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

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

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

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

differential hopf algebra structures on the universal enveloping algebra of a lie algebra we discuss a method to construct a de rham complex differential algebra of poincar e birkhoff witt type on the universal enveloping algebra of a lie algebra g we determine the cases in which this gives rise to a differential hopf algebra that naturally extends the hopf algebra structure of u g the construction of such differential structures is interpreted in terms of colour lie superalgebras

deformed noncommutative tori we recall a construction of non commutative algebras related to a one parameter family of deformed spheres and tori and show that in the case of tori the algebras can be completed into c algebras isomorphic to the standard non commutative torus as the former was constructed in the context of matrix or fuzzy geometries it provides an important link to the framework of non commutative geometry and opens up for a concrete way to study deformations of non commutative tori furthermore we show how the well known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite dimensional representations of the deformed algebras and their projective modules are described together with connections of constant curvature

toward the fourier law for a weakly interacting anharmonic crystal for a system of weakly interacting anharmonic oscillators perturbed by an energy preserving stochastic dynamics we prove an autonomous stochastic evolution for the energies at large time scale with respect to the coupling parameter it turn out that this macroscopic evolution is given by the so called conservative non gradient ginzburg landau system of stochastic differential equations the proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics

a natural basis of states for the noncommutative sphere and its moyal bracket an infinite dimensional algebra which is a nondecomposable reducible representation of su 2 is given this algebra is defined with respect to two real parameters if one of these parameters is zero the algebra is the commutative algebra of functions on the sphere otherwise it is a noncommutative analog this is an extension of the algebra normally referred to as the berezin quantum sphere or fuzzy sphere a natural indefinite inner product and a basis of the algebra orthogonal with respect to it are given the basis elements are homogeneous polynomials eigenvectors of a laplacian and related to the hahn polynomials it is shown that these elements tend to the spherical harmonics for the sphere a moyal bracket is constructed and shown to be the standard moyal bracket for the sphere

nonequilibrium statistical mechanics of hamiltonian rotators with alternated spins we consider a finite region of a d dimensional lattice of nonlinear hamiltonian rotators where neighbouring rotators have opposite alternated spins and are coupled by a small potential of size varepsilon a a ge 1 2 we weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order varepsilon then we introduce action angle variables for the system of uncoupled rotators varepsilon 0 and note that the sum of actions over all nodes is conserved by the purely hamiltonian dynamics of the system with varepsilon 0 we investigate the limiting as varepsilon rightarrow 0 dynamics of actions for solutions of the varepsilon perturbed system on time intervals of order varepsilon 1 it turns out that the limiting dynamics is governed by a certain autonomous stochastic equation for the vector of actions this equation has a completely non hamiltonian nature this is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order the varepsilon perturbed system has a unique stationary measure widetilde mu varepsilon and is mixing any limiting point of the family widetilde mu varepsilon of stationary measures as varepsilon rightarrow 0 is an invariant measure of the system of uncoupled integrable rotators there are plenty of such measures however it turns out that only one of them describes the limiting dynamics of the varepsilon perturbed system we prove that a limiting point of widetilde mu varepsilon is unique its projection to the space of actions is the unique stationary measure of the autonomous equation above which turns out to be mixing and its projection to the space of angles is the normalized lebesque measure on the torus mathbb t n the results and convergences which concern the behaviour of actions on long time intervals are uniform in the number n of rotators those concerning the stationary measures are uniform in n in some natural cases

shadow of noncommutativity we analyze the structure of the 0 limit of a family of algebras a describing noncommutative versions of space time with a parameter of noncommutativity assuming the poincare covariance of the 0 limit we show that besides the algebra of functions on minkowski space a0 must contain a nontrivial extra factor a0i which is lorentz covariant and which does not commute with the functions whenever it is not commutative we give a general description of the possibilities and analyze some representative examples

submanifolds and quotient manifolds in noncommutative geometry we define and study noncommutative generalizations of submanifolds and quotient manifolds using the derivation based differential calculus introduced by m dubois violette and p michor we give examples to illustrate these definitions

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

le langage des espaces et des groupes quantiques we study the foundations of the differential calculus in quantum geometry the notions of differential quantum space and cone are introduced generalizing a construction of manin to a quantum cone we associate the quantum group of its linear automorphisms preserving the differentials and deduce a de rham complex on this group we give examples of differential calculi on quantum hyperplanes and quantum linear groups

noncommutative differential geometry of matrix algebras the noncommutative differential geometry of the algebra mn c of complex n n matrices is investigated the role of the algebra of differential forms is played by the graded differential algebra c sl n c mn c mn c sl n c sl n c acting by inner derivations on mn c a canonical symplectic structure is exhibited for mn c for which the poisson bracket is to within a factor i the commutator also a canonical riemannian structure is described for mn c finally the analog of the maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge

approximate dissipative hamiltonian realization and construction of local lyapunov functions the key in applying energy based control approach is to be able to express the system under consideration as a dissipative hamiltonian system i e to obtain dissipative hamiltonian realization dhr for the system in general the precise dhr form is hard to obtain for nonlinear dynamic systems when a precise dhr does not exist for a dynamic system or such a precise realization is difficulty to obtain it is necessary to consider its approximate realization this paper investigates approximate dhr and construction of local lyapunov functions for time invariant nonlinear systems it is shown that every nonlinear affine system has an approximate dhr if linearization of the system is controllable based on the diagonal normal form of nonlinear dynamic systems a new algorithm is established for the approximate dhr finally we present the concept of kth degree approximate lyapunov function and provide a method to construct such a lyapunov function example studies show that the methodology presented in this paper is very effective

green kubo formula for weakly coupled systems with noise we study the green kubo formula kappa varepsilon varsigma for the heat conductivity of an infinite chain of d dimensional finite systems cells coupled by a smooth nearest neighbor potential varepsilon v the uncoupled systems evolve according to hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength varsigma noting that kappa varepsilon varsigma exists and is finite whenever varsigma 0 we are interested in what happens when the strength of the noise varsigma to 0 for this we start in this work by formally expanding kappa varepsilon varsigma in a power series in varepsilon kappa varepsilon varsigma varepsilon 2 sum n geq 2 varepsilon n 2 kappa n varsigma and investigating the formal equations satisfied by kappa n varsigma we show in particular that kappa 2 varsigma is well defined when no pinning potential is present and coincides formally with the heat conductivity obtained in the weak coupling van hove limit where time is rescaled as varepsilon 2 t for the cases where the latter has been established liverani and olla in jams 25 555 583 2012 dolgopyat and liverani in commun math phys 308 201 225 2011 for one dimensional systems we investigate kappa 2 varsigma as varsigma to 0 in three cases the disordered harmonic chain the rotor chain and a chain of strongly an harmonic oscillators moreover we formally identify kappa 2 varsigma with the conductivity obtained by having the chain between two reservoirs at temperature t and t delta t in the limit delta t to 0 n to infty varepsilon to 0

quantum group covariant anti symmetrizers tensors vielbein hodge map and laplacian glq n and soq n covariant deformations of the completely symmetric antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices the precise relation between the completely antisymmetric projectors and the completely antisymmetric tensor is determined adopting the glq n and soq n covariant differential calculi on the corresponding quantum group covariant noncommutative spaces we introduce a generalized notion of vielbein basis or frame based on differential operator valued 1 forms we then give a thorough definition of a soq n covariant bilinear hodge map acting on the bimodule of differential forms on introduce the exterior coderivative and show that the laplacian acts on differential forms exactly as in the undeformed case namely it acts on each component as it does on functions

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

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

equilibrium fluctuations for interacting ornstein uhlenbeck particles we study the hydrodynamic density fluctuations of an infinite system of interacting particles on d the particles interact between them through a two body superstable potential and with a surrounding fluid in equilibrium through a random viscous force of ornstein uhlenbeck type the stationary initial distribution is the gibbs measure associated with the potential and with a given temperature and fugacity we prove that the time dependent density fluctuation field converges in law under diffusive scaling of space and time to the solution of a linear stochastic partial differential equation driven by white noise

some remarks on nonequilibrium dynamics of infinite particle systems classical mechanics of infinitely many particles in dimensions one and two is considered particles interacting by a superstable pair potential of finite range the group of motion generated by newton s equations is constructed in the space of locally finite configurations with a logarithmic order of energy fluctuations at infinity a core of the liouville operator is also described results of dobrushin and the author and of marchioro pellegrinotti pulvirenti are improved

on the passivity based control of irreversible processes irreversible port hamiltonian systems iphs have recently been proposed for the modelling of irreversible thermodynamic systems on the other hand a classical result on the use of the second law of thermodynamics for the stabilization of irreversible processes is the celebrated thermodynamic availability function these frameworks are combined to propose a class of passivity based controller pbc for irreversible processes an alternative formulation of the availability function in terms of internal energy is proposed using iphs a matching condition which is interpreted in terms of energy shaping is derived and a specific solution that permits to assign a desired closed loop structure and entropy rate is proposed the approach can be compared with interconnection and damping assignment pbc this method however leads in general to thermodynamically non coherent closed loop systems in this paper a system theoretic approach is employed to derive a constructive method for the control design the closed loop system is in iphs form hence it can be identified with a thermodynamic system and the control parameters related with thermodynamic variables such as the reaction rates in the case of chemical reactions a generic non linear non isothermal continuous stirred tank reactor is used to illustrate the approach

stochastic stability of pollicott ruelle resonances kinetic brownian motion on the cosphere bundle of a riemannian manifold mathbb m is a stochastic process that models the geodesic equation perturbed by a random white force of size varepsilon when mathbb m is compact and negatively curved we show that the l 2 spectrum of the infinitesimal generator of this process converges to the pollicott ruelle resonances of mathbb m as varepsilon goes to 0

topologically nontrivial field configurations in noncommutative geometry in the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated fuzzy sphere the corresponding field theory actions conserve all basic symmetries of the standard commutative version space isometries and global chiral symmetry but due to the noncommutativity of the space the fields are regularized and they contain only a finite number of modes

strong connections on quantum principal bundles a gauge invariant notion of a strong connection is presented and characterized it is then used to justify the way in which a global curvature form is defined strong connections are interpreted as those that are induced from the base space of a quantum bundle examples of both strong and non strong connections are provided in particular such connections are constructed on a quantum deformation of the two sphere fibrations2 rp2 a certain class of stronguq 2 connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq dependent hermitian metric a particular form of the yang mills action on a trivialuq 2 bundle is investigated it is proved to coincide with the yang mills action constructed by a connes and m rieffel furthermore it is shown that the moduli space of critical points of this action functional is independent ofq

groupoid approach to noncommutative quantization of gravity we propose a new scheme for quantizing gravity based on a noncommutative geometry our geometry corresponds to a noncommutative algebra a gc g c of smooth compactly supported complex functions with convolution as multiplication on the groupoid g e being the semidirect product of a structured space e of constant dimension or a smooth manifold and a group in the classical case e is the total space of the frame bundle and is the lorentz group the differential geometry is developed in terms of a z a submodule v of derivations of a and a noncommutative counterpart of einstein s equation is defined a pair a where is a subset of derivations of a satisfying the noncommutative einstein s equation is called an einstein pair we introduce the representation of a in a suitable hilbert space by completing a with respect to the corresponding norm change it into a c algebra and perform quantization with the help of the standard c algebraic method hermitian elements of this algebra are interp

noncommutative geometry and gauge theory on fuzzy sphere the differential algebra on the fuzzy sphere is constructed by applying connes scheme the u 1 gauge theory on the fuzzy sphere based on this differential algebra is defined the local u 1 gauge transformation on the fuzzy sphere is identified with the left u n 1 transformation of the field where a field is a bimodule over the quantized algebra the interaction with a complex scalar field is also given

the chern simons action in noncommutative geometry a general definition of chern simons actions in noncommutative geometry is proposed and illustrated in several examples these examples are based on space times which are products of even dimensional riemannian spin manifolds by a discrete two point set if the algebras of operators describing the noncommutative spaces are generated by functions over such space times with values in certain clifford algebras the chern simons actions turn out to be the actions of topological gravity on the even dimensional spin manifolds by constraining the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from chern simons actions

graded differential geometry of graded matrix algebras we study the graded derivation based noncommutative differential geometry of the z2 graded algebra m n m of complex n m n m matrices with the usual block matrix grading for n m beside the infinite dimensional algebra of graded forms the graded cartan calculus graded symplectic structure graded vector bundles graded connections and curvature are introduced and investigated in particular we prove the universality of the graded derivation based first order differential calculus and show that m n m is a noncommutative graded manifold in a stricter sense there is a natural body map and the cohomologies of m n m and its body coincide as in the case of ordinary graded manifolds

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

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

on finite 4d quantum field theory in non commutative geometry the truncated 4 dimensional spheres4 and the action of the self interacting scalar field on it are constructed the path integral quantization is performed while simultaneously keeping theso 5 symmetry and the finite number of degrees of freedom the usual field theory uv divergences are manifestly absent

local dissipative hamiltonian description of reversible reaction networks in this letter we show that closed reversible chemical reaction networks with independent elementary reactions admit a global pseudo hamiltonian structure which is at least locally dissipative around any equilibrium point the structure matrix of the hamiltonian description reflects the graph topology of the reaction network and it is a smooth function of the concentrations of the chemical species in the positive orthant the physical interpretation of the description is briefly explained and two illustrative examples are presented for global and local dissipative hamiltonian description respectively

invariant noncommutative connections in this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle it has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections we use explicitly some geometric constructions usually introduced to classify ordinary invariant connections and we expand them using algebraic objects coming from the noncommutative setting the main result is that the classification can be performed using a reduced algebra an associated differential calculus and a module over this algebra

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

a natural basis for spinor and vector fields on the noncommutative sphere the product of two heisenberg weil algebras contains the jordan schwinger representation of su 2 this algebra is quotiented by the square root of the casimir to produce a nonassociative algebra denoted by psi this algebra may be viewed as the right module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere it is now possible to interpret other subspaces as the space of spinor or vector fields on the noncommutative sphere a natural basis of psi is given which may be interpreted as the deformed entries in the rotation matrices of su 2

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

z3 graded algebras and the cubic root of the supersymmetry translations a generalization of supersymmetry is proposed based on z3 graded algebras introducing the objects whose ternary commutation relations contain the cubic roots of unity e2 i 3 e4 i 3 and 1 the operators whose trilinear combinations yield the supersymmetric translation generators can be constructed cubic matrices forming a ternary algebra are the generalization of pauli s matrices the general properties of z3 grading some other representations of such algebras and their possible pertinence with regard to the quark model are briefly discussed

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

on curvature in noncommutative geometry a general definition of a bimodule connection in noncommutative geometry has been recently proposed for a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra the corresponding curvatures are also compared

quantum field theory on the noncommutative plane with eq 2 symmetry we study properties of a scalar quantum field theory on the two dimensional noncommutative plane with eq 2 quantum symmetry we start from the consideration of a firstly quantized quantum particle on the noncommutative plane then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the eq 2 invariant measure on the noncommutative plane with the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has contrary to the common belief even more severe ultraviolet divergences than its counterpart on the usual commutative plane finally we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the eq 2 quantum group

i q graded lie algebraic extensions of the poincare algebra constraints on i and q the constraints on the index set i and on the function q of the i q graded lie algebras over k containing the poincare lie algebra are studied by using the single grading model particular choices for i and q consistent with the found constraints are determined for k c gradings are then found for which i i z2 z4n z4n gre with n n and gre an abelian group these gradings provide a way for algebraic extensions of the poincare lie algebra beyond the z2 gradings of supersymmetry and supergravity in these algebraic extensions each other commuting space time parameter can either commute or anticommute with the further parameters of the i q graded super manifold different field representations can have with each other generalized commutative behavior beyond commutativity and anticommutativity

n 3 differential calculus and gauge theory on a reduced quantum plane we discuss the algebra of n n matrices as a reduced quantum plane a n 3 nilpotent deformed differential calculus involving a complex parameter q is constructed the two cases q 3rd and nth root of unity are completely treated as an application we establish a gauge field theory for the particular cases n 2 and n 3

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

the einstein action for algebras of matrix valued functions toy models two toy models are considered within the framework of noncommutative differential geometry in the first one the einstein action of the levi civita connection is computed for the algebra of matrix valued functions on a torus it is shown that assuming some constraints on the metric this action splits into a classical like a quantum like and a mixed term in the second model an analogue of the palatini method of variation is applied to obtain critical points of the einstein action functional for m4 r it is pointed out that a solution to the palatini variational problem is not necessarily a levi civita connection in this model no additional assumptions regarding metrics are made

on globally non trivial almost commutative manifolds within the framework of connes noncommutative geometry we define and study globally non trivial or topologically non trivial almost commutative manifolds in particular we focus on those almost commutative manifolds that lead to a description of a classical gauge theory on the underlying base manifold such an almost commutative manifold is described in terms of a principal module which we build from a principal fibre bundle and a finite spectral triple we also define the purely algebraic notion of gauge modules and show that this yields a proper subclass of the principal modules we describe how a principal module leads to the description of a gauge theory and we provide two basic yet illustrative examples

on feedback equivalence to port controlled hamiltonian systems in the last few years port controlled hamiltonian pch systems have emerged as an interesting class of nonlinear models suitable for a large number of physical applications in this paper we study the question of feedback equivalence of nonlinear systems to pch systems more precisely we give conditions under which a general nonlinear system can be transformed into a pch system via static state feedback we consider the two extreme cases where the target pch system is completely a priori fixed or completely free as well as the case where it is only partially predetermined when the energy function is free a set of partial differential equations needs to be solved on the other hand if it is fixed we have to deal with a set of algebraic equations in the former case we give some verifiable necessary and sufficient conditions for solvability as a by product of our analysis we obtain some stabilization results for nonlinear systems

correction to structure and stability of certain chemical networks and applications to the kinetic proofreading model of t cell receptor signal transduction deals with the theory of structure stability robustness and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks the results given here extend but are also heavily based upon certain previous work by feinberg 1977 1979 1987 1995 and horn and jackson 1972 of which a self contained and streamlined exposition is included the theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by mckeithan 1995 for t cell receptor signal transduction

process systems and passivity via the clausius planck inequality in this paper we define a process system to be a system which has actions with the clausius planck and conservation properties we use standard and well established results derived from macroscopic thermodynamics to show that a process system has actions which satisfy the dissipation inequality furthermore these actions have an inner product structure and a link between the thermodynamic theory of process systems and the input output passivity theory of nonlinear control is established the paper therefore represents a step towards developing a passivity based approach for distributed control system design for chemical processes a review of relevant concepts from thermodynamics is given in an appendix

coulomb problem in non commutative quantum mechanics the aim of this paper is to find out how it would be possible for space non commutativity nc to alter the quantum mechanics qm solution of the coulomb problem the nc parameter is to be regarded as a measure of the non commutativity setting 0 which means a return to the standard quantum mechanics as the very first step a rotationally invariant nc space r 3 an analog of the coulomb problem configuration space r3 with the origin excluded is introduced r 3 is generated by nc coordinates realized as operators acting in an auxiliary fock space f the properly weighted hilbert schmidt operators in f form h a nc analog of the hilbert space of the wave functions we will refer to them as wave functions also in the nc case the definition of a nc analog of the hamiltonian as a hermitian operator in h is one of the key parts of this paper the resulting problem is exactly solvable the full solution is provided including formulas for the bound states for e 0 and low energy scattering fo

chirality and dirac operator on noncommutative sphere we give a derivation of the dirac operator on the noncommutative 2 sphere within the framework of the bosonic fuzzy sphere and define connes triple it turns out that there are two different types of spectra of the dirac operator and correspondingly there are two classes of quantized algebras as a result we obtain a new restriction on the planck constant in berezin s quantization the map to the local frame in noncommutative geometry is also discussed

non equilibrium statistical mechanics of strongly anharmonic chains of oscillators we study the model of a strongly non linear chain of particles coupled to two heat baths at different temperatures our main result is the existence and uniqueness of a stationary state at all temperatures this result extends those of eckmann pillet rey bellet epr99a epr99b to potentials with essentially arbitrary growth at infinity this extension is possible by introducing a stronger version of hormander s theorem for kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains

asymptotic localization of energy in nondisordered oscillator chains we study two popular one dimensional chains of classical anharmonic oscillators the rotor chain and a version of the discrete nonlinear schrodinger chain we assume that the interaction between neighboring oscillators controlled by the parameter 0 is small we rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant and we provide strong evidence that it decays faster than any power law in as 0 the weak coupling regime also translates into a high temperature regime suggesting that the conductivity vanishes faster than any power of the inverse temperature to our knowledge it is the first time that a clear connection has been established between kam like phenomena and thermal conductivity 2015 wiley periodicals inc

stationary states of random hamiltonian systems we investigate the ergodic properties of hamiltonian systems subjected to local random energy conserving perturbations we prove for some cases e g anharmonic crystals with random nearest neighbor exchanges or independent random reflections of velocities that all translation invariant stationary states with finite entropy per unit volume are microcanonical gibbs states the results can be utilized in proving hydrodynamic behavior of such systems

the renormalization flow spaces of two dimensional field theories and connes geometry we formulate a local renormalization flow using connes non commutative geometry this formulation allows a geometric description of the renormalization flow and an intrinsic characterization of the operator product expansion we define spaces for string theory in terms of a ring of correlation functions and a renormalization flow on this ring which are the analogues of the category of riemannian manifolds with metric for general relativity the beta function is related to a differential form of relative entropy between two renormalization flow trajectories

noncommutative geometry for arbitrary braidings the noncommutative geometry corresponding to arbitrary braidings is studied in a purely algebraical way the new and original proposal of the geometry is given in terms of the theory of braided monoidal categories the quantum oscillator is considered as an example of application of our geometry

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

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part

gradient dynamics of infinite point systems nonequilibrium gradient dynamics of d dimensional particle systems is investigated the interaction is given by a superstable pair potential of finite range solutions are constructed in the well defined set of locally finite configurations with a logarithmic order of energy fluctuations if the system is deterministic and d 2 then singular potentials are also allowed for stochastic models with a smooth interaction we need d 4 in order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium we investigate smoothness of the markov semigroup and describe some properties of its generator 0 introduction the purpose of this paper is to study existence and regularity properties of solutions to the following infinite system of stochastic differential equations consider a countable set s of d dimensional particles suspended in a fluid where the interaction is given by a pair potential u rgd 00 oo in a quasi microscopic description of such systems the effect of collisions with the particles of the fluid can be represented by uncorrelated stochastic forces and the soft resistance of the liquid medium reduces the order of the equations of motion from two to one see 13 23 and 24 for some further references configurations of these systems are countable subsets of rd such that any bounded domain contains a finite number of points only particles of a configuration will be identified by labelling points by elements of s thus a

extended gauge theories a scheme inspired in lie algebra extensions is introduced that enlarges gauge models to allow some coupling between space time and gauge space everything may be written in terms of a generalized covariant derivative including usual differential plus purely algebraic terms a noncovariant vacuum appears introducing a natural symmetry breaking but currents satisfy conservation laws alike those found in gauge theories