCE APPRAISAL OF TREAP AND HEAP SORT ALGORITHMS

The task of storing items to allow for fast access to an item given its key is an ubiquitous problem in many organizations. Treap as a method uses key and priority for searching in databases. When the keys are drawn from a large totally ordered set, the choice of storing the items is usually some sort of search tree. The simplest form of such tree is a binary search tree. In this tree, a set X of n items is stored at the nodes of a rooted binary tree in which some item y ϵ X is chosen to be stored at the root of the tree. Heap as data structure is an array object that can be viewed as a nearly complete binary tree in which each node of the tree corresponds to an element of the array that stores the value in the node. Both algorithms were subjected to sorting under the same experimental environment and conditions. This was implemented by means of threads which call each of the two methods simultaneously. The server keeps records of individual search time which was the basis of the comparison. It was discovered that treap was faster than heap sort in sorting and searching for elements using systems with homogenous properties.

Captive diffusions and their applications to order-preserving dynamics

We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of càdlàg bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries—possibly in a hybrid manner over non-overlapping time periods—and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.

Zero morphemes in paradigms

This paper sheds a new light on the notion of zero morphemes in inflectional paradigms: on their formal definition (§ 1), on the way of counting them (§ 2–3) and on the way of conceptualizing them at a deeper, mathematical level (§ 4). We define (zero) morphemes in the language of cartesian set products and propose a method of counting them that applies the lexical relations of homophony, polysemy, allomorphy and synonymy to inflectional paradigms (§ 2). In this line, two homophonic or synonymous morphemes are different morphemes, while two polysemous and allomorphic morphemes count as one morpheme (§ 3). In analogy to the number zero in mathematics, zero morphemes can be thought of either as minimal elements in a totally ordered set or as neutral element in a set of opposites (§ 4). Implications for language acquisition are discussed in the conclusion (§ 5).

On some properties of LS algebras

The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.

Granularity Approach for Multi-Criteria Decision Making About Hybrid Evaluation Information

In this study, a new version of TOPSIS method is reconstructed to deal with the problem of multi-criteria decision making. Here, the data representation of all alternatives is varied according to different criteria, such as real number, interval-valued number, set-valued number and intuitionistic fuzzy-valued number, etc. Because the distinguishing ability of each criterion can be reflected by its knowledge granularity, naturally, a knowledge granularity method is constructed to measure the criteria weights. Besides, the approach of how to select the ideal solution is redefined, especially for the case that the content of criterion according to all alternatives is not a totally ordered set anymore. What is more, the decision maker’s personal preference is considered, and the concrete indicator value can be calculated by the convex combination of the distance from possible alternatives to ideal solutions. Finally, the validity of the proposed decision-making algorithm is illustrated by a synthetic example.

Reasoning with Partial Orders: Restrictions on Ignorance Inferences of Superlative Modifiers

The present study is concerned with Ignorance Inferences associated with Superlative Modifiers (SMs) like at least and at most. Experimental evidence will be presented showing that the Ignorance Inferences associated with SMs depend on their associate: when the associate of an SM is a totally ordered set (e.g. a numeral), the exhaustive interpretation of the prejacent must necessarily constitute an epistemic possibility for the speaker. However, when the associate of the SM is partially ordered, the exhaustive interpretation of the prejacent can, but need not constitute an epistemic possibility for the speaker.

On 2-Absorbing Submodules

In this paper, we introduce the concept of 2-absorbing submodules as a generalization of 2-absorbing ideals. Let R be a commutative ring and M an R-module. A proper submodule N of M is called 2-absorbing if whenever a, b ∈ R, m ∈ M and abm ∈ N, then am ∈ N or bm ∈ N or ab ∈ N:RM. Let N be a 2-absorbing submodule of M. It is shown that N:RM is a 2-absorbing ideal of R and either Ass R(M/N) is a totally ordered set or Ass R(M/N) is the union of two totally ordered sets. Furthermore, it is shown that if M is a finitely generated multiplication module over a Noetherian ring R, and Ass R(M/N) a totally ordered set, then N is 2-absorbing whenever N:RM is a 2-absorbing ideal of R. Also, the 2-absorbing avoidance theorem is proved.