Global Asymptotic Almost Periodic Synchronization of Clifford-Valued CNNs with Discrete Delays

In this paper, we are concerned with Clifford-valued cellular neural networks (CNNs) with discrete delays. Since Clifford algebra is a unital associative algebra and its multiplication is noncommutative, to overcome the difficulty of the noncommutativity of the multiplication of Clifford numbers, we first decompose the considered Clifford-valued neural network into 2m2n real-valued systems. Second, based on the Banach fixed point theorem, we establish the existence and uniqueness of almost periodic solutions of the considered neural networks. Then, by designing a novel state-feedback controller and constructing a proper Lyapunov function, we study the global asymptotic synchronization of the considered neural networks. Finally, a numerical example is presented to show the effectiveness and feasibility of our results.

Monoidal Categories, 2-Traces, and Cyclic Cohomology

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ( $\mathscr{C},\otimes$ ) endowed with a symmetric 2-trace, i.e., an $F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in $F$ ”. Furthermore, we observe that if $\mathscr{M}$ is a $\mathscr{C}$ -bimodule category and $(F,M)$ is a stable central pair, i.e., $F\in \text{Fun}(\mathscr{M},\text{Vec})$ and $M\in \mathscr{M}$ satisfy certain conditions, then $\mathscr{C}$ acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

A natural extension of the conformal Lorentz group in a field theory context

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the non-semisimple part of the automorphism group can be understood as “dressing” of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman–Mandula theorem using non-semisimple groups, without SUSY.

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

A set [Formula: see text] of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists [Formula: see text] such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

ON THE GRASSMANN T-SPACE

We study the Grassmann T-space, S3, generated by the commutator [x1,x2,x3] in the free unital associative algebra K 〈x1,x2,… 〉 over a field of characteristic zero. We prove that S3 = S2 ∩ T3, where S2 is the commutator T-space generated by [x1,x2] and T3 is the Grassmann T-ideal generated by S3. We also construct an explicit basis for each vector space S3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x1,…,xn, and deduce the recursive vector space decomposition T3 ∩ Pn = (S3 ∩ Pn) ⊕ (T3 ∩ Pn-1)xn.

On the Associative Nijenhuis Relation

We give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product.