smooth and proper noncommutative schemes and gluing of dg categories abstract in this paper we discuss different properties of noncommutative schemes over a field we define a noncommutative scheme as a differential graded category of a special type we study regularity smoothness and properness for noncommutative schemes admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes in this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules as a consequence of the main theorem we obtain that for any finite dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme moreover if the algebra has finite global dimension then the full subcategory is admissible we also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance as another application of the main theorem we obtain in characteristic 0 an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme

brauer groups and quotient stacks a natural question is to determine which algebraic stacks are quotient stacks in this paper we give some partial answers and relate it to the question of whether for a scheme x the natural map from the brauer group equivalence classes of azumaya algebras to the cohomological brauer group the torsion subgroup of h 2 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i m is surjective

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

hopf bifurcation analysis of integro differential equation with unbounded delay abstract the complexity of a nonlinear dynamical system is controllable via a selection of system parameters one representative behavior of such a complex system can be illustrated by hopf bifurcation this paper presents a hopf bifurcation analysis of a kind of integro differential equations with unbounded delay based on the hopf bifurcation principle a set of relationships among system parameters are obtained when a periodic orbit exists in the system a numerical analysis is applied to solve the integro differential delay equation this paper proves the existence of hopf bifurcation in the corresponding difference equations under the same system parameters as that in the integro differential delay equations

the hochschild cohomology ring of a global quotient orbifold abstract we study the cup product on the hochschild cohomology of the stack quotient x g of a smooth quasi projective variety x by a finite group g more specifically we construct a g equivariant sheaf of graded algebras on x whose g invariant global sections recover the associated graded algebra of the hochschild cohomology of x g under a natural filtration this sheaf is an algebra over the polyvector fields t x p o l y on x and is generated as a t x p o l y algebra by the sum of the determinants det n x g of the normal bundles of the fixed loci in x we employ our understanding of hochschild cohomology to conclude that the analog of kontsevich s formality theorem for the cup product does not hold for deligne mumford stacks in general we discuss in the case of a symplectic group action on a symplectic variety x relationships with orbifold cohomology and ruan s cohomological conjectures in describing the hochschild cohomology in the symplectic situation we employ compatible trivializations of the determinants det n x g which requires for the cup product a nontrivial normalization missing in previous literature

dynamics of a discrete food limited population model with time delay abstract in this paper we consider a discrete food limited population model with time delay firstly the stability of the equilibrium of the system is investigated by analyzing the characteristic equation by choosing the time delay as a bifurcation parameter we prove that neimark sacker bifurcations occur when the delay passes a sequence of critical values then the explicit algorithm for determining the direction of the neimark sacker bifurcations and the stability of the bifurcating periodic solutions are derived finally some numerical simulations are given to verify the theoretical analysis

stability and bifurcation analysis for a discrete time model of lotka volterra type with delay a discrete model of lotka volterra type with delay is considered and a bifurcation analysis is undertaken for the model we derive the precise conditions ensuring the asymptotic stability of the positive equilibrium with respect to two characteristic parameters of the system it is shown that for certain values of these parameters fold or neimark sacker bifurcations occur but codimension 2 fold neimark sacker double neimark sacker and resonance 1 1 bifurcations may also be present the direction and the stability of the neimark sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory

the six operations in equivariant motivic homotopy theory abstract we introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks x g where g is a linearly reductive linear algebraic group we extend to this equivariant setting the main foundational results of motivic homotopy theory the unstable purity and gluing theorems of morel voevodsky and the stable ambidexterity theorem of ayoub our proof of the latter is different than ayoub s and is of interest even when g is trivial using these results we construct a formalism of six operations for equivariant motivic spectra and we deduce that any cohomology theory for g schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology

discrete hopf bifurcation for runge kutta methods an investigation into how well real bifurcations in the family of dynamical systems are approximated as the step size varies is carried out the preservation of bifurcation structures and stability under numerical simulations is discussed in addition the behaviour of numerical solutions generated by a runge kutta method applied to a dynamical system whose analytical solution undergoes a hopf bifurcation is investigated hopf bifurcation results for the numerical solution are presented and analysed

derived azumaya algebras and generators for twisted derived categories we introduce a notion of derived azumaya algebras over ring and schemes generalizing the notion of azumaya algebras of grothendieck le groupe de brauer i algebres d azumaya et interpretations diverses dix exposes sur la cohomologie des schemas pp 46 66 north holland amsterdam 1968 we prove that any such algebra b on a scheme x provides a class b in h 1 et x mathbb z times h 2 et x mathbb g m we prove that for x a quasi compact and quasi separated scheme defines a bijective correspondence and in particular that any class in h 2 et x mathbb g m torsion or not can be represented by a derived azumaya algebra on x our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories with coefficients in any local system of reasonable dg categories generalizing the well known existence of compact generators in derived categories of quasi coherent sheaves of bondal and van den bergh mosc math j 3 1 1 36 2003 a huge part of this paper concerns the treatment of twisted derived categories as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator we present explicit examples of derived azumaya algebras that are not represented by classical azumaya algebras as well as applications of our main result to the localization for twisted algebraic k theory and to the stability of saturated dg categories by direct push forwards along smooth and proper maps

dynamics of numerical discretization in a delayed diffusive nicholson s blowflies equation in this article we study the dynamics of a diffusive nicholson s blowflies equation with a finite delay and dirichlet boundary condition by the forward and backward euler difference schemes and crank nicolson scheme the existence of hopf bifurcation at the equilibrium is obtained through analyzing the distribution of the eigenvalues we find that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions our results show that crank nicolson and the backward euler difference schemes are superior to the forward euler difference scheme under the means of describing approximately the dynamics of the original system finally numerical examples are provided to illustrate the analytical results

hopf bifurcation in numerical approximation of a class delay differential equations abstract in this paper we consider the numerical solution of a delay differential equation undergoing a hopf bifurcation we prove that if the delay differential equation has a hopf bifurcation point at then the numerical solution of the equation also has a hopf bifurcation point at h o h

global attractivity in population dynamics we established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation n t mu n t i 1m pi exp n t i t 0 m 1 for m 1 equation 1 was used by wazewska czyzewska and lasota as a model for the survival of red blood cells in an animal

matroid theory and chern simons it is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum yang mills physics but also for m theory our discussion is focused in an action consisting purely of the chern simons term but in principle the main ideas can be applied beyond such an action in our treatment the theorem due to thistlethwaite which gives a relationship between the tutte polynomial for graphs and jones polynomial for alternating knots and links plays a central role before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the fano matroid and d 11 supergravity our approach also seems to be related to loop solutions of quantum gravity based in an ashtekar formalism

stability properties of numerical methods for solving delay differential equations abstract stability properties of numerical methods for delay differential equations are considered some suitable definitions for the stability of the numerical methods are included and runge kutta type methods satisfying these properties are tested on a numerical example

equivalences of derived factorization categories of gauged landau ginzburg models abstract for a given fourier mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi projective varieties we construct fourier mukai equivalences of derived factorization categories of gauged landau ginzburg lg models as an application we obtain some equivalences of derived factorization categories of k equivalent gauged lg models this result is an equivariant version of the result of baranovsky and pecharich and it also gives a partial answer to segal s conjecture as another application we prove that if the kernel of the fourier mukai equivalence is linearizable with respect to a reductive affine algebraic group action then the derived categories of equivariant coherent sheaves on the varieties are equivalent this result is shown by ploog for finite groups case

the resolution property for schemes and stacks we prove the equivalence of two fundamental properties of algebraic stacks being a quotient stack in a strong sense and the resolution property which says that every coherent sheaf is a quotient of some vector bundle moreover we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme mathematics subject classification primary 14a20 secondary 14l30

numerical hopf bifurcation of linear multistep methods for a class of delay differential equations abstract in this paper we consider the discretization of parameter dependent delay differential equation of the form y t f y t y t 1 0 y r d it is shown that if the delay differential equation undergoes a hopf bifurcation at then the discrete scheme undergoes a hopf bifurcation at h o h p for sufficiently small step size h where p 1 is the order of the strictly stable linear multistep method the direction of numerical hopf bifurcation and stability of bifurcating invariant curve are the same as that of the corresponding delay differential equation

riemann roch for equivariant k theory abstract we show that for a linear algebraic group g acting on a smooth quasi projective scheme x over a field there is a chern character map k i g x r g r g c h x g ch g x i s g s g with rational coefficients which is an isomorphism this establishes the equivariant version of the riemann roch isomorphism between the higher algebraic k theory and the higher chow groups of smooth quasi projective schemes

soldered bundle background for the de sitter top we prove that the mathematical framework for the de sitter top system is the de sitter fiber bundle in this context the concept of soldering associated with a fiber bundle plays a central role we comment on the possibility that our formalism may be of particular interest in different contexts including macdowell mansouri theory two time physics and oriented matroid theory

a short survey of noncommutative geometry we give a survey of selected topics in noncommutative geometry with some emphasis on those directly related to physics including our recent work with dirk kreimer on renormalization and the riemann hilbert problem we discuss at length two issues the first is the relevance of the paradigm of geometric space based on spectral considerations which is central in the theory as a simple illustration of the spectral formulation of geometry in the ordinary commutative case we give a polynomial equation for geometries on the four sphere with fixed volume the equation involves an idempotent e playing the role of the instanton and the dirac operator d it is of the form e 12 d e 4 5 and determines both the sphere and all its metrics with fixed volume form the expectation x is the projection on the commutant of the algebra of 4 by 4 matrices we also show using the noncommutative analog of the polyakov action how to obtain the noncommutative metric in spectral form on the noncommutative tori fr

asymptotic stability of numerical methods for linear delay parabolic differential equations this paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay these numerical processes include forward and backward euler difference schemes and crank nicolson difference scheme which are obtained by applying step by step methods to the resulting systems of delay differential equations sufficient and necessary conditions for these difference schemes to be delay independently asymptotically stable are established it reveals that an additional restriction on time and spatial stepsizes of the forward euler difference scheme is required to preserve the delay independent asymptotic stability due to the existence of the delay term numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods

searching for a connection between matroid theory and string theory we make a number of observations about matter ghost string phase which may eventually lead to a formal connection between matroid theory and string theory in particular in order to take advantage of the already established connection between matroid theory and chern simons theory we propose a generalization of string theory in terms of some kind of kahler metric we show that this generalization is closely related to the kahler chern simons action due to nair and schiff in addition we discuss matroid string connection via matroid bundles and a schild type action and we add new information about the relationship between matroid theory d 11 supergravity and chern simons formalism

separability and triangulated categories abstract we prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one this applies in particular to etale algebras more generally we do this for exact separable monads

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

local duality in algebra and topology abstract the first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts for any stable category c together with a collection of compact objects k c we construct local cohomology and local homology functors satisfying an abstract version of local duality when specialized to the derived category of a commutative ring a and a suitable ideal in a we recover the classical local duality due to grothendieck as well as generalizations by greenlees and may more generally applying our result to the derived category of quasi coherent sheaves on a quasi compact and separated scheme x implies the local duality theorem of alonso tarrio jeremias lopez and lipman as a second objective we establish local duality for quasi coherent sheaves over many algebraic stacks in particular those arising naturally in stable homotopy theory after constructing an appropriate model of the derived category in terms of comodules over a hopf algebroid we show that in familiar cases the resulting local cohomology and local homology theories coincide with functors previously studied by hovey and strickland furthermore our framework applies to global and local stable homotopy theory in a way which is compatible with the algebraic avatars of these theories in order to aid computability we provide spectral sequences relating the algebraic and topological local duality contexts

local models of shimura varieties and a conjecture of kottwitz we give a group theoretic definition of local models as sought after in the theory of shimura varieties these are projective schemes over the integers of a p adic local field that are expected to model the singularities of integral models of shimura varieties with parahoric level structure our local models are certain mixed characteristic degenerations of grassmannian varieties they are obtained by extending constructions of beilinson drinfeld gaitsgory and the second named author to mixed characteristics and to the case of general tamely ramified reductive groups we study the singularities of local models and hence also of the corresponding integral models of shimura varieties in particular we study the monodromy inertia action and show a commutativity property for the sheaves of nearby cycles as a result we prove a conjecture of kottwitz which asserts that the semi simple trace of frobenius on the nearby cycles gives a function which is central in the parahoric hecke algebra

gersten conjecture for equivariant k theory and applications for a reductive group scheme g over a regular semi local ring a we prove the gersten conjecture for the equivariant k theory as a consequence we show that if f is the field of fractions of a then k g 0 a cong k g 0 f generalizing the analogous result for a dvr by serre inst hautes etudes sci publ math 34 37 52 1968 we also show the rigidity for the k theory with finite coefficients of a henselian local ring in the equivariant setting we use this rigidity theorem to compute the equivariant k theory of algebraically closed fields

on the differential graded eilenberg moore construction abstract the eilenberg moore construction for modules over a differential graded monad is used to study a question of balmer regarding existence of an exact adjoint pair representing an exact monad a bousfield like localization for differential graded categories is realized as a special case of this construction using drinfeld quotients as applications we study some example coming from g equivariant triangulated categories and twisted derived categories

uniqueness of enhancement for triangulated categories the paper contains general results on the uniqueness of a dg enhancement for trian gulated categories as a consequence we obtain such uniqueness for the unbounded categories of quasi coherent sheaves for the triangulated categories of perfect complexes and for the bounded de rived categories of coherent sheaves on quasi projective schemes if a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de rived categories of coherent sheaves these results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product

numerical hopf bifurcation for a class of delay differential equations abstract in this paper we consider discretization of parameter dependent delay differential equations of the form x t f x t x t r we show that if the delay differential equation undergoes a hopf bifurcation then the discrete scheme undergoes a hopf bifurcation of the same type the results of this paper extend the results of our previous analysis relating to the discretization of the delay logistic equation to a wider class of problems

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

on the derived categories of degree d hypersurface fibrations we provide descriptions of the derived categories of degree d hypersurface fibrations which generalize a result of kuznetsov for quadric fibrations and give a relative version of a well known theorem of orlov using a local generator and morita theory we re interpret the resulting matrix factorization category as a derived equivalent sheaf of dg algebras on the base then applying homological perturbation methods we obtain a sheaf of a infty algebras which gives a new description of homological projective duals for relative d veronese embeddings recovering the sheaf of clifford algebras obtained by kuznetsov in the case when d 2

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases explicit algorithm for determining the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold global existence of periodic solutions are established using a global hopf bifurcation result of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27