Projective congruent symmetric matrices enumeration

2019 ◽  
pp. 204-210
Author(s):  
Olga. A Starikova
Keyword(s):  
Normal Form ◽  
Quadratic Form ◽  
Local Ring ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Symmetric Matrix ◽  
Projective Spaces ◽  
Symmetric Matrices ◽  
Matrix Classes

Projective spaces over local ring R = 2R with principal maximal ideal J; 1+J ⊆ R*2 have been investigated. Quadratic forms and corresponding symmetric matrices A and B are projectively congruent if kA = UBU T for a matrix U ∈ GL(n;R) and for some k ∈ R * : In the case of k = 1 quadratic forms (corresponding symmetric matrices) are called congruent. The problem of enumerating congruent and projective congruent quadratic forms is based on the identification of the (unique) normal form of the corresponding symmetric matrices and is related to the theory of quadratic form schemes. Over the local ring R on conditions R * =R *2 ={1;-1; p;-p} and D(1; 1)=D(1; p)={1; p}; D(1;-1)=D(1;-p)={1;-1; p;-p} (unique) normal form of congruent symmetric matrices over ring R is detected. Quantities of congruent and projective congruent symmetric matrix classes is found when maximal ideal is nilpotent.

scholarly journals Estimating the Quadratic Form xTA−mx for Symmetric Matrices: Further Progress and Numerical Computations

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1432
Author(s):  
Marilena Mitrouli ◽  
Athanasios Polychronou ◽  
Paraskevi Roupa ◽  
Ondřej Turek
Keyword(s):  
Quadratic Form ◽  
Projection Method ◽  
Quadratic Forms ◽  
Absolute Error ◽  
Upper Bounds ◽  
Symmetric Matrices ◽  
Numerical Computations ◽  
Numerical Examples ◽  
Minimization Procedure

In this paper, we study estimates for quadratic forms of the type xTA−mx, m∈N, for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper bounds for the corresponding absolute error. Specifically, we consider three different approaches for estimating the quadratic form xTA−mx. The first approach is based on a projection method, the second is a minimization procedure, and the last approach is heuristic. Numerical examples showing the effectiveness of the estimates are presented. Furthermore, we compare the behavior of the proposed estimates with other methods that are derived in the literature.

scholarly journals Strengthening of a theorem on Coxeter–Euclidean type of principal partially ordered sets

2018 ◽  
pp. 8-15
Author(s):  
V. Bondarenko ◽  
M. Styopochkina
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Symmetric Matrix ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Dynkin Diagrams ◽  
Dynkin Quivers ◽  
Euclidean Type ◽  
Modern Mathematics

Among the quadratic forms, playing an important role in modern mathematics, the Tits quadratic forms should be distinguished. Such quadratic forms were first introduced by P. Gabriel for any quiver in connection with the study of representations of quivers (also introduced by him). P. Gabriel proved that the connected quivers with positive Tits form coincide with the Dynkin quivers. This quadratic form is naturally generalized to a poset. The posets with positive quadratic Tits form (analogs of the Dynkin diagrams) were classified by the authors together with the P-critical posets (the smallest posets with non-positive quadratic Tits form). The quadratic Tits form of a P-critical poset is non-negative and corank of its symmetric matrix is 1. In this paper we study all posets with such two properties, which are called principal, related to equivalence of their quadratic Tits forms and those of Euclidean diagrams. In particular, one problem posted in 2014 is solved.

scholarly journals Asymmetric minima of indefinite ternary quadratic forms

1967 ◽  
Vol 7 (2) ◽  
pp. 191-238 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Symmetric Matrix ◽  
Real Symmetric Matrix ◽  
Ternary Quadratic Forms

Let f = f(x) = f(x1, x2,…, xn) be an indefinite n-ary quadratic form of determinant det (f); that is, f(x) = x' Ax where A is a real symmetric matrix with determinant det (f). Such a form is said to take the value v if there exists integral x ≠ 0 such that f(x) = v.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Toward a theory of generalized Cohen-Macaulay modules

1986 ◽  
Vol 102 ◽  
pp. 1-49 ◽  
Author(s):  
Ngô Viêt Trung
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

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