scholarly journals When every multilinear mapping is multiple summing

2009 ◽  
Vol 282 (10) ◽  
pp. 1414-1422 ◽  
Author(s):  
Geraldo Botelho ◽  
Daniel Pellegrino
Keyword(s):  
Multilinear Mapping

ALGEBRAIC APPROACH TO DYNAMICS OF MULTIVALUED NETWORKS

2010 ◽  
Vol 20 (03) ◽  
pp. 561-582 ◽  
Author(s):  
ZHIQIANG LI ◽  
DAIZHAN CHENG
Keyword(s):  
Discrete Time ◽  
Transition Matrix ◽  
Algebraic Approach ◽  
Bilinear System ◽  
Logical Function ◽  
Control Networks ◽  
Logical Network ◽  
Multilinear Mapping ◽  
Network Transition ◽  
Matrix Expression

Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is expressed as a multilinear mapping. Under this framework, the dynamics of a multivalued logical network is converted into a standard discrete-time linear system. Analyzing the network transition matrix, easily computable formulas are obtained to show (a) the number of equilibriums; (b) the numbers of cycles of different lengths; (c) transient period, the minimum time for all points to enter the set of attractors, respectively. A method to reconstruct the logical network from its network transition matrix is also presented. This approach can also be used to convert the dynamics of a multivalued control network into a discrete-time bilinear system. Then, the structure and the controllability of multivalued logical control networks are revealed.

MULTILINEAR MAPPING AND STRUCTURE OF MULTIPARTITE STATES

2008 ◽  
Vol 06 (06) ◽  
pp. 1149-1154
Author(s):  
HOSHANG HEYDARI
Keyword(s):  
Tensor Product ◽  
Multilinear Mapping ◽  
Multilinear Mappings ◽  
Multipartite States

We investigate the relation between multilinear mappings and multipartite states. We show that the isomorphism between multilinear mapping and tensor product completely characterizes decomposable multipartite states in a mathematically well-defined manner.

scholarly journals Extension of the Kantorovich inequality for positive multilinear mappings

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6473-6481 ◽  
Author(s):  
Mohsen Kian ◽  
Mahdi Dehghani
Keyword(s):  
Power Function ◽  
Condition Number ◽  
Matrix Inequalities ◽  
Real Numbers ◽  
Positive Real ◽  
Kantorovich Inequality ◽  
Multilinear Mapping ◽  
Multilinear Mappings ◽  
Positive Matrices

It is known that the power function f (t) = t2 is not matrix monotone. Recently, it has been shown that t2 preserves the order in some matrix inequalities. We prove that if A = (A1,...,Ak) and B = (B1,...,Bk) are k-tuples of positive matrices with 0 < m ? Ai; Bi ? M (i = 1,...,k) for some positive real numbers m < M, then ?2 (A-11,...,A-1k) ? (1+vk)2/4vk)2 ?-2(A1,...,Ak) and ?2 (A1+B1/2,..., Ak+Bk/2)? (1+vk)2/4vk)2 ?2 (A1#B1,...Ak#Bk), where ? is a unital positive multilinear mapping and v = M/m is the condition number of each Ai.

scholarly journals ALGEBRAIC STRUCTURES OF MULTIPARTITE QUANTUM SYSTEMS

2011 ◽  
Vol 09 (01) ◽  
pp. 555-561
Author(s):  
HOSHANG HEYDARI
Keyword(s):  
Tensor Product ◽  
Quantum Systems ◽  
Algebraic Structures ◽  
Multipartite Quantum Systems ◽  
Multilinear Mapping ◽  
Multilinear Mappings ◽  
Multipartite States

We investigate the relation between multilinear mappings and multipartite states. We show that the isomorphism between multilinear mapping and tensor product completely characterizes decomposable multipartite states in a mathematically well-defined manner.

Topologies Extending Valuations

1978 ◽  
Vol 30 (01) ◽  
pp. 164-169 ◽  
Author(s):  
Thomas Rigo ◽  
Seth Warner
Keyword(s):  
Vector Space ◽  
Cartesian Product ◽  
Field Extension ◽  
Valuation Theory ◽  
Topological Isomorphism ◽  
Product Topology ◽  
Hausdorff Topology ◽  
Absolute Value ◽  
Finite Dimensional ◽  
Multilinear Mapping

Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from K n to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from E m to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.

scholarly journals Extensions of multilinear mappings to powers of linear spaces

2016 ◽  
Vol 8 (2) ◽  
pp. 211-214
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Linear Spaces ◽  
Cartesian Power ◽  
Multilinear Mapping ◽  
Multilinear Mappings

We consider the question of the possibility to recover a multilinear mapping from the restriction to the diagonal of its extension to a Cartesian power of a space.

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