change of rings and singularity categories abstract we investigate the behavior of singularity categories and stable categories of gorenstein projective modules along a morphism of rings the natural context to approach the problem is via change of rings that is the classical adjoint triple between the module categories in particular we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of gorenstein projective modules moreover we study this problem at the level of big singularity categories in the sense of krause 30 along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories this is achieved by introducing the notion of 0 cocompact objects in triangulated categories and proving a dual version of bousfield s localization lemma we provide applications and examples illustrating our main results

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

on a ferromagnetic spin chain the quotient s 1 s of riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperatures

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

stable homotopy as a triangulated functor let j i be the homotopy category of all finite spectra the homotopy category of all spectra and y 89 be the homotopy category of all prime to 2 spectra let h y abelian groups be a homological functor i e a generalized homology theory a natural question is whether h can be lifted to a triangulated functor is there a functor t y d abelian groups so that composed with the usual homology functor on d abelian groups is h here d abelian groups d z is the derived category of z modules if h is ordinary homology the answer is clearly yes after all the ordinary homology of a space is given by the singular chain complex almost by definition one has a lifting by the brown representability theorem the general statement follows from the special case h h the stable homotopy functor let us be a little more precise

morita theory and singularity categories abstract we propose an analogue of the bounded derived category for an augmented ring spectrum defined in terms of a notion of noether normalization in many cases we show this category is independent of the chosen normalization based on this we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are koszul dual in the style of the bgg correspondence examples of interest include koszul algebras and ginzburg dg algebras c b g for finite groups or for compact lie groups with orientable adjoint representation cochains in rational homotopy theory and various examples from chromatic homotopy theory

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

the finite cardinalities of level sets of the takagi function abstract let t be takagi s continuous but nowhere differentiable function it is known that almost all level sets with respect to lebesgue measure on the range of t are finite we show that the most common cardinality of the level sets of t is two and investigate in detail the set of ordinates y such that the level set at level y has precisely two elements as a by product we obtain a simple iterative procedure for solving the equation t x y we show further that any positive even integer occurs as the cardinality of some level set and conjecture that all even cardinalities occur with positive probability if an ordinate y is chosen at random from the range of t the key to the results is a system of set equations for the level sets which are derived from the partial self similarity of t these set equations yield a system of linear relationships between the cardinalities of level sets at various levels from which all the results of this paper flow

elementary duality of modules let r be a ring a formula x in the language of left r modules is called a positive primitive formula ppf if it is of the form y ab x y 0 where a and b are matrices of appropriate size with entries in r we apply prest s notion of d x the ppf in the language of right r modules dual to to show that the model theory of left r modules as developed by ziegler z is in some sense dual to the model theory of right r modules we prove that the topologies on the left and right ziegler spectra are isomorphic proposition 4 4 when the lattice of ppfs is well behaved there is a homeomorphism d between the left and right ziegler spectra which assigns to a given pure injective indecomposable left r module u the dual pure injective indecomposable right r module du

an inequality for sums of binary digits with application to takagi functions abstract let x 2 inf x n n z and define for 0 the function f x j 0 1 2 j 2 j x tabor and tabor j tabor j tabor takagi functions and approximate midconvexity j math anal appl 356 2 2009 729 737 recently proved the inequality f x y 2 f x f y 2 x y for 1 2 by developing an explicit expression for f at dyadic rational points it is shown in this paper that the above inequality can be reduced to a simple inequality for weighted sums of binary digits that inequality which seems of independent interest is used to give an alternative proof of the result of tabor and tabor which captures the essential structure of f

statistical properties of a nonuniformly hyperbolic map of the interval we prove a power law upper bound for the decay of the correlations for holder observables in the case of a nonuniformly hyperbolic map of the interval introduced by gaspard and wang as a piecewise linear approximation of the intermittent map of manneville pomeau the result is then applied to compute the central limit theorem for the same class of observables

on the hausdorff dimension of some graphs consider the functions wb x b cn 1 bnx on 1 0n n oo where b 1 0 0 such that if b is large enough then the hausdorff dimension of the graph of wb is bounded below by 2a c ln b we also show that if a function f is convex lipschitz of order a then the graph of f has a finite measure with respect to hausdorff s measure in dimension 2 a the convex lipschitz functions of order a include zygmund s class a our analysis shows that the graph of the classical van der waerdentagaki nowhere differentiable function has a finite measure with respect to h t t in 1 t we consider the hausdorff dimension of the graphs of various continuous functions we introduce a new geometric property of a function convex lipschitz of some order and obtain an upper bound on the dimension of a graph with this property in particular our analysis includes functions of the form

on asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics let let n be a positive integer and define n as the number of matrices c which are products of a and b where both a and b must occur such that the trace tr c n it has been conjectured that see 10 in this note we consider the summatory function and show that

grothendieck categories of enriched functors abstract it is shown that the category of enriched functors c v is grothendieck whenever v is a closed symmetric monoidal grothendieck category and c is a category enriched over v localizations in c v associated to collections of objects of c are studied also the category of chain complexes of generalized modules ch c r is shown to be identified with the grothendieck category of enriched functors mod r ch mod r over a commutative ring r where the category of finitely presented r modules mod r is enriched over the closed symmetric monoidal grothendieck category ch mod r as complexes concentrated in zeroth degree as an application it is proved that ch c r is a closed symmetric monoidal grothendieck model category with explicit formulas for tensor product and internal hom objects furthermore the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of hovey palmieri strickland 14 is introduced it is shown that the derived category of generalized modules d c r over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category

a farey fraction spin chain we introduce a new number theoretic spin chain and explore its thermodynamics and connections with number theory the energy of each spin configuration is defined in a translation invariant manner in terms of the farey fractions and is also expressed using pauli matrices we prove that the free energy exists and a phase transition occurs for positive inverse temperature 2 the free energy is the same as that of related non translation invariant number theoretic spin chain using a number theoretic argument the low temperature 3 state is shown to be completely magnetized for long chains the number of states of energy e log n summed over chain length is expressed in terms of a restricted divisor problem we conjecture that its asymptotic form is n log n consistent with the phase transition at 2 and suggesting a possible connection with the riemann function the spin interaction coefficients include all even many body terms and are translation invariant computer results indicate that all the interaction coefficients except the constant term are ferromagnetic

generalized number theoretic spin chain connections to dynamical systems and expectation values we generalize the number theoretic spin chain a one dimensional statistical model based on the farey fractions by introducing a parameter x geqslant 0 this allows us to write recursion relations in the length of the chain these relations are closely related to the lewis three term equation which is useful in the study of the selberg function we then make use of these relations and spin orientation transformations we find a simple connection with the transfer operator of a model of intermittency in dynamical systems in addition we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length some of these expectation values appear to be directly connected with the mechanism of the phase transition

a new generalization of the takagi function abstract we consider a one parameter family of functions f t x t on 0 1 and partial derivatives t k f t x with respect to the parameter t each function of the class is defined by a certain pair of two square matrices of order two the class includes the lebesgue singular functions and other singular functions our approach to the takagi function is similar to hata and yamaguti the class of partial derivatives t k f t x includes the original takagi function and some generalizations we consider real analytic properties of t k f t x as a function of x specifically differentiability the hausdorff dimension of the graph the asymptotics around dyadic rationals variation a question of local monotonicity and a local modulus of continuity our results are extensions of some results for the original takagi function and some generalizations

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

totally acyclic complexes over noetherian schemes abstract we define a notion of total acyclicity for complexes of flat quasi coherent sheaves on a semi separated noetherian scheme and study these complexes using the pure derived category of flat quasi coherent sheaves we prove that a scheme is gorenstein if and only if every acyclic complex of flat quasi coherent sheaves is totally acyclic our formalism also removes the need for a dualising complex in several known results for rings including jorgensen s proof of the existence of gorenstein projective precovers

the twisted inverse image pseudofunctor over commutative dg rings and perfect base change abstract let k be a gorenstein noetherian ring of finite krull dimension and consider the category of cohomologically noetherian commutative differential graded rings a over k such that h 0 a is essentially of finite type over k and a has finite flat dimension over k we extend grothendieck s twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup we prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps and prove a perfect base change result for f in this setting as application we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings our results generalize and solve some recent conjectures of yekutieli

rigid complexes via dg algebras let a be a commutative ring b a commutative a algebra and m a complex of b modules we begin by constructing the square sq b a m which is also a complex of b modules the squaring operation is a quadratic functor and its construction requires differential graded dg algebras if there exists an isomorphism p m sq b a m then the pair m p is called a rigid complex over b relative to a there are some finiteness conditions there is an obvious notion of rigid morphism between rigid complexes we establish several properties of rigid complexes including their uniqueness existence under some extra hypothesis and formation of pullbacks f b m p resp f m p along a finite resp essentially smooth ring homomorphism f b c in the subsequent paper rigid dualizing complexes over commutative rings we consider rigid dualizing complexes over commutative rings building on the results of the present paper the project culminates in our paper rigid dualizing complexes and perverse sheaves on schemes where we give a comprehensive version of grothendieck duality for schemes the idea of rigid complexes originates in noncommutative algebraic geometry and is due to van den bergh 1997

perverse t structure on artin stacks in this paper we develop the theory of perverse sheaves on artin stacks continuing our earlier study of lisse etale sheaves on stacks in laszlo and olsson the six operations for sheaves on artin stacks i finite coefficients publ math ihes 2008 the six operations for sheaves on artin stacks ii adic coefficients publ math ihes 2008

on the level sets of the takagi van der waerden functions abstract this paper examines the level sets of the continuous but nowhere differentiable functions f r x n 0 r n r n x where x is the distance from x to the nearest integer and r is an integer with r 2 it is shown by using properties of a symmetric correlated random walk that almost all level sets of f r are finite with respect to lebesgue measure on the range of f but that for an abscissa x chosen at random from 0 1 the level set at level y f r x is uncountable almost surely as a result the occupation measure of f r is singular

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

infinite derivatives of the takagi van der waerden functions abstract in this paper we characterize the set of points where the lateral derivatives of the takagi van der waerden functions are infinite we also prove that the set of points with infinite derivative has hausdorff dimension one and lebesgue measure zero

cluster approximation for the farey fraction spin chain we consider the farey fraction spin chain in an external field h utilising ideas from dynamical systems the free energy of the model is derived by means of an effective cluster energy approximation this approximation is valid for divergent cluster sizes and hence appropriate for the discussion of the magnetizing transition we calculate the phase boundaries and the scaling of the free energy at h 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy for h 0 our results are largely consistent with those found previously using mean field theory and renormalization group arguments

monoidal infinity category of complexes from tannakian viewpoint in this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infty categories of quasi coherent complexes it can be viewed as a derived analogue of tannaka duality as a consequence we deduce that an algebraic stack satisfying a certain condition can be recovered from the symmetric monoidal stable infty category of quasi coherent complexes with tensor operation

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

the number theoretical spin chain and the riemann zeroes it is an empirical observation that the riemann zeta function can be well approximated in its critical strip using the number theoretical spin chain a proof of this would imply the riemann hypothesis here we relate that question to the one of spectral radii of a family of markov chains this in turn leads to the question whether certain graphs are ramanujan

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

gorenstein homological dimensions for triangulated categories abstract let c be a triangulated category with a proper class e of triangles asadollahi and salarian introduced and studied e g projective and e g injective objects and developed a relative homological algebra in c in this paper we further study gorenstein homological dimensions for triangulated categories first we discuss the finiteness of e g projective and e g injective dimensions then we define e gorenstein derived functors ge xt e i and characterize e g projective and e g injective dimensions of objects in c by vanishing of such functors consequently we can prove the equality sup e g pd m for any m c sup e g id m for any m c which is used to define the global e gorenstein dimension of c finally e g phantom tower and e g cellular tower for objects in c with finite e g projective dimension are constructed

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

towards a complete determination of the spectrum of a transfer operator associated with intermittency the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point for an example of such an intermittent map the farey map we give a simple proof that the transfer operator is self adjoint on a suitably defined hilbert space and characterize its spectrum using a suitable first return map we present a highly efficient numerical method for the determination of all the eigenvalues including those embedded in the continuous spectrum

intermittency and regularized fredholm determinants we consider real analytic maps of the interval i 0 1 which are expanding everywhere except for a neutral fixed point at 0 we show that on a certain function space the associated perron frobenius operator has a continuous and residual spectrum contained in the line segment c 0 1 and a point spectrum p which has no points of accumulation outside 0 and 1 furthermore points in p 0 1 are eigenvalues of finite multiplicity we construct a regularized fredholm determinant d which has a holomorphic extension to c and can be analytically continued from each side of c to an open neighborhood of c 0 1 on different riemann sheets in c the zero set of d is in one to one correspondence with the point spectrum of through the conformal transformation the function d z extends to a holomorphic function in a domain which contains the unit disc

a singular function with a non zero finite derivative on a dense set abstract the authors had exhibited in a previous paper a continuous strictly increasing singular function from 0 1 into 0 1 with a derivative that takes non zero finite values at the points of an uncountable set in this article the construction is improved to encompass a dense set

the hausdorff dimension of the level sets of takagi s function abstract recently maddock 2006 12 has conjectured that the hausdorff dimension of each level set of takagi s function is at most 1 2 we prove this conjecture using the self affinity of the function of takagi and the existing relationship between the hausdorff and box counting dimensions

on deformations of triangulated models abstract this paper is the first part of a project aimed at understanding deformations of triangulated categories and more precisely their dg and a models and applying the resulting theory to the models occurring in the homological mirror symmetry setup in this first paper we focus on models of derived and related categories based upon the classical construction of twisted objects over a dg or a algebra for a hochschild 2 cocycle on such a model we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects we unravel the construction in the specific cases of derived a and abelian categories homotopy categories and categories of graded free qdg modules we identify a purity condition on our models which ensures that the structure of the model is preserved under deformation this condition is typically fulfilled for homotopy categories but not for unbounded derived categories

the improper infinite derivatives of takagi s nowhere differentiable function abstract let t be takagi s continuous but nowhere differentiable function using a representation in terms of rademacher series due to n kono acta math hungar 49 1987 315 324 we give a complete characterization of those points where t has a left sided right sided or two sided infinite derivative this characterization is illustrated by several examples a consequence of the main result is that the sets of points where t x have hausdorff dimension one as a byproduct of the method of proof some exact results concerning the modulus of continuity of t are also obtained