Some new results in differential algebraic control theory

In the differential algebra framework, static state and dynamicstate feedback linearization are considered. The relationshipbetween dynamic feedback and flatness defined by Fliess is discussed.The existence of an equivalent proper differential I–Osystem of a given differential I–O system is discussed,which is closely related to the choice of a proper fictitiousoutput in control design. The concept flatness and its relationwith dynamic feedback linearizability, controllability, observability,invertibility and minimal realization are discussed. Finally,it is demonstrated that many fundamental control concepts andtheir interrelationships can be incorporated into an extendedcontrol diagram.     

Affine algebraic sets relative to an algebraic theory

For an arbitrary algebraic theory T and a prescribed T-algebraL, the geometrical category AfAlgSet(L) of affine algebraic sets over the affine lineL is build up. It is proved to be dually equivalent to the category FcAlg(L) of functional T-algebras overL. The Galois theory ofL is defined and a Galois duality established. These geometrical categories are characterized up to an equivalence of categories.     

Differential inequalities for algebraic polynomials

We sharpen and supplement the results by V. I. Smirnov, A. Aziz, and Q. M. Dawood for algebraic polynomials which generalize the classical Bernstein and Erdos-Lax inequalities.     

Differential algebraic equations with after-effect

In this paper, we are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research.     

Intersection theory of algebraic obstructions

Let A be a noetherian commutative ring of dimension d and L be a rank one projectiveA-module. For 1≤r≤d, we define obstruction groups E^r(A,L). This extends the original definition due to Nori, in the case r=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) E^r(A,A)×E^s(A,A)→E^r+s(A,A). For a projective A-module Q of rank n≤d, with an orientation , we define a Chern class like homomorphism

w(Q,χ):E^d−n(A,L^′)→E^d(A,LL^′),     

An application of algebraic sieve theory

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieris theorem on primes in arithmetic progressions.     

An algebraic theory of graph factorization

An algebraic theory of graph factorization is introduced. For a factor h, a graph G(h) is constructed whose structure contains information about h-factorability. The 1-factorable and cycle factorable graphs over $\text{Z}$₂ are characterized, and properties of the corresponding graph G(h) are obtained.     

On the algebraic theory of scattering

Within the context of the algebraic theory for simple quantum mechanical potential scattering, we establish existence and completeness of generalized wave operators for a large class of long-range potential scattering problems. Our approach utilizes a priori knowledge of the spectral properties of the scattering Hamiltonian and an algebraic theorem on the uniqueness of a particular Euclidean group representation.     

The Hodge theory of algebraic maps

We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.     

An algebraic theory of sequence enumeration

A general method is provided for enumerating sequences σ₁σ₂ … σ_n with respect to length, number of inversions, and the relationship between σ_i and σ_j for all i and j.     

Morse theory from an algebraic viewpoint

Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.

     

Homological methods in algebraic map theory

The primary intent of the present paper is to adapt familiar notions of algebraic topology, such as the fundamental group, homology, and cohomology to the context of algebraic maps and their (ramified) covering projections. The notion of a principal derived map is already fairly well understood in terms of the defining voltages; however, once it is recognized that voltages are essentially cohomological in nature, the functorial interplay among homology provides a very tractible methodology for studying such properties as connectivity or regularity of the covering, or for obtaining explicit constructions of the voltages affording the given covering map.     

Cluster algebras in algebraic lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac–Moody groups. We also review a quantized version of these results.     

微分代数多项式系统的可解集

本文对一般微分代数方程给出了几类可解集的概念，并讨论了多项式微分代数方程的可解集，得到子多项式系统具有唯一解。特别是有多于一个的解的条件，并得到了低次多项式在一定意义下的充分必要条件。     

Differential games,dynamic optimization,and generalized control theory

In this paper, we examine the problem of control in a generalized framework in which there are more than one criterion, more than one intelligent controller, each of which has access to different information. It is argued that optimal control, differential games, dynamic team theory, etc., can all be viewed as special cases of this generalized control theory. Useful concepts and potential difficulties are discussed.The work reported in this paper is a result of research support extended over a period of years by Joint Services Electronic Contract No. N00014-67-A-0298-0006, NASA Grant No. NGR-22-007-068, and AFOSR Grant No. 69-1768 administered through Harvard University. The author has benefited from discussions with numerous colleagues, in particular, K. C. Chu, A. Starr, W. Willman, H. Witsenhausen, and S. Mitter.