on the discrete time version of the brussels formalism we develop a discrete time version of the so called brussels formalism in nonequilibrium statistical mechanics for continuous endomorphisms of a banach space we show that if the evolution operator u and the projector p are such that pu is a compact operator and the spectral radius of i p u i p is strictly less than the spectral radius of u then the formalism holds and the evolution operator is quasicompact

structure arising from induction and jacquet modules of representations of classical p adic groups jacquet modules are very useful in the study of parabolically induced representations of reductive groups over a p adic field f we shall assume char f 2 in this paper it is very hard to describe explicitly the structure of jacquet modules of parabolically induced representations particularly in the most interesting cases there exists a description of factors of certain filtrations on them that description was done by j bernstein and a v zelevinsky in bz2 and by w casselman in c in the case of general linear groups the functor of parabolic induction and the jacquet functor induce a structure of z graded hopf algebra on the sum r of grothendieck groups of categories of smooth representations of gl n f s of finite length n 0 z1 the multiplication m r r r is defined using parabolic induction while the comultiplication m r r r is defined in terms of jacquet modules the most interesting part of the structure is the property that m r r r is a ring homomorphism hopf axiom this enables one to compute composition series of parabolically induced representations in a very simple way it is interesting to note that the existence of this strong structure did not have serious impact on the development of the representation theory of gl n f one of the reasons for that may lay in the fact that for gl n f there existed a very powerful theory of gelfandkazhdan derivatives and the main results of z1 were obtained using them nevertheless a v zelevinsky showed in z2 that some interesting parts of the representation theory of gl n over a finite field can be obtained as a structure theory of such hopf algebra defined in this setting besides that one of the main tools for the study of representation theory of gl n over a central division f algebra in t1 is such hopf algebra structure in this situation are not available gelfand kazhdan derivatives it is natural to ask does some structure of this kind exist for other simple split classical p adic groups since levi factors of classical groups are isomorphic to direct products of general linear groups and smaller groups from the same series one can expect for such structure to have some relation with r if it exists in the third section we define the direct sum of grothendieck groups r s which corresponds either to the series sp n f n 0 or so 2n 1 f n 0 in a similar way as r was defined for general linear groups the action of r on r s is defined using the parabolic induction in this way r s becomes z graded r module we can make r s a z graded comodule over r the comodule structure map r s r r s is again defined using the jacquet modules similarly as in the case of gl n f it is not hard to see that r s is not a hopf module over r see remark 7 3 in this paper we

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

jacquet modules of strongly positive representations of the metaplectic group strongly positive discrete series represent a particularly important class of irreducible square integrable representations of p adic groups indeed these representations are used as basic building blocks in known constructions of general discrete series in this paper we explicitly describe jacquet modules of strongly positive discrete series the obtained description of jacquet modules which relies on the classification of strongly positive discrete series given in 10 is valid in both classical and metaplectic case we expect that our results besides being interesting by themselves should be relevant to some potential applications in the theory of automorphic forms where both representations of metaplectic groups and structure of jacquet modules play an important part

a multi point boundary value problem with two critical conditions abstract we apply a coincidence degree theorem due to mawhin to show the existence of at least one solution of the nonlinear second order multi point boundary value problem u t f t u t u t t 0 1 u 0 u i 1 n i u i u 1 where 0 1 0 i 1 i 1 n i 1 n i i 1 n i i 1 n 2 and f 0 1 r 2 r satisfying the caratheodory conditions in our setting both boundary conditions are responsible for resonance

reply to comments by george and prigogine on the paper on the validity of the brussels formalism in statistical mechanics for original paper see coveney et al ibid vol 25 p 4947 1992 george and prigogine ibid vol 26 p 3905 1993 interpret the authors previous paper as destructive criticism of the brussels formalism however they argue that their aim was to discover conditions under which this method might be supported by rigorous analysis and that their results give a set of sufficient conditions admittedly quite restrictive for the validity of a part of the formalism the brussels school has never promulgated any agreed choice for the function space in which the formalism s operators are to act in their work they chose a hilbert space this structure sufficed for the theorems they could prove which concern only the p subspace they agree that additional structure such as that of a rigged hilbert space may be necessary to give effect to the brussels formalism outside the p subspace but since this additional structure would not have affected their results they do not see that its omission was a misrepresentation further comments are made

on supports of induced representations for symplectic and odd orthogonal groups let g be sp 2 n f resp so 2 n 1 f where f is a p adic field of characteristic zero in this paper we give a correspondence which associates to an irreducible representation of g an m tuple of irreducible representations of lower rank symplectic resp orthogonal groups based on the supercuspidal support of we show that this correspondence respects the induction and jacquet module functors in a sense to be made precise as well as verifying a number of other useful properties in essence this correspondence allows one to isolate the effects of the different families of supercuspidal representations of general linear groups which appear in the support of

on exponential long time evolution in statistical mechanics penrose coveney 1994 recently introduced an invertible discrete time dynamical system called the pastry cook s transformation for which they constructed a canonical non equilibrium ensemble in the present paper we apply the brussels formalism of non equilibrium statistical mechanics to this system the use of the formalism is justified rigorously and the operators which arise in the theory are calculated exactly the set of ensembles for which the theory is valid is a banach space of functions satisfying a certain smoothness condition this condition ensures that ensembles show a decay towards equilibrium in agreement with the time asymmetry observed in thermodynamics we also calculate the decay of time correlation functions using ruelle s theory of dynamical resonances we find that all three methods furnish essentially the same description of the exponential decay to equilibrium in this system

kinetic time irreversible evolution of the unstable π meson abstract in liouville formalism the expression for density matrix determining the time evolution of unstable meson in the framework of unified formulation of quantum and kinetic dynamics is defined the eigenvalues problem is investigated in the framework of prigogine s principles of description of nonequilibrium processes at microscopic level the problem was solved on the basis of complex spectral representation it was shown that the approach leads to pauli master equation for the weakly interacting system

quantum systems with diagonal singularity their logic and intrinsic irreversibility abstract the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables and the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity

the extension of classical dynamics for unstable hamiltonian systems abstract classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the liouville equation it is shown that for the class of large nonintegrable poincare systems lps the two descriptions are not equivalent practically all dynamical systems studied in statistical mechanics belong to this class the basic step is the extension of the liouville operator l h outside the hilbert space to functions singular in their fourier transformation this function space plays an important role in statistical mechanics as functions of the hamiltonian and therefore equilibrium distribution functions belong to this class physically these functions correspond to situations characterized by persistent interactions as they are realized in macroscopic physics persistent interactions are introduced in contrast to transient interactions studied in quantum mechanics by the s matrix approach asymptotically free in and out states the eigenvalue problem for the liouville operator l h is solved in this generalized function space for lps we obtain a complex irreducible spectral representation complex means that the eigenvalues are complex numbers whose imaginary parts refer to the various irreversible processes such as relaxation times diffusion irreducible means that these representations cannot be implemented by trajectory theory as the result the dynamical group of evolution splits into two semigroups moreover the laws of classical dynamics take a new form as they have to be formulated on the statistical level they express possibilities and no more certitudes two examples of typical classical systems i e interacting particles and anharmonic lattices are studied

on the existence of dynamical systems with exponentially decaying collision operators the paper is concerned with the time domain collision operator of the brussels school of non equilibrium statistical mechanics t plqexp tqlq qlp where l is the skew adjoint liouville operator of a dynamical system and p and q are complementary orthogonal projectors under the assumption that p is finite rank we prove that if is norm bounded by a decreasing exponential then l must have a certain spectral property and that conversely this spectral property guarantees the existence of a projector p for which the corresponding decays exponentially we use this characterization to show that k systems admit exponentially decaying collision operators we also show that this property is enjoyed by the collision operator of the pietenpol model for a large class of interactions this answers in the affirmative a question raised by coveney and penrose coveney p v and penrose o 1992 j phys a math gen 25 4947 66

existence of solutions to functional boundary value problem of second order nonlinear differential equation abstract in this paper a functional boundary value problem is studied based on mawhin s coincidence degree theory some existence theorems are obtained in the case of non resonance and the cases of dim ker l 1 and dim ker l 2 at resonance the results not only generalize and improve some known results of multi point and integral boundary value problems but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x

subdynamics in the time dependent formalism abstract by means of the recently derived analytical continuation procedure of coveney and george together with henin s theorem in the dynamics of correlations we show that the complete correlation subdynamics of a large dynamical system may be obtained within an explicitly time dependent formalism this result represents a significant extension of the earlier work of balescu and misguich insofar as the time dependent problem is concerned and bring us a stage nearer to a subdynamics based description of irreversible processes in open systems which are described by a time dependent liouville operator

poincare s theorem and subdynamics for driven systems abstract large dynamical systems with explicit time dependence will be studied for these systems which we call driven systems there exist resonances between internal frequencies and or between internal and external frequencies as in the time independent case the usual canonical or unitary perturbation theory leads to divergences due to the resonances as a result there exist no trajectories analytic in both the coupling constant and the initial data this is a generalization of poincare s theorem on non integrability and extends the notion of large poincare systems lps i e systems with a continuous spectrum and a continuous set of resonances here the resonances involve external frequencies along the line of the subdynamics theory developed by prigogine and his co workers we study lps within the liouville space formalism we construct projection operators which decompose the equations of motion and are analytic in the coupling constant our approach recovers coveney s theory of time dependent subdynamics but is based on recursion formulas which significantly simplify the construction of the projection operators these projectors are non hermitian and provide a description with broken time symmetry as an application we study the modification of the well known three stages of the decay process from a time dependent perturbation

comments on the paper by coveney and penrose on the validity of the brussels formalism in statistical mechanics for original paper see coveney et al ibid vol 25 p 4947 1992 the previous authors state in the introductory section of their paper that they are looking only at a small part of the brussels formalism the title of their paper does not reflect this restriction and refers to the general validity of the brussels formalism the criticisms are focused on a few papers published about twenty years ago however even restricted to these papers their claims do not apply the present authors point out two major misrepresentations

canonical nonequilibrium ensembles and subdynamics a method for constructing a canonical nonequilibrium ensemble for systems in which correlations decay exponentially has recently been proposed by coveney and penrose in this paper we show that the method is equivalent to the subdynamics formalism developed by prigogine and others when the dimension of the subdynamic kinetic subspace is finite the comparison between the two approaches helps to clarify the nature of the various operators used in the brussels formalism we discuss further the relationship between these two approaches with particular reference to a simple discrete time dynamical system based on the baker s transformation which we call the baker s urn

strongly positive subquotients in a class of induced representations of classical p adic groups we determine under which conditions the induced representation of the form 1 2 where 1 2 are irreducible essentially square integrable representations of a general linear group and is a discrete series representation of classical p adic group contains an irreducible strongly positive subquotient

subdynamics and nonintegrable systems work on the application of poincare s theorem to large classical or quantum systems with a continuous spectrum is continued in situations where it is applicable poincare s theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant in contrast the theory of subdynamics as developed by the brussels group permits the construction of a unique set of projectors for t 0 giving up the requirement of hermiticity which is replaced by star hermiticity the theory of subdynamics is presented in a new self contained way starting from the commutation relation lh lh where lh is the liouvillian this presentation is far more direct and avoids some of the lengthy discussions associated with previous presentations based mainly on the resolvent of the liouvillian subdynamics appears to be of interest from many points of view it generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed hamiltonian in contrast degrees of freedom are lost when going to the spectral representation e g in the friedrichs model subdynamics permits us to solve the initial value problem associated with the liouville equation retaining the non markovian contributions which appear in the standard presentation finally it introduces a classification of large dynamical systems classical or quantum into integrable and nonintegrable ones it is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems such as the interaction of matter with light the applications of this technique to these problems will be worked out in subsequent papers

multi point boundary value problems on time scales at resonance abstract we study the nonlinear second order differential equation on a time scale u t f t u t u t e t t 0 1 t subject to the multi point boundary conditions u 0 0 u 1 i 1 m i u i and u 0 0 u 1 i 1 m i u i where t is a time scale such that 0 1 t i 0 1 t i 1 m and f is a continuous function satisfying a caratheodory type growth condition and e is a lebesgue integrable function our existence results are obtained by applying a coincidence degree theorem due to mawhin

existence and multiplicity of solutions for four point boundary value problems at resonance abstract this paper deals with the second order four point boundary value problem x t f t x t 0 t 1 x 0 ax x 1 bx where f 0 1 r r is continuous 0 1 0 a 1 1 0 b 1 a 1 b 1 a s 1 b 0 by using the coincidence degree theory the method of lower and upper solutions is established as applications some existence and multiplicity results are obtained

on the existence of a three point boundary value problem at resonance in rn abstract we apply a coincidence degree theorem due to mawhin to obtain a sufficient condition for the existence of at least one solution for a class of nonlinear second order differential equations in r n associated with three point boundary conditions

exact markovian kinetic equation for a quantum brownian oscillator we derive an exact markovian kinetic equation for an oscillator linearly coupled to a heat bath describing quantum brownian motion our work is based on the subdynamics formulation developed by prigogine and collaborators the space of distribution functions is decomposed into independent subspaces that remain invariant under liouville dynamics for integrable systems in poincar e s sense the invariant subspaces follow the dynamics of uncoupled renormalized particles in contrast for nonintegrable systems the invariant subspaces follow a dynamics with broken time symmetry involving generalized functions this result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces we comment on the relation between our markovian kinetic equation and the hu paz zhang equation

construction of discrete series for classical p adic groups in this paper the authors complete the classification of irreducible square integrable representations of classical p adic groups assuming a natural assumption which is expected to hold in general this classification implies a parameterization of irreducible tempered representations of these groups and it implies a classification of the non unitary duals of these groups modulo cuspidal data

a note on multi point boundary value problems abstract multi point boundary value problems for a second order ordinary differential equation are considered in this note an existence result is obtained with the help of coincidence degree theory

is there a canonical non equilibrium ensemble we consider a simple dynamical system related to the baker s transformation this pastry cook s transformation operating in a space consisting of two congruent squares is performed by cutting both squares into q vertical slices squashing each to 1 q of its previous height rearranging the squashed slices to make two new squares as in the baker s transformation and finally interchanging the lowest p slices between the two squares this transformation approximately mimics the ergodic behaviour of a particle in a vessel consisting of two compartments separated by a diaphragm with a small hole in it occupying a fraction p q of the total surface area of either compartment we show for large classes of initial ensembles and observable dynamical variables that the expectation value of any such dynamical variable at large times t is asymptotically the same as it would be in a canonical non equilibrium ensemble in which expectations are defined by e g ast phi e text eq phi langle h x g rangle langle h y phi rangle here e g ast denotes an expectation taken with respect to the initial phase space density g e text eq denotes an expectation in the equilibrium ensemble h x h y are certain linear functionals and phi u t f where u is the evolution operator and f is an observable

generalized spectral decomposition of the β adic baker s transformation and intrinsic irreversibility abstract we construct a generalized spectral decomposition of the frobenius perron operator of the adic baker s transformation using a general iterative operator method applicable in principle for any mixing dynamical system the eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one we explicitly define appropriate generalized function spaces which provide mathematical meaning to the formally obtained spectral decomposition the unitary frobenius perron evolution of densities when extended to the generalized function spaces splits into two semigroups one decaying in the future and the other in the past this split which reflects the asymptotic evolution of the forward and backward k partitions shows the instrinsic irreversibility of the baker s transformation

alternative formulation of classical and quantum dynamics for non integrable systems we continue our work on large poincare systems lps characterized by a continuous spectrum and a continuous set of resonances poincare s theorem prevents the existence of solutions of the eigenvalue problem associated with the hamiltonian as well as the liouville operator which would be analytic in the coupling constant in simple cases such as the friedrichs model studied in a recent paper poincare s divergences can be avoided using a natural time ordering of the dynamical states in the hilbert space however in general the time ordering has to be performed in the liouville space of density matrices

on the long time behaviour of ensembles in a model of deterministic diffusion we describe a large model system based on the baker transformation where deterministic diffusion occurs the model is similar to one recently considered by gaspard and by hasegawa and driebe we point out the close relationship between this system and a simple random walk and analyse the evolution with time of ensembles in the system using the resolvent based version of the subdynamics formalism developed by prigogine and his collaborators we obtain an exact and rigorous description of the long time behaviour of ensembles including the irreversible approach to equilibrium for the case where the system has finite size we also consider the thermodynamic limit where the size of the system becomes infinite and derive a description of the long time behaviour in this case where correlations decay non exponentially with time

strongly positive representations of metaplectic groups abstract in this paper we obtain a classification of irreducible strongly positive square integrable genuine representations of metaplectic groups over p adic fields using a purely algebraic approach our results parallel those of m glin and tadic for classical groups but their work relies on certain conjectures on the other hand our results are complete and there are no additional conditions or hypotheses the important point to note here is that our results and techniques can be used in the case of classical p adic groups in a completely analogous manner

on the validity of the brussels formalism in statistical mechanics the so called brussels approach to the derivation of kinetic equations usually proceeds by representing the liouville operator l in the form l0 lambda l1 where lambda is a perturbation parameter it can be formulated in terms of an operator p or a set of such operators commuting with l0 and projecting from a hilbert space h spanned by all square integrable phase space densities or density matrices rho into a subspace h1 in which the reduced or kinetic description is to apply for the case where h1 is finite dimensional the authors prove two main results i if the time domain collision operator defined by psi t pl1q exp qlqt ql1p where q 1 p is bounded above in norm by a decreasing exponential function of mod t mod and satisfies the condition that the hermitian part of integral 0infinity eiyt psi t dt be invertible for all real y then for sufficiently small positive lambda the long time asymptotic approach to equilibrium in the subspace h1 is an exponential decay or exponentially decaying oscillation and is correctly given by the brussels perturbation method ii if the norm of the collision operator decays to zero at t to infinity more slowly than any exponential then regardless of the value of lambda the asymptotic behaviour does not have the exponential form predicted by the brussels method

intrinsic irreversibility of quantum systems with diagonal singularity the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity we illustrate the general ideas for the friedrichs model

quantum theory of non integrable systems in 1889 h poincare introduced a basic distinction between integrable and non integrable dynamical systems this distinction refers to the role of resonances which may lead to divergences a specially important class of non integrable systems are large poincare systems lps which have a continuous spectrum and present continuous sets of resonances as has been shown earlier lps play an essential role both in classical and quantum mechanics essentially all nontrivial problems of field theory as well as kinetic theory belong this class we consider here the well known friedrichs model in which an unstable discrete level is coupled to a continuum poincare s theorem prevents the existence of solutions of the eigenvalue problem associated to the hamiltonian which would be analytic in the coupling constant

monotone positive solutions of second order multi point boundary value problems abstract in this paper by using guo krasnosel skii fixed point theorem and schauder s fixed point theorem we establish sufficient conditions for the existence and nonexistence of monotone increasing positive solution for a class of second order differential equation with multi point nonhomogeneous boundary conditions

solvability of a second order multi point boundary value problem at resonance abstract based on the coincidence degree theory of mawhin we get a general existence result for the following second order multi point boundary value problem at resonance x t f t x t x t e t t 0 1 x 0 i 1 m i x i x 1 j 1 n j x j where f 0 1 r 2 r is a caratheodory function e l 1 0 1 0 1 2 m 1 i r i 1 2 m m 2 and 0 1 n 1 j r j 1 n n 1 in this paper both of the boundary value conditions are responsible for resonance