on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

on the negative k theory of schemes in finite characteristic abstract we show that if x is a d dimensional scheme of finite type over an infinite perfect field k of characteristic p 0 then k i x 0 and x is k i regular for i d 2 whenever the resolution of singularities holds over k this proves the k dimension conjecture of weibel c weibel k theory and analytic isomorphisms invent math 61 1980 177 197 2 9 except for d 1 i d 2 in all characteristics assuming the resolution of singularities

algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

the novikov conjecture for algebraic k theory of the group algebra over the ring of schatten class operators abstract in this paper we prove the algebraic k theory novikov conjecture for group algebras over the ring of schatten class operators the main technical tool in the proof is an explicit construction of the connes chern character

topological hochschild homology of l and ko we calculate the integral homotopy groups of thh ell at any prime and of thh ko at p 2 where ell is the adams summand of the connective complex p local k theory spectrum and ko is the connective real k theory spectrum

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

hochschild witt complex abstract in a previous paper we have defined polynomial witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups in this paper we use polynomial witt vectors to construct a functorial hochschild witt complex w c h a for any associative unital k algebra a with homology groups w h h a we prove that the group w h h 0 a coincides with the group of non commutative witt vectors defined by hesselholt while if a is commutative finitely generated and smooth the groups w h h i a are naturally identified with the terms w a i of the de rham witt complex of the spectrum of a

equivariant novikov conjecture for groups acting on euclidean buildings we prove the equivariant novikov conjecture for groups acting on euclidean buildings by using an equivariant f tilsum skandalis methocl we also obtain an equivariant version of the connes gromov moscovic th eorem for almost flat c algebra bundles

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

oscillation analysis of numerical solutions for delay differential equations with real coefficients abstract the paper is mainly concerned with the oscillations of numerical solutions by the linear methods for a kind of linear delay differential equations with positive and negative coefficients some conditions under which the numerical solutions are oscillatory are obtained and it is proved that oscillations of the analytical solutions are preserved by the numerical solutions under mild conditions numerical experiments are given to demonstrate our results

on the topological cyclic homology of the integers this article provides a computation of the mod p homotopy groups of the fixed points of the topological hochschild homology of the ring of integers under the action of any finite subgroup of the cirle group whose order is a power of an odd prime p this leads to a computation of the topological cyclic homology groups of the ring of integers and determines also the 7 adic completion of the algebraic at theory of the p adic integers 1 introduction the topological hochschild homology j is a spec trum or infinite loop space associated functorially to each algebraic ring r or more generally to each strictly associative ring spectrum tl it was introduced originally by bokstedt in b the spectrum thh r comes equipped with a natural action of the circle group t the topological cyclic homology of r tc r is the homotopy inverse limit of a diagram of spectra with vertices the fixed point spectra thh r c of thh r under the action of the finite cyclic subgroups c of the circle the maps in this diagram are either inclusions of fixed points or certain frobenius maps particular to the topological hochschild homology construction see g the topological hochschild and cyclic homology of r are both related to the algebraic jt their relation can be depicted by a commutative diagram of spectra

new approximations and policy implications from a delayed dynamic model of a fast pandemic abstract we study an seiqr susceptible exposed infectious quarantined recovered model due to young et al 2019 for an infectious disease with time delays for latency and an asymptomatic phase for fast pandemics where nobody has prior immunity and everyone has immunity after recovery the seiqr model decouples into two nonlinear delay differential equations ddes with five parameters one parameter is set to unity by scaling time the simple subcase of perfect quarantining and zero self recovery before quarantine with two free parameters is examined first the method of multiple scales yields a hyperbolic tangent solution and a long wave short delay approximation yields a first order ordinary differential equation ode with imperfect quarantining and nonzero self recovery the long wave approximation is a second order ode these three approximations each capture the full outbreak from infinitesimal initiation to final saturation low dimensional dynamics in the ddes is demonstrated using a six state non delayed reduced order model obtained by galerkin projection numerical solutions from the reduced order model match the dde over a range of parameter choices and initial conditions finally stability analysis and numerics show how a well executed temporary phase of social distancing can reduce the total number of people affected the reduction can be by as much as half for a weak pandemic and is smaller but still substantial for stronger pandemics an explicit formula for the greatest possible reduction is given

mac lane co homology of the second kind and wieferich primes abstract in this paper we investigate the connection between the mac lane co homology and wieferich primes in finite localizations of global number rings following the ideas of polishchuk positselski 29 we define the mac lane co homology of the second kind of an associative ring with a central element we compute these invariants for finite localizations of global number rings with an element w and obtain that the result is closely related to the wieferich primes to the base w in particular for a given non zero integer w the infiniteness of wieferich primes to the base w turns out to be equivalent to the following for any positive integer n we have h m l ii 0 z 1 n w q as an application of our technique we identify the ring structure on the mac lane cohomology of a global number ring and compute the adams operations introduced in this case by mccarthy 26 on its mac lane homology

coexistence of a general elliptic system in population dynamics this paper is concerned with a strongly coupled elliptic system representing a competitiveinteraction between two species we give a sufficient condition for the existence of positive solutions an example is also given to show that there is a coexistence of a steady state if the cross diffusions or the intraspecific competitions are weak

the homotopy theory of cyclic sets the aim of this note is to show that the homotopy theory of the cyclic sets of connes 3 is equivalent to that of s0 2 spaces i e spaces with a circle action and hence to that of spaces over k z 2

the big de rham witt complex this paper gives a new and direct construction of the multi prime big de rham witt complex which is defined for every commutative and unital ring the original construction by madsen and myself relied on the adjoint functor theorem and accordingly was very indirect the construction given here also corrects the 2 torsion which was not quite correct in the original version the new construction is based on the theory of modules and derivations over a ring which is developed first the main result in this first part of the paper is that the universal derivation of a ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the ring structure in question in the case of the ring of big witt vectors this additional structure gives rise to divided frobenius operators on the module of kahler differentials it is the existence of these divided frobenius operators that makes the new construction of the big de rham witt complex possible it is further shown that the big de rham witt complex behaves well with respect to etale maps and finally the big de rham witt complex of the ring of integers is explicitly evaluated

the topological hochschild homology of the gaussian integers the calculation in this paper gives the 2 torsion in the topological hochschild homology of rings of integers in quadratic extensions of the rationals in general it is not hard to deduce the p torsion in the topological hochschild homology of rings of integers which do not ramify at p from the corresponding torsion for the integers thus for the ring of gaussian integers and also for the integers with the square root of 2 or 2 adjoined this paper completes the calculation of the topological hochschild homology spectrum which is a priori known to be a product of eilenberg maclane spectra precisely because of this a priori knowledge the homotopy groups can be found by doing a homology calculation a spectral sequence arising from the filtration of topological hochschild homology by simplicial skeleta converges to the desired homology if one reduces the rings in question modulo 2 the spectral sequence collapses at its second term this comparison bounds the nontriviality of higher differentials in the spectral sequence for the original ring it is explicitly demonstrated using simplicial calculations that the higher differentials are as nontrivial as they could be given this bound 0 introduction this paper gives an explicit calculation of the 2 torsion in the topological hochschild homology thh of rings of integers in quadratic extensions of the rationals which are ramified at the prime 2 theorem 1 14 the calculation of the torsion in thh of rings of integers in extensions unramified at p can be deduced from bokstedt s calculation of this invariant for the integers see 2 2 this means that given the results of this paper the 2 torsion of thh is known for any quadratic number ring it also means that for rings of integers in extensions which ramify only at the prime 2 specifically the gaussian integers z a 2 and z 2 the homotopy type of the thh spectrum is completely known and given in

detecting k theory by cyclic homology we discuss which part of the rationalized algebraic k theory of a group ring is detected via trace maps to hochschild homology cyclic homology periodic cyclic or negative cyclic homology

coassembly and the k theory of finite groups abstract we study the k theory and swan theory of the group ring r g when g is a finite group and r is any ring or ring spectrum in this setting the well known assembly map for k r g has a companion called the coassembly map we prove that their composite is the equivariant norm of k r this gives a splitting of both assembly and coassembly after k n localization a new map between whitehead torsion and tate cohomology and a partial computation of k theory of representations in the category of spectra

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

p adic deformation of algebraic cycle classes we study the p adic deformation properties of algebraic cycle classes modulo rational equivalence we show that the crystalline chern character of a vector bundle on the closed fibre lies in a certain part of the hodge filtration if and only if rationally the class of the vector bundle lifts to a formal pro class in k theory on the p adic scheme

algebraic k theory of topological k theory we are interested in the arithmetic of ring spectra to make sense of this we must work with structured ring spectra such as s algebras ekmm symmetric ring spectra hss or rings ly we will refer to these as salgebras the commutative objects are then commutative s algebras the category of rings is embedded in the category of s algebras by the eilenberg maclane functor r hr we may therefore view an s algebra as a generalization of a ring in the algebraic sense the added flexibility of s algebras provides room for new examples and constructions which may eventually also shed light upon the category of rings itself in algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its picard group its unit group and its brauer group these are

numerical bifurcation analysis of immunological models with time delays in recent years a large number of mathematical models that are described by delay differential equations ddes have appeared in the life sciences to analyze the models dynamics numerical methods are necessary since analytical studies can only give limited results in turn the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling which may lead to more realistic models we outline recently developed numerical methods for bifurcation analysis of ddes and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis b virus infection

transfer maps and the cyclotomic trace we analyze the equivariant restriction or transfer maps in topological hochschild homology associated to inclusions of group rings of the form r h r g where r is a symmetric ring spectrum g is a discrete group and h g is a subgroup of finite index this leads to a complete description of the associated restriction or transfer maps in topological cyclic homology

higher topological cyclic homology and the segal conjecture for tori abstract we investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory higher topological cyclic homology is constructed from the fixed points of a version of topological hochschild homology based on the n dimensional torus and we propose it as a computationally tractable cousin of n fold iterated algebraic k theory the fixed points of toral topological hochschild homology are related to one another by restriction and frobenius operators we introduce two additional families of operators on fixed points the verschiebung indexed on self isogenies of the n torus and the differentials indexed on n vectors we give a detailed analysis of the relations among the restriction frobenius verschiebung and differentials producing a higher analog of the structure hesselholt and madsen described for 1 dimensional topological cyclic homology we calculate two important pieces of higher topological cyclic homology namely topological restriction homology and topological frobenius homology for the sphere spectrum the latter computation allows us to establish the segal conjecture for the torus which is to say to completely compute the cohomotopy type of the classifying space of the torus

on the vanishing of negative k groups we show that for a d dimensional scheme x essentially of finite type over an infinite perfect field k of characteristic p 0 the negative k groups k q x vanish for q d provided that strong resolution of singularities holds over the field k

introduction to the numerical analysis of stochastic delay differential equations abstract we consider the problem of the numerical solution of stochastic delay differential equations of ito form d x t f x t x t d t g x t x t d w t t 0 t and x t t for t 0 with given f g wiener noise w and given 0 with a prescribed initial function we indicate the nature of the equations of interest and give a convergence proof for explicit single step methods some illustrative numerical examples using a strong euler maruyama scheme are provided

continuously controlled algebraic k theory of spaces and the novikov conjecture here the l polynomial is a certain rational polynomial in the pontrjagin classes this theorem is surprising from many points of view the left hand side is obviously a homotopy invariant and an integer whereas the right hand side a priory is only a smooth invariant and a rational number this led novikov to the following conjecture if m has fundamental group and x is an element in h b f the classifying map could it be that the higher signatures x l m m are homotopy invariants of the manifold m from this point of view hirzebruch s theorem is a verification of the novikov conjecture for simply connected manifolds later wall 17 section 17h realized that the novikov conjecture can be expressed using the assembly map h b l z l z the novikov conjecture is equivalent to the assembly map being a rational monomorphism over the years it has turned out that there are lots of assembly maps in algebraic ktheory in c theory and in a theory among others it has become common practice to call the statement that the assembly map is a rational monomorphism the novikov conjecture in that theory in the case of c theory monicity of the assembly map implies but is not equivalent to the classical novikov conjecture in this paper we treat the a theory case we wish to extend the results in 6 and 7 to a theory using a variation of the continuously controlled a theory in 13 15 to replace the continuously controlled k theory in 1 one of the main problems here is that as a computational device slightly discontinuous maps were allowed in 1 and 6 and atheory does not respond nicely to that the answer is to work with spaces that are so locally contractible that the slightly discontinuous maps eventually continuous maps can be replaced by continuous maps otherwise the strategy is to follow 7 6 and 13 15 and we shall assume the reader has some familiarity with these papers we prove the following theorem

finite complex reflection arrangements are k pi 1 let v be a finite dimensional complex vector space and w gl v be a finite complex reflection group let v reg be the complement in v of the reflecting hyperplanes we prove that v reg is a k 1 space this was predicted by a classical conjecture originally stated by brieskorn for complexified real reflection groups the complexified real case follows from a theorem of deligne and after contributions by nakamura and orliksolomon only six exceptional cases remained open in addition to solving these six cases our approach is applicable to most previously known cases including complexified real groups for which we obtain a new proof based on new geometric objects we also address a number of questions about 1 w v the braid group of w this includes a description of periodic elements in terms of a braid analog of springer s theory of regular elements

cyclic polytopes and the k theory of truncated polynomial algebras this paper calculates the relative algebraic k theory k k x x n x of a truncated polynomial algebra over a perfect field k of positive characteristic p since the ideal generated by x is nilpotent we can apply mccarthy s theorem the relative algebraic k theory is isomorphic to the relative topological cyclic homology mc and it is the latter groups we actually evaluate the result is best expressed in terms of big witt vectors let wm k denote the big witt vectors in k of length m i e the multiplicative group wm k 1 xk x 1 x k x

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

on the k theory of truncated polynomial algebras revisited abstract we revisit the computation due to hesselholt and madsen of the k theory of truncated polynomial algebras for perfect fields of positive characteristic the resulting k groups are expressed in terms of big witt vectors of the field the original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid using the nikolaus scholze framework for topological cyclic homology we achieve the same result using only the homology of said cyclic bar construction as well as the action of connes operator

an etale approach to the novikov conjecture we show that the rational novikov conjecture for a group of finite homological type follows from the mod 2 acyclicity of the higson compactification of an e we then show that for groups of finite asymptotic dimension the higson compactification is mod p acyclic for all p and deduce the integral novikov conjecture for these groups 2007 wiley periodicals inc

on the k theory of groups with finite asymptotic dimension it is proved that the assembly maps in algebraic kand l theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for e the result also applies to certain groups that admit only a finite dimensional model for e in particular it applies to discrete subgroups of virtually connected lie groups

ring completion of rig categories we o er a solution to the long standing problem of group completing within the context of rig categories also known as bimonoidal categories given a rig category r we construct a natural additive group completion r that retains the multiplicative structure hence has become a ring category if we start with a commutative rig category r also known as a symmetric bimonoidal category the additive group completion r will be a commutative ring category in an accompanying paper we show how to use this construction to prove the conjecture that the algebraic k theory of the connective topological k theory ring spectrum ku is equivalent to the algebraic k theory of the rig category v of complex vector spaces

the cyclic homology of crossed product algebras in their article 9 on cyclic homology feigin and tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra generalizing burghelea s calculation 4 of the cyclic homology of a group algebra for an analogous spectral sequence for the hochschild homology of a crossed product algebra see brylinski 2 3 in this article we give a new derivation of this spectral sequence and generalize it to negative and periodic cyclic homology hc a and hp a the method of proof is itself of interest since it involves a natural generalization of the notion of a cyclic module in which the condition that the morphism n n is cyclic of order n 1 is relaxed to the condition that it be invertible we call this category the paracyclic category given a paracyclic module p we can define a chain complex c p with differentials b and b which respectively lower and raise degree the condition that the module p is paracyclic translates to the condition on c p that 1 bb bb is invertible our main result is to show that there is an analogue of the eilenberg zilber theorem for bi paracyclic modules it is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of feigin and tsygan if m is a module over a commutative ring k we will denote by m k the iterated tensor product defined by m 0 k and m k 1 m k m if m and n are graded modules we will denote by m n their graded tensor product we would like to thank j block j l brylinski and c kassel for a number of interesting discussions on the results presented here

on the k theory of local fields in this paper we establish a connection between the quillen k theory of certain local fields and the de rham witt complex of their rings of integers with logarithmic poles at the maximal ideal the fieldsk we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p 2 when k contains the pth roots of unity the relationship between the k theory with z p coefficients and the de rham witt complex can be described by a sequence

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

equivariant evaluation on free loop spaces there is a natural evaluation map lambda x to x k on the free loop space which sends a loop to its values at the kth roots of unity this map is equivariant with respect to the action of the cyclic group on k elements c k we study the induced map in c k equivariant cohomology with mod two coefficients in the cases where k 2 m for m geq 1

the witt vectors for green functors abstract we define twisted hochschild homology for green functors this construction is the algebraic analogue of the relative topological hochschild homology thh c n and under flatness conditions it describes the e 2 term of the kunneth spectral sequence for relative thh applied to ordinary rings we obtain new algebraic invariants extending hesselholt s construction of the witt vectors of noncommutative rings we interpret our construction as providing witt vectors for green functors

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

on the algebraic k theory of the complex k theory spectrum let p 5 be a prime let ku be the connective complex k theory spectrum and let k ku be the algebraic k theory spectrum of ku in this paper we study the p primary homotopy type of the spectrum k ku by computing its mod p v1 homotopy groups we show that up to a finite summand these groups form a finitely generated free module over the polynomial algebra mathbb f p b where b is a class of degree 2p 2 defined as a higher bott element

periodic solutions and chaos in a non linear model for the delayed cellular immune response abstract we model the cellular immune response using a set of non linear delayed differential equations we observe that the stationary solution becomes unstable above a critical immune response time the exponents characterizing the approach to this bifurcation point as well as the critical slow dynamics are obtained in the periodic regime the minimum virus load is substantially reduced with respect to the stationary solution further increasing the delay time the dynamics display a series of bifurcations evolving to a chaotic regime characterized by a set of 2d portraits