Krull Domains and Monoids, Their Sets of Lengths, and Associated Combinatorial Problems

2017 ◽  
pp. 73-112 ◽  
Author(s):  
Scott Chapman ◽  
Alfred Geroldinger
Keyword(s):  
Combinatorial Problems ◽  
Sets Of Lengths ◽  
Krull Domains

scholarly journals Long Sets of Lengths With Maximal Elasticity

2018 ◽  
Vol 70 (6) ◽  
pp. 1284-1318 ◽  
Author(s):  
Alfred Geroldinger ◽  
Qinghai Zhong
Keyword(s):  
Finite Type ◽  
Algebraic Number ◽  
Class Group ◽  
Number Fields ◽  
Additive Combinatorics ◽  
Algebraic Number Fields ◽  
Krull Monoids ◽  
Sets Of Lengths ◽  
Positive Integers ◽  
Krull Domains

AbstractWe introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid H, let Δρ(H) be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity ρ(H). We study Δρ(H) for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

Divisibility Properties of Graded Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 196-215 ◽  
Author(s):  
D. D. Anderson ◽  
David F. Anderson
Keyword(s):  
Integral Domain ◽  
Homogeneous Element ◽  
Carry Over ◽  
Divisibility Properties ◽  
Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

scholarly journals Q-Learnheuristics: Towards Data-Driven Balanced Metaheuristics

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1839
Author(s):  
Broderick Crawford ◽  
Ricardo Soto ◽  
José Lemus-Romani ◽  
Marcelo Becerra-Rozas ◽  
José M. Lanza-Gutiérrez ◽  
...  
Keyword(s):  
Combinatorial Problems ◽  
Set Covering ◽  
Covering Problem ◽  
Integration Framework ◽  
Q Learning ◽  
Whale Optimization ◽  
Sine Cosine Algorithm ◽  
Exploration Exploitation ◽  
Selection Of

One of the central issues that must be resolved for a metaheuristic optimization process to work well is the dilemma of the balance between exploration and exploitation. The metaheuristics (MH) that achieved this balance can be called balanced MH, where a Q-Learning (QL) integration framework was proposed for the selection of metaheuristic operators conducive to this balance, particularly the selection of binarization schemes when a continuous metaheuristic solves binary combinatorial problems. In this work the use of this framework is extended to other recent metaheuristics, demonstrating that the integration of QL in the selection of operators improves the exploration-exploitation balance. Specifically, the Whale Optimization Algorithm and the Sine-Cosine Algorithm are tested by solving the Set Covering Problem, showing statistical improvements in this balance and in the quality of the solutions.

scholarly journals $$v_c$$-Noetherian domains and Krull domains

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Dedekind Domain ◽  
Closed Domain ◽  
One Dimensional ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Krull Domains

AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

scholarly journals Inversion-free geometric mapping construction: A survey

2021 ◽  
Vol 7 (3) ◽  
pp. 289-318
Author(s):  
Xiao-Ming Fu ◽  
Jian-Ping Su ◽  
Zheng-Yu Zhao ◽  
Qing Fang ◽  
Chunyang Ye ◽  
...  
Keyword(s):  
Mesh Generation ◽  
Optimization Problem ◽  
Texture Mapping ◽  
Deformation Texture ◽  
Research Field ◽  
Combinatorial Problems ◽  
Open Problems ◽  
Nonlinear Constrained Optimization ◽  
Construction Methods ◽  
Geometric Mapping

AbstractA geometric mapping establishes a correspondence between two domains. Since no real object has zero or negative volume, such a mapping is required to be inversion-free. Computing inversion-free mappings is a fundamental task in numerous computer graphics and geometric processing applications, such as deformation, texture mapping, mesh generation, and others. This task is usually formulated as a non-convex, nonlinear, constrained optimization problem. Various methods have been developed to solve this optimization problem. As well as being inversion-free, different applications have various further requirements. We expand the discussion in two directions to (i) problems imposing specific constraints and (ii) combinatorial problems. This report provides a systematic overview of inversion-free mapping construction, a detailed discussion of the construction methods, including their strengths and weaknesses, and a description of open problems in this research field.

scholarly journals Feature Selection and Parameter Optimization of Support Vector Machines Based on Modified Artificial Fish Swarm Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Kuan-Cheng Lin ◽  
Sih-Yang Chen ◽  
Jason C. Hung
Keyword(s):  
Feature Selection ◽  
Parameter Optimization ◽  
Combinatorial Problem ◽  
Combinatorial Problems ◽  
Support Vector ◽  
Local Optimum ◽  
Artificial Fish Swarm Algorithm ◽  
Swarm Algorithms ◽  
Artificial Fish Swarm ◽  
Information And Communication

Rapid advances in information and communication technology have made ubiquitous computing and the Internet of Things popular and practicable. These applications create enormous volumes of data, which are available for analysis and classification as an aid to decision-making. Among the classification methods used to deal with big data, feature selection has proven particularly effective. One common approach involves searching through a subset of the features that are the most relevant to the topic or represent the most accurate description of the dataset. Unfortunately, searching through this kind of subset is a combinatorial problem that can be very time consuming. Meaheuristic algorithms are commonly used to facilitate the selection of features. The artificial fish swarm algorithm (AFSA) employs the intelligence underlying fish swarming behavior as a means to overcome optimization of combinatorial problems. AFSA has proven highly successful in a diversity of applications; however, there remain shortcomings, such as the likelihood of falling into a local optimum and a lack of multiplicity. This study proposes a modified AFSA (MAFSA) to improve feature selection and parameter optimization for support vector machine classifiers. Experiment results demonstrate the superiority of MAFSA in classification accuracy using subsets with fewer features for given UCI datasets, compared to the original FASA.

Lower bounds for combinatorial problems on graphs

1985 ◽  
Vol 6 (3) ◽  
pp. 393-399 ◽  
Author(s):  
Hiroyuki Nakayama ◽  
Takao Nishizeki ◽  
Nobuji Saito
Keyword(s):  
Lower Bounds ◽  
Combinatorial Problems

scholarly journals On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

2008 ◽  
Vol 17 (2) ◽  
pp. 203-224 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
CSABA D. TÓTH
Keyword(s):  
Unit Volume ◽  
Time Algorithm ◽  
Combinatorial Problems ◽  
List Type ◽  
Type Number ◽  
Point Sets ◽  
Minimum Number ◽  
Three Space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined bynpoints in 3-space, and in general inddimensions.(i)The number of tetrahedra of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$3is at most$\frac{2}{3}n^3-O(n^2)$, and there are point sets for which this number is$\frac{3}{16}n^3-O(n^2)$. We also present anO(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every$k,d\in \mathbb{N}, 1\leq k \leq d$, the maximum number ofk-dimensional simplices of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$dis Θ(nk).(ii)The number of unit volume tetrahedra determined bynpoints in$\mathbb{R}$3isO(n7/2), and there are point sets for which this number is Ω(n3log logn).(iii)For every$d\in \mathbb{N}$, the minimum number of distinct volumes of all full-dimensional simplices determined bynpoints in$\mathbb{R}$d, not all on a hyperplane, is Θ(n).

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