Workshop on Matrices and Operators 2009

Labelling Neural Matrix and Collective Dynamics Mau-Hsiang Shih 施茂祥 National Normal University, Taiwan Email: [email protected] Abstract The brain is considered to be a complex, self-organizing system; it consists of enormous numbers of interacting neurons and perpetually weaves its intricate web. Scientists believe that collective dynamics of the brain is deeply entwined with the neural circuits. But working out how neural circuits affect collective dynamics has been a great mystery. From the mathematical perspective, we wish to introduce a novel conception of “labeling neural circuits” and unlock how neural circuits affect collective dynamics.The brain is considered to be a complex, self-organizing system; it consists of enormous numbers of interacting neurons and perpetually weaves its intricate web. Scientists believe that collective dynamics of the brain is deeply entwined with the neural circuits. But working out how neural circuits affect collective dynamics has been a great mystery. From the mathematical perspective, we wish to introduce a novel conception of “labeling neural circuits” and unlock how neural circuits affect collective dynamics. Characterizations of derivations on prime rings: additive maps derivable at an idempotent Jinchuan Hou 侯晉川 Taiyuan University of Technology, China Email: [email protected] Abstract A given element Z in a ring A is called an additive full-derivable point of A if every additive map δ from A into itself derivable at Z (i.e. δ(A)B +Aδ(B) = δ(Z) for every A,B ∈ A with AB = Z) is a derivation. Let A be a unital prime ring containing a nontrivial idempotent P . It is shown that if δ is derivable at 0 and δ(I) belongs to the center of A, then there exists an additive derivation τ such that δ(A) = τ(A) + δ(I)A for all A ∈ A; if, for every A ∈ A, there is some integer n such that nI − A is invertible, then the idempotent P is an additive full-derivable point of A; if, in addition, the characteristic of A is not 2, then the unit I is an additive full-derivable point of A, too. These are applied to some operator algebras such as Banach algebras and von Neumann algebras. For instance, it is shown that every nonzero idempotent in a factor von Neumann algebra is a full-derivable point.A given element Z in a ring A is called an additive full-derivable point of A if every additive map δ from A into itself derivable at Z (i.e. δ(A)B +Aδ(B) = δ(Z) for every A,B ∈ A with AB = Z) is a derivation. Let A be a unital prime ring containing a nontrivial idempotent P . It is shown that if δ is derivable at 0 and δ(I) belongs to the center of A, then there exists an additive derivation τ such that δ(A) = τ(A) + δ(I)A for all A ∈ A; if, for every A ∈ A, there is some integer n such that nI − A is invertible, then the idempotent P is an additive full-derivable point of A; if, in addition, the characteristic of A is not 2, then the unit I is an additive full-derivable point of A, too. These are applied to some operator algebras such as Banach algebras and von Neumann algebras. For instance, it is shown that every nonzero idempotent in a factor von Neumann algebra is a full-derivable point. Generalized numerical ranges and multiplicities Tin Yau Tam 譚天祐 Auburn University, USA Email: [email protected]