permutation arrays under the chebyshev distance an n d permutation array pa is a subset of sn with the property that the distance under some metric between any two permutations in the array is at least d they became popular recently for communication over power lines motivated by an application to flash memories in this paper the metric used is the chebyshev metric a number of different constructions are given as well as bounds on the size of such pa

strong limit point criteria for a class of singular discrete linear hamiltonian systems this paper is concerned with the limit point case for a class of singular discrete linear hamiltonian systems the limit point case is divided into the strong and the weak limit point cases several sufficient conditions for the strong limit point case are established in consequence two criteria of the strong limit point case for second order formally self adjoint vector difference equations are obtained

spectra of modules abstract this manuscript solves the problem that the so called stable category mod of an artin algebra is in general not triangulated the method is to mimick topology and hence first form the spanier whitehead category stab mod and then construct a category spectra of spectra of modules which completes the compact part of stab mod under small coproducts spectra is then a triangulated substitute for mod the main results are that spectra is a compactly generated triangulated category which contains the compact part of stab mod as a full subcategory and even admits a precise description of its compact objects which only form a small set of isomorphism classes as an application it is proved that over an artin algebra the gorenstein projective modules form a pre covering class this was previously only known for rings satisfying strong homological conditions

precovers and orthogonality in the stable module category abstract we show that any module admits a presentation as the quotient of a gorenstein projective module by a submodule which is itself right orthogonal with respect to the standard ext 1 pairing to the class of gorenstein projective modules of type fp for that purpose we use the concept of orthogonality in the stable module category and examine the orthogonal pair which is induced therein by the class of completely finitary gorenstein projective modules

titchmarsh sims weyl theory for complex hamiltonian systems on sturmian time scales abstract we study non self adjoint hamiltonian systems on sturmian time scales defining weyl sims sets which replace the classical weyl circles and a matrix valued m function on suitable cone shaped domains in the complex plane furthermore we characterize realizations of the corresponding dynamic operator and its adjoint and construct their resolvents even order scalar equations and the orr sommerfeld equation on time scales are given as examples illustrating the theory which are new even for difference equations these results unify previous discrete and continuous theories to dynamic equations on sturmian time scales

inapproximability results for the weight problems of subgroup permutation codes a subgroup permutation code is a set of permutations on n symbols with the property that its elements are closed under the operation of composition in this paper we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well known metrics based on previous works we prove that under hamming lee cayley kendall s tau ulam s and lp distance metrics 1 there is no polynomial time 2log1 en approximation algorithm for the minimum weight problem for any constant e 0 unless np dtime 2polylog n quasi polynomial time and 2 there is no polynomial time r approximation algorithm for the minimum weight problem for any constant r 1 unless p np under l metric we prove that it is np hard to approximate the minimum weight problem within factor 2 e for any constant e 0 we also prove that for any constant e 0 it is np hard to approximate the maximum weight within p 3 2 e under lp distance metric and within 3 2 e under hamming lee cayley kendall s tau and ulam s distance metrics

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

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

dependence of eigenvalues on the boundary conditions of sturm liouville problems with one singular endpoint abstract this paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self adjoint sturm liouville problems with one singular endpoint locally continuous dependence of eigenvalues on the boundary conditions is proved especially in the limit point case the continuous dependence of isolated eigenvalues is shown by the relationships between weyl titchmarsh m function and the spectrum of the singular problems then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed it is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in l w 2 norm its proof is different from that in the regular sturm liouville problems the derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given and thus its monotonicity with respect to some parameters is derived

hardness of approximating the shortest vector problem in lattices let p 1 be any fixed real we show that assuming np n rp there is no polynomial time algorithm that approximates the shortest vector problem svp in e p norm within a constant factor under the stronger assumption np n rtime 2poly log n we show that there is no polynomial time algorithm with approximation ratio 2 log n 1 2 e where n is the dimension of the lattice and e 0 is an arbitrarily small constant we first give a new randomized reduction from closest vector problem cvp to svp that achieves some constant factor hardness the reduction is based on bch codes its advantage is that the svp instances produced by the reduction behave well under the augmented tensor product a new variant of tensor product that we introduce this enables us to boost the hardness factor to 2 log n 1 2 e

on the absolutely continuous spectrum of a vector matrix dirac system a dirac system is considered which has a matrix valued long range short range and oscillatory potentials the system has one singular endpoint at infinity additional conditions on the potential are given which guarantee particular asymptotic behaviour of an energy functional associated with a certain set of solutions this asymptotic behaviour guarantees the existence of a purely absolutely continuous spectrum outside a gap containing the origin

self adjoint extensions for linear hamiltonian systems with two singular endpoints abstract this paper is concerned with self adjoint extensions for a linear hamiltonian system with two singular endpoints the domain of the closure of the corresponding minimal hamiltonian operator h 0 is described by properties of its elements at the endpoints of the discussed interval decompositions of the domains of the corresponding left and right maximal hamiltonian operators are provided and expressions of the defect indices of h 0 in terms of those of the left and right minimal operators are given based on them characterizations of all the self adjoint extensions for a hamiltonian system are obtained in terms of square integrable solutions as a consequence the characterizations of all the self adjoint extensions are given for systems in several special cases

decoding frequency permutation arrays under chebyshev distance a frequency permutation array fpa of length n m and distance d is a set of permutations on a multiset over m symbols where each symbol appears exactly times and the distance between any two elements in the array is at least d fpa generalizes the notion of permutation array in this paper under the chebyshev distance we first prove lower and upper bounds on the size of fpa then we give several constructions of fpas and some of them come with efficient encoding and decoding capabilities moreover we show one of our designs is locally decodable i e we can decode a message bit by reading at most 1 symbols which has an interesting application to private information retrieval

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

phantom maps and purity in modular representation theory iii abstract in the third installment of this paper we exhibit two phantom maps between kg modules whose composite is not projective whenever k is an uncountable field of characteristic p and g is a finite group of p rank at least 2

j self adjoint extensions of a class of hamiltonian differential systems abstract this paper is concerned with j self adjoint extensions for a class of hamiltonian differential systems the domains of the corresponding minimal and maximal operators are described a classification of limit types of the systems at each of endpoints is given a property of the regularity field of the corresponding minimal operator is derived and an expression of the j defect index of the minimal operator by those of the left and right minimal operators is presented based on them a characterization of all j self adjoint extensions for the hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this hamiltonian system and its formal adjoint as a consequence characterizations of all j self adjoint extensions are given for the systems in some special cases in particular all j self adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases

the derived category of an exact category there is some confusion in the literature regarding the derived category of an exact category 8 thomason gives a satisfactory treatment of the bounded derived category in ct 1 11 6 see also appendix a the result is that provided all weakly split epimorphisms in 8 are admissible the bounded derived category may be defined as usual although thomason does not say it the categories d 6 d d may also be defined in the same way the definition of the unbounded derived category is more difficult and is very poorly treated in the literature for instance in bbd this derived category is only defined provided every morphism in 6 admits a kernel see bbd 1 1 41 this is very unsatisfactory since very few exact categories satisfy the condition in this article we will show that d b may be defined whenever 6 is saturated karoubian in thomason s terminology an exact category is saturated if every idempotent splits i e 6 contains all direct summands of its objects for the purpose of comparing with thomason s result the bounded derived category is defined for more 8 s all that is required to define db is that whenever an idempotent e a a factors as a f ba a with fog lb then e is split thomason calls such idempotents weakly split and we will honor his notation we will show here that these constructions are in some sense best possible remark 1 8 for d 6 remarks 1 9 and 1 10 for d d d b db 6 there are two reasons why i wrote this note one is to correct the misconceptions in the literature but more importantly the proof of the key result lemma 1 2 depends on an important characterization of tpaisse subcategories due to rickard and this article is intended to highlight rickard s criterion a full triangulated subcategory y of a triangulated

the dual of brown representability for homotopy categories of complexes abstract we call product generator of an additive category a fixed object satisfying the property that every other object is a direct factor of a product of copies of it in this paper we start with an additive category with products and images e g a module category and we are concerned with the homotopy category of complexes with entries in that additive category we prove that the brown representability theorem is valid for the dual of the homotopy category if and only if the initial additive category has a product generator

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

relative homological algebra in the category of quasi coherent sheaves abstract in this paper we prove the existence of a flat cover and of a cotorsion envelope for any quasi coherent sheaf over a scheme x o x indeed we prove something more general we define what it is understood by the category of quasi coherent r modules where r is a representation by rings of a quiver q and we prove the existence of a flat cover and a cotorsion envelope for quasi coherent r modules then we use the fact that the category of quasi coherent sheaves on x o x is equivalent to the category of quasi coherent r modules for some q and r to get our result

singular second order operators the maximal and minimal operators and selfadjoint operators in between differential operators arising from the differential equation py qy lambda wy are put in the hilbert space setting l 2 a b w a new and simpler characterization of the minimal operator is exhibited selfadjoint operators which lie in between the minimal and maximal operator are easily described in terms of conditions on boundary coefficients which look like and indeed are the same as those imposed on regular problems examples drawn from mathematical physics include the legendre laguerre hermite and bessel problems

a note on eigenvalues of a class of singular continuous and discrete linear hamiltonian systems abstract in this paper we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear hamiltonian systems are equal where both endpoints are in the limit circle cases the proof is fundamental and is given for both continuous and discrete hamiltonian systems the method used in this paper also works for both endpoints are regular or one endpoint is regular and the other is in the limit circle case

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

brown representability for the dual let t be the homotopy category of all spectra brown proved that a homological functor h t o p ab is representable if it takes coproducts to products that is the functors h may be characterised as the homological functors taking coproducts to products in this article we will prove the dual a covariant functor h t ab which takes products to products is representable it is of the form h

cohomology theories based on flats abstract let a be an associative ring with identity k flat a the homotopy category of flat modules and k p flat a the full subcategory of pure complexes the quotient category k flat a k p flat a called here the pure derived category of flats was introduced by neeman in this category flat resolutions are unique up to homotopy and so can be used to compute cohomology we develop theories of tate and complete cohomology in the pure derived category of flats these theories extend naturally to sheaves over semi separated noetherian schemes where there are not always enough projectives but we do have enough flats as applications we characterize rings with finite sfli and schemes which are locally gorenstein

a deterministic reduction for the gap minimum distance problem determining the minimum distance of a linear code is one of the most important problems in algorithmic coding theory the exact version of the problem was shown to be np complete by vardy the gap version of the problem was shown to be np hard for any constant factor under a randomized reduction in an earlier work it was shown in the same paper that the minimum distance problem is not approximable in randomized polynomial time to the factor 2log1 n unless np rtime 2polylog n in this paper we derandomize the reduction and thus prove that there is no deterministic polynomial time algorithm to approximate the minimum distance to any constant factor unless p np we also prove that the minimum distance is not approximable in deterministic polynomial time to the factor 2log1 n unless np dtime 2polylog n as the main technical contribution for any constant 2 3 1 we present a deterministic algorithm that given a positive integer s runs in time poly s and constructs a code c of length poly s with an explicit hamming ball of radius d c such that the projection at the first s coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension s where d c denotes the minimum distance of c the codes are obtained by concatenating reed solomon codes with hadamard codes

the homotopy category of complexes of projective modules abstract the homotopy category of complexes of projective left modules over any reasonably nice ring is proved to be a compactly generated triangulated category and a duality is given between its subcategory of compact objects and the finite derived category of right modules

defect indices and definiteness conditions for a class of discrete linear hamiltonian systems abstract this paper is concerned with a class of discrete linear hamiltonian systems in finite or infinite intervals a definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given a precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established in particular they are equal if and only if the definiteness condition is satisfied finally two criteria for the limit point case and one criterion for the limit circle case are obtained

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

homotopy category of projective complexes and complexes of gorenstein projective modules abstract let r be a ring with identity and c r denote the category of complexes of r modules in this paper we study the homotopy categories arising from projective resp injective complexes as well as gorenstein projective resp gorenstein injective modules we show that the homotopy category of projective complexes over r denoted k prj c r is compactly generated provided k prj r is so based on this result it will be proved that the class of gorenstein projective complexes is precovering whenever r is a commutative noetherian ring of finite krull dimension furthermore it turns out that over such rings the inclusion functor k gprj r k r has a right adjoint where k gprj r is the homotopy category of gorenstein projective r modules similar or rather dual results for the injective resp gorenstein injective complexes will be provided if r has a dualising complex a triangle equivalence between homotopy categories of projective and of injective complexes will be provided as an application we obtain an equivalence between the triangulated categories k gprj r and k ginj r that restricts to an equivalence between k prj r and k inj r whenever r is commutative noetherian and admits a dualising complex

cartesian modules over representations of small categories we introduce the new concept of cartesian module over a pseudofunctor r from a small category to the category of small preadditive categories already the case when r is a strict functor taking values in the category of commutative rings is sucient

on pure acyclic complexes abstract in this paper we study the pure acyclic complexes of modules we obtain several characterizations of these complexes extending results that are known for the pure acyclic complexes of flat modules in particular we extend neeman s characterization 18 of the pure acyclic complexes of flat modules in terms of complexes of projective modules by considering more generally complexes of pure projective modules as a consequence we obtain simson s result 22 on the pure projectivity of pure periodic modules

singular hamiltonian system with several spectral parameters abstract in this paper the weyl titchmarsh theory has been constructed for the singular 2n dimensional even order hamiltonian system with several spectral parameters in particular we consider that the left end point of the interval is regular and the right end point of the interval is singular for the hamiltonian system with several parameters using the nested circles approach we prove that at least n linearly independent solutions are squarly integrable with respect to some matrix functions

on the rank of the matrix radius of the limiting set for a singular linear hamiltonian system this paper is concerned with the rank of the matrix radius of the limiting set for a singular hamiltonian system with one singular end point the exact relationship between the rank of the matrix radius and the number of square integrable solutions is obtained and then the defect index of the corresponding minimal operator can be represented in terms of the rank of the matrix radius so two results obtained by allan m krall siam j math anal 20 1989 664 are improved in addition it is discussed that the rank of the matrix radius is independent of the spectral parameter and a certain matrix especially the classification of singular linear hamiltonian systems is present and several sufficient and necessary conditions for the limit point and limit circle cases are established

semi boundedness of ordinary differential operators abstract the boundedness from above and from below of very general symmetric quasi differential operators s is studied here we show that i if s is regular of even order and has positive leading coefficient then it is bounded below ii if the order of s which may be regular or singular is odd then s is unbounded both above and below and iii if s is of even order regular or singular but with a leading coefficient which changes sign then s is unbounded both above and below

self adjoint extensions for discrete linear hamiltonian systems abstract this paper is concerned with self adjoint extensions for a class of discrete linear hamiltonian systems by applying the generalized von neumann theory and the gkn theory for hermitian subspaces a complete characterization of all the self adjoint subspace extensions for the systems are obtained in terms of boundary conditions via linear independent square summable solutions as a consequence characterizations of all the self adjoint subspace extensions are given in the limit point and limit circle cases in addition some sufficient conditions for the corresponding minimal subspace to be an operator or a densely defined operator are given and consequently characterizations of all the self adjoint operator extensions for the system are obtained in particular even order formally self adjoint vector difference equations are discussed as a special case

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

on the relation between k flatness and k projectivity abstract krause 19 has proved that the homotopy category k r pureproj of pure projective modules over an associative ring r is compactly generated and equivalent to the verdier quotient k r t r where t r is the triangulated subcategory of the homotopy category k r consisting of the pure acyclic complexes we provide an alternative proof of this result when restricted to the homotopy category of projective modules this equivalence reduces to that obtained by neeman 26 as an application of this description of k r pureproj we show that the k flat complexes defined by spaltenstein 29 are up to homotopy equivalence the filtered colimits of k projective complexes of pure projective modules

powers of the phantom ideal the mono epi me exact structure on the morphisms of an exact category a e is introduced and used to prove ideal versions of salce s lemma christensen s ghost lemma and wakamatsu s lemma for an exact category salce s lemma establishes a bijective correspondence i i between the class of special precovering ideals of a e and that of its special preenveloping ideals me extensions of morphisms are used to define an extension i j of ideals christensen s lemma asserts that the class of special precovering respectively special preenveloping ideals is closed under products and extensions and that the bijective correspondence of salce s lemma satisfies ij j i and i j j i wakamatsu s lemma asserts that if a covering ideal i is closed under me extensions then it is a special precovering ideal as an application it is proved that if g is a finite group and is the ideal of phantom morphisms in the category k g mod then n 1 is the object ideal generated by projective modules where n is the nilpotency index of the jacobson radical j if r is a semiprimary ring with jn 0 then n is generated by projective modules for a right coherent ring r over which every cotorsion left r module has a coresolution of length n by pure injective modules n 1 is generated by flat modules

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

correcting limited magnitude errors in the rank modulation scheme we study error correcting codes for permutations under the infinity norm motivated by a novel storage scheme for flash memories called rank modulation in this scheme a set of n flash cells are combined to create a single virtual multilevel cell information is stored in the permutation induced by the cell charge levels spike errors which are characterized by a limited magnitude change in cell charge levels correspond to a low distance change under the infinity norm we define codes protecting against spike errors called limited magnitude rank modulation codes lmrm codes and present several constructions for these codes some resulting in optimal codes these codes admit simple recursive and sometimes direct encoding and decoding procedures we also provide lower and upper bounds on the maximal size of lmrm codes both in the general case and in the case where the codes form a subgroup of the symmetric group in the asymptotic analysis the codes we construct outperform the gilbert varshamov like bound estimate

the popescu gabriel theorem for triangulated categories abstract the popescu gabriel theorem states that each grothendieck abelian category is a localization of a module category in this paper we prove an analogue where grothendieck abelian categories are replaced by triangulated categories which are well generated in the sense of neeman and algebraic in the sense of keller the role of module categories is played by derived categories of small differential graded categories an analogous result for topological triangulated categories has recently been obtained by a heider

weyl titchmarsh theory for a class of discrete linear hamiltonian systems abstract this paper is concerned with establishing the weyl titchmarsh theory for a class of discrete linear hamiltonian systems over a half line fundamental properties of solutions regular spectral problems and the corresponding maximal and minimal operators are first studied matrix disks are constructed and proved to be nested and converge to a limiting set some precise relationships among the rank of the matrix radius of the limiting set the number of linearly independent square summable solutions and the defect indices of the minimal operator are established based on the above results a classification of singular discrete linear hamiltonian systems is given in terms of the defect indices of the minimal operator and several equivalent conditions on the cases of limit point and limit circle are obtained respectively especially several problems in the limit point case are more carefully investigated including fundamental properties of square summable solutions properties of the weyl function which is the unique element in the limiting set in this case and inhomogeneous boundary problems self adjointness of the corresponding hamiltonian operator relationship between the spectrum of the hamiltonian operator and the analyticity of the weyl function as well as the dependence of the spectrum on the boundary data in which some interesting separation results for the spectrum are obtained finally another set of four equivalent conditions on the limit point case are established

on essential spectra of singular linear hamiltonian systems abstract the paper is concerned with essential spectra of singular linear hamiltonian systems of arbitrary order with arbitrary equal defect indices several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding hamiltonian system and a sufficient and necessary condition for the essential spectral points is obtained for hamiltonian systems of even order with one singular endpoint an advantage of these results is that one can determine the essential spectral points of hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations in addition two illustrative examples are provided to show how to get some information about the essential spectrum by our results

minimal semi flat cotorsion replacements and cosupport abstract over a commutative noetherian ring r of finite krull dimension we show that every complex of flat cotorsion r modules decomposes as a direct sum of a minimal complex and a contractible complex moreover we define the notion of a semi flat cotorsion complex as a special type of semi flat complex and provide functorial ways to construct a quasi isomorphism from a semi flat complex to a semi flat cotorsion complex consequently every r complex can be replaced by a minimal semi flat cotorsion complex in the derived category over r furthermore we describe structure of semi flat cotorsion replacements by which we recover classic theorems for finitistic dimensions in addition we improve some results on cosupport and give a cautionary example we also explain that semi flat cotorsion replacements always exist and can be used to describe the derived category over any associative ring

totally acyclic complexes abstract it is known that over an iwanaga gorenstein ring the gorenstein injective gorenstein projective gorenstein flat modules are simply the cycles of acyclic complexes of injective projective flat modules we consider the question are these characterizations only working over iwanaga gorenstein rings we prove that if r is a commutative noetherian ring of finite krull dimension then the following are equivalent 1 r is an iwanaga gorenstein ring 2 every acyclic complex of injective modules is totally acyclic 3 the cycles of every acyclic complex of gorenstein injective modules are gorenstein injective 4 every acyclic complex of projective modules is totally acyclic 5 the cycles of every acyclic complex of gorenstein projective modules are gorenstein projective 6 every acyclic complex of flat modules is f totally acyclic 7 the cycles of every acyclic complex of gorenstein flat modules are gorenstein flat thus we improve slightly on a result of iyengar and krause in 22 they proved that for a commutative noetherian ring r with a dualizing complex the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if r is gorenstein we replace the dualizing complex hypothesis by the finiteness of the krull dimension and add more equivalent conditions in the second part of the paper we focus on the noncommutative case we prove that for a two sided noetherian ring r of finite finitistic flat dimension that satisfies the auslander condition the following are equivalent 1 every complex of injective left and respectively right r modules is totally acyclic 2 r is iwanaga gorenstein

classification of sturm liouville differential equations with complex coefficients and operator realizations in this paper a new classification of sturm liouville differential equations with complex coefficients is given compared with the corresponding result of brown et al this classification reveals the great effects of rotation angle and it is independent of the rotation angles moreover the asymptotic behaviours of functions in the maximal domain are presented and j self adjoint extensions associated with the differential equations are characterized

the derived category of quasi coherent sheaves and axiomatic stable homotopy we prove in this paper that for a quasi compact and semi separated nonnecessarily noetherian scheme x the derived category of quasi coherent sheaves over x is a stable homotopy category in the sense of hovey palmieri and strickland answering a question posed by strickland moreover we show that it is unital and algebraic we also prove that for a noetherian semi separated formal scheme its derived category of sheaves of modules with quasi coherent torsion homologies is a stable homotopy category it is algebraic but if the formal scheme is not a usual scheme it is not unital therefore its abstract nature differs essentially from that of the derived category which is equivalent to in the case of a usual scheme

ghosts in modular representation theory a ghost over a finite p group g is a map between modular representations of g which is invisible in tate cohomology motivated by the failure of the generating hypothesis the statement that ghosts between finite dimensional g representations factor through a projective we define the ghost number of kg to be the smallest integer l such that the composite of any l ghosts between finite dimensional g representations factors through a projective in this paper we study ghosts and the ghost numbers of p groups we begin by showing that a weaker version of the generating hypothesis where the target of the ghost is fixed to be the trivial representation k holds for all p groups we then compute the ghost numbers of all cyclic p groups and all abelian 2 groups with c2 as a summand we obtain bounds on the ghost numbers for abelian p groups and for all 2 groups which have a cyclic subgroup of index 2 using these bounds we determine the finite abelian groups which have ghost number at most 2 our methods involve techniques from group theory representation theory triangulated category theory and constructions motivated from homotopy theory

renormalized oscillation theory for hamiltonian systems abstract we extend a result on renormalized oscillation theory originally derived for sturm liouville and dirac type operators on arbitrary intervals in the context of scalar coefficients to the case of general hamiltonian systems with block matrix coefficients in particular this contains the cases of general sturm liouville and dirac type operators with block matrix valued coefficients as special cases the principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate wronskians of solutions oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

universal toda brackets of ring spectra we construct and examine the universal toda bracket of a highly structured ring spectrum r this invariant of r is a cohomology class in the mac lane cohomology of the graded ring of homotopy groups of r which carries information about r and the category of r module spectra it determines for example all triple toda brackets of r and the first obstruction to realizing a module over the homotopy groups of r by an r module spectrum for periodic ring spectra we study the corresponding theory of higher universal toda brackets the real and complex k theory spectra serve as our main examples

limit point and limit circle classification for symplectic systems on time scales abstract in this paper we study the limit point and limit circle classification for symplectic systems on time scales which depend linearly on the spectral parameter in a broader context we develop a unified weyl titchmarsh theory for continuous and discrete linear hamiltonian and symplectic systems both separated and coupled boundary conditions are allowed our results include the study of the weyl disks and circles and their limiting behavior as well as a precise analysis of the number of linearly independent square integrable solutions we also prove an analogue of the famous weyl alternative we connect and unify many known results in the weyl titchmarsh theory for continuous discrete and special time scales systems and explain the differences between them some of our statements in particular those connected with coupled endpoints or the weyl alternative are new even in the continuous time setting

realizability of modules over tate cohomology let k be a field and let g be a finite group there is a canonical element in the hochschild cohomology of the tate cohomology g hh 3 h1 h g k with the following property given a graded h g k module x the image of g in ext 3 1 h g k x x vanishes if and only if x is isomorphic to a direct summand of h g m for some kg module m the description of the realizability obstruction works in any triangulated category with direct sums we show that in the case of the derived category of a differential graded algebra a there is also a canonical element of hochschild cohomology hh 3 1 h a which is a predecessor for these obstructions

the galois group of a stable homotopy theory abstract to a stable homotopy theory a presentable symmetric monoidal stable category we naturally associate a category of finite etale algebra objects and using grothendieck s categorical machine a profinite group that we call the galois group we then calculate the galois groups in several examples for instance we show that the galois group of the periodic e algebra of topological modular forms is trivial and that the galois group of k n local stable homotopy theory is an extended version of the morava stabilizer group we also describe the galois group of the stable module category of a finite group a fundamental idea throughout is the purely categorical notion of a descendable algebra object and an associated analog of faithfully flat descent in this context

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

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

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