fast algorithms for n dimensional restrictions of hard problems let m be a parallel ram with p processors and arithmetic operations addition and subtraction recognizing l n n in t steps inputs for m are given integer by integer not bit by bit then l can be recognized by a sequential linear search algorithm lsa in o n 4 log n t log p steps thus many n dimensional restrictions of np complete problems binary programming traveling salesman problem etc and even that of the uniquely optimum traveling salesman problem which is p 2 complete can be solved in polynomial time by an lsa this result generalizes the construction of a polynomial lsa for the n dimensional restriction of the knapsack problem previously shown by the author and destroys the hope of proving nonpolynomial lower bounds on lsas for any problem that can be recognized by a pram as above with 2 poly n processors in poly n time

the projective geometry of freudenthal s magic square abstract we connect the algebraic geometry and representation theory associated to freudenthal s magic square we give unified geometric descriptions of several classes of orbit closures describing their hyperplane sections and desingularizations and interpreting them in terms of composition algebras in particular we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial

computing over the reals with addition and order abstract we provide a fairly complete picture of the complexity theory of additive real machines this model of computation is a restriction of the real turing machine of blum et al 1989 since addition and subtraction are the only legal arithmetic operations removing the order relation r yields an even weaker class of machines which is also studied our main results are characterization of the classes of recognizable boolean languages equivalence of real and digital nondeterminism a simpler proof of meer s plin nplin result

local rigidity of quasi regular varieties for a g variety x with an open orbit we define its boundary x as the complement of the open orbit the action sheaf s x is the subsheaf of the tangent sheaf made of vector fields tangent to x we prove for a large family of smooth spherical varieties the vanishing of the cohomology groups h i x s x for i 0 extending results of bien and brion compos math 104 1 26 1996 we apply these results to study the local rigidity of the smooth projective varieties with picard number one classified in pasquier math ann in press

on fano complete intersections in rational homogeneous varieties complete intersections inside rational homogeneous varieties provide interesting examples of fano manifolds for example if x cap i 1 r d i subset g p is a smooth complete intersection of r ample divisors such that k g p otimes mathcal o g p sum i d i is ample then x is fano we first classify these fano complete intersections which are locally rigid it turns out that most of them are hyperplane sections we then classify general hyperplane sections which are quasi homogeneous

particle reservoir correlations in quantal brownian motion abstract a general study of particle reservoir correlations in quantal brownian motion is developed the non markovian character of the analysis requires the application of analytic continuation techniques which yield as a remarkable result general formulae for the analytic continuation of the reduced weak coupling collision operator the study focuses upon the mean value of the interaction energy which is shown to evolve according to the particle relaxation frequencies in addition to a set of non markovian correlation frequencies the formalism is applied to a spin j coupled to a magnetic field and in contact weak coupling to a phonon reservoir showing that i the reservoir coordinates generally fluctuate on an average in opposite direction to the spin angular momentum and ii the set of non markovian correlation frequencies is absent

topological lower bounds on algebraic random access machines we prove general lower bounds for set recognition on random access machines rams that operate on real numbers with algebraic operations times as well as rams that use the operations times lfloor rfloor we do it by extending a technique formerly used with respect to algebraic computation trees in the case of algebraic computation trees the complexity was related to the number of connected components of the set w to be recognized for rams four similar results apply to the number of connected components of w circ the topological interior of w two results use overline w circ the interior of the topological closure of w we present theorems that can be applied to a variety of problems and obtain lower bounds many of them tight for the following models 1 a ram which operates on real numbers using integers to address memory and either the operations times or times lfloor rfloor 2 a ram of each of the above instruction sets extended by allowing arbitrary real numbers to be used as memory addresses and adding a test for integer instruction 3 a ram of each of the above instruction sets which can compute with arbitrary real numbers as well as use them for memory addressing while the input is restricted to the integers for one result on this model we require that all program constants be rational

separation of complexity classes in koiran s weak model abstract we continue the study of complexity classes over the weak model introduced by p koiran in particular we provide several separations of complexity classes the most remarkable being the strict inclusion of p in np other separations concern classes defined by weak polynomial time over parallel or alternating machines as well as over nondeterministic machines whose guesses are required to be 0 or 1

lower bounds for solving linear diophantine equations on random access machines the problem of recognizing the language l n l n k of solvable diophantine linear equations with n variables and solutions from o k n is considered the languages nen l n nen l n l the knapsack problem are np complete the o n 2 lower bound for l n 1 on linear search algorithms due to dobkin and lipton is generalized to an o n 2 log k 1 lower bound for l n k the method of klein and meyer auf der heide is further improved to carry over the o n 2 lower bound for l n 1 to random access machines rams in such a way that it holds for a large class of problems and for very small input sets by this method lower bounds that depend on the input size as is necessary for l n are proved thereby an o n 2 log k 1 lower bound is obtained for rams recognizing l n or l n k for inputs from 0 nk 0 n 2 n

analytic continuation of generalized susceptibility and system reservoir correlation effects a fully non markovian calculation of the generalized susceptibility of a quantum system weakly coupled to a heat bath is presented in contrast to most of the treatments we provide a complete solution in the sense that it includes the analytic continuation to the lower half plane various applications are discussed including the consideration of boson and fermion heat baths it is shown that our result takes into account system reservoir correlations which are neglected in the markovian approximation

a polynomial linear search algorithm for the n dimensional knapsack problem presentation d un algorithme de recherche lineaire qui reconnait le probleme de sac a dos a n dimensions par etapes de 2n 4 logn o n 3

on the complexity of quadratic programming in real number models of computation abstract the complexity of linearly constrained nonconvex quadratic programming is analyzed within the framework of real number models namely the one of blum shub and smale and its modification recently introduced by koiran weak bss model in particular we show that this problem is not np complete in the koiran setting applications to the full bss model are discussed

memory effects upon the relaxation dynamics of the wigner distribution function the solutions of the generalized fokker planck equation that arises in the semiclassical description of a quantal harmonic coordinate immersed in an arbitrary heat bath are investigated in the non markovian regime the treatment consists of retaining a finite number of terms of the analytic expansion of the laplace transformed memory kernel around the origin which leads to approximate relaxation frequencies involving different degrees of memory for a regime of highly inelastic collisions the authors extend previous findings concerning the relation between markovian and non markovian relaxation frequencies and for the usual elastic coupling they consider fermion and phonon environments showing that the former involves a much more complex dynamics with a non exponential time decay focusing upon the exponential part of the decay they find that for both classes of reservoirs the first two non markovian corrections have well defined signs that increase the relaxation frequencies

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

on some smooth projective two orbit varieties with picard number 1 we classify all smooth projective horospherical varieties with picard number 1 we prove that the automorphism group of any such variety x acts with at most two orbits and that this group still acts with only two orbits on x blown up at the closed orbit we characterize all smooth projective two orbit varieties with picard number 1 that satisfy this latter property

on the limits of computations with the floor function abstract up to now few models of computation with the power of evaluating discontinuous functions have been analyzed and few of their lower bounds or results on the decidability of languages are known in this paper we present a model of an analytic computation tree act these trees operate on real numbers and are able to compare real numbers to evaluate functions on real numbers and to evaluate certain discontinuous functions like the floor function this model generalizes the model of algebraic computation trees introduced by ben or we show by topological arguments that by acts one cannot decide certain classes of languages examples of which are q n and the set of tuples x1 xn r n that have components which are z linearly or algebraically dependent

non markovian relaxation of a quantum system a non markovian treatment of the relaxation of a spin j is developed the corresponding reduced collision operator admits a discrete spectral representation whose eginevalues are extracted in a high temperature limit then assuming a phonon reservoir with a debye like density of phonon states the time evolution of the collision operator is explicitly given showing that the corresponding time decay arises from the branch points of its laplace transform the extraction of the analytic continuation of the transform allows the memory effects on the relaxation frequencies resolvent poles to be analyzed and the validity of the rotating wave approximation to be tested

simulating probabilistic by deterministic algebraic computation trees a probabilistic algebraic computation tree probabilistic act which recognizes l rn in expected time t and which gives the wrong answer with probability 12 can be simulated by a deterministic act in o t2n steps the same result holds for linear search algorithms lsas the result for acts establishes a weaker version of results previously shown by the author for lsas namely that lsas can only be slightly sped up by their nondeterministic versions this paper shows that acts can only be slightly sped up by their probabilistic versions the result for lsas solves a problem posed by snir 1983 he found an example where probabilistic lsas are faster than deterministic ones and asked how large this gap can be

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of picard number 1 under kahler deformation

classification of non degenerate projective varieties with non zero prolongation and application to target rigidity the prolongation mathfrak g k of a linear lie algebra mathfrak g subset mathfrak gl v plays an important role in the study of symmetries of g structures cartan and kobayashi nagano have given a complete classification of irreducible linear lie algebras mathfrak g subset mathfrak gl v with non zero prolongations

generalized finite automata over real and complex numbers in a recent work gandhi khoussainov and liu 7 introduced and studied a generalized model of finite automata able to work over arbitrary structures as one relevant area of research for this model the authors identify studying such automata over particular structures such as real and algebraically closed fields in this paper we start investigations into this direction we prove several structural results about sets accepted by such automata and analyze decidability as well as complexity of several classical questions about automata in the new framework our results show quite a diverse picture when compared to the well known results for finite automata over finite alphabets

lower time bounds for solving linear diophantine equations on several parallel computational models we consider parallel random access machines pram s with p processors and distributed systems of random access machines dram s with p processors being partially joint by wires according to a communication graph for these computational models we prove lower bounds for testing the solvability of linear diophantine equations and related problems including the knapsack problem these bounds are achieved by generalizing and simplifying a lower bound for parallel computation trees due to yao introducing a new type of computation trees which models computations of dram s and by generalizing a technique used by paul and simon and klein and meyer auf der heide to carry over lower bounds from computation trees to ram s thereby we prove that for many problems p processors cannot speed up a computation by a factor o p but only by a factor o log p 1 and in the case of dram s whose communication network has degree c by a factor o log c 1 only

analysis of the projection operator method inconsistencies and their removal it is pointed out that the application of the projection operator method leads to the violation of a consistency condition if the so called born approximation is made in the nonmarkovian master equation since only the validity of this consistency condition guarantees obtaining non negative probabilities its violation makes the applicability of the whole method in this approximation meaningless therefore it is demonstrated that an additional markov approximation restores the validity of the consistency condition in the master equation and thus re establishes the applicability of the projection operator method furthermore the possibility of obtaining a self consistent projection operator method is also examined