Hierarchy for groups acting on hyperbolic ℤn-spaces

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic [Formula: see text]-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic [Formula: see text]-metric spaces, where [Formula: see text] is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case [Formula: see text] taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic [Formula: see text]-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy[Formula: see text][Formula: see text]AMS-[Formula: see text], Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for [Formula: see text]-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in [Formula: see text], Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

An approach to optimize the cost of transportation problem based on triangular fuzzy programming problem

In this article, we address a fully fuzzy triangular linear fractional programming (FFLFP) problem under the condition that all the parameters and decision variables are characterized by triangular fuzzy numbers. Utilizing the computation of triangular fuzzy numbers and Lexicographic order (LO), the FFLFP problem is changed over to a multi-objective function. Consequently, the problem is changed into a multi-objective crisp problem. This paper outfits another idea for diminishing the computational complexity, in any case without losing its viability crisp LFP issues. Lead from real-life problems, a couple of mathematical models are considered to survey the legitimacy, usefulness and applicability of our method. Finally, some mathematical analysis along with one case study is given to show the novel strategies are superior to the current techniques.

Asymmetrization of a Set of Degressively Proportional Allocations with Respect to Lexicographic Order. An Algorithmic Approach

In the case of the proportional allocation of goods and burdens, the shares of all agents with respect to their values are equal, i.e., they form a constant sequence. In a degressively proportional allocation this sequence is nondecreasing when agents are increasingly ordered according to their values. The division performed according to this principle is ambiguous, and its selection requires many negotiations among participants. The aim of this paper is to limit the range of such negotiations when the problem is complex, i.e., the set of feasible solutions has high cardinality. It can be done thanks to a numerical analysis of the set of all feasible solutions, and eliminating allocations favoring or disfavoring some coalitions of agents. The problem is illustrated by the case study of allocating seats in the European Parliament in its 2019–2024 term.

The COVID-19 Vaccine Preference for Youngsters Using PROMETHEE-II in the IFSS Environment

Extensive decision-making during the vaccine preparation period is unpredictable. An account of the severity of the disease, the younger people with COVID-19 comorbidities and other chronic diseases are also at a higher risk of the COVID-19 pandemic. In this research article, the preference ranking structure for the COVID-19 vaccine is recommended for young people who have been exposed to the effects of certain chronic diseases. Multiple Criteria Decision-Making (MCDM) approach effectively handles this vague information. Furthermore, with the support of the Intuitionistic Fuzzy Soft Set (IFSS), the entries under the new extension of the Preference Ranking Organization Method for Enrichment Evaluation-II (PROMETHEE-II) is suggested for Preference Ranking Structure. The concept of intuitionistic fuzzy soft sets is parametric in nature. IFSS suggests how to exploit an intuitionistic ambiguous input from a decision-maker to make up for any shortcomings in the information provided by the decider. The weight of the inputs is calculated under the Intuitionistic Fuzzy Weighted Average (IFWA) operator, the Simply Weighted Intuitionistic Fuzzy Average (SWIFA) operator, and the Simply Intuitionistic Fuzzy Average (SIFA) operator. An Extended PROMETHEE-based ranking, outranking approach is used, and the resultant are recommended under the lexicographic order. Its sustainability and feasibility are explored for three distinct priority structures and the possibilities of the approach. To demonstrate the all-encompassing intuitionistic fuzzy PROMETHEE approach, a practical application regarding COVID-19 severity in patients is given, and then it is compared to other existing approaches to further explain its feasibility, and the sensitivity of the preference structure is examined according to the criteria.

A Vectorial Tree Distance Measure

A vectorial distance measure for trees is presented. Given two trees, we align the trees from their centers outwards, starting from the root-branches, to make the next level as similar as possible. The algorithm is recursive; condition on the alignment of the root-branches we align the sub-branches, thereafter each alignment is conditioned on the previous one. We define a minimal alignment under a lexicographic order which follows the intuition that the differences between the two trees closer to their cores dominate their differences at a higher level. Given such a minimal alignment, the difference in the number of branches calculated at any level defines the entry of the distance vector at that level. We compare our algorithm to other well-known tree distance measures in the task of clustering sets of phylogenetic trees. We use the TreeSimGM simulator for generating stochastic phylogenetic trees. The vectorial tree distance can successfully separate symmetric from asymmetric trees, and hierarchical from non-hierarchical trees.

Lexicographic Unranking of Combinations Revisited

In the context of combinatorial sampling, the so-called “unranking method” can be seen as a link between a total order over the objects and an effective way to construct an object of given rank. The most classical order used in this context is the lexicographic order, which corresponds to the familiar word ordering in the dictionary. In this article, we propose a comparative study of four algorithms dedicated to the lexicographic unranking of combinations, including three algorithms that were introduced decades ago. We start the paper with the introduction of our new algorithm using a new strategy of computations based on the classical factorial numeral system (or factoradics). Then, we present, in a high level, the three other algorithms. For each case, we analyze its time complexity on average, within a uniform framework, and describe its strengths and weaknesses. For about 20 years, such algorithms have been implemented using big integer arithmetic rather than bounded integer arithmetic which makes the cost of computing some coefficients higher than previously stated. We propose improvements for all implementations, which take this fact into account, and we give a detailed complexity analysis, which is validated by an experimental analysis. Finally, we show that, even if the algorithms are based on different strategies, all are doing very similar computations. Lastly, we extend our approach to the unranking of other classical combinatorial objects such as families counted by multinomial coefficients and k-permutations.

Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube

The problem “Given a Boolean function [Formula: see text] of [Formula: see text] variables by its truth table vector. Find (if exists) a vector [Formula: see text] of maximal (or minimal) weight, such that [Formula: see text].” is considered here. It is closely related to the problem of computing the algebraic degree of Boolean functions which is an important cryptographic parameter. To solve this problem efficiently, we explore the orders of the vectors of the [Formula: see text]-dimensional Boolean cube [Formula: see text] according to their weights. The notion of “[Formula: see text]th layer” of [Formula: see text] is involved in the definition and examination of the “weight order” relation. It is compared with the known relation “precedes”. Several enumeration problems concerning these relations are solved and the relevant notes were added to three sequences in the on-line encyclopedia of integer sequences (OEIS). One special weight order is defined and examined in detail. In it, the lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called weight-lexicographic order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. The results obtained by them were used in creating two new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed — the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results from numerous tests confirm the efficiency of these algorithms. Other applications of the obtained algorithms are also discussed — when representing, generating and ranking other combinatorial objects.