Extension of the Kantorovich inequality for positive multilinear mappings

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.

Extensions of multilinear mappings to powers of linear spaces

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.

ALGEBRAIC STRUCTURES OF MULTIPARTITE QUANTUM SYSTEMS

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.

ALGEBRAIC APPROACH TO DYNAMICS OF MULTIVALUED NETWORKS

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

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

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.