cellular networks as models for planck scale physics starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the planck scale one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics we base our own approach on what we call cellular networks consisting of cells nodes interacting with each other via bonds figuring as elementary interactions according to a certain local law geometrically our dynamical networks are living on graphs hence a substantial amount of the investigation is devoted to the development of various versions of discrete functional analysis and geometry on such almost random webs another important topic we address is a suitable concept of intrinsic fractal dimension on erratic structures of this kind in the course of the investigation we make comments concerning both different and related approaches to quantum gravity as say the spin network framework it may perhaps be said that certain parts of our programme seem to be a realization of ideas sketched by smolin some time ago

abstract volterra integrodifferential equations with applications to parabolic models with memory in this paper we are concerned with local existence regularity and continuous dependence upon the initial data of epsilon regular mild solutions for the abstract integrodifferential equation 1 2 we also present a result on unique continuation and a blow up alternative for an epsilon regular mild solution of 1 2 finally we apply our results to three interesting models navier stokes equations with memory diffusion equations with memory and a strongly damped plate equation with memory

noncommutative differential geometry and new models of gauge theory the noncommutative differential geometry of the algebra c v mn c of smooth mn c valued functions on a manifold v is investigated for n 2 the analog of maxwell s theory is constructed and interpreted as a field theory on v it describes a u n yang mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential the euclidean action which is positive vanishes on exactly two distinct gauge orbits which are interpreted as two vacua of the theory in one of the corresponding vacuum sectors the su n part of the yang mills field is massive for the case n 2 analogies with the standard model of electroweak theory are pointed out finally a brief description is provided of what happens if one starts from the analog of a general yang mills theory instead of maxwell s theory which is a particular case

optimal birth control for an age dependent n dimensional food chain model abstract in this paper we investigate optimal policies for an age dependent n dimensional food chain model which is controlled by fertility by using dubovitskii milyutin s general theory the maximum principles are obtained for problems with free terminal states infinite horizon and target sets respectively

differential calculi and linear connections a method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra as an example the generalized quantum plane is studied it is found that there is a strong correlation but not a one to one correspondence between the module structure of the 1 forms and the metric torsion free connections on it in the commutative limit the connection remains as a shadow of the algebraic structure of the 1 forms

soliton equations and the zero curvature condition in noncommutative geometry familiar nonlinear and in particular soliton equations arise as zero curvature conditions for connections with noncommutative differential calculi the burgers equation is formulated in this way and the cole hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework the kdv modified kdv equation and the miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the kp and a modified kp equation furthermore a differential calculus associated with the boussinesq equation is derived from the kp calculus

dynamical evolution in non commutative discrete phase space and the derivation of classical kinetic equations by considering a lattice model of extended phase space and using techniques of non commutative differential geometry we are led to a the concept of vector fields as generators of motion and transition probability distributions on the lattice b the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure c the general prescription for the evolution of the observables in analogy with classical dynamics we show that in the limit of a continuous description these results lead to the time evolution of observables in terms of the adjoint of generalized fokker planck equations having 1 a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion 2 a drift term given by the microscopic average of the dynamical equations in the present context these results are applied to one and two dimensional problems specifically we derive i the equations of diffusion smoluchowski and fokker planck in velocity space thus indicating the way random walk models are incorporated in the present context ii kramers equation by further assuming that motion is deterministic in coordinate space

heat kernel regularization of quantum fields we discuss some consequences of the existence of a heat kernel regularization hkr for quantum fields we demonstrate that hkr applies in certain examples using methods which should be useful more generally

numerical approximations of algebraic riccati equations for abstract systems modelled by analytic semigroups and applications this paper provides a numerical approximation theory of algebraic riccati operator equations with unbounded coefficient operators a and b such as arise in the study of optimal quadratic cost problems over the time interval 0 oo for the abstract dynamics j ay bu here a is the generator of a strongly continuous analytic semigroup and b is an unbounded operator with any degree of unboundedness less than that of a convergence results are provided for the riccati operators as well as for all the other relevant quantities which enter into the dynamic optimization problem the present numerical theory is the counterpart of a known continuous theory several examples of partial differential equations with boundary point control where all the required assumptions are verified illustrate the theory they include parabolic equations with l2 dirichlet control as well as plate equations with a strong degree of damping and point control

non commutative geometry and kinetic theory of open systems the basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated leading to the conclusion that if they are differential equations on its phase space m they are of at most second order for open systems interacting with a bath at canonical equilibrium they have the particular form of an equation of a generalized fokker planck type we show that it is possible to obtain them as liouville equations of hamiltonian dynamics on m with a particular non commutative differential structure provided that certain conditions geometric in character are fulfilled to this end symplectic geometry on m is developed in this context and an outline of the required tensor analysis and differential geometry is given certain questions as regards the possible mathematical interpretation of this structure are also discussed

on the mathematical model for linear elastic systems with analytic damping in the present paper we investigate linear elastic systems with damping ddot y b dot y ay 0 in hilbert spaces where a is a positive definite unbounded linear operator and b is a closed linear operator related in various ways to a alpha frac 1 2 leqq alpha leqq 1 we discuss the spectral property of these systems and obtain some fundamental results for the holomorphic property and the exponential stability of the semigroups associated with these systems

linear wave equations with time dependent propagation speed and strong damping abstract we consider a second order linear equation with a time dependent coefficient c t in front of the elastic operator for these equations it is well known that a higher space regularity of initial data compensates a lower time regularity of c t in this paper we investigate the influence of a strong dissipation namely a friction term which depends on a power of the elastic operator what we discover is a threshold effect when the exponent of the elastic operator in the friction term is greater than 1 2 the damping prevails and the equation behaves as if the coefficient c t were constant when the exponent is less than 1 2 the time regularity of c t comes into play if c t is regular enough once again the damping prevails on the contrary when c t is not regular enough the damping might be ineffective and there are examples in which the dissipative equation behaves as the non dissipative one as expected the stronger is the damping the lower is the time regularity threshold we also provide counterexamples showing the optimality of our results

strongly damped wave problems bootstrapping and regularity of solutions abstract the aim of the article is to present a unified approach to the existence uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in banach spaces our results are applied next to a number of examples appearing in literature which fall into the class of strongly damped semilinear wave equations the present work essentially extends the results on the existence and regularity of solutions to such problems previously these problems have been considered mostly within the hilbert space setting and with the main part operators being selfadjoint in this article we present a more general approach involving sectorial operators in reflexive banach spaces

discrete riemannian geometry within a framework of noncommutative geometry we develop an analog of pseudo riemannian geometry on finite and discrete sets on a finite set there is a counterpart of the continuum metric tensor with a simple geometric interpretation the latter is based on a correspondence between first order differential calculi and digraphs the vertices of the latter are given by the elements of the finite set arrows originating from a vertex span its co tangent space if the metric is to measure length and angles at some point it has to be taken as an element of the left linear tensor product of the space of 1 forms with itself and not as an element of the nonlocal tensor product over the algebra of functions as considered previously by several authors it turns out that linear connections can always be extended to this left tensor product so that metric compatibility can be defined in the same way as in continuum riemannian geometry in particular in the case of the universal differential calculus on a finite set the euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion in our rather general framework which also comprises structures which are far away from continuum differential geometry there is in general nothing like a ricci tensor or a curvature scalar because of the nonlocality of tensor products over the algebra of functions of forms corresponding components with respect to some module basis turn out to be rather nonlocal objects but one can make use of the parallel transport associated with a connection to localize such objects and in certain cases there is a distinguished way to achieve this in particular this leads to covariant components of the curvature tensor which allow a contraction to a ricci tensor several examples are worked out to illustrate the procedure furthermore in the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the vacuum einstein equations

optimal control for age structured population dynamics of incomplete data we are concerned with the control question for linear age structured population dynamics of incomplete initial data more precisely the initial population age distribution is supposed to be unknown we here generalize the notion of no regret control of lions 1992 10 to such singular population dynamics following the method by nakoulima omrane and velin 2000 16 we prove that the problem we are considering has a unique no regret control that we characterize by a singular optimality system

optimal control of averaged state of a parabolic equation with missing boundary condition we consider the optimal control of general heat governed by an operator depending on an unknown parameter and with missing boundary condition using the notion of no regret and low regret control

a finite difference scheme with a leibniz rule in a finite difference scheme for numerical approximation the usual notion of product is replaced by a convolution to circumvent the failure of the product rule for differentiation the solution displays a complementarity the more localised is the product the more extended is the approximation to the derivative and vice versa

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

optimal birth control of population dynamics abstract we study optimal birth control policies for an age structured population of mckendrick type which is a distributed parameter system involving first order partial differential equations with nonlocal bilinear boundary control the functional analytic approach of dubovitskii and milyutin is adopted in the investigation maximum principles for problems with a free end condition a fixed final horizon are developed and the time optimal control problem the problem with target sets and the infinite planning horizon case are investigated

markov dilations and quantum detailed balance we construct quantum stochastic processes whose multi time correlation functions with suitable time ordering can be obtained from a quantum dynamical semigroup we prove that such a process defines a stationary markov dilation of the associated semigroup if and only if up to technicalities the semigroup satisfies the quantum detailed balance condition with respect to its stationary state

bicovariant differential calculi on glp q 2 and quantum subgroups in a previous paper all bicovariant differential calculi on the two parameter quantum group glp q 2 were determined in this work the authors elaborate some of their properties and discuss relations with work by other authors furthermore they show that there are two different ways in which bicovariant differential calculi on glq 2 where p q induce corresponding calculi on the quantum subgroup slq 2 it is shown that on slq 2 there are only two different bicovariant differential calculi from these one obtains the 4d or calculi on suq 2 the classical limit of the two differential calculi on slq 2 is investigated revealing a relation with recent work on deformed differential calculi on commutative function algebras

pointwise pressure observations of a canonical structural acoustics model in this paper we study regularity properties of a physically motivated observation associated with a controlled partial differential equation pde model on a rectangular domain this particular pde describes the active control of acoustic pressure in a chamber by means of piezoelectric activation and involves a wave coupled to a structurally damped beam equation with the coupling being accomplished by boundary trace terms the observation here consists of pointwise in space measurements of the velocity of the wave component with this velocity representing the acoustic pressure of the system as defined then the observation of the system like the control imput is modeled by an unbounded operator the main result of this paper is the discovery of an energy space such that for initial data taken from this space the desired observation of the acoustic pressure is well defined

kinetic theory in the weak coupling approximation ii a classical harmonic oscillator model abstract the kinetic theory of open systems presented in a previous paper is applied to a classical harmonic oscillator model it leads to a kinetic equation very different from the conventionally used linearized landau equation that shows explicitly the influence of the field on the interaction of the harmonic particle with its surroundings contrary to what is often assumed a self consistent approach based on our equation suggests a one dimensional generalization of kramers model equation the limit of our equation when the harmonic frequency vanishes contains a divergent term whose nature is briefly discussed

discrete differential calculus graphs topologies and gauge theory differential calculus on discrete sets is developed in the spirit of noncommutative geometry any differential algebra on a discrete set can be regarded as a reduction of the universal differential algebra and this allows a systematic exploration of differential algebras on a given set associated with a differential algebra is a di graph where two vertices are connected by at most two antiparallel arrows the interpretation of such a graph as a hasse diagram determining a locally finite topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure it is shown that field theories and in particular gauge theories can be formulated on a discrete set in close analogy with the continuum case the framework presented generalizes ordinary lattice theory which is recovered from an oriented hypercubic lattice graph it also includes e g the two point space used by connes and lott and others in models of elementary particle physics the formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an internal discrete space a la connes et al to models of dimensionally reduced gauge fields furthermore a symmetric lattice is also studied which in a certain continuum limit turns out to be related to a noncommutative differential calculus on manifolds

diffusion processes quantum dynamical semigroups and the classical kms condition we study some properties of the generator of a diffusion process describing the reduced dynamics of a classical system weakly coupled to an external reservoir we show that such processes may be equivalently obtained as classical limits of quantum dynamical semigroups derived in the weak coupling limit we prove that the stationary state is independent of the coupling if and only if the reservoir is in a state of thermal equilibrium

local and global smoothing effects for some linear hyperbolic equations with a strong dissipation we consider an abstract second order linear equation with a strong dissipation namely a friction term which depends on a power of the elastic operator in the homogeneous case we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions in the nonhomogeneous case we study how the regularity of solutions depends on the regularity of forcing terms and we characterize the spaces where a bounded forcing term yields a bounded solution what we discover is a variety of different regimes with completely different behaviors depending on the exponent in the friction term we also provide counterexamples in order to show the optimality of our results mathematics subject classification 2010 msc2010 35l10 35l15 35l20

differential calculus and gauge theory on finite sets we develop differential calculus and gauge theory on a finite set g an elegant formulation is obtained when g is supplied with a group structure and in particular for a cyclic group connes two point model 1986 which is an essential ingredient of his reformulation of the standard model of elementary particle physics is recovered in our approach reductions of the universal differential calculus to lower dimensional differential calculi are considered the complete reduction leads to a differential calculus on a periodic lattice

controllability of second order infinite dimensional systems in the paper the approximate controllability of linear abstract second order infinite dimensional dynamical systems is considered it is proved using the frequency domain method that approximate controllability of second order system can be verified by the approximate controllability conditions for the corresponding simplified first order system general results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure some special cases are also considered moreover remarks and comments on the relationships between different concepts of controllability are given the paper extends earlier results on approximate controllability of second order abstract dynamical systems

simplified conditions of initial observability for infinite dimensional second order damped dynamical systems abstract in the paper the initial observability of abstract second order infinite dimensional dynamical systems is considered two cases of observation operator are explored first the observation operator is assumed to be bounded then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation using the frequency domain method it is proved that initial observability of second order system under discussion can be verified by the initial observability conditions for the corresponding simplified first order system general results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures some special cases are also discussed and practical comments and remarks are given the paper extends earlier results on initial observability for a class of infinite dimensional second order abstract dynamical systems

on the dual boundary conditions and the nonnegative square root of the second derivative operator abstract a dual relationship between the boundary conditions of the second derivative operator au u is established some dual properties determined by the dual boundary conditions are considered furthermore the singularities of the nonnegative square root of the above second derivative operator are studied our results show that although a 1 2 u is not u it is a harmonic conjugate of u in some sense

regularity of solutions on the global attractor for a semilinear damped wave equation we consider attractors a 0 1 corresponding to a singularly perturbed damped wave equation utt 2 a 1 2 ut aut au f u

noncommutative differential calculus and lattice gauge theory the authors study consistent deformations of the classical differential calculus on algebras of functions and more generally commutative algebras such that differentials and functions satisfy nontrivial commutation relations for a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice these differential calculi generate a lattice on a space continuum the whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus in this framework one just has to express the lagrangian for the continuum theory in terms of differential forms this expression then also makes sense for the deformed differential calculus there is a natural integral associated with the latter integration of the lagrangian over a space continuum then produces the correct lattice action for a large class of theories this is explicitly shown for the scalar field action and the action for su m gauge theory

regularity of some structurally damped problems with point control and with boundary control abstract we study the regularity of structurally damped abstract systems with application to point control and boundary control problems for plate equations in general multidimensional domains in rn our treatment includes in particular both the cases where the damping operator is equal to the square root of the original differential operator and the case where the damping operator is only comparable to such a square root

carleman estimate for an adjoint of a damped beam equation and an application to null controllability abstract in this article we consider a control problem of a linear euler bernoulli damped beam equation with potential in dimension one with periodic boundary conditions we derive a new carleman estimate for an adjoint of the equation under consideration then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state

a non autonomous strongly damped wave equation existence and continuity of the pullback attractor abstract in this paper we consider the strongly damped wave equation with time dependent terms u t t u t u t e t u t f u in a bounded domain r n under some restrictions on e t t and growth restrictions on the nonlinear term f the function e t depends on a parameter e e t e 0 0 we will prove under suitable assumptions local and global well posedness using the uniform sectorial operators theory the existence and regularity of pullback attractors a e t t r uniform bounds for these pullback attractors characterization of these pullback attractors and their upper and lower semicontinuity at 0

discrete differential manifolds and dynamics on networks a discrete differential manifold is a countable set together with an algebraic differential calculus on it this structure has already been explored in previous work and provides a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time several examples are presented and a notion of differentiability of maps between discrete differential manifolds is introduced particular attention is given to differentiable curves in such spaces every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity

non commutative geometry of finite groups a finite set can be supplied with a group structure which can then be used to select classes of differential calculi on it via the notions of left right and bicovariance a corresponding framework has been developed by woronowicz more generally for hopf algebras including quantum groups a differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and moreover for the formulation of field theories and dynamics on finite sets associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures for a covariant calculus there are notions of invariance for linear connections and tensors all these concepts are explored for finite groups and illustrated with examples some results are formulated more generally for arbitrary associative hopf algebras in particular the problem of extension of a connection on a bimodule over an associative algebra to tensor products is investigated leading to the class of extensible connections it is shown that invariance properties of an extensible connection on a bimodule over a hopf algebra are carried over to the extension furthermore an invariance property of a connection is also shared by a dual connection which exists on the dual bimodule as defined in this work

on the pareto control and no regret control for distributed systems with incomplete data we discuss the control of distributed systems with incomplete data following the notion of no regret control or equivalently pareto control used by lions in c r acad sci paris ser i math 302 1986 pp 223 227 and c r acad sci paris ser i math 302 1992 pp 1253 1257 we associate with the no regret control a sequence of low regret controls defined by a quadratic perturbation previously used by nakoulima omrane and velin in c r acad sci paris ser i math 330 2000 pp 801 806 in the first part we prove that the perturbed system corresponds to a sequence of standard control problems and converges to the no regret or pareto control for which we obtain a singular optimality system we give also some applications in the second part we show how the method can be extended to the evolution case equations of parabolic type petrowsky type or hyperbolic type are considered

deformations of super kms functionals we investigate the stability of the super kms property under deformations we show that a family of continuous deformations of the super derivation in the quantum algebra yields a continuous family of deformed super kms functionals these functionals define a family of cohomologous entire cocycles in this paper we investigate the super kms skms property of functionals co on a quantum algebra our interest in skms functionals was inspired by work of kastler and by conversations with alain connes ci k jlo2 the skms construction relies on the existence of a super derivation d acting on a dense subalgebra of a c algebra d the square of d is the infinitesimal generator of a continuous one parameter automorphism group c of the quantum algebra the usual kms property relates the cyclicity of a state co to the analytic continuation of a group et of automorphisms the skms property also involves invariance under the super derivation d whose square generates the automorphism group er it is known that an skms functional on a quantum algebra defines an entire cyclic cocycle z this is just the chern character which jaffe lesniewski and osterwalder defined in the context ofsupertrace functionals on a quantum algebra jlo 1 the skms property ensures that the functional co and the cocycle z which is derived from it are invariant under this group action in this paper we study the stability of this structure under perturbations of d we study only bounded perturbations which arise from the graded super commutator with an odd element q of the algebra d we show that such perturbations dq of d can be used to define a deformation coq of co which satisfies the skms property furthermore the corresponding family of cocycles z are cohomologous of course more singular unbounded perturbations can lead to

quantum mechanics as noncommutative symplectic geometry the authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation this leads to a mathematical framework in which quantum mechanics can be understood as noncommutative symplectic geometry they point out the possibility to describe interactions geometrically as deformations of the differential calculus

kinetic equation for a classical test particle model in the weak coupling approximation abstract a kinetic equation for a harmonically bound classical particle weakly coupled to a thermal reservoir is presented the derivation based on the formalism of master equations developed in brussels is outlined and compared to other approaches some comments are also made as to the consistency of the so called kramers equation in the presence of an external field

all bicovariant differential calculuses on glq 3 c and slq 3 c all bicovariant first order differential calculuses on the quantum group glq 3 c are determined there are two distinct one parameter families of calculuses in terms of a suitable basis of 1 forms the commutation relations can be expressed with the help of the r matrix of glq 3 c some calculuses induce bicovariant differential calculuses on slq 3 c and on real forms of glq 3 c for the generic deformation parameter q there are six calculuses on slq 3 c on suq 3 there are only two the classical limit q to 1 of bicovariant calculuses on slq 3 c is not the ordinary calculus on sl 3 c one obtains a deformation of it which involves the cartan killing metric

a mechanical model of brownian motion we consider a dynamical system consisting of one large massive particle and an infinite number of light point particles we prove that the motion of the massive particle is in a suitable limit described by the ornstein uhlenbeck process this extends to three dimensions previous results by holley in one dimension

differential geometry of group lattices in a series of publications we developed differential geometry on discrete sets based on concepts of noncommutative geometry in particular it turned out that first order differential calculi over the algebra of functions on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set a particular class of digraphs are cayley graphs also known as group lattices they are determined by a discrete group g and a finite subset s there is a distinguished subclass of bicovariant cayley graphs with the property ad s s s we explore the properties of differential calculi which arise from cayley graphs via the above correspondence the first order calculi extend to higher orders and then allow us to introduce further differential geometric structures furthermore we explore the properties of discrete vector fields which describe deterministic flows on group lattices a lie derivative with respect to a discrete vector field and an inner

on a population dynamics control problem with age dependence and spatial structure we consider the control problem for a population dynamics model with age dependence spatial structure and a nonlocal birth process arising as a boundary condition we examine the controllability at a given time t and show that approximate controllability holds for every fixed finite time t as a consequence a new uniqueness condition continuation result is proved

kinetic theory in the weak coupling approximation iii classical systems extension of the theory and further applications abstract the kinetic theory of open classical systems in the weak coupling approximation presented in a previous paper paper i is extended here to a wider class of systems so that it is applicable when the liouville operator of the open system has a continuous spectrum it is applied i to an anharmonic oscillator and ii to a free test particle model thus leading to a generalization of the conventionally used linearized landau equation problems related to the divergences that appear when the kinetic equations obtained by the formalism of paper i are considered in the limit of a continuous spectrum of the open system s liouville operator are also discussed

on the nonnegative square root of the fourth derivative operators abstract the singularities of the nonnegative square root of the fourth derivative operator au u 4 are studied

differential calculi on commutative algebras a differential calculus on an associative algebra a is an algebraic analogue of the calculus of differential forms on a smooth manifold it supplies a with a structure on which dynamics and field theory can to some extent be formulated in very much the same way we are used to in the geometrical arena underlying classical physical theories and models in previous work certain differential calculi on a commutative algebra exhibited relations with lattice structures stochastics and parametrized quantum theories this motivated the present systematic investigation of differential calculi on commutative and associative algebras various results about their structure are obtained in particular it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of i forms an example of such a product is provided by the ito calculus of stochastic differentials for the case where the algebra a is freely generated by coordinates xi i 1 n we study calculi for which the differentials dxi constitute a basis of the space of 1 forms as a left a module these may be regarded as deformations of the ordinary differential calculus on rn for n or 3 a classification of all orbits under the general linear group of such calculi with constant structure functions is presented we analyse whether these calculi are reducible i e a skew tensor product of lower dimensional calculi or whether they are the extension as defined in this paper of a one dimension lower calculus furthermore generalizations to arbitrary n are obtained for all these calculi

existence results for some damped second order volterra integro differential equations abstract in this paper we make a subtle use of operator theory techniques and the well known schauder fixed point principle to establish the existence of pseudo almost automorphic solutions to some second order damped integro differential equations with pseudo almost automorphic coefficients in order to illustrate our main results we will study the existence of pseudo almost automorphic solutions to a structurally damped plate like boundary value problem

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

the stationary state and the heat equation for a variant of davies model of heat conduction we study a variant of davies model of heat conduction consisting of a chain of classical or quantum harmonic oscillators whose ends are coupled to thermal reservoirs at different temperatures and where neighboring oscillators interact via intermediate reservoirs in the weak coupling limit we show that a unique stationary state exists and that a discretized heat equation holds we give an explicit expression of the stationary state in the case of two classical oscillators the heat equation is obtained in the hydrodynamic limit and it is proved that it completely describes the macroscopic behavior of the model

automorphisms of associative algebras and noncommutative geometry a class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra several examples are presented in particular differential calculi on the quantum plane the h deformed plane and the quantum group glp q 2 are recovered in this way geometric structures such as metrics and compatible linear connections are introduced

integrable discretizations of chiral models via deformation of the differential calculus a construction of conservation laws for chiral models generalized models on a two dimensional spacetime continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions this is achieved via a deformation of the ordinary differential calculus in particular the nonlinear toda lattice results in this way from the linear continuum wave equation the method is applied to several further examples we also construct lax pairs and backlund transformations for the class of models considered in this work

ergodic boundary point control of stochastic semilinear systems a controlled markov process in a hilbert space and an ergodic cost functional are given for a control problem that is solved where the process is a solution of a parameter dependent semilinear stochastic differential equation and the control can occur only on the boundary or at discrete points in the domain the linear term of the semilinear differential equation is the infinitesimal generator of an analytic semigroup the noise for the stochastic differential equation can be distributed boundary and point some ergodic properties of the controlled markov process are shown to be uniform in the control and the parameter the existence of an optimal control is verified to solve the ergodic control problem the optimal cost is shown to depend continuously on the system parameter

the commutative limit of a matrix geometry the noncommutative geometry of complex n n matrices is considered as an approximation to the geometry of the two sphere s2 and the differential forms defined over the matrices are mapped into the ordinary differential forms on su2 considered as a fiber bundle over s2 it is found that the extra gauge invariant vacuum studied in previous publications has as image a constant multiple of the connection on su2 that defines the fibration the dirac monopole with unit magnetic charge

characterization of domains of fractional powers of certain operators arising in elastic systems and applications with x a hilbert space where a the elastic operator and b the damping operator are closed operators in x with dense domains q a and 9 b respectively moreover a is throughout assumed to be self adjoint and strictly positive stimulated by two conjectures raised in crl the most general of which referred to the case lx f described below we have already studied the operators and sb defined respectively by

linear connections on graphs in recent years discrete spaces such as graphs have attracted much attention as models for physical spacetime or as models for testing the spirit of noncommutative geometry in this work we construct the differential algebras for graphs by extending the work of dimakis et al and discuss linear connections and curvatures on graphs especially we calculate connections and curvatures explicitly for the general nonzero torsion case there is a metric but no metric compatible connection in general except the complete symmetric graph with two vertices