GRADED MORITA EQUIVALENCES FOR GEOMETRIC AS-REGULAR ALGEBRAS

AbstractClassification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

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A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000302 ◽

2020 ◽

pp. 1-19

Author(s):

Masaki Matsuno

Keyword(s):

Algebraic Geometry ◽

Complete Classification ◽

Complete List ◽

Graded Algebra ◽

Regular Algebra ◽

3 Dimensional ◽

Noncommutative Algebraic Geometry ◽

Regular Algebras ◽

Point Scheme

Abstract Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$ -dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$ . In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$ -dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

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AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788721000070 ◽

2021 ◽

pp. 1-25

Author(s):

AYAKO ITABA ◽

MASAKI MATSUNO

Keyword(s):

Elliptic Curves ◽

Three Dimensional ◽

Preceding Paper ◽

Regular Algebra ◽

Noncommutative Algebraic Geometry ◽

Defining Relations ◽

Morita Equivalent ◽

Cubic Curve ◽

Regular Algebras ◽

Geometric Algebras

Abstract In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

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Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

Revista Integración ◽

10.18273/revint.v39n1-2021007 ◽

2021 ◽

Vol 39 (1) ◽

Author(s):

Héctor Suárez ◽

Duban Cáceres ◽

Armando Reyes

Keyword(s):

Differential Geometry ◽

Algebraic Geometry ◽

Regular Algebra ◽

Noncommutative Algebraic Geometry ◽

Noncommutative Differential Geometry ◽

Regular Algebras ◽

Skew Pbw Extensions ◽

Finitely Presented

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

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Some Constructions of Bi-Koszul Algebras

Algebra Colloquium ◽

10.1142/s1005386713000126 ◽

2013 ◽

Vol 20 (01) ◽

pp. 141-154

Author(s):

Junru Si ◽

Jiafeng Lü

Keyword(s):

Global Dimension ◽

Koszul Algebras ◽

Natural Question ◽

Graded Algebras ◽

Regular Algebras ◽

Ore Extensions

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed.

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Phonetic iconicity in the evaluative morphology of a sample of Indo-European, Niger-Congo and Austronesian languages

WORD Structure ◽

10.3366/word.2010.0003 ◽

2010 ◽

Vol 3 (2) ◽

pp. 156-180 ◽

Cited By ~ 3

Author(s):

Renáta Gregová ◽

Lívia Körtvélyessy ◽

Július Zimmermann

Keyword(s):

Three Dimensional ◽

Sound Symbolism ◽

Three Dimensional Model ◽

Austronesian Languages ◽

Place Of Articulation ◽

European Languages ◽

Phonetic Symbolism ◽

The Individual ◽

Core Vocabulary

Universals Archive (Universal #1926) indicates a universal tendency for sound symbolism in reference to the expression of diminutives and augmentatives. The research ( Štekauer et al. 2009 ) carried out on European languages has not proved the tendency at all. Therefore, our research was extended to cover three language families – Indo-European, Niger-Congo and Austronesian. A three-step analysis examining different aspects of phonetic symbolism was carried out on a core vocabulary of 35 lexical items. A research sample was selected out of 60 languages. The evaluative markers were analyzed according to both phonetic classification of vowels and consonants and Ultan's and Niewenhuis' conclusions on the dominance of palatal and post-alveolar consonants in diminutive markers. Finally, the data obtained in our sample languages was evaluated by means of a three-dimensional model illustrating the place of articulation of the individual segments.

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Three-Dimensional Analysis and Classification of Arteries in the Skin and Subcutaneous Adipofascial Tissue by Computer Graphics Imaging

Plastic & Reconstructive Surgery ◽

10.1097/00006534-199809010-00020 ◽

1998 ◽

Vol 102 (3) ◽

pp. 748-760 ◽

Cited By ~ 36

Author(s):

Hideo Nakajima ◽

Toshiharu Minabe ◽

Nobuaki Imanishi

Keyword(s):

Computer Graphics ◽

Dimensional Analysis ◽

Three Dimensional

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How Can I Grab That?

i-com ◽

10.1515/icom-2020-0011 ◽

2020 ◽

Vol 19 (2) ◽

pp. 67-85

Author(s):

Matthias Weise ◽

Raphael Zender ◽

Ulrike Lucke

Keyword(s):

Virtual Reality ◽

User Experience ◽

Dimensional Space ◽

Three Dimensional ◽

Advantages And Disadvantages ◽

Multiple Dimensions ◽

Application Developers ◽

Three Dimensional Space

AbstractThe selection and manipulation of objects in Virtual Reality face application developers with a substantial challenge as they need to ensure a seamless interaction in three-dimensional space. Assessing the advantages and disadvantages of selection and manipulation techniques in specific scenarios and regarding usability and user experience is a mandatory task to find suitable forms of interaction. In this article, we take a look at the most common issues arising in the interaction with objects in VR. We present a taxonomy allowing the classification of techniques regarding multiple dimensions. The issues are then associated with these dimensions. Furthermore, we analyze the results of a study comparing multiple selection techniques and present a tool allowing developers of VR applications to search for appropriate selection and manipulation techniques and to get scenario dependent suggestions based on the data of the executed study.

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Research on cargo-loading optimization based on genetic and fuzzy integration

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-189669 ◽

2021 ◽

Vol 40 (4) ◽

pp. 8493-8500

Author(s):

Yanwei Du ◽

Feng Chen ◽

Xiaoyi Fan ◽

Lei Zhang ◽

Henggang Liang

Keyword(s):

Genetic Algorithm ◽

Mathematical Model ◽

Three Dimensional ◽

Distribution Center ◽

Long Time ◽

Low Efficiency ◽

Decision Making Model ◽

The Mathematical Model ◽

Loading Scheme

With the increase of the number of loaded goods, the number of optional loading schemes will increase exponentially. It is a long time and low efficiency to determine the loading scheme with experience. Genetic algorithm is a search heuristic algorithm used to solve optimization in the field of computer science artificial intelligence. Genetic algorithm can effectively select the optimal loading scheme but unable to utilize weight and volume capacity of cargo and truck. In this paper, we propose hybrid Genetic and fuzzy logic based cargo-loading decision making model that focus on achieving maximum profit with maximum utilization of weight and volume capacity of cargo and truck. In this paper, first of all, the components of the problem of goods stowage in the distribution center are analyzed systematically, which lays the foundation for the reasonable classification of the problem of goods stowage and the establishment of the mathematical model of the problem of goods stowage. Secondly, the paper abstracts and defines the problem of goods loading in distribution center, establishes the mathematical model for the optimization of single car three-dimensional goods loading, and designs the genetic algorithm for solving the model. Finally, Matlab is used to solve the optimization model of cargo loading, and the good performance of the algorithm is verified by an example. From the performance evaluation analysis, proposed the hybrid system achieve better outcomes than the standard SA model, GA method, and TS strategy.

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