scholarly journals Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 666
Author(s):  
Raúl M. Falcón
Keyword(s):  
Algebraic Geometry ◽  
Computer Algebra System ◽  
Latin Squares ◽  
Combinatorial Structures ◽  
New Approach ◽  
Algebraic Set ◽  
Computational Algebraic Geometry ◽  
Isomorphism Classes ◽  
Partial Transpose ◽  
Partial Latin Squares

With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic.

scholarly journals Switching in One-Factorisations of Complete Graphs

10.37236/3606 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Petteri Kaski ◽  
André de Souza Medeiros ◽  
Patric R.J. Östergård ◽  
Ian M. Wanless
Keyword(s):  
Degree Sequence ◽  
Clique Number ◽  
Latin Squares ◽  
Complete Graphs ◽  
Switching Operation ◽  
Isomorphism Classes ◽  
Switching Graph

We define two types of switchings between one-factorisations of complete graphs, called factor-switching and vertex-switching. For each switching operation and for each $n\le 12$, we build a switching graph that records the transformations between isomorphism classes of one-factorisations of $K_{n}$.  We establish various parameters of our switching graphs, including order, size, degree sequence, clique number and the radius of each component.As well as computing data for $n\le12$, we demonstrate several properties that hold for one-factorisations of $K_{n}$ for general $n$. We show that such factorisations have a parity which is not changed by factor-switching, and this leads to disconnected switching graphs. We also characterise the isolated vertices that arise from an absence of switchings. For factor-switching the isolated vertices are perfect one-factorisations, while for vertex-switching the isolated vertices are closely related to atomic Latin squares.

scholarly journals A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Jakob Cimprič
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Representation Theorem ◽  
Commutative Rings ◽  
Real Algebraic Geometry ◽  
New Approach ◽  
Noncommutative Version ◽  
Quadratic Modules ◽  
Rings With Involution ◽  
Associative Rings

AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

Numerical irreducible decomposition over a number field

2018 ◽  
Vol 17 (10) ◽  
pp. 1850195
Author(s):  
Timothy M. McCoy ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Number Field ◽  
Polynomial System ◽  
Field Extension ◽  
General Point ◽  
Numerical Algebraic Geometry ◽  
Irreducible Decomposition ◽  
Algebraic Set ◽  
Field Of Definition ◽  
Witness Set

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].

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