central extensions and h1 l l of the graded lie algebras of cartan type let f be an algebraically closed field of characteristic p 0 l i rsl i a graded lie algebra of cartan type over f we give a description of the dual adjoint module l by means of the mixed product and the coinduced module such that the computation of h1 l l can be reduced to that of the first cohomology of l 0 then we determine the central extensions and h2 l f by computing h1 l l

parameter identification problems for an extensible beam equation parameter identification problems are studied for extensible beam equations the problem is formulated by minimizations of quadratic cost functionals by distributive measurements the existence of optimal parameters and necessary conditions for the parameters are proved

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

on the cohomology of ring extensions in this article we are primarily concerned with the relativized cohomology theory of ring extensions as initiated by g hochschild 141 our main objective is to lay the foundation for a general vanishing theory of relativized extension functors the significance of such a theory rests on the unification it provides for the various classical results from the cohomology theories of groups associative algebras and lie algebras the approach chosen here heavily emphasizes locally nilpotent operators as an underlying principle for vanishing and isomorphism theorems although in algebraic homology theory these concepts have appeared only recently cf 5 they occur implicitly in singular homology theory of topological spaces the conventional proof of the excision theorem for instance resorts by means of barycentric subdivision to the construction of a locally nilpotent operator that is chain homotopic to the identity in section 1 we study the question to what an extent properties of operators are inherited by resolutions it is shown that resolutions defined by injective envelopes and projective covers are suitable tools for ordinary homology in the framework of relative homological algebra of a given ring extension r s the availability of standard resolutions allows the application of more computational techniques we show that the commutator algebra of the centralizer c s of s in r operates on these resolutions in a natural fashion this result in conjunction with a certain homotopy is subsequently utilized to show that the associated action on the spaces ext s m n is trivial the succeeding section combines these techniques with basic facts concerning locally finite operators to establish various vanishing theorems for extension functors the third section deals with the study of the cohomology theory of frobenius extensions we address questions emerging from recent developments in the cohomology theories of finite dimensional modular 9 and finite dimensional restricted lie algebras 7 93 our results are closely

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

generalized restricted lie algebras and representations of the zassenhaus algebra abstract letfbe a field of characteristicp 0 la generalized restricted lie algebra overf andp l the primitivep envelope ofl a close relation betweenl representations andp l representations is established in particular the irreducible reduced modules oflfor any l coincide with the irreducible 0 reduced modules ofp l where 0 p l is a trivial extension of from this result the determination of all irreducible representations of the zassenhaus algebra is completed and the dimensions of the corresponding modules are also given

representations of cartan type lie algebras in characteristic p in this paper we discuss the representations of cartan type lie algebras in characteristic p 2 from the viewpoint of reducing rank when the character is regular semisimple for generalized witt algebras we can essentially reduce higher rank representations to lower rank representations

split abelian chief factors and first degree cohomology for lie algebras abstract in this paper we investigate the relation between the multiplicities of split abelian chief factors of finite dimensional lie algebras and first degree cohomology in particular we obtain a characterization of modular solvable lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors the analogues of these results are well known in the modular representation theory of finite groups an important tool in the proof of these results is a refinement of a non vanishing theorem of seligman for the first degree cohomology of non solvable finite dimensional lie algebras in prime characteristic as an application we derive several results in the representation theory of restricted lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite dimensional restricted lie algebra in particular we obtain a characterization of solvable restricted lie algebras in terms of the second loewy layer of the projective cover of the trivial irreducible module

infinitesimal deformations of restricted simple lie algebras i abstract we compute the infinitesimal deformations of two families of restricted simple modular lie algebras of cartan type the witt jacobson and the special lie algebras

a further expansion of the riemann ζ function a further explicit expansion is presented here for the riemann function in terms of the basis set involving the diagamma or function via the expansion of the tricomi function

on the vanishing of homology and cohomology groups of associative algebras this paper establishes sufficient conditions for the vanishing of the homology and cohomology groups of an associative algebra with coefficients in a two sided module 0 introduction algebraic homology and cohomology theory may be considered an extension of ordinary representation theory certain problems of the latter discipline initially motivated the definition of low dimensional cohomology groups which subsequently was generalized to arbitrary dimension cf 2 8 the theory of derived functors set forth by cartan and eilenberg in 2 provided the appropriate unifying concept for the various ramifications of cohomology theory and also layed the foundation for the study of homology groups among the most notable applications of cohomology theory are weyl s theorem for finite dimensional semisimple nonmodular lie algebras the theorem concerning the complete reducibility of finite dimensional modules of compact topological groups as well as maschke s theorem the proofs of these classical results are primarily based on certain averaging techniques which produce central elements whose action forces the vanishing of the pertinent homology and cohomology groups this paper is concerned with various generalizations of the above mentioned classical vanishing theorems it is conceivable that our new results may be applicable in some nonclassical contexts notably in the cohomology theory of modular and zgraded lie algebras of arbitrary dimension in 3 and 4 we present in accordance with the philosophy of the theory of algebraic complexes a treatment of the most general case namely the homology and cohomology of associative algebras the third section establishes conditions for the vanishing of cohomology groups of an associative algebra a in terms of the action of the enveloping algebra ae on the a bimodule m by emphasizing the role of central elements of the augmentation ideal this approach naturally extends previous results cf 6 theorem 1 2 7 proposition c 6 the methods of 4 sharply contrast those employed in 3 the algebra a is considered a lie algebra which operates on hn a m in a natural fashion in particular we shall demonstrate that the results of 6 retain their validity if one utilizes elements that act locally nilpotently on a via their adjoint representation incidentally the above mentioned lie action appears to be useful in the study of restricted cohomology groups which hitherto were not amenable to methods from ordinary lie cohomology theory received by the editors february 2 1987 1980 mathematics subject classification 1985 revision primary 18h15 r 1988 american mathematical society 0002 9947 88 1 00 25 per page

parameter identification problem for the equation of motion of membrane with strong viscosity the parameter identification problem of constant parameters in the equation of membrane with strong viscosity is studied the problem is formulated by a minimization of quadratic cost functionals by distributive measurements the existence of optimal parameters and necessary optimality conditions for the parameters are proved

on the theory of frobenius extensions and its application to lie superalgebras by using an approach to the theory of frobenius extensions that emphasizes notions related to associative forms we obtain results concerning the trace and corestriction mappings and transitivity these are employed to show that the extension of enveloping algebras determined by a subalgebra of a lie superalgebra is a frobenius extension and to study certain questions in representation theory

optimal control problems for an extensible beam equation optimal control problems are studied for extensible beam equations the g teaux differentiability of solution mapping on control variables is proved and the various types of necessary optimality conditions corresponding to the distributive and terminal value observations are established

on ad semisimple lie algebras the question whether any ad semisimple lie algebra is commutative is related to the existence of simple semiabelian lie algebras these algebras determine the brauer group of a field and are in addition to that significant for the study of lie algebras all whose proper subalgebras are nilpotent in this paper we consider a class of lie algebras related to several topics of lie algebra theory and field theory we begin with the study of finite dimensional lie algebras g having the property that the adjoint representation sends every element of g to a semisimple derivation we call such a lie algebra ad semisimple our study of these may be regarded as a complement to the classical investigation of ad nilpotent lie algebras due to engel however the results of the first section show that in contrast with engels theory the theory of ad semisimple lie algebras is highly sensitive to the choice of the base field this feature becomes more evident if we confine our studies to the subclass of simple semiabelian lie algebras sections 3 5 are devoted to a brief discussion of some aspects of the structure of this interesting class a more thorough investigation can be found in 15 1 theorem 5 2 and section 6 are closely related to a paper by towers 11 i after discussing the relationship between simple semiabelian lie algebras and restricted lie algebras having a nonsingular p map i e a p map whose only zero is 0 we illustrate in the final section some aspects of the linkage between the brauer group of the underlying field and the isomorphism classes of simple semiabelian lie algebras the brauer group of the field admitting no simple semiabelian lie algebras is trivial it is conceivable that the brauer group determines the isomorphism classes of simple semiabelian lie algebras the cases of ir and 6 for example indicate the existence of a more general yet discerning relationship