a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

an sln stable homotopy type for matched diagrams abstract there exists a simplified bar natan khovanov complex for open 2 braids the khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation we lift this idea to the level of the lipshitz sarkar stable homotopy type and use it to make new computations similarly there exists a simplified khovanov rozansky sl n complex for open 2 braids with oppositely oriented strands and an even number of crossings diagrams made by gluing tangles of this type are called matched diagrams and knots admitting matched diagrams are called bipartite knots to a pair consisting of a matched diagram and a choice of integer n 2 we associate a stable homotopy type in the case n 2 this agrees with the lipshitz sarkar stable homotopy type of the underlying knot in the case n 3 the cohomology of the stable homotopy type agrees with the sl n khovanov rozansky cohomology of the underlying knot we make some consistency checks of this sl n stable homotopy type and show that it exhibits interesting behaviour for example we find a c p 2 in the sl 3 type for some diagram and show that the sl 4 type can be interesting for a diagram for which the lipshitz sarkar type is a wedge of moore spaces

continuity properties of lqg optimal controllers it is shown that the lqg optimal controller is a continuous function of the plant the result is proved for a class of plants which contains the class of strictly proper finite dimensional plants the topology employed is the one generated by convergence of the closed loop transfer functions in an induced l sense this topology is slightly stronger than the usual h2 gap metric convergence on transfer functions

convenient categories of smooth spaces a chen space is a set x equipped with a collection of plots i e maps from convex sets to x satisfying three simple axioms while an individual chen space can be much worse than a smooth manifold the category of all chen spaces is much better behaved than the category of smooth manifolds for example any subspace or quotient space of a chen space is a chen space and the space of smooth maps between chen spaces is again a chen space souriau s diffeological spaces share these convenient properties here we give a unified treatment of both formalisms following ideas of penon and dubuc we show that chen spaces diffeological spaces and even simplicial complexes are examples of concrete sheaves on a concrete site as a result the categories of such spaces are locally cartesian closed with all limits all colimits and a weak subobject classifier for the benefit of differential geometers our treatment explains most of the category theory we use 2011 john c baez and alexander e hoffnung

shock formation in a multidimensional viscoelastic diffusive system we examine a model for non fickian sorption overshoot behavior in diffusive polymer penetrant systems the equations of motion proposed by cohen and white siam j appl math 51 1991 pp 472 483 are solved for two dimensional problems using matched asymptotic expansions the phenomenon of shock formation predicted by the model is examined and contrasted with similar behavior in classical reaction diffusion systems mass uptake curves produced by the model are examined and shown to compare favorably with experimental observations

beta elements and divided congruences the f invariant is an injective homomorphism from the 2 line of the adams novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms we compute the f invariant for two infinite families of elements and explain the relation of the arithmetic of divided congruences with the kervaire invariant one problem

on the robustness of output feedback control methods to modeling errors singular perturbation analysis is employed to study the robustness of output feedback control schemes for linear time invariant systems the effect of designing the control strategy using a model that neglects unknown parasitic elements is considered it is shown that such a strategy may in general be destabilizing even when the neglected modes are asymptotically stable special structures that will not permit this situation and will guarantee asymptotic stability are illustrated

weak solvability for equations of viscoelastic diffusion in polymers with variable coefficients abstract diffusion of a penetrant liquid in a polymer does not satisfy fick s law but requires viscoelastic mechanisms for its description we study the system of differential equations for such a diffusion in the presence of convection we construct weak solutions to the initial boundary value problem for this system in a bounded domain

on cobordism of manifolds with corners this work sets up a cobordism theory for manifolds with corners and gives an identication with the homotopy of a certain limit of thom spectra it thereby creates a geometrical interpretation of adams novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized as an application lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting chern numbers the work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

higher dimensional temperley lieb algebras a category which generalizes to higher dimensions many of the features of the temperley lieb category is introduced

feedback system design the tracking and disturbance rejection problems the problem of designing a compensator for a specified plant which simultaneously stabilizes the resultant feedback system and causes it to track a prescribed family of inputs and or reject prescribed disturbances is considered a set of linear design equations in the space of stable systems is formulated in a general linear systems setting and an explicit parameterization of the resultant solution space is obtained for a class of generalized multivariate systems the theory is illustrated with several single and multivariate examples

local structures on stratified spaces abstract we develop a theory of conically smooth stratified spaces and their smooth moduli including a notion of classifying maps for tangential structures we characterize continuous space valued sheaves on these conically smooth stratified spaces in terms of tangential data and we similarly characterize 1 excisive invariants of stratified spaces these results are based on the existence of open handlebody decompositions for conically smooth stratified spaces an inverse function theorem a tubular neighborhood theorem an isotopy extension theorem and functorial resolutions of singularities to smooth manifolds with corners

corners in m theory m theory can be defined on closed manifolds as well as on manifolds with boundary as an extension we show that manifolds with corners appear naturally in m theory we illustrate this with four situations the lift to bounding 12 dimensions of m theory on anti de sitter spaces ten dimensional heterotic string theory in relation to 12 dimensions and the two m branes within m theory in the presence of a boundary the m2 brane is taken with or as a boundary and the worldvolume of the m5 brane is viewed as a tubular neighborhood we then concentrate on the variant of the heterotic theory as a corner and explore analytical and geometric consequences in particular we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner s the analysis involves considering m theory on disconnected manifolds and makes use of the extension of the atiyah patodi singer index theorem to manifolds with corners and the b calculus of melrose

stabilizability of linear systems with uncertain gains abstract the problem is considered of stabilizing a linear time invariant plant by means of a linear time invariant feed back controller some of the gains of the plant are uncertain it is proved that robust stabilization is possible if and only if every unstable pole of the nominal plant has maximum degree

robust control of flexible structures a case study a comparison is made between three partial differential equation models for a flexible beam with different types of damping and varying parameter values robust controllers can be designed to stabilize all linear systems whose transfer functions lie within a ball in the l norm given a nominal model for our flexible beam we calculate which sets of models can be stabilized by the same finite dimensional robust controller by stabilize we mean here that the poles of the system are moved to the left of the line re s 0

dissipative solutions for equations of viscoelastic diffusion in polymers we study the system of differential equations which describes the diffusion of a penetrant liquid in a polymer we construct dissipative solutions to the cauchy problem for this system in the space

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

the construction of special coprime factorizations in discrete time given a rational discrete time transfer matrix it is shown how to construct two special kinds of coprime factorizations for it normalized and all pass denominator this is done in two ways first by carrying the existing continuous time construction results to discrete time via a bilinear transformation this is shown to be messy and computationally burdensome second direct constructions are derived in discrete time the derivations point out interesting differences between discrete and continuous time for these constructions

a fitness driven cross diffusion system from population dynamics as a gradient flow we consider a fitness driven model of dispersal of n interacting populations which was previously studied merely in the case n 1 n 1 based on some optimal transport distance recently introduced we identify the model as a gradient flow in the metric space of radon measures we prove existence of global non negative weak solutions to the corresponding system of parabolic pdes which involves degenerate cross diffusion under some additional hypotheses and using a new multicomponent poincare beckner functional inequality we show that the solutions converge exponentially to an ideal free distribution in the long time regime

brief paper algebraic and topological aspects of the regulator problem for lumped linear systems the regulator problem with robustness is solved for systems modelled by rational transfer matrices a topology for possibly unstable plants is presented a necessary and sufficient condition is derived for the existence of a proper controller which provides internal stability and output regulation throughout an open neighbourhood of the plant a characterization of all such controllers is determined

proper stable transfer matrix factorizations and internal system descriptions the exact relations between coprime proper stable factorizations of p s and coprime polynomial matrix factorizations are derived and they directly lead to relations with internal descriptions of the plant in differential operator or state space form it is shown that obtaining any right or left proper stable coprime factorization is equivalent to state feedback stabilization or to designing a full order full state observer respectively solving the diophantine equation is shown to be equivalent to designing a full or reduced order observer of a linear functional of the state and to designing a stable inverse system and this suggests new computational methods to solve the diophantine

a comparative study of input output stability results at present many similar but disparate input output i o stability criteria exist without means for comparison it is unclear which result is best used in any given application this paper proposes a means for comparison between i o stability results involving norms and inner products of inputs and outputs the extended conic sector theorem provides a framework for determining which results are least conservative and most broadly applicable in so doing numerous existing stability results are unified and revealed as more powerful than previously thought

framed m branes corners and topological invariants we uncover and highlight relations between the m branes in m theory and various topological invariants the hopf invariant over q z and z2 the kervaire invariant the f invariant and the invariant this requires either a framing or a corner structure the canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies we characterize the flux quantization condition on the c field and the topological action of the m5 brane via the hopf invariant and the dual of the c field as a refinement of an element of hopf invariant two in the signature formulation the contribution to the m brane effective action is given by the maslov index of the corner the kervaire invariant implies that the effective action of the m5 brane is quadratic our study leads to viewing the self dual string which is the boundary of the m2 brane on the m5 brane worldvolume as a string theory in the sense of cobordism of manifolds with corners we show that the dynamics of the c field and its dual are encoded in a unified way in the 4 sphere which suggests the corresponding spectrum as the generalized cohomology theory describing the fields the effective action of the corner is captured by the f invariant which is an invariant at chromatic level two finally considering m theory on manifolds with g2 holonomy we show that the canonical g2 structure minimizes the topological part of the m5 brane action this is done via the invariant and a variant that we introduce related to the one loop polynomial we uncover and highlight relations between the m branes in m theory and various topological invariants the hopf invariant over q z and z2 the kervaire invariant the f invariant and the invariant this requires either a framing or a corner structure the canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies we characterize the flux quantization condition on the c field and the topological action of the m5 brane via the hopf invariant and the dual of the c field as a refinement of an element of hopf invariant two in the signature formulation the contribution to the m brane effective action is given by the maslov index of the corner the kervaire invariant implies that the effective action of the m5 brane is quadratic our study leads to viewing the self dual string which is the boundary of the m2 brane on the m5 brane worldvolume as a string theory in the sense of cobordism of manifolds with corners

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

brief paper a velocity algorithm for the implementation of gain scheduled controllers a new method is proposed to implement gain scheduled controllers for nonlinear plants given a family of linear feedback controllers designed for linearizations of a nonlinear plant about constant operating points a nonlinear gain scheduled controller is derived that preserves the input output properties of the linear closed loop systems locally about each equilibrium point the key procedures in the proposed method are to provide integral action at the inputs to the plant and differentiate some of the measured outputs before they are fed back to the scheduled controller for a fairly general class of systems the nonlinear gain scheduled controllers are easy to obtain and their structure is similar to that of the original linear controllers

discontinuous galerkin finite element approximation of nonlinear non fickian diffusion in viscoelastic polymers we consider discrete schemes for a nonlinear model of non fickian diffusion in viscoelastic polymers the model is motivated by but not the same as that proposed by cohen et al in siam j appl math 55 pp 348 368 1995 the spatial discretisation is effected with both the symmetric and non symmetric interior penalty discontinuous galerkin finite element method and the time discretisation is of crank nicolson type we also discuss two means of handling the nonlinearity either implicitly which requires the solution of nonlinear equations at each time level or through a linearisation based on extrapolating from previous time levels the same optimal orders of convergence are proven in both cases and to verify this some numerical results are also given for the linearised scheme

desorption overshoot in polymer penetrant systems asymptotic and computational results many practically relevant polymers undergoing desorption change from the rubbery saturated to the glassy nearly dry state the dynamics of such systems cannot be described by the simple fickian diffusion equation due to viscoelastic effects the mathematical model solved numerically is a set of two coupled pdes for concentration and stress asymptotic solutions are presented for a moving boundary value problem for the two states in the short time limit the solutions exhibit desorption overshoot where the penetrant concentration in the interior is less than that on the surface in addition it is shown that if the underlying time scale of the equations is ignored when postulating boundary conditions nonphysical solutions can result

the stack of formal groups in stable homotopy theory we construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of m hovey and n strickland on comodule categories for landweber exact algebras this leads to a geometric understanding of their results as well as to a generalisation