gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

lyapunov theory for 2 d nonlinear roesser models application to asymptotic and exponential stability this technical note deals with a general class of discrete 2 d possibly nonlinear systems based on the roesser model we first motivate the introduction of lyapunov type definitions of asymptotic and exponential stability this will allow us to introduce and discuss several particularities that cannot be found in 1 d systems once this background has been carefully designed we develop different lyapunov theorems in order to check asymptotic and exponential stability of nonlinear 2 d systems finally we propose the first converse lyapunov theorem in the case of exponential stability

eigenvalue inequalities and poincar e duality in noncommutative geometry in the context of connes noncommutative geometry eigenvalue inequalities of the type discovered by vafa and witten are shown to be a characteristic feature of those spectral geometric spaces of finite topological type that satisfy rational poincare duality in k theory

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

polynomially parameter dependent lyapunov functions for robust stability of polytopic systems an lmi approach in this note robust stability of state space models with respect to real parametric uncertainty is considered specifically a new class of parameter dependent quadratic lyapunov functions for establishing stability of a polytope of matrices is introduced i e the homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs the choice of this class which contains parameter dependent quadratic lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form is motivated by the property that a polytope of matrices is stable if and only there exists an hpd qlf the main result of the note is a sufficient condition for determining the sought hpd qlf which amounts to solving linear matrix inequalities lmis derived via the complete square matricial representation csmr of homogeneous matricial forms and the lyapunov matrix equation numerical examples are provided to demonstrate the effectiveness of the proposed approach

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

existence results for some nonuniformly elliptic equations with irregular data abstract in this paper we prove the existence of solutions of some nonlinear elliptic equations with degenerate coercivity of the type formula when the datum f belongs to the space lr where r 2 and 0 1

stability of two dimensional linear systems with singularities on the stability boundary using lmis this paper gives results on stability and asymptotic stability of two dimensional systems using linear matrix inequalities lmis despite a long history of research in this area systems with singularities on the stability boundary ssb have received limited attention because they cannot produce a sign definite solution to the required lmi however 2d systems describing some classes of models of vehicle platoons generically involve an ssb therefore commonly used definitions for asymptotic stability and strict lmi conditions are not suitable to discuss the stability of these systems it is shown that the existence of a negative semidefinite solution together with simple additional conditions is sufficient to guarantee asymptotic stability thus the stability conditions discussed here can be used to study a wider range of dynamical systems including systems with singularities on the stability boundary ssb which cannot be exponentially stable a unified framework is used to analyse continuous continuous continuous discrete and discrete discrete systems simultaneously

necessary and sufficient lmi conditions for stability and performance analysis of 2 d mixed continuous discrete time systems this paper proposes necessary and sufficient conditions for stability and performance analysis of 2 d mixed continuous discrete time systems that can be checked with convex optimization in particular linear matrix inequalities lmis specifically the first contribution of the paper is a condition for exponential stability based on the introduction of a complex lyapunov function depending polynomially on a parameter and on the use of the gram matrix method it is shown that this condition is sufficient for any chosen degree of the complex lyapunov function and necessary for an a priori known degree the second contribution is a non lyapunov condition for exponential stability based on eigenvalue products this condition is necessary and sufficient and has the advantage of requiring a significantly smaller computational burden for achieving necessity lastly the third contribution is to show how upper bounds on the h norm of 2 d mixed continuous discrete time systems can be obtained through a semidefinite program based on complex lyapunov functions a necessary and sufficient condition is provided for establishing the tightness of the found upper bounds

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

on very weak solutions to a class of double obstacle problems abstract we prove the uniqueness of very weak solutions by hodge decomposition and obtain the existence by an approximation argument moreover we prove that the solution quasiminimizes the r dirichlet integral in k r and finally the stability of very weak solutions is established for varying obstacle functions

lossless h synthesis for 2d systems special issue jcw abstract we study 2d systems with a continuous and discrete time axis we embed known results about the stability and h performance properties of such systems into multiplier theory from robust control it is shown that this opens the way for applying recently developed gain scheduled controller synthesis techniques in order to solve the h design problem for 2d systems without any conservatism this is presented if the discrete time axis is either one or two sided the latter leading to non conservative h synthesis result for infinite string interconnections of identical linear time invariant systems

exponential stability of 2 d systems exponential stability of 2 d systems is defined and characterized the definition of exponential stability works also with unbounded data as for the case of 1 d systems we show in particular that exponential stability and asymptotic stability are equivalent properties

noncommutative geometry and reality we introduce the notion of real structure in our spectral geometry this notion is motivated by atiyah s kr theory and by tomita s involution j it allows us to remove two unpleasant features of the connes lott description of the standard model namely the use of bivector potentials and the asymmetry in the poincare duality and in the unimodularity condition

exponential stability of nonlinear differential repetitive processes with applications to iterative learning control abstract this paper studies exponential stability properties of a class of two dimensional 2d systems called differential repetitive processes drps since a distinguishing feature of drps is that the problem domain is bounded in the time direction the notion of stability to be evaluated does not require the nonlinear system defining a drp to be stable in the typical sense in particular we study a notion of exponential stability along the discrete iteration dimension of the 2d dynamics which requires the boundary data for the differential pass dynamics to converge to zero as the iterations evolve our main contribution is to show under standard regularity assumptions that exponential stability of a drp is equivalent to that of its linearized dynamics in turn exponential stability of this linearization can be readily verified by a spectral radius condition the application of this result to iterative learning control ilc is discussed theoretical findings are supported by a numerical simulation of an ilc algorithm

robust stability and performance analysis of 2d mixed continuous discrete time systems with uncertainty this paper investigates 2d mixed continuous discrete time systems whose coefficients are polynomial functions of an uncertain vector constrained into a semialgebraic set it is shown that a nonconservative linear matrix inequality lmi condition for ensuring robust stability can be obtained by introducing complex lyapunov functions depending polynomially on the uncertain vector and a frequency moreover it is shown that nonconservative lmi conditions for establishing upper bounds of the robust h and h 2 norms can be obtained by introducing analogous lyapunov functions depending rationally on the frequency some numerical examples illustrate the proposed methodology

on rank one convex and polyconvex conformal energy functions with slow growth we make some remarks about rank one convex and polyconvex functions on the set of all real n n matrices that vanish on the subset k n consisting of all conformal matrices and grow like a power function at infinity we prove that every non negative rank one convex function that vanishes on k n and grows below a power of degree n 2 must vanish identically in odd dimensions n 3 we prove that every non negative polyconvex function that vanishes on k n must vanish identically if it grows below a power of degree n while in even dimensions such polyconvex functions can exist that also grow like a power of half dimension degree

extended h 2 and h norm characterizations and controller parametrizations for discrete time systems this paper presents new synthesis procedures for discrete time linear systems it is based on a recently developed stability condition which contains as particular cases both the celebrated lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters these new synthesis conditions have some nice properties a they can be expressed in terms of lmi linear matrix inequalities and b the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic lyapunov function used to test stability this second feature is important for several reasons first structural constraints as those appearing in the decentralized and static output feedback control design can be addressed less conservatively second parameter dependent lyapunov function can be considered with a very positive impact on the design of robust h 2 and h x control problems third the design of controller with mixed

a new short proof of the local index formula and some of its applications we give a new short proof of the index formula of atiyah and singer based on combining getzler s rescaling with greiner s approach of the heat kernel asymptotics as an application we can easily compute the cm cyclic cocycle of even and odd dirac spectral triples and then recover the atiyah singer index formula even case and the atiyah patodi singer spectral flow formula odd case

cal h _ infty and cal h _ 2 norms of 2 d mixed continuous discrete time systems via rationally dependent complex lyapunov functions this paper addresses the problem of determining the cal h infty and cal h 2 norms of 2 d mixed continuous discrete time systems the first contribution is to propose a novel approach based on the use of complex lyapunov functions with even rational parametric dependence which searches for upper bounds on the sought norms via linear matrix inequalities lmis the second contribution is to show that the upper bounds provided are nonconservative by using lyapunov functions in the chosen class with sufficiently large degree the third contribution is to provide conditions for establishing the tightness of the upper bounds the fourth contribution is to show how the numerical complexity of the proposed approach can be significantly reduced by proposing a new necessary and sufficient lmi condition for establishing positive semidefiniteness of even hermitian matrix polynomials this result is also exploited to derive an improved necessary and sufficient lmi condition for establishing exponential stability of 2 d mixed continuous discrete time systems some numerical examples illustrate the proposed approach it is worth remarking that nonconservative lmi methods for determining the cal h infty and cal h 2 norms of 2 d mixed continuous discrete time systems have not been proposed yet in the literature

equivariant spectral triples and poincare duality for su_q 2 let a be the c algebra associated with su q 2 let be the representation by left multiplication on the l 2 space of the haar state and let d be the equivariant dirac operator for this representation constructed by the authors earlier we prove in this article that there is no operator other than the scalars in the commutant a that has bounded commutator with d this implies that the equivariant spectral triple under consideration does not admit a rational poincare dual in the sense of moscovici which in particular means that this spectral triple does not extend to a k homology fundamental class for suq 2 we also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements poincare duality

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

a finite element model of two layer ocean circulation a mixed finite element formulation is applied to solve the quasi geostrophic equations in a two layer ocean domain enclosed by solid boundaries a combination of leap frog and schemes is used to integrate the equations in time wood s formula of the wall vorticity is derived in our formulation for the case of square elements adjacent to the boundaries long term computations with and without lumping have been carried out

analytic formulas for the topological degree of non smooth mappings the odd dimensional case the notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo convex domain in a stein manifold into a manifold in terms of index theory of toeplitz operators on the hardy space the index formalism of non commutative geometry is used to derive analytic integral formulas for the index of a toeplitz operator with holder continuous symbol the index formula gives an analytic formula for the degree of a holder continuous mapping from the boundary of a strictly pseudo convex domain

brief homogeneous lyapunov functions for systems with structured uncertainties the problem addressed in this paper is the construction of homogeneous polynomial lyapunov functions hplfs for linear systems with time varying structured uncertainties a sufficient condition for the existence of an hplf of given degree is formulated in terms of a linear matrix inequalities lmi feasibility problem this condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the lyapunov function the maximum norm of the parametric uncertainty for which there exists a homogeneous polynomial lyapunov function is computed by solving a generalized eigenvalue problem the construction of such lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in lmi form comparisons with other classes of lyapunov functions through numerical examples taken from the literature show that hplfs are a powerful tool for robustness analysis

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

control systems analysis for the fornasini marchesini 2d systems model progress after four decades the development of a control systems theory for multidimensional linear systems i e systems with more than one independent variable is an area where a very significant volume of research is based

noncommutative geometry and conformal geometry iii vafa witten inequality and poincare duality abstract this paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint in this paper we reformulate the inequality of vafa witten 42 in the setting of twisted spectral triples this involves a notion of poincare duality for twisted spectral triples our main results have various consequences in particular we obtain a version in conformal geometry of the original inequality of vafa witten in the sense of an explicit control of the vafa witten bound under conformal changes of metrics this result has several noncommutative manifestations for conformal deformations of ordinary spectral triples spectral triples associated with conformal weights on noncommutative tori and spectral triples associated with duals of torsion free discrete cocompact subgroups satisfying the baum connes conjecture

global integrability theorems for a harmonic tensors abstract we extend global integrability theorems for the gradients of a harmonic functions to the exterior derivative of differential forms satisfying rather general nonhomogeneous elliptic equations these include the usual a harmonic equations geometric conditions on the boundary of the domains of integration imply a corresponding exponent of integrability in the process we generalize the weak reverse holder inequality to such differential forms

on the role of mass operators in the group finite element formulation abstract the influence of the mass operators on the accuracy economy and computational efficiency of the time split group finite element formulation is investigated for the viscous flow over a backward facing step on a coarse grid it is found that mass operators must be retained adjacent to the computational boundaries to obtain the correct steady state solution the present time split or approximate factorisation group finite element formulation is only 18 less economical than a fully lumped equivalent to a finite difference formulation if mass lumping is introduced in the interior only the solution is almost as accurate as if the full mass operators are retained but there is little gain in economy in two dimensions an operation count estimate indicates that interior mass lumping would be about 40 more economical than retaining the mass operators in three dimensions

kronecker product based stability tests and performance bounds for a class of 2d continuous discrete linear systems abstract this paper reports further development of the so called 1d lyapunov equation based approach to the stability analysis of differential linear repetitive processes which are a distinct class of 2d continuous discrete linear systems of both practical and theoretical interest in particular it is shown that this approach leads to stability tests which can be implemented by computations with matrices which have constant entries also if the example under consideration is stable then physically meaningful information concerning one key aspect of transient performance is available for no extra cost

d branes rr fields and duality on noncommutative manifolds we develop some of the ingredients needed for string theory on noncommutative spacetimes proposing an axiomatic formulation of t duality as well as establishing a very general formula for d brane charges this formula is closely related to a noncommutative grothendieck riemann roch theorem that is proved here our approach relies on a very general form of poincare duality which is studied here in detail among the technical tools employed are calculations with iterated products in bivariant mathsf k theory and cyclic theory which are simplified using a novel diagram calculus reminiscent of feynman diagrams

group actions and equivariant lipschitz analysis the idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by singer the concrete realization of this in the case of rational pontrjagin classes was the result of works by sullivan who showed that topological manifolds have essentially unique lipschitz structures and teleman who showed that lipschitz manifolds are the type of object for which one can analyze the signature operator a refinement of the same method proves the ko l 2 orientability of topological bundles much of this can be done for manifolds with group actions the main implication is that for certain sorts of topological group actions including all smooth and pl actions one can construct an equivariant signature operator whose class in equivariant if homology is a topological invariant this class satisfies the g signature theorem the topological method has implications for various precise classification problems e g semifree topological actions on the sphere but here we shall present only the consequences of these lipschitzian ideas a topologically locally linear g manifolds for odd order g have kog 1 2 orientations madsen rothenberg for all finite g there are lipschitz structures but the signature class is a zero divisor for even order groups the same method produces upon application of the localization theorem atiyah singer classes for some actions with nonmanifold fixed set weinberger extending earlier joint work with cappell b for odd order groups topological and linear conjugacy of linear representations are equivalent hsiang pardon madsen rothenberg this is a consequence of a alternatively one can return to the original argument of atiyah bott milnor for semifree representations a stronger result is true the atiyah bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations this distinguishes many pairs of representations of even order groups c r top g r lip g for all finite g where r cat is the grothendieck group of representations up to cat conjugacy weinberger after inverting 2 this implies that there is no analytic definition of reidemeister torsion that works well in the lipschitz setting the reason is that following