2,251 results
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2. Rupture or continuity: The arithmetico-algebraic thinking as an alternative in a modelling process in a paper and pencil and technology environment.
- Author
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Hitt, Fernando, Saboya, Mireille, and Zavala, Carlos
- Subjects
- *
ALGEBRA education in elementary schools , *ALGEBRAIC geometry , *COGNITIVE structures , *POLYGONAL numbers , *ELEMENTARY education , *CHILDREN - Abstract
Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
3. The points and localisations of the topos of M-sets.
- Author
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Pirashvili, Ilia
- Subjects
IDEMPOTENTS ,ALGEBRAIC geometry ,MONOIDS - Abstract
In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos Qcoh(X). This was achieved using topos points and localisations of Qcoh(X). With this philosophy in mind, the aim of this paper is to study the topos of M-sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M-sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M, respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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4. SUPPLEMENT TO PAPER "ON SOME PROBLEMS OF EXPANSIONS INVESTIGATED BY P. ERDÖS ET AL".
- Author
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Kallós, Gábor
- Subjects
NUMBER systems ,MATHEMATICAL sequences ,OPERATOR product expansions ,SET theory ,ALGEBRAIC geometry - Abstract
Based on a seminal theorem of P. Erdös et al and using the theory of interval-filling sequences, in this paper we present a generalized result, which ensures the existence of continuum many different fraction expansions, for numbers x ∊ (0, L), with digit set {0,1}. [ABSTRACT FROM AUTHOR]
- Published
- 2017
5. GMJ volume 64 issue 3 Cover and Back matter.
- Subjects
RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,HYPERBOLIC groups ,REPRESENTATION theory ,ORBIFOLDS ,RIEMANN surfaces - Published
- 2022
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- View/download PDF
6. Dimensions of the hull of generalized Reed-Solomon codes.
- Author
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Jing Huang, Jingge Liu, and Dong Yu
- Subjects
REED-Solomon codes ,DECODING algorithms ,ALGEBRAIC codes ,ALGEBRAIC geometry - Abstract
Let GRSk(α, υ) be a k-dimensional generalized Reed-Solomon (GRS) code over Fq associated with α = (α
1 ,. . ., αn ) and υ = (υ1 , . . ., υn ). In this paper, we determined the dimension of the Euclidean hull GRSk(α, υ) ∩ GRSk(α, υ)⊥, which addresses an open problem posed in [Chen et al., IEEE-TIT, 2023]. We also presentd a new approach to generating all self-dual RS codes. [ABSTRACT FROM AUTHOR]- Published
- 2024
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7. Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings.
- Author
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Bagchi, Susmit
- Subjects
POLYNOMIAL rings ,ALGEBRAIC topology ,ALGEBRAIC geometry ,TOPOLOGICAL property ,MIRROR symmetry ,HOMEOMORPHISMS - Abstract
The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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- View/download PDF
8. GMJ volume 65 issue 3 Cover and Front matter.
- Subjects
FUNCTOR theory ,RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,REPRESENTATION theory ,CAYLEY graphs ,ABELIAN categories - Published
- 2023
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9. Noncommutative Nullstellensätze and Perfect Games.
- Author
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Bene Watts, Adam, Helton, J. William, and Klep, Igor
- Subjects
ALGEBRAIC geometry ,OPERATOR theory ,STRATEGY games ,GAMES ,LINEAR systems - Abstract
The foundations of classical algebraic geometry and real algebraic geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades, the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged. This paper concerns commuting operator strategies for nonlocal games, recalls NC Nullstellensatz which are helpful, extends these, and applies them to a very broad collection of games. In the process, it brings together results spread over different literature studies, hence rather than being terse, our style is fairly expository. The main results of this paper are two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. Previously, Paulsen and others translated the study of perfect synchronous games to problems entirely involving a ∗ -algebra. The characterization we present is analogous, but works for all games. The second characterization is based on a new Nullstellensatz we derive in this paper. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games, we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem and, for linear systems games, we further show this subgroup membership characterization is equivalent to the standard characterization of perfect commuting operator strategies in terms of solution groups. Both the general and torically determined games characterizations are amenable to computer algebra techniques, which we also develop. For context, we mention that Positivstellensätze are behind the standard NPA upper bound on the score players can achieve for a game using a commuting operator strategy. This paper develops analogous NC real algebraic geometry which bears on perfect games. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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10. PICARD GROUPOIDS AND г-CATEGORIES.
- Author
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Sharma, Amit
- Subjects
PICARD groups ,MATHEMATICAL equivalence ,HOMOTOPY groups ,TOPOLOGICAL spaces ,ALGEBRAIC geometry - Abstract
Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2023
11. GMJ volume 65 issue S1 Cover and Back matter.
- Subjects
RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,REPRESENTATION theory ,APPLIED mathematics ,FUNCTIONAL analysis - Published
- 2023
- Full Text
- View/download PDF
12. On a family of linear MRD codes with parameters [8×8,16,7]q.
- Author
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Timpanella, Marco and Zini, Giovanni
- Subjects
ALGEBRAIC geometry ,ALGEBRAIC varieties ,FINITE fields ,FINITE geometries ,PROJECTIVE spaces ,LINEAR codes ,FAMILIES - Abstract
In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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13. Spheres over fields, their entire rational maps and applications.
- Author
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Golasiński, Marek
- Subjects
HOMOTOPY groups ,ALGEBRAIC geometry ,HOMOTOPY theory - Abstract
The paper summarizes some results on algebraic geometry presence in the homotopy theory. For the homotopy group π
m (Sn ), denote by πm alg (Sn ) its subset of homotopy classes represented by R-entire rational maps Sm → Sn of spheres. The main result of this paper concerns to the study of πn+k alg (Sn ) for k = 0, . . ., 7. [ABSTRACT FROM AUTHOR]- Published
- 2024
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14. de Broglie, General Covariance and a Geometric Background to Quantum Mechanics.
- Author
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Hiley, Basil and Dennis, Glen
- Subjects
QUANTUM mechanics ,WAVE-particle duality ,QUANTUM theory ,ALGEBRAIC geometry ,WAVE functions ,KLEIN-Gordon equation - Abstract
What is striking about de Broglie's foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ 2 Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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15. Foreword.
- Author
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Ghorpade, Sudhir R. and Sathaye, Avinash M.
- Subjects
ALGEBRA periodicals ,PERIODICAL editors ,ALGEBRAIC geometry ,MATHEMATICAL research ,PUBLISHING - Published
- 2015
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16. Bifurcation theory for Fredholm operators.
- Author
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López-Gómez, Julián and Sampedro, Juan Carlos
- Subjects
- *
FREDHOLM operators , *BIFURCATION theory , *OPERATOR theory , *BOUNDARY value problems , *BIFURCATION diagrams , *ALGEBRAIC geometry - Abstract
This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. A non-iterative formula for straightening fillings of Young diagrams.
- Author
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Hodges, Reuven
- Subjects
ALGEBRAIC geometry ,REPRESENTATION theory ,GENERALIZATION - Abstract
Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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18. GMJ volume 65 issue 1 Cover and Back matter.
- Subjects
RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,REPRESENTATION theory ,COXETER groups ,ENDOMORPHISMS ,FUNCTIONAL analysis - Published
- 2023
- Full Text
- View/download PDF
19. GMJ volume 65 issue 2 Cover and Back matter.
- Subjects
RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,REPRESENTATION theory ,BRAID group (Knot theory) ,APPLIED mathematics ,SHEAF theory - Published
- 2023
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20. Commutative Algebra : Expository Papers Dedicated to David Eisenbud on the Occasion of His 75th Birthday
- Author
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Irena Peeva and Irena Peeva
- Subjects
- Commutative algebra, Commutative rings, Algebraic geometry, Associative rings, Associative algebras
- Abstract
This contributed volume is a follow-up to the 2013 volume of the same title, published in honor of noted Algebraist David Eisenbud's 65th birthday. It brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Category Theory, Combinatorics, Computational Algebra, Homological Algebra, Hyperplane Arrangements, and Non-commutative Algebra.The book aims to showcase the area and aid junior mathematicians and researchers who are new to the field in broadening their background and gaining a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.
- Published
- 2022
21. GMJ volume 63 issue 2 Cover and Back matter.
- Subjects
NILPOTENT Lie groups ,APPLIED mathematics ,ALGEBRAIC geometry ,RING theory ,NONCOMMUTATIVE algebras ,REPRESENTATION theory - Published
- 2021
- Full Text
- View/download PDF
22. POSET-BLOWDOWNS OF GENERALIZED QUATERNION GROUPS.
- Author
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RYOTA HIRAKAWA, KENJIRO SASAKI, and SHIGERU TAKAMURA
- Subjects
QUATERNIONS ,ALGEBRAIC geometry ,FINITE groups ,PARTIALLY ordered sets - Abstract
Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: tame, wild, and hybrid depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups Q
2n (n ≥ 3). They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of Q2n . [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
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23. Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold.
- Author
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Cai, Shi-Fan, Chaubey, Sudhakar Kumar, Xu, Xin, Zhang, Pan, and Zhang, Zhi-Heng
- Subjects
ANALYTIC geometry ,ALGEBRAIC geometry ,METRIC geometry - Abstract
The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ , τ) -Hermite–Yang–Mills metric in differential geometry and the analytic (σ , τ) -stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau's continuity method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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24. Preface.
- Subjects
ALGEBRAIC geometry ,NONCOMMUTATIVE rings ,CODING theory ,DIFFERENTIAL algebra ,COMMUTATIVE rings - Abstract
The call for papers welcomed any original paper within the scope and not simultaneously submitted to another journal or conference. Cristina Bertone, Universitá di Torino, Italy Michela Ceria, Politecnico di Bari, Italy Fatma Karaoglu, Gebze Technical University, Turkey Ilias Kotsireas, Wilfrid Laurier University, Canada Teo Mora, University of Genova, Italy Dimitris Simos, SBA Research Center, Austria Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This volume contains the refereed papers related to the 26th International Conference on Applications of Computer Algebra, which has been held online, from the 23rd to the 27th of July 2021. [Extracted from the article]
- Published
- 2022
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25. On Nonlinear Kloosterman Sums.
- Author
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Korolev, M. A.
- Subjects
EXPONENTIAL sums ,ALGEBRAIC geometry ,NUMBER theory - Abstract
Exponential sums of a special type—the so-called Kloosterman sums—play a key role in numerous number-theoretic problems concerning the distribution of inverse residues in residue rings modulo a given q. In many cases, estimates of such sums are based on A. Weil's bound for complete Kloosterman sums over primes. This bound allows one to estimate Kloosterman sums of length for any fixed with a power-saving factor. Weil's bound was originally proved using methods of algebraic geometry. Later, S.A. Stepanov gave an elementary proof of this bound, but this proof was also complicated enough. The aim of this paper is to give an elementary proof of an estimate for the Kloosterman sum of length , which also leads to a power-saving factor. This proof is based on the trick of additive shift of the summation variable, which is widely used in various number theory problems. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
26. On a relation between GAG codes and AG codes.
- Author
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Şenel, Engin and Öke, Figen
- Subjects
ALGEBRAIC geometry ,ALGEBRAIC codes ,AUTOMORPHISM groups ,ISOMORPHISM (Mathematics) ,ALGEBRAIC fields ,ALGEBRAIC functions - Abstract
In this paper, we first give a relationship between generalized algebraic geometry codes (GAG codes) and algebraic geometry codes (AG codes). More precisely, we show that a GAG code is contained (up to isomorphism) in a suitable AG code. Next we recall the concept of an N
1 N2 -automorphism group, a subgroup of the automorphism group of a GAG code. With the use of the relation we obtained between these two classes of codes, we show that the N1 N2 -automorphism group is a subgroup of the automorphism group of an AG code. [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
27. A Topological Approach to the Bézout' Theorem and Its Forms.
- Author
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Bagchi, Susmit
- Subjects
ALGEBRAIC curves ,ALGEBRAIC topology ,ALGEBRAIC geometry ,TOPOLOGICAL spaces ,TOPOLOGY - Abstract
The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
28. GMJ volume 61 issue 2 Cover and Back matter.
- Subjects
SYMPLECTIC spaces ,APPLIED mathematics ,RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry - Published
- 2019
- Full Text
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29. Convex plumbings in closed hyperbolic 4-manifolds
- Author
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Leone Slavich, Bruno Martelli, Stefano Riolo, Martelli B., Riolo S., and Slavich L.
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Hyperbolic geometry ,Hyperbolic 4-manifold ,Intersection form ,Plumbing ,Algebraic geometry ,01 natural sciences ,Mathematics - Geometric Topology ,Simple (abstract algebra) ,Hyperbolic set ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Original Paper ,010102 general mathematics ,Regular polygon ,Geometric Topology (math.GT) ,Submanifold ,Mathematics::Geometric Topology ,57M50 ,Differential geometry ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry - Abstract
We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold., 18 pages, 11 figures
- Published
- 2021
30. Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic
- Author
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Michelle Bucher, Caterina Campagnolo, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Surface (mathematics) ,Pure mathematics ,Hyperbolic geometry ,0102 computer and information sciences ,Algebraic geometry ,Signature ,01 natural sciences ,symbols.namesake ,Mathematics - Geometric Topology ,Mathematics::Algebraic Geometry ,Simplicial volume ,Euler characteristic ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,FOS: Mathematics ,Secondary 57R22 ,0101 mathematics ,[MATH]Mathematics [math] ,ddc:510 ,Mathematics::Symplectic Geometry ,Mathematics ,Projective geometry ,Surface bundle ,Original Paper ,010102 general mathematics ,Geometric Topology (math.GT) ,55R10, 57R22 ,Primary 55R10 ,16. Peace & justice ,Differential geometry ,010201 computation theory & mathematics ,symbols ,Geometry and Topology ,Signature (topology) - Abstract
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature., 14 pages. Simplified the proof of Proposition 1.2. This is the final version, accepted in Geometriae Dedicata
- Published
- 2021
31. GMJ volume 64 issue 1 Cover and Back matter.
- Subjects
CLUSTER algebras ,SHEAF theory ,APPLIED mathematics ,RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,MODULES (Algebra) - Published
- 2022
- Full Text
- View/download PDF
32. Lefschetz duality for local cohomology.
- Author
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Varbaro, Matteo and Yu, Hongmiao
- Subjects
- *
ALGEBRAIC geometry , *MATHEMATICAL connectedness , *ALGEBRAIC topology - Abstract
Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the first author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Defects of codes from higher dimensional algebraic varieties.
- Author
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Can, Mahir Bilen, Joshua, Roy, and Ravindra, G. V.
- Subjects
ALGEBRAIC codes ,ALGEBRAIC geometry ,ALGEBRAIC varieties - Abstract
An MDS code is a code which achieves equality in the singleton bound. The defect of a code measures how far it is from an MDS code. Amplifying on the relationship between the weight distribution of a code and its dual code as in the well-known MacWilliams identities, we show in this paper that there are indeed strong lower bounds on the defects of codes or the dual codes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Cohomology of flag supervarieties and resolutions of determinantal ideals.
- Author
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Sam, Steven V. and Snowden, Andrew
- Subjects
SHEAF cohomology ,DETERMINANTAL varieties ,GRASSMANN manifolds ,ALGEBRAIC geometry ,ALGEBRA - Abstract
We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated with the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers.
- Author
-
Sho Ejiri
- Subjects
FIBER spaces (Mathematics) ,ALGEBRAIC geometry ,LOGICAL prediction ,HYPOTHESIS ,ALGEBRA - Abstract
In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber F has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when F is F-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases. We also prove an analog of Fujino's result regarding his Fujita-type conjecture. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. The Properties of Topological Manifolds of Simplicial Polynomials.
- Author
-
Bagchi, Susmit
- Subjects
TOPOLOGICAL property ,ALGEBRAIC topology ,POLYNOMIALS ,ALGEBRAIC geometry ,TOPOLOGICAL spaces ,HOMEOMORPHISMS ,GROBNER bases - Abstract
The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. Reduction of general one-loop integrals using auxiliary vector.
- Author
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Feng, Bo, Li, Tingfei, Wang, Hongbin, and Zhang, Yaobo
- Subjects
INTEGRALS ,INTEGRAL equations ,DIFFERENTIAL equations ,SCATTERING amplitude (Physics) ,ALGEBRAIC geometry ,MASS transfer - Abstract
As a key method to deal with loop integrals, Integration-By-Parts (IBP) method can be used to do reduction as well as establish the differential equations for master integrals. However, when talking about tensor reduction, the Passarino-Veltman (PV) reduction method is also widely used for one-loop integrals. Recently, we have proposed an improved PV reduction method, i.e., the PV reduction method with auxiliary vector R, which can easily give analytical reduction results for any tensor rank. However, our results are only for integrals with propagators with power one. In this paper, we generalize our method to one-loop integrals with general tensor structures and propagators with general powers. Our ideas are simple. We solve the generalised reduction problem by combining differentiation over masses and proper limit of reduction with power-one propagators. Finally, we demonstrate our method with several examples. With the result in this paper, we have shown that our improved PV-reduction method with auxiliary vector is a self-completed reduction method for one-loop integrals. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
38. Method of Verbal Operations and Automorphisms of the Category of Free Algebras.
- Author
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Aladova, E. V.
- Subjects
MATHEMATICAL logic ,ALGEBRAIC geometry ,ALGEBRAIC varieties ,ALGEBRA - Abstract
Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. This paper is the second in a series of papers started in [Algebra and Logic, 61, No. 1, 1-15 (2022)] where we deal with automorphisms of the category of free finitely generated algebras. Here we describe in detail a method of verbal operations. The method provides a characterization of automorphisms of the category of all free finitely generated algebras in a given variety. The characterization plays a crucial role in universal algebraic geometry. We supply the reader with illuminating examples which clarify the method. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
39. AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES.
- Author
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ITABA, AYAKO and MATSUNO, MASAKI
- Subjects
PLANE curves ,CUBIC curves ,ALGEBRAIC geometry ,ALGEBRA ,RELATION algebras ,ELLIPTIC curves ,ARTIN 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. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
40. Reconfiguration Analysis and Characteristics of a Novel 8-Link Variable-DOF Planar Mechanism with Five Motion Modes.
- Author
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Kong, Xianwen and Wang, Jieyu
- Subjects
ALGEBRAIC geometry ,PARALLELOGRAMS ,DEGREES of freedom ,ANGLES - Abstract
Variable-DOF (or kinematotropic) mechanisms are a class of reconfigurable mechanisms that have varying degrees of freedom (DOF) in different motion modes and can be reconfigured without disassembly. However, the number of proposed variable-DOF multi-loop planar mechanisms is currently limited. This paper introduces a new 8-link variable-DOF planar mechanism that has five motion modes. Firstly, the 8-link variable-DOF planar mechanism is described. Then, reconfiguration analysis of the mechanism is performed using a hybrid approach that combines elimination and computer algebraic geometry methods. The analysis reveals that the 8-link mechanism has one 2-DOF motion mode and four 1-DOF motion modes. It can switch among three motion modes at four transition configurations and between two motion modes at the remaining four transition configurations. The paper also highlights the geometric characteristics of the mechanism in different motion modes. In contrast to variable-DOF planar mechanisms presented in the literature, the proposed 8-link mechanism has two inactive joints in one of its 1-DOF motion modes. Moreover, both closed-loop 4R kinematic sub-chains of the mechanism must appear as either a pair of parallelograms or a pair of anti-parallelograms in the same motion mode. As a by-product of this research, a method for factoring trigonometric functions in two angles is also proposed. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. The influence of test mode and visuospatial ability on mathematics assessment performance.
- Author
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Logan, Tracy
- Subjects
MATHEMATICS ,ACADEMIC achievement ,REASONING ,VISUAL extinction (Neurology) ,ALGEBRAIC geometry - Abstract
Mathematics assessment and testing are increasingly situated within digital environments with international tests moving to computer-based testing in the near future. This paper reports on a secondary data analysis which explored the influence the mode of assessment-computer-based (CBT) and pencil-and-paper based (PPT)-and visuospatial ability had on students' mathematics test performance. Data from 804 grade 6 Singaporean students were analysed using the knowledge discovery in data design. The results revealed statistically significant differences between performance on CBT and PPT test modes across content areas concerning whole number algebraic patterns and data and chance. However, there were no performance differences for content areas related to spatial arrangements geometric measurement or other number. There were also statistically significant differences in performance between those students who possess higher levels of visuospatial ability compared to those with lower levels across all six content areas. Implications include careful consideration for the comparability of CBT and PPT testing and the need for increased attention to the role of visuospatial reasoning in student's mathematics reasoning. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
42. GMJ volume 64 issue 2 Cover and Back matter.
- Subjects
RING theory ,NONCOMMUTATIVE algebras ,ALGEBRAIC geometry ,APPLIED mathematics ,REPRESENTATION theory ,FUNCTIONAL analysis ,COMMUTATORS (Operator theory) - Published
- 2022
- Full Text
- View/download PDF
43. Special Issue: "Algebraic Geometry from an Algorithmic Point of View", ACA 2019, Montréal, Canada, July 16–20, 2019.
- Author
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Bertone, Cristina, Cioffi, Francesca, and Pardo, Luis M.
- Subjects
ALGEBRAIC geometry ,PLANE curves ,GROBNER bases ,ALGORITHMS ,APPLIED mathematics ,TORIC varieties - Published
- 2020
- Full Text
- View/download PDF
44. Machine learning from a mathematical perspective.
- Author
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Rajendra, P., Ravi, Pusuluri V. N. H., and Meenakshi, K.
- Subjects
- *
VECTOR spaces , *SINGULAR value decomposition , *ALGEBRAIC geometry , *FUNCTION algebras , *ALGEBRAIC varieties - Abstract
Mathematics plays an essential role in the domain of machine learning by providing the tools necessary for representing, analyzing, and understanding data. In this paper, we will delve into several mathematical concepts that are commonly used in machine learning, including linear algebra, calculus, probability and statistics, optimization, algebraic geometry, and topology. Linear algebra is a mathematical discipline concerned with the study of vector spaces and linear transformations. Within the framework of machine learning, linear algebra is used to represent data as vectors and perform operations such as vector addition, scalar multiplication, and dot products. Algorithms such as principal component analysis (PCA), singular value decomposition (SVD), and matrix factorization (MF) rely on linear algebra to function. The field of calculus, which focuses on the analysis of rates of change and the accumulation of quantities, plays a crucial role in machine learning by facilitating the optimization of models through the identification of function minima or maxima. Optimization methods, such as gradient descent and stochastic gradient descent, commonly employed in the training of neural networks, leverage principles from calculus to determine the most favourable solution. Probability and statistics are critical for understanding and analyzing data. Probability is used to model uncertainty, while statistics are used to describe and summarize data. In machine learning, probability and statistics are used to estimate the performance of models, analyze bias and variance, and validate experiment results. The concept of optimisation, which involves the identification of the most favourable solution to a given problem, holds significant importance in the field of machine learning. The purpose of this technique is to identify the parameters of a model that result in the lowest possible value for the error or loss function. Commonly employed optimisation strategies in the field of machine learning encompass gradient descent, stochastic gradient descent, and genetic algorithms. Algebraic geometry is a field within mathematics concerned with the investigation of algebraic varieties, which are the solutions to systems of polynomial equations. Algebraic geometry is employed in the field of machine learning to examine the geometric characteristics of neural networks and evaluate the efficacy of deep learning models. Topology is an academic discipline concerned with the examination of spatial features that remain invariant under continuous transformations. under continuous deformations, is used in machine learning to analyze the structure of data and identify patterns and relationships. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. GMJ volume 61 issue 1 Cover and Back matter.
- Subjects
APPLIED mathematics ,METRIC spaces ,ALGEBRAIC geometry ,RING theory ,NONCOMMUTATIVE algebras - Published
- 2019
- Full Text
- View/download PDF
46. Algebraic Morphology of DNA–RNA Transcription and Regulation.
- Author
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Planat, Michel, Amaral, Marcelo M., and Irwin, Klee
- Subjects
GENETIC transcription regulation ,INFINITE groups ,ALGEBRAIC geometry ,GENERATORS of groups ,FREE groups ,GENE enhancers - Abstract
Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π , we compute the S L (2 , C) character variety, where S L (2 , C) is simultaneously a 'space-time' (a Lorentz group) and a 'quantum' (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group F r ( r = 1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group F r in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in { A , T / U , G , C } . We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. Falsification of Einstein's relativity.
- Author
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Taylor, Emory
- Subjects
ALGEBRAIC geometry ,FALSIFICATION ,FIELD theory (Physics) ,GERMAN language ,GENERAL relativity (Physics) ,MATHEMATICAL continuum - Abstract
Copyright of Physics Essays is the property of Physics Essays Publication and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2021
- Full Text
- View/download PDF
48. Singular spin structures and superstrings.
- Author
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Matone, Marco
- Subjects
RIEMANN surfaces ,QUADRICS ,ALGEBRAIC geometry ,DIFFERENTIAL geometry - Abstract
There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙ
g−1 . Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric. [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
49. Totally invariant divisors of non trivial endomorphisms of the projective space.
- Author
-
Mabed, Yanis
- Abstract
It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. Exploring tropical differential equations.
- Author
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Cotterill, Ethan, López, Cristhian Garay, and Luviano, Johana
- Subjects
DIFFERENTIAL equations ,ALGEBRAIC geometry ,DIFFERENTIAL geometry ,POWER series ,DIFFERENTIAL algebra ,NEWTON diagrams - Abstract
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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