Lexicographic Order Research Articles

A simple refined DNA minimizer operator enables 2-fold faster computation.

The minimizer concept is a data structure for sequence sketching. The standard canonical minimizer selects a subset of k-mers from the given DNA sequence by comparing the forward and reverse k-mers in a window simultaneously according to a predefined selection scheme. It is widely employed by sequence analysis such as read mapping and assembly. k-mer density, k-mer repetitiveness (e.g. k-mer bias), and computational efficiency are three critical measurements for minimizer selection schemes. However, there exist trade-offs between kinds of minimizer variants. Generic, effective, and efficient are always the requirements for high-performance minimizer algorithms. We propose a simple minimizer operator as a refinement of the standard canonical minimizer. It takes only a few operations to compute. However, it can improve the k-mer repetitiveness, especially for the lexicographic order. It applies to other selection schemes of total orders (e.g. random orders). Moreover, it is computationally efficient and the density is close to that of the standard minimizer. The refined minimizer may benefit high-performance applications like binning and read mapping. The source code of the benchmark in this work is available at the github repository https://github.com/xp3i4/mini_benchmark.

A robust LP-based approach for a dynamic surgical case scheduling problem with sterilisation constraints

The purpose of this article is to investigate a practical scheduling problem in which a group of elective surgical cases are scheduled over time, while considering their unpredictable durations and potential delays in the sterilisation of surgical instruments. The primary objectives were to schedule the maximum number of surgeries and decrease overtime for the surgical staff, as well as limit the number of instruments requiring emergency sterilisation. The study was conducted in collaboration with the University Hospital of Angers in France, which also contributed historical data for the experiments. We propose two robust mixed integer linear programming models, which are then solved iteratively through a rolling horizon approach, in which the objective functions are taken into account in lexicographic order. Experiments on randomly generated instances indicated which of the two approaches had better performance. Comparison of the results for a real-world scenario involving actual planning at the hospital indicated a greater than 69% decrease in overtime, and a minimum of 92% fewer stressful situations in the sterilising unit.

Shadow Minimization Boolean Function Reconstruction

A special class of monotone Boolean functions coming from shadow minimization theory of finite set-systems – KK-MBF functions – is considered. These functions are a descriptive model for systems of compatible groups of constraints, however, the class of all functions is unambiguously complex and it is sensible to study relatively simple subclasses of functions such as KK-MBF. Zeros of KK-MBF functions correspond to initial segments of lexicographic order on hypercube layers. This property is used to simplify the recognition. Lexicographic order applies priorities over constraints which is applicable property of practices. Query-based algorithms for KK-MBF functions are investigated in terms of their complexities.

Towards Practical Multi-Client Order-Revealing Encryption: Improvement and Application

Order-revealing encryption (ORE) enables the untrusted server to perform greater-than-comparison over ciphertext without compromising data privacy, which allows anyone to evaluate the lexicographic ordering of two arbitrary ciphertexts with a public comparison algorithm. However, most ORE constructions merely support ciphertext comparison for single-user. Recently, a variant of ORE named delegatable ORE has been introduced, which achieves cross-user ciphertext comparison by employing token mutual authorization technique at the cost of weak security, i.e., reveals the most significant differing bit of underlying plaintexts. To tackle this problem, we first present a deterministic property-preserving hash called DPPH with short-size hash value, and then propose a novel multi-client ORE scheme (m-ORE) from DPPH that supports ciphertext comparison among multiple users while hiding the most significant differing bits. Furthermore, we present an enhanced construction dubbed m-H-ORE by introducing a two-phase comparison method, which can achieve supper-efficient comparison in some cases, i.e., two ciphertexts with different bit-length. Finally, we provide formal security proofs of the proposed schemes and run extensive experiments to evaluate their performance on real-world and synthetic datasets. The results demonstrate that both of the proposed schemes can achieve a speedup of 47× and 138× in comparison cost to that of parameter-hiding ORE, respectively.

The Archimedean Origin of Modern Positional Number Systems

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics.

Lexicographic Ordering and Loss Aversion among Low-Income Farmers

As Richard Day explained, expected utility theory suffers from procedural irrationality. This and other problems are illustrated here in the context of decision-making among low-income farmers. Farmers in developing countries are commonly thought to underinvest in modern techniques because their low incomes make them especially risk averse. In addition to the procedural leap of faith, highly restrictive assumptions are needed to apply expected utility theory to the problem. Nor does expected utility theory, as usually prescribed, fit the narrative of loss aversion. The reader is introduced to a procedurally rational substitute called lexicographic safety first. The model is illustrated for the case of rice fertilization in the Philippines, and policy implications are drawn. To illustrate the potential appeal of lexicographic ordering for other applications involving thresholds, a lexicographic model of rational addiction is also provided.

The multivariate Serre conjecture ring

It is well-known that for any integral domain R, the Serre conjecture ring R〈X〉, i.e., the localization of the univariate polynomial ring R[X] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R〈X1,…,Xn〉:=(R〈X1,…,Xn−1〉)〈Xn〉, the ring R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R〈X1,…,Xn〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R[X1,…,Xn]-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n∈N>0, and any finitely generated ideal I of R[X1,…,Xn], the ideal LT(I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R〈X1,…,Xn〉 can also be defined directly as the localization of the multivariate polynomial ring R[X1,…,Xn] at polynomials whose leading coefficients according to the lexicographic monomial order with X1<X2<⋯<Xn is 1, we propose to generalize the fact that R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case.

Quantum Algorithm for Lexicographically Minimal String Rotation

Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an O(n3/4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^{3/4})$$\\end{document} quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\sqrt{n} \\log n)$$\\end{document}, which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its Ωn/logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\sqrt{n/\\log n}\\right) $$\\end{document} lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.

Conformal Geometric Algebra–Based Geometric Modeling Method for Forward Displacement Analysis of 6-4 Stewart Platforms

Abstract This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on conformal geometric algebra (CGA). First, a geometric constraint relationship of four lines and a plane intersecting at a point is published. Second, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Third, five polynomial equations are established by CGA theory. Fourth, based on the above six equations, a 5 × 5 Sylvester’s matrix is formulated by using Sylvester’s Dialytic elimination method and Gröbner bases method under the graded reverse lexicographical order. Finally, the coordinates of four points on the moving platform are revealed. Besides, a numerical example is used to prove the validity of the proposed method. The novelty of this study is that a whole geometric modeling method by geometric constraint relationship of four lines and a plane intersecting at a point is put forward under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. At the same time, Sylvester’s matrix constructed by this method is the smallest one in the known literature for forward displacement analysis of 6-4 Stewart platforms.

Beyond the Erdős discrepancy problem in function fields

We characterize the limiting behavior of partial sums of multiplicative functions f:Fq[t]→S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {F}_q[t]\\rightarrow S^1$$\\end{document}. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative f:Fq[t]→{-1,+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {F}_q[t]\\rightarrow \\{-1,+1\\}$$\\end{document} with q odd has bounded short interval sums if and only if f coincides with a “modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document} that such modified characters are extremal with respect to partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of Fq[t]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{q}[t]$$\\end{document}. This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on Fq[t]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_q[t]$$\\end{document}. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.

Effective and Efficient Lexicographical Order Dependency Discovery

Lexicographical order dependencies state relationships of order between lists of attributes. They naturally model the order-by clauses in SQL queries, and are proven useful in query optimizations concerning sorting. Despite their importance, order dependencies on a dataset are typically unknown and are too costly, if not impossible, to design or discover manually. Techniques for automatic order dependency discovery are recently studied. It is challenging for order dependency discovery to scale well, since it is by nature factorial in the number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> of attributes and quadratic in the number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> of tuples. In this paper, we adopt a strategy that decouples the impact of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> from that of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> , and that still finds all minimal and valid lexicographical order dependencies. We present carefully designed data structures, a host of algorithms and optimizations, and an enhanced strategy combined with multithreaded parallelism, for an efficient implementation. Using a host of real-life and synthetic datasets, we experimentally verify our approach is up to orders of magnitude faster than the state-of-the-art methods, and can deliver better results with an improved definition of minimal attribute lists.

Equations of the multi-Rees algebra of fattened coordinate subspaces

In this paper, we describe the equations defining the multi-Rees algebra [Formula: see text] ([Formula: see text]), where the ideals are generated by subsets of [Formula: see text]. We also show that a family of binomials whose leading terms are squarefree, form a Gröbner basis for the defining equations with lexicographic order. We show that if we remove binomials that include [Formula: see text]’s, then remaining binomials form a Gröbner basis for the toric ideal associated to the multi-fiber ring. However binomials, including [Formula: see text]’s, in Gröbner basis of defining equations of the multi-Rees algebra are not necessarily defining equations of corresponding symmetric algebra. Despite this fact, we show that this family of ideals is of multi-fiber type.

Domain-driven multiple-criteria decision-making for flight crew decision support tool

During the flight, the crew might consider modifying their planned trajectory, taking into account currently available information, such as an updated weather forecast report or the already accrued amount of delay. This modified planned trajectory translates into changes on expected fuel and flying time, which will impact the airline’s relevant performance indicators leading to a complex multiple-criteria decision-making problem. Pilot3, a project from the Clean Sky Joint Undertaking 2 under European Union’s Horizon 2020 research and innovation programme, aims to develop an objective optimisation engine to assist the crew on this process. This article presents a domain-driven approach for the selection of the most suitable multiple-criteria decision-making methods to be used for this optimisation framework. The most relevant performance indicators, based on airline’s objectives and policies, are identified as: meeting on-time performance, leading to a binary value in a deterministic scenario; and total cost, which can be disaggregated into sub-cost components. The optimisation process consists of two phases: first, Pareto optimal solutions are generated with a multi-objective optimisation method (lexicographic ordering); second, alternative trajectories are filtered and ranked using a combination of multi-criteria decision analysis methods (analytic hierarchy process and VIKOR). A realistic example of use shows the applicability of the process and studies the sensibility of the optimisation framework.

Gröbner Basis Approach for Solving Fuzzy Complex System of Linear Equations

The foundation of Gröbner’s work is a potent approach for separating polynomial systems of equations. In this study, we solve complex fuzzy systems of linear equations utilizing Gröbner’s basic technique. First, a system of explicit polynomial equations is generated by applying fuzzy computations and fuzzy complex computations to the fundamental system. Taking into account the lexicographical order, Gröbner’s basis is then determined for the ideal given by a definite equivalent polynomial system. The Gröbnerbase that was obtained has a high triangular structure. Thus, the system solution may be readily implemented by proceeding. Hence, all of the system’s answers are readily attainable. Next, a criterion is established to identify whether or not a solution exists for the primary system. The advantage of the proposed approach is that it acquires all of the problem’s solutions simultaneously. Using this technique, a mathematical example and an application issue based on an electrical circuit are resolved. Comparing the generated findings to the known outcomes reveals a high level of congruence.

Codesign of Quantized Dynamic Output Feedback MPC for the Takagi–Sugeno Model

We present a co-design of measurement quantized dynamic output feedback model predictive control (DOFMPC) for Takagi-Sugeno model with bounded disturbance. The system output is quantized by a dynamic quantizer before it is transmitted to the DOFMPC controller. Hence, we utilize the dynamic output feedback control law with quantized output signal and consider the mixed input and quantized output constraint for the controller design. By optimizing the quantizer and controller paramters on-line, the control performance is enhanced. Moreover, we formulate a 2-leveled optimizations, with the upper level optimizing the performance index and the lower level optimizing the soft constraint in a lexicographic order, for the co-design of the DOFMPC controller and dynamic quantizer. Thus, there are more degrees of freedom for tighting the soft constraints. The recursive feasibility and stability of the proposed approaches are guaranteed. The applicability of the proposed approach is illustrated by a simulation example.

Consistency and Consensus Analysis for Group Decision Making With Fuzzy Preference Relations Based on Cooperative Game Theory

Consistency and consensus study for group decision making (GDM) with fuzzy preference relations (FPRs) are necessary, which have a decisive impact on the quality of decision results. Consistency can prevent self-contradictory judgments, and consensus limits the deviation among individual judgments within the preset threshold. Because acceptably multiplicative consistency still cannot avoid self-contradictory judgments, this paper considers the ordinal consistency and acceptably multiplicative consistency simultaneously. In view of these aspects, we introduce a new optimal model-based method for GDM with FPRs. As minimum total adjustment and minimum individual adjustment may be inconsistent, we introduce the Solidarity consistency and consensus adjustment mechanism (So-CCAM) to allocate the minimum total adjustment, which fully considers the cooperation and solidarity among decision makers. When the So-CCAM does not exhibit stability, we present the nucleolus consistency and consensus adjustment mechanism (Nu-CCAM), which is the best element in the core according to lexicographical order. After that, an algorithm is offered. To demonstrate the application and efficiency of the new method, an illustrative example is offered, and a comparison is carried out. It offers the first optimization models-based method for GDM with FPRs in view of ordinal consistency, acceptably multiplicative consistency, and consensus analysis. Further, it provides two rational consistency and consensus adjustment mechanisms by cooperative game theory.

A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis

Given n⩾2 and a1,…,an∈N. Let S=〈a1,…,an〉 be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on a1, which computes the Apéry set and the Frobenius number of S. We also find the Gröbner basis of the toric ideal defined by S, for the weighted degree reverse lexicographical order ≺w to x1,…,xn, without using Buchberger's algorithm. As an application we introduce and study some special classes of semigroups. Namely, when S is generated by generalized arithmetic progressions and generalized almost arithmetic progressions with the ratio a positive or a negative number. We determine symmetric and almost symmetric semigroups generated by a generalized arithmetic progression.

Backtesting Systemic Risk Forecasts Using Multi-Objective Elicitability

Systemic risk measures such as CoVaR, CoES, and MES are widely-used in finance, macroeconomics and by regulatory bodies. Despite their importance, we show that they fail to be elicitable and identifiable. This renders forecast comparison and validation, commonly summarized as “backtesting,” impossible. The novel notion of multi-objective elicitability solves this problem by relying on bivariate scores equipped with the lexicographic order. Based on this concept, we propose Diebold–Mariano type tests with suitable bivariate scores to compare systemic risk forecasts. We illustrate the test decisions by an easy-to-apply traffic-light approach. Finally, we apply our traffic-light approach to DAX 30 and S&P 500 returns, and infer some recommendations for regulators.

On split extensions of preordered groups

We investigate the behaviour of split extensions in the category OrdGrp of (pre)- ordered groups. Namely we show that the lexicographic order plays a key role on the existence of compatible orders for semidirect products, establishing necessary and sufficient conditions for such existence; we prove that the Split Short Five Lemma holds for (stably) strong split extensions, and identify classes of split extensions which admit a classifier.

Minimum gradation in greyscales of graphs

In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches.