model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

higher traces noncommutative motives and the categorified chern character abstract we propose a categorification of the chern character that refines earlier work of toen and vezzosi and of ganter and kapranov if x is an algebraic stack our categorified chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over x which we introduce to s 1 equivariant perfect complexes on the derived free loop stack l x as an application of the theory we show that toen and vezzosi s secondary chern character factors through secondary k theory our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal n categories which is of independent interest

rings modules and algebras in infinite loop space theory we give a new construction of the algebraic k theory of small permutative categories that preserves multiplicative structure and therefore allows us to give a unified treatment of rings modules and algebras in both the input and output this requires us to define multiplicative structure on the category of small permutative categories the framework we use is the concept of multicategory elsewhere also called colored operad a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction our method ends up in the hovey shipley smith category of symmetric spectra with an intermediate stop at a category of functors out of a particular wreath product

uniqueness of the multiplicative cyclotomic trace abstract making use of the theory of noncommutative motives we characterize the topological dennis trace map as the unique multiplicative natural transformation from algebraic k theory to topological hochschild homology thh and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology tc moreover we prove that the space of all multiplicative structures on algebraic k theory is contractible we also show that the algebraic k theory functor from small stable categories to spectra is lax symmetric monoidal which in particular implies that e n ring spectra give rise to e n 1 ring algebraic k theory spectra along the way we develop a multiplicative morita theory establishing a symmetric monoidal equivalence between the category of small idempotent complete stable categories and the morita localization of the category of spectral categories

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

a genetic based framework for solving multi criteria weighted matching problems abstract the purpose of this paper is to present a flexible genetic based framework for solving the multi criteria weighted matching problem mc wmp in the first part of this paper we design a genetic based framework for solving the ordinary weighted matching problem we present an extensive analysis of the quality of the results and introduce a methodology for tuning its parameters in the second part we develop a modified genetic based algorithm for solving the mc wmp the algorithm generates a significant and representative portion of the pareto optimal set to assess the performance of the algorithm we conduct computational experiments with two and three criteria the potential of the proposed algorithm is demonstrated by comparing to a multi objective simulated annealing algorithm

the smash product for derived categories in stable homotopy theory abstract an e 1 or a ring spectrum r has a derived category of modules d r an e 2 structure on r endows d r with a monoidal product r an e 3 structure on r endows r with a braiding if the e 3 structure extends to an e 4 structure then the braided monoidal product r is symmetric monoidal

a model category structure on the category of simplicial categories in this paper we put a cofibrantly generated model category struc ture on the category of small simplicial categories the weak equivalences are a simplicial analogue of the notion of equivalence of categories

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

the homotopy theory of dg categories and derived morita theory the main purpose of this work is to study the homotopy theory of dg categories up to quasi equivalences our main result is a description of the mapping spaces between two dg categories c and d in terms of the nerve of a certain category of c d bimodules we also prove that the homotopy category ho dg cat possesses internal hom s relative to the derived tensor product of dg categories we use these two results in order to prove a derived version of morita theory describing the morphisms between dg categories of modules over two dg categories c and d as the dg category of c d bi modules finally we give three applications of our results the first one expresses hochschild cohomology as endomorphisms of the identity functor as well as higher homotopy groups of the classifying space of dg categories i e the nerve of the category of dg categories and quasi equivalences between them the second application is the existence of a good theory of localization for dg categories defined in terms of a natural universal property our last application states that the dg category of continuous morphisms between the dg categories of quasi coherent resp perfect complexes on two schemes resp smooth and proper schemes is quasi equivalent to the dg category of quasi coherent resp perfect complexes on their product

colocalizations of noncommutative spectra and bootstrap categories we construct a compactly generated and closed symmetric monoidal stable category nsp and show that hnsp op contains the suspension stable homotopy category of separable c algebras hoc constructed by cuntz meyer rosenberg as a fully faithful triangulated subcategory then we construct two colocalizations of nsp namely nsp k 1 and nsp z 1 both of which are shown to be compactly generated and closed symmetric monoidal we prove that kasparov kk category of separable c algebras sits inside the homotopy category of kk nsp k 1 op as a fully faithful triangulated subcategory hence kk should be viewed as the stable categorical incarnation of kasparov kk category for arbitrary pointed noncommutative spaces including nonseparable c algebras as an application we find that the bootstrap category in hnsp k 1 admits a completely algebraic description we also construct a k theoretic bootstrap category in hkk that extends the construction of the uct class by rosenberg schochet motivated by the algebraization problem we finally analyze a couple of equivalence relations on separable c algebras that are introduced via the bootstrap categories in various colocalizations of nsp

control of minimally persistent formations in the plane this paper studies the problem of controlling the shape of a formation of point agents in the plane a model is considered where the distance between certain agent pairs is maintained by one of the agents making up the pair if enough appropriately chosen distances are maintained with the number growing linearly with the number of agents then the shape of the formation will be maintained the detailed question examined in the paper is how one may construct decentralized nonlinear control laws to be operated at each agent that will restore the shape of the formation in the presence of small distortions from the nominal shape using the theory of rigid and persistent graphs the question is answered as it turns out a certain submatrix of a matrix known as the rigidity matrix can be proved to have nonzero leading principal minors which allows the determination of a stabilizing control law

fundamental properties and performance of conventional bearings only target motion analysis this paper considers the problem of estimating the position and velocity of an object from noise corrupted bearing measurements obtained by a single moving observation platform the process is inherently nonlinear and exhibits unusual observability properties that are geometry dependent a maximum likelihood estimate mle of the target motion analysis solution is developed and its performance analyzed a comparison is drawn between the mle and two previously reported methods a nonlinear modified instrumental variable estimate miv and the pseudo linear estimate ple both the miv and ple are shown to derive from approximations to the nonlinear measurement equation and therefore share some common properties with the mle the limits on performance that can be expected from processing bearing data are detailed specifically for long range to baseline geometries approximate expressions for the cramer rao bound are derived extension of the results to the practical filters approximately predicts numerically observed behavior for less restrictive geometries bounds are presented incorporation of a target speed constraint on the mle results in a transition to a lower dimensional problem as noise level and range increases monte carlo experimental results are presented and the improvements realized by the mle techniques are evident

control of minimally persistent leader remote follower and coleader formations in the plane this paper solves an n agent formation shape control problem in the plane the objective is to design decentralized control laws so that the agents cooperatively restore a prescribed formation shape in the presence of small perturbations from the prescribed shape we consider two classes of directed cyclic information architectures associated with so called minimally persistent formations leader remote follower and coleader in our framework the formation shape is maintained by controlling certain interagent distances only one agent is responsible for maintaining each distance we propose a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance the resulting nonlinear closed loop system has a manifold of equilibria which implies that the linearized system is nonhyperbolic we apply center manifold theory to show local exponential stability of the desired formation shape the result circumvents the non compactness of the equilibrium manifold choosing stabilizing gains is possible if a certain submatrix of the rigidity matrix has all leading principal minors nonzero and we show that this condition holds for all minimally persistent leader remote follower and coleader formations with generic agent positions simulations are provided

formation shape control based on bearing rigidity distance measurements are not the only geometric quantities that can be used for multi agent formation shape control bearing measurements can be used in conjunction with distances this article employs bearing rigidity for mobile formations which was developed for robot and sensor network localisation so that bearings can be used for shape control in mobile formations the first part of this article examines graph theoretical models for formation network analysis and control law design that are needed to maintain the shape of a formation in two dimensional space while the formation moves as a cohesive whole bearing based shape control for a formation of mobile agents involves the design of distributed control laws that ensure the formation moves so that bearing constraints maintain some desired values the second part of this article focuses on the design of a distributed control scheme for nonholonomic agents to solve the bearing based formation shape control problem in particular a control law u

a model for the homotopy theory of homotopy theory we describe a category the objects of which may be viewed as models for homotopy theories we show that for such models functors between two homotopy theories form a homotopy theory or more precisely that the category of such models has a well behaved internal hom object

finite time convergent gradient flows with applications to network consensus this paper introduces the normalized and signed gradient dynamical systems associated with a differentiable function extending recent results on nonsmooth stability analysis we characterize their asymptotic convergence properties and identify conditions that guarantee finite time convergence we discuss the application of the results to consensus problems in multi agent systems and show how the proposed nonsmooth gradient flows achieve consensus in finite time

distributed control of angle constrained cyclic formations using bearing only measurements abstract this paper studies distributed control of multi vehicle formations with angle constraints using bearing only measurements it is assumed that each vehicle can only measure the local bearings of their neighbors and there are no wireless communications among the vehicles the desired formation is a cyclic one whose underlying information flow is described by an undirected cycle graph we propose a distributed bearing only formation control law that ensures local exponential or finite time stability collision avoidance between any vehicles can be locally guaranteed in the absence of inter vehicle distance measurements

homotopy theory of spectral categories abstract we construct a quillen model structure on the category of spectral categories where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories

derived azumaya algebras and twisted k theory abstract we construct a relative version of topological k theory in the sense of 11 over an arbitrary quasi compact quasi separated c scheme x this has as input a perf x linear stable category and output a sheaf of spectra on x c the space of complex points of x we then characterize the values of this functor on inputs of the form perf x a for a a derived azumaya algebra over x in such cases we show that this coincides with the twisted topological k theory of x c for some appropriately defined twist of k theory we use this to provide a topological analogue of a classical result of quillen s on the algebraic k theory of severi brauer varieties