scholarly journals ANALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES

2020 ◽  
pp. 1-26
Author(s):  
CARLES BIVIÀ-AUSINA ◽  
JONATHAN MONTAÑO
Keyword(s):  
Computer Algebra ◽  
Maximal Rank ◽  
Integral Closure ◽  
Simple Computer ◽  
Direct Sum ◽  
Newton Polyhedra ◽  
Analytic Spread ◽  
The Family

Abstract We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

Split Lie algebras of order 3

2019 ◽  
Vol 19 (04) ◽  
pp. 2050070
Author(s):  
Antonio J. Calderón ◽  
Rosa M. Navarro ◽  
José M. Sánchez
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Subspace ◽  
Type Theorem ◽  
Natural Generalization ◽  
Lie Superalgebras ◽  
Direct Sum ◽  
The Family ◽  
Split Lie Algebra

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.

scholarly journals Combining the Runge Approximation and the Whitney Embedding Theorem in Hybrid Imaging

2020 ◽  
Author(s):  
Giovanni S Alberti ◽  
Yves Capdeboscq
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Hybrid Imaging ◽  
Approximation Property ◽  
Elliptic Partial Differential Equation ◽  
Maximal Rank ◽  
Embedding Theorem ◽  
Vanishing Constraints ◽  
The Family

Abstract This paper addresses enforcing non-vanishing constraints for solutions to a 2nd-order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension $d\geq 2$, under suitable regularity assumptions, the family of $2d$ solutions such that their Jacobian has maximal rank in the domain is both open and dense. The case of less regular coefficients is also addressed, together with other constraints, which are relevant for applications to recent hybrid imaging modalities. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [ 44]. The method is very general and can be used in other settings.

A FAMILY OF DISCRETE INTEGRABLE COUPLING SYSTEMS AND ITS LIOUVILLE INTEGRABILITY

2009 ◽  
Vol 23 (13) ◽  
pp. 1671-1685
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Liouville Integrability ◽  
Discrete Integrable Systems ◽  
Direct Sum ◽  
The Family ◽  
Matrix Spectral Problem ◽  
Hamiltonian Forms ◽  
Integrable Coupling

A discrete matrix spectral problem and corresponding family of discrete integrable systems are discussed. A semi-direct sum of Lie algebras of four-by-four matrices is introduced, and the related integrable coupling systems of resulting discrete integrable systems are derived. The obtained discrete integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, Liouville integrability of the family of obtained integrable coupling systems is demonstrated.

scholarly journals Preorders on canonical families of modules of finite length

1992 ◽  
Vol 53 (1) ◽  
pp. 64-77 ◽  
Author(s):  
Uri Fixman ◽  
Frank Okoh
Keyword(s):  
Finite Length ◽  
Artinian Ring ◽  
Isomorphism Type ◽  
Type Ii ◽  
Dimensional Algebra ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Kronecker’S Theorem ◽  
The Family ◽  
Type C

AbstractLet R be an artinian ring. A family, ℳ, of isomorphism types of R-modules of finite length is said to be canonical if every R-module of finite length is a direct sum of modules whose isomorphism types are in ℳ. In this paper we show that ℳ is canonical if the following conditions are simultaneously satisfied: (a) ℳ contains the isomorphism type of every simple R-module; (b) ℳ has a preorder with the property that every nonempty subfamily of ℳ with a common bound on the lengths of its members has a smallest type; (c) if M is a nonsplit extension of a module of isomorphism type II1 by a module of isomorphism type II2, with II1, II2 in ℳ, then M contains a submodule whose type II3 is in ℳ and II1 does not precede II3. We use this result to give another proof of Kronecker's theorem on canonical pairs of matrices under equivalence. If R is a tame hereditary finite-dimensional algebra we show that there is a preorder on the family of isomorphism types of indecomposable R-modules of finite length that satisfies Conditions (b) and (c).

Lie superalgebras with a set grading

2020 ◽  
pp. 2150028 ◽  
Author(s):  
Valiollah Khalili
Keyword(s):  
Linear Subspace ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Minimal Set ◽  
Direct Sum ◽  
Simple Set ◽  
The Family

This paper studies Lie superalgebras graded by an arbitrary set [Formula: see text] (set grading). We show that the set-graded Lie superalgebra [Formula: see text] decomposes as the sum of well-described set-graded ideals plus a certain linear subspace. Under certain conditions, the simplicity of [Formula: see text] is characterized and it is shown that the above decomposition is exactly the direct sum of the family of its minimal set-graded ideals, each one being a simple set-graded Lie superalgebra.

On the Decomposition of Nonsingular CS-Modules

1996 ◽  
Vol 39 (3) ◽  
pp. 257-265 ◽  
Author(s):  
John Clark ◽  
Nguyen Viet Dung
Keyword(s):  
Direct Sum ◽  
The Family

AbstractIt is shown that if M is a nonsingular CS-module with an indecomposable decomposition M = ⊕i∊I Mi, then the family {Mi | i € I} is locally semi-T"- nilpotent. This fact is used to prove that any nonsingular self-generator Σ-CS module is a direct sum of uniserial Noetherian quasi-injective submodules. As an application, we provide a new proof of Goodearl's characterization of non-singular rings over which all nonsingular right modules are projective.

scholarly journals Multipartite Circulant States with Positive Partial Transposes

2008 ◽  
Vol 15 (03) ◽  
pp. 189-212 ◽  
Author(s):  
Dariusz Chruściński ◽  
Andrzej Kossakowski
Keyword(s):  
Hilbert Space ◽  
Large Class ◽  
Quantum States ◽  
Direct Sum ◽  
Circular Structure ◽  
The Family ◽  
Direct Sum Decomposition ◽  
New States

We construct a large class of multipartite qudit states which are positive under the family of partial transpositions. The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure and hence generalizes a class of bipartite circulant states proposed recently by the authors. This class contains many well-known examples of multipartite quantum states from the literature and gives rise to a huge family of completely new states.

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