Lexicographic Order Research Articles

Variety and multiplicity for partitioned factors in Christoffel and Sturmian words

The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equals Variety (CV) and Structure yields Multiplicity (SM), correspond to statements about Christoffel words. These results may be stated and proved in a purely word-theoretic environment by means of the concept of a partitioned factor. In this setting, the generalized circle of fifths of Clough and Myerson is replaced by the lexicographic order of word theory, and their notion of the generic description of a musical line is replaced by that of a partitioned factor of a circular Christoffel word. Sturmian words are the infinite counterparts to Christoffel words, characterized as aperiodic but of minimal complexity, i.e. words such that for all there are n+1 factors of length n. Berthé showed that the factors of a given length have at most three frequencies (probabilities). Drawing on previous work, we extend to results on factors under a fixed partitioning (decompositions of factors of length n into concatenations of words whose lengths are given by a composition of n into k parts). Any factor of a Sturmian word thus partitioned into k components belongs to one of k+1 types (varieties). We show how to compute the frequencies of the varieties. These results recapture CV and SM in Sturmian words, and we suggest musical interpretations, parallel to those in the finitary case.

Naming and ordering the modes, in light of combinatorics on words

One of the most telling features of the application of combinatorics on words in mathematical music theory is the extent to which aspects of the history of theory are captured by the word theory model. Well-formed modes, including the usual diatonic modes, are modeled by Christoffel words and their conjugates. A class of Sturmian morphisms produces words representing the Glarean modes, and gives rise to standard factorizations, which in turn model the authentic and plagal divisions of the diatonic modes. The principal discussions are the renaming of the ancient Greek modes in the tenth-century Alia musica, and the reordering of the authentic modes of Glarean's Dodecachordon by Zarlino in 1571. The latter exemplifies a recently demonstrated property in combinatorics on words, the matching of the conjugation order with the lexicographic order for the conjugation class of a given special Sturmian morphism. The two main topics are unified in this presentation.

A computer verification for the value of the Topological Entropy for some special subshifts in the Lexicographical Scenario

The Lorenz Attractor has been a source for many mathematical studies. Most of them deal with lower dimensional representations of its first return map. An one dimensional scenario can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics, in this case, can be modeled by a subshift in the Lexicographical model. The Lexicographical model is the set of two symbols with the topology induced by the lexicographical metric and with the lexicographical order. These subshifts are the maximal invariant set for the shift map in some interval. For some of them, the extremes of the interval are a minimal periodic sequence and a maximal periodic sequence which is an iteration of the lower extreme (by the shift map). For some of these subshifts the topological entropy is zero. In this case the dynamics (of the respective Lorenz map) is simple.Associated to any of these subshifts (let call it Λ) we consider an extension (let call it Γ) that contains Λ which also can be constructed by using an interval whose extremes can be defined by the extremes of Λ. For these extensions we present here a computer verification of the result that compute its topological entropy. As a consequence, of our results, we can say: the longer the period of the periodic sequence is then the lower complexity in the dynamics of the extension the associated map has.

Inverse Lyndon words and inverse Lyndon factorizations of words

Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that is Lyndon words with respect to the inverse lexicographic order. The Lyndon factorization of a nonempty word w is unique but w may have several inverse Lyndon factorizations. We prove that any nonempty word w admits a canonical inverse Lyndon factorization, named ICFL(w), that maintains the main properties of the Lyndon factorization of w: it can be computed in linear time, it is uniquely determined, and it preserves a compatibility property for sorting suffixes. In particular, the compatibility property of ICFL(w) is a consequence of another result: any factor in ICFL(w) is a concatenation of consecutive factors of the Lyndon factorization of w with respect to the inverse lexicographic order.

On the Polymatroidal Property of Monomial Ideals with a View Towards Orderings of Minimal Generators

We prove that a monomial ideal I generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.

An Approximate Algorithm for Lexicographic Search in Multiple Orders for the Solution of the Multidimensional Boolean Knapsack Problem

A new scheme of approximate lexicographic search is proposed for the solution of the multidimensional Boolean knapsack problem. The main idea of the algorithm is to gradually define the lexicographic order (ordering of variables) in which “qualitative” solutions of the problem belong to a direct two-sided lexicographic constraint whose upper bound is the lexicographic maximum of the set of feasible solutions of the problem in this order. Since the search for “qualitative” solutions in each order is carried out on a bounded lexicographic interval, the proposed algorithm is called a bounded lexicographic search. The quality of the approximate method of bounded lexicographic search is investigated by solving test tasks from the well-known Beasley and Glover–Kochenberger sets.

A reply to Ponte et al (2016) Supply chain collaboration: Some comments on the nucleolus of the beer game

Purpose: The aim of the paper is to pick up the result of a previously published paper in order to deepen the discussion. We analyze the solution against the background of some well-known concepts and we introduce a newer one. In doing so we would like to inspire the further discussion of supply chain collaborationDesign/methodology/approach: Based on game theoretical knowledge we present and compare seven properties of fair profit sharing.Findings: We show that the nucleolus is a core-solution, which does not fulfil aggregate monotonicity. In contrast the Shapley value is an aggregate monotonic solution but does not belong to the core of every cooperative game. Moreover, we present the Lorenz dominance as an additional fairness criteria.Originality/value: We discuss the very involved procedure of establishing lexicographic orders of excess vectors for games with many players.

On the structure of subsets of the discrete cube with small edge boundary

On the structure of subsets of the discrete cube with small edge boundary, Discrete Analysis 2018:9, 29 pp. An isoperimetric inequality is a statement that tells us how small the boundary of a set can be given the of the set, for suitable notions of size and boundary. For example, one formulation of the classical isoperimetric inequality in $\mathbb R^n$ is as follows. Given a subset $X$ of $\mathbb R^n$, define the $\epsilon$-_expansion_ of $X$ to be the open set $X_\epsilon=\{y\in\mathbb R^n: d(y,X)<\epsilon\}$. If in addition $X$ is measurable, define the of its boundary to be $\lim\inf_{\epsilon\to 0} \epsilon^{-1}\mu(X_\epsilon\setminus X)$. (If $X$ is a set with a suitably smooth topological boundary $\partial X$, then this turns out to equal the surface measure of $\partial X$.) Then amongst all sets $X$ of a given measure, the one with the smallest boundary is an $n$-dimensional ball. Isoperimetric inequalities have been the focus of a great deal of research, partly for their intrinisic interest, but also because they have numerous applications. One particularly useful one is the _edge-isoperimetric inequality in the discrete cube_. This concerns subsets $X$ of the $n$-dimensional cube $\{0,1\}^n$, which we turn into a graph by joining two points $x$ and $y$ if they differ in exactly one coordinate. The of a set $X$ is simply its cardinality, the _edge-boundary_ of $X$ is defined to be the set of edges between $X$ and its complement, and the of the edge-boundary is the number of such edges. If $|X|=2^d$, then it is known that the edge-boundary is minimized when $X$ is a $d$-dimensional subspace of $\mathbb F_2^n$ generated by $d$ standard basis vectors. More generally, if $|X|=m$, then the edge-boundary is minimized when $X$ is an initial segment in the lexicographical ordering, which is the ordering where we set $x<y$ if $x_i<y_i$ for the first coordinate $i$ where $x_i$ and $y_i$ differ. (This coincides with the ordering we obtain if we think of the sequences as binary representations of integers.) For the two isoperimetric inequalities just mentioned, as well as many others, it is known that the extremal examples provided are essentially the only ones: a subset of $\mathbb R^n$ with a boundary that is as small as possible has to be an $n$-dimensional ball, and a subset of the discrete cube with edge-boundary that is as small as possible has to be an initial segment of the lexicographical ordering, up to the symmetries of the graph. Furthermore, there are _stablity_ results: a set with a boundary that is _almost_ as small as possible must be close to an extremal example. Such a result tells us that the isoperimetric inequalities are robust, in the sense that if you slightly perturb the condition on the set, then you only slightly perturb what the set has to look like. This paper is about an extremely precise stability result for the edge isoperimetric inequality in the discrete cube. There have been a number of papers on such results (see the introduction to the paper for details), but they have been mainly for sets of $2^d$ for some $d$, where the goal is to prove that they must be close to $d$-dimensional subcubes -- that is, subspaces (or their translates) generated by $d$ standard basis vectors. This paper considers sets of arbitrary and proves the following result. Suppose that $X$ is a subset of $\{0,1\}^n$ of $m$. Suppose that the of the edge-boundary of $X$ is at most $g_n(m)+l$, where $g_n(m)$ is the of the edge-boundary of the initial segment $I_m$ of $m$ in the lexicographical order. Then there is an automorphism $\phi$ of $\{0,1\}^n$ (meaning a bijection that takes neighbouring points to neighbouring points) such that $|X\Delta\phi(I_m)|\leq Cl$, where $C$ is an absolute constant. They give an example to show that $C$ must be at least 2, and thus that their result is best possible up to the value of the constant $C$. Previous proofs of stability versions of the edge-isoperimetric inequality in the cube have used Fourier analysis. The proof in this paper uses purely combinatorial methods, such as induction on the dimension, and compressions. To get these methods to work, several interesting ideas are needed, including some new results about the influence of variables.

A semilinear system with positivity conditions

This paper studies a semi-linear system of equations in $$ {\mathbb{R}}^{N} $$ , which comes from a mathematical model for a new tax system proposed in Chile’s 2014 Tax Reform. The system of equations involves a non negative coefficients matrix and simultaneously relates the unknown vector with its positive part, and hence the nonlinear nature of the problem. In addition to find appropriate conditions for the existence and the uniqueness of a solution, in this paper an algorithm is proposed to obtain it by solving at most $$ N $$ linear systems of size $$ N \times N $$ . The proof of the results is based on a monotony method with respect to the usual lexicographic order in $$ {\mathbb{R}}^{N} $$ .

Constructing de Bruijn sequences with co-lexicographic order: The [formula omitted]-ary Grandmama sequence

A k-ary de Bruijn sequence of order n is a circular k-ary string of length kn which contains every k-ary string of length n exactly once as a substring. It is well-known that a k-ary de Bruijn sequence of order n can be constructed by concatenating the aperiodic prefixes of the k-ary necklaces of length n in lexicographic order. In this article we prove that an alternate de Bruijn sequence is created by replacing lexicographic order with co-lexicographic order. We also provide a simple successor rule for generating each successive symbol in O(n)-time.

ИСПОЛЬЗОВАНИЕ КОМПОНЕНТНО-КОМПЕТЕНТНОСТНОГО ПОДХОДА ДЛЯ ФОРМИРОВАНИЯ КООПЕРАЦИИ ИСПОЛНИТЕЛЕЙ ИННОВАЦИОННОГО ЗАКАЗА ПРИ СОЗДАНИИ СЛОЖНОЙ АЭРОКОСМИЧЕСКОЙ ТЕХНИКИ

The task of creation of innovative order executors’ cooperation to create the new aerospace technique (AST) on the basis of the component-competence approach has been set and solved. In order to do so at the initial design stages the architecture of future AST article is represented as a component (both innovative and from the past development experience). Each created component is evaluated in terms of competence, in the form of competence components (knowledge, skills, attainments, experience). For that to happen with the help of experts competence requirements are formed for each component of the article. In order to find the responsive competencies among the potential candidates to implement the project precedent base (PB) has been created. PB contains the assessments of independent experts on specific competencies of the candidates. Further, the customer's requirements, formalized in the form of a control "word" are compared with the competencies of candidates in PB for each component of a complex article. Quantitative and qualitative expert assessments of competencies based on quantitative linear programming methods and lexicographic ordering of alternatives were used. The proposed approach is advisable to be used at the initial stages of the innovation project, when it is necessary to create the executors’ cooperation and, at the same time, to consider the fulfillment of the requirements for the competence on the customer side, that are based on the analysis of the component architecture of a complex article, in which both the components of past development and new, innovative components are highlighted

A generic framework to include belief functions in preference handling and multi-criteria decision

Modelling the preferences of a decision maker about alternatives having multiple criteria usually starts by collecting preference information (comparisons of alternatives, importance of criteria, …), which are then used to fit a preference model issued from some set of hypothesis (weighted average, CP-net, lexicographic orderings, AHP, …). In practice, this process often leads to inconsistencies that may be due to inaccurate information provided by the decision maker, who can be unsure about the provided information, or to a poor choice of hypothesis set, which can be too restrictive or not well adapted to the decision process. In this paper, we propose to use belief functions as a way to quantify and resolve such inconsistencies, notably by allowing the decision maker to express her/his certainty about the provided preferential information. Our framework is generic, in the sense that it does not assume a given set of hypothesis a priori, and is consistent with precise methods, in the sense that in the absence of uncertainty and inconsistencies in the information, precise models are ultimately retrieved.

A Lexicographic 0.5-Approximation Algorithm for the Multiple Knapsack Problem

We present a 0.5-approximation algorithm for the Multiple Knapsack Problem (MKP). The algorithm uses the ordering of knapsacks according to the nondecreasing of size and the two orderings of items: in nonincreasing utility order and in nonincreasing order of the utility/size ratio. These orderings create two lexicographic orderings on A × B (here A is the set of knapsacks and B is the set of indivisible items). Based on each of these lexicographic orderings, the algorithm creates a feasible solution to the MKP by looking through the pairs (a, b) ∈ A × B in the corresponding order and placing item b into knapsack a if this item is not placed yet and there is enough free space in the knapsack. The algorithm chooses the best of the two obtained solutions. This algorithm is 0.5-approximate and has runtime O(mn) (without sorting), where mand n are the sizes of A and B correspondingly.

Negations With Respect to Admissible Orders in the Interval-Valued Fuzzy Set Theory

Admissible orders have brought the structure of chains in the framework of interval-valued fuzzy sets. However, a deeper study of functions monotone with respect to admissible orders is still missing in the literature. In this work, we consider the construction of negations and strong negations on intervals with respect to admissible orders, in particular, for the Xu and Yager and lexicographical orders, as well as for those based on $K_\alpha$ operators. We introduce and discuss an approach to the construction of strong negations on intervals with respect to $K_{\alpha,\beta }$ orders based on an arbitrary couple of strong negations defined over the standard real interval $[0,1]$ . The introduced strong negations have a deep impact on all fields exploiting fuzzy methods dealing with intervals, allowing to introduce complements, dual aggregations, implications, entropies, etc.

Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture

The symmetric group Sn acts on the polynomial ring Q[xn]=Q[x1,…,xn] by variable permutation. The invariant ideal In is the ideal generated by all Sn-invariant polynomials with vanishing constant term. The quotient Rn=Q[xn]In is called the coinvariant algebra. The coinvariant algebra Rn has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization In,k⊆Q[xn] of the ideal In indexed by two positive integers k≤n. The corresponding quotient Rn,k:=Q[xn]In,k carries a graded action of Sn and specializes to Rn when k=n. We generalize many of the nice properties of Rn to Rn,k. In particular, we describe the Hilbert series of Rn,k, give extensions of the Artin and Garsia–Stanton monomial bases of Rn to Rn,k, determine the reduced Gröbner basis for In,k with respect to the lexicographic monomial order, and describe the graded Frobenius series of Rn,k. Just as the combinatorics of Rn are controlled by permutations in Sn, we will show that the combinatorics of Rn,k are controlled by ordered set partitions of {1,2,…,n} with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of Rn,k is (up to a minor twist) the t=0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded Sn-module Vn,k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module Rn,k solves this problem in the specialization t=0.

On Dantzig figures from graded lexicographic orders

We construct two families of Dantzig figures, which are d-dimensional polytopes with 2d facets and an antipodal vertex pair, from convex hulls of initial subsets for the graded lexicographic (grlex) and graded reverse lexicographic (grevlex) orders on Z≥0d. These two polytopes have the same number of vertices, O(d2), and the same number of edges, O(d3), but are not combinatorially equivalent. We provide an explicit description of the vertices and the facets for both families and describe their graphs along with analyzing their basic properties such as the radius, diameter, existence of Hamiltonian circuits, and chromatic number. Moreover, we also analyze the edge expansions of these graphs.

"On the Two Machine Permutation Flowshop Scheduling Problems Using Variable Neighborhood Search Algorithms "

"This paper considers the two machine permutation flow shop scheduling problem PFSSP. We study some special cases of the problem 2||( , Σ , , Σ ) (P). In the first one, the special cases to find the exact non dominated solutions for problem () are based on the relation between processing times , and the due dates on machine A and machine B, ( = 1,2, … , ). In the second one, the special cases are based on the lexicographical order minimization to solve three problems, in each one the primary objective function is maximum completion time ( ), while the secondary objective function is one of the objective functions: total completion times (Σ ), maximum tardiness ( ) and total tardiness (Σ ), the resulting problems are (P1), (P2), (P3). We propose three algorithms: Variable neighborhood simulated annealing(VNSA), Variable neighborhood descent method(VNDM) and Variable neighborhood with modefied NEH algorithm(VNMNEH). The comparison results show the efficiency of the proposed algorithms. All algorithmes were coded in Matlab programe."

ВЫБОР И ОБОСНОВАНИЕ АРХИТЕКТУРЫ ВИРТУАЛЬНОЙ ПРОИЗВОДСТВЕННОЙ СИСТЕМЫ ДЛЯ ПЛАНИРУЕМОГО ПОРТФЕЛЯ ЗАКАЗОВ

The task to form and research the virtual production system (VPS) operation for the planned portfolio of orders (PPO) has been stated and solved. Due to the complexity of the proposed task, the solution is implemented in several stages. At the first stage the research of the main indicators of the formed VPS has been made. First of all, this concerns the composition of the VPS that according to its characteristics should meet the basic requirements of the PPO. For this purpose a set of potential enterprises – VPS candidates – have been formed using a precedent base (PB). A qualitative scale in the form of Latin alphabet was used to evaluate each enterprise in terms of key indicators and requirements of PPO. By lexicographic ordering of options in the PB, the most suitable enterprises are selected to make the VPS. To adapt the enterprises (equipment, technologies) to the specific requirements of PPO, the choice of activities for the modernization of production is made. The adaptation costs have been selected as an objective function to evaluate the set of activities. The time to make the activities and the acceptable risks are taken as limitations for the task. In order to study the VPS operation an agent simulation model has been constructed. The analysis of the VPS "supply-production-marketing" logistics chain has been carried out on the basis of this model. The results of the simulation make it possible to estimate the time of PPO implementation, the impact of the risks and, if necessary, to make the further restructuring of the VPS architecture. The proposed approach is expedient to be used at the initial stage of VPS architecture development taking into account the basic requirements associated with the planned portfolio of orders

Lexicodes over finite principal ideal rings

Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module $$R^n$$ is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.

Shared parking problem: A novel truthful double auction mechanism approach

Considering a parking platform with multiple slot demanders and multiple slot suppliers, this paper addresses two truthful double auction mechanisms for shared parking problem, where the demanders targeted by the proposed mechanisms are with flexible schedules and for non-work activities, and the suppliers are with the typical daily ‘driving go out early and come back at dusk’ pattern. To provide flexible match schemes and increase the trading scale, we allow “all-or-nothing” principle for demanders and “one-to-many” principle for suppliers. Based on the parking slot allocation rule and the transaction payment rule, we first relax the single output restriction (suppliers can only submit bids on a single unit of one commodity) and propose a “demander competition padding method (DC-PM)” auction mechanism, which is used to solve potential budget deficit. To avoid the likely distorted social welfare resulted by the DC-PM auction mechanism and to add budget surplus, we further modify the parking slot allocation rule and the transaction payment rule in the DC-PM auction mechanism, and propose a “modified demander competition padding method (MDC-PM)” auction mechanism. Three economical properties of both auction mechanisms, i.e. incentive compatibility, individual rationality and budget balance, are proved. Numerical experiments show that the proposed two auction mechanisms can realize asymptotic efficiency as both demanders and suppliers approach infinity. Moreover, the DC-PM auction mechanism is superior to the MDC-PM auction mechanism with respect to participants’ utilities and can maintain the strict lexicographic allocation order, and the MDC-PM auction mechanism outperforms the DC-PM auction mechanism in terms of welfare efficiency and platform's payoff. Issues of managerial implications for shared parking problem are further discussed in this paper.