some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

ideal regular languages and strongly connected synchronizing automata abstract we introduce the notion of a reset left regular decomposition of an ideal regular language and we prove that the category formed by these decompositions with the adequate set of morphisms is equivalent to the category of strongly connected synchronizing automata we show that every ideal regular language has at least one reset left regular decomposition as a consequence every ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton furthermore this one to one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal from which follows a reformulation of cern s conjecture in purely language theoretic terms finally we present and characterize a subclass of ideal regular languages for which a better upper bound for the reset decomposition complexity holds with respect to the general case

reset words for commutative and solvable automata abstract a reset word takes all states of a finite automaton to a single state in this paper it is shown that the length of the shortest reset word for a solvable automaton with n states is at most n 1 and this bound is reachable

nilpotent orbits and some small unitary representations of indefinite orthogonal groups abstract for 2 m l 2 let g be a simply connected lie group with g 0 so 2m 2l 2m as lie algebra let g k p be the complexification of the usual cartan decomposition let k be the analytic subgroup with lie algebra k g 0 and let u g be the universal enveloping algebra of g this work examines the unitarity and k spectrum of representations in the analytic continuation of discrete series of g relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g the roots with respect to the usual compact cartan subalgebra are all ei ej with 1 i q l u be the parabolic subalgebra of g for which em em 1 contributes to u and the other simple roots contribute to l let l be the analytic subgroup of g with lie algebra l g 0 let l c int g l let 2 u be the sum of the roots contributing to u and let q l u be the parabolic subalgebra opposite to q the members of u p are nilpotent members of g the group l c acts on u p with finitely many orbits and the topological closure of each orbit is an irreducible algebraic variety if y is one of these varieties let r y be the dual coordinate ring of y this is a quotient of the algebra of symmetric tensors on u p that carries a fully reducible representation of l c for s z let s k 1 m l s 2 e k then s defines a one dimensional l l module c s extend this to a q l module by having u act by 0 and define n s 2 u u g q c s 2 u let n s 2 u be the unique irreducible quotient of n s 2 u the representations under study are s s n s 2 u and s s n s 2 u where s dim u k and s is the sth derived bernstein functor for s 2l 2 it is known that s s and that s is in the discrete series enright parthsarathy wallach and wolf showed for m s 2l 2 that s s and that s is still unitary the present paper shows that s is unitary for 0 s m 1 even though s s and it relates the k spectrum of the representations s to the representation of l c on a suitable r y with y depending on s use of a branching formula of d e littlewood allows one to obtain an explicit multiplicity formula for each k type in s the variety y is indispensable in the proof the chief tools involved are an idea of b gross and wallach a geometric interpretation of littlewood s theorem and some estimates of norms it is shown further that the natural invariant hermitian form on s does not make s unitary for s l c on any r y a final section of the paper treats in similar fashion the simply connected lie group with lie algebra g 0 so 2m 2l 2m 1 2 m l 2

synchronizing automata preserving a chain of partial orders we present a new class of automata which strictly contains the class of aperiodic automata and shares with the latter certain synchronization properties in particular every strongly connected automaton in this new class is synchronizing and has a synchronizing word of length n n 1 6 where n is the number of states of the automaton

composition sequences for functions over a finite domain diverse problems ranging from many valued logics to finite automata can be expressed as questions concerning compositions of functions over a finite domain we develop a theory dealing with the depth and complete depth of such functions interconnections with synchronizable finite automata are also discussed many of the very basic problems turn out to be np hard also several open problems are pointed out

unipotent representations of complex semisimple groups algebraic group over r or c in this paper we restrict attention to c we generalize arthur s definition slightly or perhaps simply make it more precise all of the resulting representations except for a finite set are then unitarily induced from representations of the same kind on proper parabolic subgroups we call the finite set remaining special unipotent representations a precise definition will be given later definition 1 17 our main result theorem iii of this introduction is a character formula for any special unipotent representation of course such a formula can be deduced from the kazhdan lusztig conjecture cf v3 the advantages of theorem iii are that it is in closed form and that it lends itself to verification of some conjectures of arthur in a so let g be a connected complex semisimple lie group and q its lie algebra choose

synchronizing generalized monotonic automata in an earlier paper we have studied reset words for synchronizing automata whose states admit a stable linear order here we show that the same bound on the length of the shortest reset word persists for synchronizing automata satisfying much weaker stability restriction this result supports our conjecture concerning the length of reset words for synchronizing automata accepting only star free languages

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

one dimensional representations of u p q and the howe correspondence abstract we explicitly determine the theta lifts of all one dimensional representations of u p q in terms of langlands parameters and determine exactly which lifts are unitary moreover we show that such a lift is unitary if and only if it is a weakly fair derived functor module of the form a q finally we show that the correspondence of unitary representations behaves well with respect to associated cycles

reset sequences for monotonic automata natarajan reduced the problem of designing a certain type of mechanical parts orienter to that of finding reset sequences for monotonic deterministic finite automata he gave algorithms that in polynomial time either find such sequences or prove that no such sequence exists in this paper a new algorithm based on breadth first search is presented that runs in faster asymptotic time than natarajan s algorithms and in addition finds the shortest possible reset sequence if such a sequence exists tight bounds on the length of the minimum reset sequence are given the time and space bounds of another algorithm given by natarajan are further improved that algorithm finds reset sequences for arbitrary deterministicfinite automata when all states are initially possible

convergence of the trinomial tree method for pricing european american options in this paper we present a trinomial tree method for pricing options and show its equivalence to certain explicit difference scheme using the numerical analysis and the notion of viscosity solution we prove the uniform convergence of the trinomial tree method for european american options the accuracy and efficiency are shown through numerical simulations

cells of harish chandra modules for real classical groups we compute the w module structure of a large number of cells of harish chandra modules for a classical group g where w is the complex weyl group

complexity of problems concerning reset words for cyclic and eulerian automata a word is called a reset word for a deterministic finite automaton if it maps all states of this automaton to one state we consider two classes of automata cyclic automata and eulerian automata for these classes we study the computational complexity of the following problems does there exist a reset word of given length for a given automaton what is the minimal length of the reset words for a given automaton