canonical p dimension of algebraic groups we describe a way to compute the p relative version of the berhuy reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic p is any prime integer the canonical p dimension is computed for all split simple groups of classical types

chaotic functions with zero topological entropy on caracterise la classe m c i i des applications chaotiques en ce sens on montre que m contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis on montre que le complement de m consiste en applications qui ont seulement des trajectoires approchables par des cycles

excellent algebraic groups i abstract the notion of an excellent quadratic form has been introduced by m knebusch proc london math soc 33 1976 65 93 34 1977 1 31 it has an obvious analogue for a semisimple linear algebraic group g over an arbitrary field k the group g is excellent if for any field extension k of k the anisotropic kernel of the k group g k k can be defined by a k group the aim of this paper is to investigate excellence properties of special linear and orthogonal groups

clifford algebras of hyperbolic involutions for as a central simple algebra of even degree with hyperbolic orthogonal involution we describe the canonically induced involution underline sigma on the even clifford algebra c 0 a sigma underline sigma of a sigma when deg a equiv 0 mod 8 a cong m 2 b and the interesting part of underline sigma is isomorphic to the canonical involution on an exterior power algebra of b as a corollary we get some properties of the involution on the exterior power algebra

cohomological invariants and r triviality of adjoint classical groups using a cohomological obstruction we construct examples of absolutely simple adjoint classical groups of type 2a n with n equivalent to 3 mod 4 c n or 1d n with n equivalent to 0 mod 4 which are not r trivial hence not stably rational

ω chaos and topological entropy we present a new concept of chaos chaos and prove some properties of chaos then we prove that chaos is equivalent to positive entropy on the interval we also prove that chaos is equivalent to the definition of chaos given by devaney on the interval

r equivalence in adjoint classical groups over fields of virtual cohomological dimension 2 let f be a field of characteristic not 2 whose virtual cohomological dimension is at most 2 let g be a semisimple group of adjoint type defined over f let rg f denote the normal subgroup of g f consisting of elements r equivalent to identity we show that if g is of classical type not containing a factor of type d n g f rg f 0 if g is a simple classical adjoint group of type d n we show that if f and its multi quadratic extensions satisfy strong approximation property then g f rg f 0 this leads to a new proof of the r triviality of f rational points of adjoint classical groups defined over number fields

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

torsion in cohomology of compact lie groups and chow rings of reductive algebraic groups the purpose of the present note is to give a general procedure for calculating the singular cohomology h k ifp with coefficients in an arbitrary finite field ifv of a connected compact lie group k simultaneously we calculate the chow ring a g ifp a g p of the corresponding complex algebraic group g and the degrees of the basic generalized invariants of the weyl group w in arbitrary characteristic p a mysterious connection between the degrees of generators of a g ifp and the kernel of the principal nilpotent element in characteristic p is pointed out the computation of real cohomology of compact lie groups started by e cartan in 1929 was completed for classical groups by r brauer and by l pontryagin in 1935 h hopf and h samelson showed in 1941 that h k f is an exterior algebra on generators of degrees 2 d 1 1 2dr l where l r a n k k but only in 1949 c t yen managed to compute the d for all exceptional groups case by case on the other hand c chevalley and j leray showed that the d i are nothing else but the degrees of the basic invariants over of the weyl group w a simple procedure for calculating the dl was given in the late fifties by a j coleman and by b kostant nowadays this is common knowledge the problem of computing the torsion in h k z proved to be more recalcitrant this was done by several authors c e miller settled so in 1953 23 a borel settled spin ad spin and the quotients of su and sp in 1954 6 p f baum and w browder settled the remaining classical group the semispin group in 1965 4 g 2 and f 4 were settled in 1954 6 and the p 3 torsion for e 6 and p 5tors ion for ev e s in 1961 7 by a borel s araki computed the 2 torsion for e 6 e 7 1 and the 3 torsion for e 7 e s 2 and announced the 2 torsion for e 8 in a joint paper with y shikata in 1961 3 the details of the proof for the last result have appeared only recently 18 there was an error in 1 in the case of ad e 7 which has been fixed in 13

opacity of nondeterministic transition systems a bi simulation relation approach in this paper we propose several opacity preserving bi simulation relations for nondeterministic transition systems ntss in terms of initial state opacity current state opacity k step opacity and infinite step opacity we also show how one can leverage quotient constructions to compute such relations as a result although the opacity verification problem for infinite ntss is generally undecidable if one can find such an opacity preserving relation from an infinite nts to a finite one the lack of opacity of the infinite nts can be easily verified over the finite one which is decidable

chaotic sets of shift and weighted shift maps abstract in this paper li yorke chaotic sets generated by shift and weighted shift maps are studied the characterization of li yorke chaotic sets by p scrambled sets maximal scrambled sets and orbit invariants are proved for the general shift maps for weighted shift maps on infinite dimensional spaces the necessary and sufficient conditions for li yorke chaotic are proved both in the abstract sequence setting and in the eigenfunction setting besides a constructive proof is provided for the devaney chaos of weighted shift maps on the schwartz space

chaotic sets of continuous and discontinuous maps abstract this paper discusses li yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps scrambled sets and maximal scrambled sets are introduced to characterize li yorke chaotic sets the orbit invariant for a scrambled set is discussed some properties about maximality equivalence and uniqueness of maximal scrambled sets are also discussed it is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps such as the gauss map interval exchange transformations and a class of planar piecewise isometries are studied finally some open problems on scrambled sets are listed and remarked

examples of non rational varieties of adjoint groups let k be a field of characteristic 2 and k a separable closure of k s we say that an algebraic variety xrk is stably k rational if there exist two m n m n w x affine spaces a a and a k rational map a x f a merkurjev 9 k k gave a criterion of stable k rationality for the adjoint classical groups with absolute rank f 3 which covers the case of the variety pso q for any quadratic form qrk of rank f 6 this criterion gives examples of field k and quadratic form q of rank 6 with non trivial signed discriminant such that the variety pso q is not stably k rational the main result of this paper is the following

symbolic models for nonlinear control systems without stability assumptions finite state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications most techniques for the construction of such symbolic models have two main drawbacks either they can only be applied to restrictive classes of systems or they require the exact computation of reachable sets in this paper we propose a new abstraction technique that is applicable to any nonlinear sampled data control system as long as we are only interested in its behavior in a compact set moreover the exact computation of reachable sets is not required the effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle

decidability of opacity verification problems in labeled petri net systems abstract a system is said to be opaque if an intruder that observes its evolution through a mask cannot infer that the system s evolution belongs to a given secret behavior opacity verification is the problem of determining whether the system is opaque with respect to a given secret or not in this paper we address the decidability of the opacity verification problem using reduction approaches we show that verification of initial state current state and language opacity is undecidable in labeled petri nets

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

a new approach for the verification of infinite step and k step opacity using two way observers abstract in the context of security analysis for information flow properties where a potentially malicious observer intruder tracks the observed behavior of a given system infinite step opacity respectively k step opacity holds if the intruder can never determine for sure that the system was in a secret state for any instant within infinite steps respectively k steps prior to that particular instant we present new algorithms for the verification of the properties of infinite step opacity and k step opacity for partially observed discrete event systems modeled as finite state automata our new algorithms are based on a novel separation principle for state estimates that characterizes the information dependence in opacity verification problems and they have lower computational complexity than previously proposed ones in the literature specifically we propose a new information structure called the two way observer that is used for the verification of infinite step and k step opacity based on the two way observer a new upper bound for the delay in k step opacity is derived which also improves previously known results

a fully automated framework for control of linear systems from temporal logic specifications we consider the following problem given a linear system and a linear temporal logic ltl formula over a set of linear predicates in its state variables find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula our solution to this problem consists of three main steps first we partition the state space in accordance with the predicates in the formula and construct a transition system over the partition quotient which captures our capability of designing controllers second using a procedure resembling model checking we determine runs of the transition system satisfying the formula third we generate the control strategy illustrative examples are included

generically split projective homogeneous varieties let g be an exceptional simple algebraic group over a field k and x a projective g homogeneous variety such that g splits over k x we classify such varieties x this classification allows to relate the rost invariant of groups of type e7 and their isotropy and to give a two line proof of the triviality of the kernel of the rost invariant for such groups apart from this it plays a crucial role in the solution s08 of a problem posed by j p serre for groups of type e8

unramified degree three invariants for reductive groups of type a abstract let g be a reductive group over an algebraically closed field f of characteristic zero such that the dynkin diagram of g is the disjoint union of diagrams of type a we prove that the third unramified cohomology group of the classifying space of g with coefficients in q z is trivial

degree three unramified cohomology of adjoint semisimple groups we compute degree 3 unramified cohomology of adjoint semisimple group new classes of non rational adjoint semisimple groups are found

shells of twisted flag varieties and the rost invariant we introduce two new general methods to compute the chow motives of twisted flag varieties and settle a 20 year old conjecture of markus rost about the rost invariant for groups of type e 7

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

essential dimension of finite p groups we prove that the essential dimension and p dimension of a p group g over a field f containing a primitive p th root of unity is equal to the least dimension of a faithful representation of g over f

decomposable and indecomposable algebras of degree 8 and exponent 2 we study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras we consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension let b be a biquaternion algebra over f a with trivial corestriction a degree 3 cohomological invariant is defined and we show that it determines whether b has a descent to f this invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2 cohomological dimension 3 and over a field m t where the u invariant of m is 8 and t is an indeterminate the construction of these indecomposable algebras uses chow group computations provided by merkurjev in appendix

verification of initial state opacity in security applications of discrete event systems in this paper we formulate and analyze methodologies for verifying the notion of initial state opacity in discrete event systems that are modeled as non deterministic finite automata with partial observation on their transitions a system is initial state opaque if the membership of its true initial state to a set of secret states remains opaque i e uncertain to an intruder who observes system activity through some projection map initial state opacity can be used to characterize security requirements in a variety of applications including tracking problems in sensor networks in order to model and analyze the intruder capabilities regarding initial state opacity we first address the initial state estimation problem in a non deterministic finite automaton via the construction of an initial state estimator we analyze the properties and complexity of the initial state estimator and show how the complexity of the verification method can be greatly reduced in the special case when the set of secret states is invariant we also establish that the verification of initial state opacity is a pspace complete problem

operators with dense invariant cyclic vector manifolds abstract we study a class of banach space operators patterned after the weighted backward shifts on hilbert space and show that any non scalar operator in the commutant of one of these generalized backward shifts has a dense invariant linear manifold whose non zero members are cyclic vectors under appropriate extra hypotheses on the commuting operator stronger forms of cyclicity are possible the most extreme being hypercyclicity density of an orbit motivated by these results we examine the cyclic behavior of two seemingly unrelated classes of operators adjoint multiplications on hilbert spaces of holomorphic functions and differential operators on the frechet space of entire functions we show that each of these operators other than the scalar multiples of the identity possesses a dense invariant linear submanifold each of whose non zero elements is hypercyclic finally we explore some connections with dynamics many of the hypercyclic operators discussed here are in at least one of the commonly accepted senses of the word chaotic

motivic decomposition of a compactification of a merkurjev suslin variety we provide a motivic decomposition of a twisted form of a smooth hyperplane section of gr 3 6 this twisted form is a norm variety corresponding to a symbol in the milnor k theory k3 3 as an application we construct a torsion element in the chow group of this variety msc2000 20g15 14f43