A Faà di Bruno Hopf algebra for a group of Fliess operators with applications to feedback

A Faà di Bruno type Hopf algebra is developed for a group of integral operators known as Fliess operators, where operator composition is the group product. Such operators are normally written in terms of generating series over a noncommutative alphabet. Using a general series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. The result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product, that is, the generating series for the Fliess operator representation of the closed-loop system written in terms of the generating series of the Fliess operator component systems. This formula is employed to provide a proof that local convergence is preserved under feedback.     

三角代数上的ξ-Lie可导映射

设U=Tri(A,M,B)是含单位元I的三角代数并且φ:U→U是线性映射.利用代数分解的方法,证明了当三角代数U满足适当条件时,如果U,V∈U且UV=VU=I,有φ([U,V]ξ)=[φ(U),V]ξ+[U,φ(V)]ξ(ξ≠±1),则φ是导子.并得到了套代数上ξ-Lie可导映射的一个刻画.     

ON TEMPORAL DEDUCTIVE DATABASES

This article introduces a temporal deductive database system featuring a logic programming language and an algebraic front-end. The language, called Temporal DATALOG, is an extension of DATALOG based on a linear-time temporal logic in which the flow of time is modeled by the set of natural numbers. Programs of Temporal DATALOG are considered as temporal deductive databases, specifying temporal relationships among data and providing base relations to the algebraic front-end. The minimum model of a given Temporal DATALOG program is regarded as the temporal database the program models intensionally. The algebraic front-end, called TRA, is a point-wise extension of the relational algebra upon the set of natural numbers. When needed during the evaluation of TRA expressions, slices of temporal relations over intervals can be retrieved from a given temporal deductive database by bottom-up evaluation strategies.
A modular extension of Temporal DATALOG is also proposed, through which temporal relations created during the evaluation of TRA expressions may be fed back to the deductive part for further manipulation. Modules therefore enable the algebra to have full access to the deductive capabilities of Temporal DATALOG and to extend it with nonstandard algebraic operators. This article also shows that the temporal operators of TRA can be simulated in Temporal DATALOG by program clauses.     

An approach to the parallel solution of a high-dimensional basic flow problem

The body of mathematics of modified systems of algorithmic algebras (SAA-M) is used to formalize the Edmonds–Karp algorithm of finding the maximum flow in a network. With allowance made for the distributed system features that are usually used for solving complicated problems, optimization criteria are formulated and used to obtain parallel SAA-M-schemes. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 146–152, March–April 2009.     

Constructing algebraic groups from their Lie algebras

A connected algebraic group in characteristic 0$0$ is uniquely determined by its Lie algebra. In this paper algorithms are given for constructing an algebraic group in characteristic 0$0$, given its Lie algebra. In particular, this leads to an algorithm for finding a maximal reductive subgroup and the unipotent radical of an algebraic group.     

Formal Performance Modelling and Evaluation of an Adaptive Mechanism for Packetised Audio over the Internet

A case study is presented which concerns the design of an adaptive mechanism for packetised audio for use over the Internet. During the design process, the audio mechanism was modelled with the stochastically timed process algebra EMPA and analysed via simulation by the EMPA based software tool TwoTowers in order to predict the percentage of packets that are received in time for being played out. The predicted performance figures obtained from the algebraic model illustrated in advance the adequacy of the approach adopted in the design of the audio playout delay control mechanism. Based on these performance figures, it was possible to implement and develop the complete mechanism without incurring additional costs due to the late discovery of unexpected errors or inefficiency. Performance results obtained from experiments conducted on the field confirmed the predictive simulative results. Received March 1997 / Accepted in revised form July 1998     

Transformational reducibility and synthesis of algorithms and programs of symbolic processing

This paper deals with the algebra of algorithmics in the context of developing modern means of transformation and synthesis of algorithms and programs in object-oriented environments. This report was first presented by the author in English at the conference ECI-2005 (Kosice, Slovakia). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 165–173, September–October 2006.     

Asymptotic stabilization of planar systems

Sufficient conditions for the existence of a C¹ asymptotically stabilizing feedback control for planar, affine systems are given. The methods are constructive and yield homogeneous feedback controls for homogeneous systems which allow the results to be extended to more general systems.