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

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

fast base extension using a redundant modulus in rns a technique to extend the base of a residue number system rns based on the chinese remainder theorem crt and the use of a redundant modulus is proposed the technique obtains the residue s of a given number in the extended moduli without resorting to the traditional mixed radix conversion mrc algorithm the base extension can be achieved in log sub 2 n table lookup cycles where n is the number of moduli in the rns the superiority of the technique compared in terms of latency and hardware requirements to the traditional szabo tanaka method is demonstrated

on the least eigenvalue of cacti with pendant vertices abstract a connected graph is a cactus if any two of its cycles share at most one common vertex we determine the unique graph with minimum least eigenvalue among the cacti with fixed numbers of vertices and pendant vertices

on theory and fast algorithms for error correction in residue number system product codes the authors develop a coding theory approach to error control in residue number system product codes based on this coding theory framework computationally efficient algorithms are derived for correcting single errors double errors and multiple errors and simultaneously detecting multiple errors and additive overflow these algorithms have lower computational complexity than previously known algorithms by at least an order of magnitude in addition it is noted that all the literature published thus far deals almost exclusively with single error correction

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

the ziegler spectrum for derived discrete algebras let be a derived discrete algebra we show that the krull gabriel dimension of the homotopy category of projective modules and therefore the cantor bendixson rank of its ziegler spectrum is 2 thus extending a result of bobinski and krause 8 we also describe all the indecomposable pure injective complexes and hence the ziegler spectrum for derived discrete algebras extending a result of z han 17 using this we are able to prove that all indecomposable complexes in the homotopy category of projective modules are pure injective so obtaining a class of algebras for which every indecomposable complex is pure injective but which are not derived pure semisimple

the least eigenvalue of a graph with cut vertices in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices during the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound

the least eigenvalue of the complements of trees let tnc be the set of the complements of trees of order n in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in tnc

the spectrum of prime ideals in tensor triangulated categories we define the spectrum of a tensor triangulated category k as the set of so called prime ideals endowed with a suitable topology in this very generality the spectrum is the universal space in which one can define supports for objects of k this construction is functorial with respect to all tensor triangulated functors several elementary properties of schemes hold for such spaces e g the existence of generic points or some quasi compactness locally trivial morphisms are proved to be nilpotent we establish in complete generality a classification of thick ideal subcategories in terms of arbitrary unions of closed subsets with quasi compact complements thomason s theorem for schemes mutatis mutandis we also equip this spectrum with a sheaf of rings turning it into a locally ringed space we compute examples and show that our spectrum unifies the schemes of algebraic geometry and the support varieties of modular representation theory

single residue error correction in residue number systems we present a new method to correct single errors in an n residue number system through the use of r redundant moduli the method requires 2n r 2 recombinations of n residues in the worst case this is of lower complexity than any other known method

on λ pure acyclic complexes in a grothendieck category abstract let a be a grothendieck category we prove that there is an infinite regular cardinal such that a has pure injective envelopes furthermore we show that any pure acyclic complex in a is a directed union of contractible subcomplexes this provides an answer to a question which is conjectured by simson in 14 conjecture 3 4

further results on the least eigenvalue of connected graphs in this paper we identify within connected graphs of order n and size n k with 0 k 4 and n k 5 the graphs whose least eigenvalue is minimal it is also observed that the same graphs have the largest spectral spread if n is large enough

the spectral radius of trees on k pendant vertices abstract in this paper we consider the following problem of all trees of order n with k pendant vertices n k fixed which achieves the maximal spectral radius we show that the maximal spectral radius is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex of k 1 k to an end vertex of one path we also discuss the spectral radius of t n k and get some results

a new residue arithmetic error correction scheme automatic detection and correction of errors in the residue number system involves the conversion of residue representations to integers and base extension the residue number system is generally restricted to moduli that are pairwise relatively prime in this paper we consider error detection and correction using a moduli set with common factors a method to construct a moduli set that leads to simplified error detection and correction is presented error detection can now be performed by computing residues in parallel error correction does not involve base extension any more it is also shown that removing all restrictions on the moduli set leads to more complex error detection correction algorithms

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

flat complexes pure periodicity and pure acyclic complexes abstract the paper can be viewed as an addition and extension of the recent paper of emmanouil 2016 5 among others an alternative functor category approach to emmanouil s results is presented by applying an idea of neeman 2008 16 and the main results obtained there we prove that a chain complex f in a locally finitely presented grothendieck category a is pure acyclic if and only if any chain map f p f from a complex p of pure projective objects in a to f is null homotopic as a consequence we prove that any pure periodic object in a is pure projective moreover we show that a is pure semisimple if and only if a has the pure qf property that is every pure injective object in a is pure projective

extended closed form expressions for the robust symmetrical number system dynamic range and an efficient algorithm for its computation the robust symmetrical number system rsns is a number theoretic transform based on n 2 sequences that can extract the maximum amount of information from symmetrical folding waveforms the sequences based on coprime moduli exhibit an integer gray code property making the rsns well suited for many applications that benefit from an inherent error detection and correction capability such as analog to digital converters direction finding arrays and radar waveform design to use the rsns it is necessary to know the greatest length of combined sequences without ambiguities called the dynamic range m for which only a few closed form expressions currently exist in this paper an efficient algorithm for computing m and its position within the combined set of sequences is presented and shown to be independent of the size of the moduli the algorithm is used to generate the equations for several groups of additional moduli arrangements closed form expressions for m are conjectured and proved using the obtained congruence equations that define the ambiguity locations

sharp lower bounds of the least eigenvalue of planar graphs let g be a simple graph with n 3 vertices and orientable genus g and non orientable genus h we define the euler characteristic g of a graph g by g max 2 2g 2 h let g be the least eigenvalue of the adjacency matrix a of g in this paper we obtain the following lower bounds of g g 2 n g in particular if g is the planar graph then g 2n 4 the equality holds if and only if g k2 n 2 further we have same result of series parallel graph

error correction in polynomial remainder codes with non pairwise coprime moduli and robust chinese remainder theorem for polynomials this paper investigates polynomial remainder codes with non pairwise coprime moduli we first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small namely the robust chinese remainder theorem crt for polynomials it basically says that a polynomial can be reconstructed from its erroneous residues such that the degree of the reconstruction error is upper bounded by tau whenever the degrees of all residue errors are upper bounded by tau where a sufficient condition for tau and a reconstruction algorithm are obtained by relaxing the constraint that all residue errors have small degrees another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues we finally obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result some errors with small degrees can be also corrected in the residues with this newly obtained result improvements in uncorrected error probability and burst error correction capability in data transmission are illustrated

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

high resolution phase sampled interferometry using symmetrical number systems this paper identifies a new phase sampling interferometer approach that can he easily incorporated into the established techniques to provide a high resolution small baseline array with a fewer number of phase sampling comparators the approach is based on preprocessing the received signal using symmetrical number systems sns antennas based on both an optimum symmetrical number system osns and a robust symmetrical number system rsns are investigated the sns preprocessing is used to decompose the spatial filtering operation into a number of parallel suboperations moduli that are of smaller computational complexity a much higher direction finding df spatial resolution is achieved after the n different moduli are used and the results of these low precision suboperations are recombined by incorporating the osns or rsns preprocessing concept the field of view of a specific configuration of interferometers and phase sampling comparator arrangements can be analyzed exactly the osns gives the maximum dynamic range or number of spatial resolution bins while the rsns reduces considerably the number of possible encoding errors experimental results for both a 5 bit osns and a 6 bit rsns array are compared the errors in the encoding of the direction of arrival are quantified for both architectures

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

on certain homological invariants of groups recently the notions of sfli the supremum of the flat lengths of injective modules and silf the supremum of the injective lengths of flat modules have been studied by some authors these homological invariants are based on spli and silp invariants of gedrich and gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups in this paper we will study these invariants it turns out that for any group the finiteness of silf implies the finiteness of sfli but the converse is not known we investigate the situation in which sfli implies silf the statement holds for example for groups with the property that flat modules have finite projective dimension moreover we show that the gorenstein flat dimension of the trivial z module z that will be called gorenstein homological dimension of denoted ghd is completely related to these invariants

homotopy equivalences induced by balanced pairs abstract we introduce the notion of balanced pair of additive subcategories in an abelian category we give sufficient conditions under which a balanced pair of subcategories gives rise to a triangle equivalence between two homotopy categories of complexes as an application we prove that for a left gorenstein ring there exists a triangle equivalence between the homotopy category of its gorenstein projective modules and the homotopy category of its gorenstein injective modules which restricts to a triangle equivalence between the homotopy category of projective modules and the homotopy category of injective modules in the case of commutative gorenstein rings we prove that up to a natural isomorphism our equivalence extends iyengar krause s equivalence

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

homological dimensions and regular rings abstract a question of avramov and foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings a key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules analogous results for flat dimension and projective dimension are also established

systematic redundant residue number system codes analytical upper bound and iterative decoding performance over awgn and rayleigh channels the novel family of redundant residue number system rrns codes is studied rrns codes constitute maximum minimum distance block codes exhibiting identical distance properties to reed solomon codes binary to rrns symbol mapping methods are proposed in order to implement both systematic and nonsystematic rrns codes furthermore the upper bound performance of systematic rrns codes is investigated when maximum likelihood ml soft decoding is invoked the classic chase algorithm achieving near ml soft decoding is introduced for the first time for rrns codes in order to decrease the complexity of the ml soft decoding furthermore the modified chase algorithm is employed to accept soft inputs as well as to provide soft outputs assisting in the turbo decoding of rrns codes by using the soft input soft output chase algorithm

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

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

implementation issues of the two level residue number system with pairs of conjugate moduli one of the most important considerations when designing residue number systems rnss is the choice of the moduli set this is due to the fact that the dynamic range of the system its speed as well as its hardware complexity depend on both the forms as well as the number of moduli chosen in this paper a new class of multimoduli rnss based on sets of forms 2 sup n 1 1 2 sup n 1 1 2 sup n2 1 2 sup n 2 1 spl middot spl middot spl middot 2 sup n l 1 2 sup n l 1 is presented the moduli 2 sup n i 1 and 2 sup n i 1 are called conjugates of each other the new rnss that rely on pairs of conjugate moduli result in hardware efficient two level implementations for the weighted to rns and rns to weighted conversions achieve very large dynamic ranges and imply fast and efficient rns processing when compared with conventional systems of the same number of moduli and the same dynamic range the proposed new systems offer the following benefits 1 hardware savings of 25 to 40 for the weighted to rns conversion and 2 a reduction of over 80 in the complexity of the final chinese remainder theorem crt involved in the rns to weighted conversion thus significant compromises between large dynamic ranges fast internal processing and low complexity are achieved by the new systems

a non iterative multiple residue digit error detection and correction algorithm in rrns error detection and correction code based on redundant residue number system rrns has a unique advantage that arithmetical processing errors can also be corrected existing algorithms for multiple residue digit error correction in rrns require either large modulo operations to decode the magnitude of the residue digits or variable number of iterative computations to identify the locations of erroneous residue digits this paper presents an efficient syndrome based multiple residue digit error detection and correction algorithm in rrns the received residue digits are divided into three groups from which seven error location categories are defined for all combinations of residue digit errors of any legitimate moduli set their error magnitudes are abstracted into three syndromes which are used to identify the exact error location category and retrieve the residue digit errors concurrently from six lookup tables in no more than three cycles the syndromes can be computed in parallel by small modulo subtractors and their uniqueness criteria are proved to be easily fulfilled the redundant residue space due to the uniqueness criteria can also be relaxed by increasing the number of syndrome computations and lookup tables used for the error decoding according to the number of information moduli

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

design of residue generators and multioperand modular adders using carry save adders residue generator is an essential building block of encoding decoding circuitry for arithmetic error detecting codes and binary to residue number system rns converter in either case a residue generator is an overhead for a system and as such it should be built with minimum amount of hardware and should not compromise the speed of a system multioperand modular adder moma is a computational element used to implement various operations in digital signal processing systems using rns a comprehensive study of new residue generators and moma s is presented the design methods given here take advantage of the periodicity of the series of powers of 2 taken module a a is a module four design schemes of the n input residue generators mod a which are best suited for various pairs of n and a are proposed their pipelined versions can be clocked with the cycle determined by the delay of a full adder and a latch a family of design methods for parallel and word serial using similar concepts is also given both classes of circuits employ new highly parallel schemes using carry save adders with end around carry and a minimal amount of rom and are well suited for vlsi implementation they are faster and use less hardware than similar circuits known to date one of the moma s can be used to build a high speed residue to binary converter based on the chinese remainder theorem

the least eigenvalue of unicyclic graphs with n vertices and k pendant vertices abstract let u n k be the set of unicyclic graphs with n vertices and k pendant vertices in this paper we determine the unique graph with the minimal least eigenvalue among all graphs in u n k the work is related with that of guo s g guo the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices linear algebra appl 408 2005 78 85 which determined the unicyclic graph with the maximal spectral radius in u n k we can observe that the extremal graph on the least eigenvalue is different from that on the spectral radius

the least eigenvalue of graphs with given connectivity abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of g are those of a g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity

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

minimizing the least eigenvalues of unicyclic graphs with application to spectral spread the spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph let u k denote the set of connected unicyclic graphs of order n and girth k and let un denote the set of connected unicyclic graphs of order n in this paper we determine the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in u k we finally characterize the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in un

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

fault tolerant convolution via chinese remainder codes constructed from non coprime moduli this paper develops a framework for performing fault tolerant convolution via error correcting codes based on the chinese remainder theorem crt with non coprime moduli in contrast to convolution that is protected through crt codes with coprime moduli our scheme allows errors to be detected and located in a highly parallel manner and has the advantage of being robust to faults that occur during the error handling stage in addition for certain important classes of errors the codes developed in this paper require less redundancy than crt codes constructed from coprime moduli we demonstrate the applicability and advantages of our codes by comparing them to both crt codes with coprime moduli and repetition codes we focus on codes over polynomial rings but our constructions apply naturally to integer codes we discuss this extension and show that it generalizes the distance properties and error correction strategies associated with previously developed approaches for codes based on non coprime integer moduli

on purely generated α smashing weight structures and weight exact localizations abstract this paper is dedicated to new methods of constructing weight structures and weight exact localizations our arguments generalize their bounded versions considered in previous papers of the authors we start from a class of objects p of a triangulated category c that satisfies a certain countable negativity condition there are no c extensions of positive degrees between elements of p we actually need a somewhat stronger condition of this sort to obtain a weight structure both halves of which are closed either with respect to c coproducts of less than objects where is a fixed regular cardinal or with respect to all coproducts provided that c is closed with respect to coproducts of this sort this construction gives all reasonable weight structures satisfying the latter conditions in particular one can obtain certain weight structures on spectra in sh consisting of less than cells and on certain localizations of sh these results are new

use of the symmetrical number system in resolving single frequency undersampling aliases the relationship between the discrete fourier transform dft and the symmetrical number system sns is examined as a means of resolving single frequency undersampling aliases f f sub s 2 it is shown that the dft naturally encodes the frequency information of a signal in a format that is in the same form as the sns consequently aliases that result from undersampling a signal can be resolved exactly using r spl ges 2 channels to demonstrate the concept two and three channel examples are presented using the sns properties of the dft the undersampling aliases can be easily resolved to a much greater extent than previously possible

exact model structures and recollements abstract we show how to obtain recollements of triangulated categories using the theory of exact model structures from 13 after noting how the theory relates to well known notions in the simplest case of frobenius categories we apply these ideas to recollements of homotopy categories of chain complexes in short we give model categorical explanations for the classical verdier localization recollement as well as several recollement situations due to neeman and murfet

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

optimal estimates of two common remainders for a robust generalized chinese remainder theorem estimation of multiple common remainders from a sequence of erroneous residue sets is an important step for the robust generalized chinese remainder theorem crt this paper considers the problem of how to estimate the two common remainders from their residue sets modulo a set of moduli to measure the errors properly under the modular operation we introduce two type circular distances based on these circular distances two estimation methods are proposed by properly grouping the erroneous remainders into two ordered clusters both of the two methods perform better with lower computational complexities than the existing methods in this paper theoretical analysis as well as analytical results for the two proposed methods are obtained for the first method the two optimal estimates are proved to be in a finite set with no more than l candidates where l is the number of the given moduli the second method have closed forms simulation results show that the two proposed methods have nearly the same performance these optimal estimates can improve the performance of the robust generalized crt significantly

bicyclic graphs for which the least eigenvalue is minimum abstract the spread of a graph is defined to be the difference between the greatest eigenvalue and the least eigenvalue of the adjacency matrix of the graph in this paper we determine the unique graph with minimum least eigenvalue among all connected bicyclic graphs of order n also we determine the unique graph with maximum spread in this class for each n 28

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

on the least eigenvalue of cacti among the cacti with n vertices and k cycles we determine a unique cactus whose least eigenvalue is minimal we also explore cacti with n vertices and among them we find a unique cactus whose least eigenvalue is minimal

flat covers of complexes abstract in this article we define and study flat complexes over any ring also we prove that any complex over a commutative noetherian ring with finite krull dimension has a flat cover and a dg flat cover

graphs for which the least eigenvalue is minimal ii let g be a connected graph whose least eigenvalue g is minimal among the connected graphs of prescribed order and size we show first that either g is complete or g is a simple eigenvalue in the latter case the sign pattern of a corresponding eigenvector determines a partition of the vertex set and we study the structure of g in terms of this partition we find that g is either bipartite or the join of two graphs of a simple form

n sequence rsns ambiguity analysis the robust symmetrical number system rsns is a modular system formed using nges2 integer sequences and ensures that two successive rsns vectors paired terms from all n sequences differ by only one integer this integer gray code property reduces the possibility of encoding errors and makes the rsns useful in applications such as folding analog to digital converters adcs direction finding antenna architectures and photonic processors this paper determines the length of combined sequences that contain no vector ambiguities this length or longest run of distinct vectors we call the rsns dynamic range mcirc the position of mcirc which is the starting point in the sequence is also derived computing mcirc and the position of mcirc allows the integer gray code properties of the rsns to be used in practical applications we first extend our two sequence results to develop a closed form expression for mcirc for a three sequence rsns with moduli of the form 2r 1 2r 2r 1 we then extend the results to solving the n sequence rsns ambiguity locations in general

approximations and adjoints in homotopy categories we provide a criterion for the existence of right approximations in cocomplete additive categories it is a straightforward generalisation of a result due to el bashir this criterion is used to construct adjoint functors in homotopy categories applications include the study of pure derived categories for instance it is shown that the pure derived category of any module category is compactly generated

the vertex edge independence number vertex edge cover number and the least eigenvalue of a graph in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex edge independence number or vertex edge cover number and get some bounds for the vertex edge independence number vertex edge cover number of a graph in terms of the least eigenvalue of the graph

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

on the homotopy categories of projective and injective representations of quivers abstract let r be a ring and q be a quiver we study the homotopy categories k prj q and k inj q consisting respectively of projective and injective representations of q by r modules we show that for certain quivers these triangulated categories are compactly generated and provide explicit descriptions of compact generating sets moreover in case r is commutative and noetherian with a dualizing complex d the dualizing functor d r k prj r k inj r is extended to a triangulated functor k prj q k inj op q which is an equivalence of triangulated categories this functor establishes an equivalence on k prj q and k inj q whenever q is finite

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

chinese remaindering with errors the chinese remainder theorem states that a positive integer m is uniquely specified by its remainder module k relatively prime integers p sub 1 spl middot spl middot spl middot p sub k provided m spl pi sub i 1 sup k p sub i thus the residues of m module relatively prime integers p sub 1 p sub 2 spl middot spl middot spl middot p sub n form a redundant representation of m if m spl pi sub i 1 sup k p sub i and k n this gives a number theoretic construction of an error correcting code that has been considered often in the past in this code a message integer m spl pi sub i 1 sup k p sub i is encoded by the list of its residues module p sub 1 spl middot spl middot spl middot p sub n by the chinese remainder theorem if a codeword is corrupted in e n k 2 coordinates then there exists a unique integer m whose corresponding codesword differs from the corrupted word in at most e places furthermore mandelbaum 1976 1978 shows how m can be recovered efficiently given the corrupted word provided that the p sub i s are very close to one another to deal with arbitrary p sub i s we present a variant of his algorithm that runs in almost linear time and recovers from e log p sub 1 log p sub 1 log p sub n spl middot n k errors our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than n k 2 specifically given n residues r sub 1 spl middot spl middot spl middot r sub n and an agreement parameter t we find a list of all integers m spl pi sub i 1 sup k p sub i such that m mod p sub i r sub i for at least t values of i spl isin 1 spl middot spl middot spl middot n provided t spl omega spl radic kn log p sub n log p sub 1 for n spl gt k and p sub n spl les p sub 1 sup o 1 the fraction of error corrected by the algorithm is almost twice that corrected by the previous work more significantly the algorithm recovers the message even when the amount of agreement between the received word and the codeword is much smaller than the number of errors

error detection and correction in communication channels using inverse gray rsns codes a novel number theoretic transform called inverse gray robust symmetrical number system igrsns is proposed for error control coding in this paper igrsns is obtained by modifying robust symmetrical number system rsns that was proposed earlier using inverse gray code property due to ambiguities present in each residue rsns has a short dynamic range dr compared to that in other number systems the short dr of rsns enables it to be effectively used for error detection without the addition of any redundant modulus as in redundant residue number system although rsns has a large redundant range its detection ability is not optimal due to the gray code property associated with it in an attempt to overcome this limitation we have proposed inverse gray coding to be combined with rsns in increasing its effectiveness in error detection the resulting igrsns thus inherits properties of both rsns and gray code analysis and simulations show that igrsns has a near optimal error detection ability the potential of igrsns for error correction is also investigated further a novel error correction algorithm using one redundant modulus is proposed simulations show that the proposed algorithm performs well under all cases of single bit errors

multiple error detection and correction based on redundant residue number systems this paper presents some results on multiple error detection and correction based on the redundant residue number system rrns rrns is often used in parallel processing environments because of its ability to increase the robustness of information passing between the processors the proposed multiple error correction scheme utilizes the chinese remainder theorem crt together with a novel algorithm that significantly simplifies the error correcting process for integers an extension of the scheme further reduces the computational complexity without compromising its error correcting capability proofs and examples are provided for the coding technique

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

a surprising property of the least eigenvalue of a graph abstract let g be the least eigenvalue of a graph g a real number r has the induced subgraph property provided g r implies g has an induced subgraph h with h r it is shown that the only numbers with the induced subgraph property are 0 1 2 and 2

presheaves of triangulated categories and reconstruction of schemes to any triangulated category with tensor product k otimes we associate a topological space rm spc k otimes by means of thick subcategories of k a la hopkins neeman thomason moreover to each open subset u of this space spc k otimes we associate a triangulated category cal k u producing what could be thought of as a presheaf of triangulated categories applying this to the derived category k otimes rm d rm perf x otimes l of perfect complexes on a noetherian scheme x the topological space rm spc k otimes turns out to be the underlying topological space of x moreover for each open u subset x the category cal k u is naturally equivalent to rm d rm perf u as an application we give a method to reconstruct any reduced noetherian scheme x from its derived category of perfect complexes rm d rm perf x considering the latter as a tensor triangulated category with otimes l

on exact categories and applications to triangulated adjoints and model structures abstract we show that quillen s small object argument works for exact categories under very mild conditions this has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures in particular the interplay of different exact structures on the category of complexes of quasi coherent sheaves leads to a streamlined and generalized version of recent results obtained by estrada gillespie guil asensio hovey jorgensen neeman murfet prest trlifaj and possibly others

koszul duality for lie algebroids abstract this paper studies the role of dg lie algebroids in derived deformation theory more precisely we provide an equivalence between the homotopy theories of formal moduli problems and dg lie algebroids over a commutative dg algebra of characteristic zero at the level of linear objects we show that the category of representations of a dg lie algebroid is an extension of the category of quasi coherent sheaves on the corresponding formal moduli problem we describe this extension geometrically in terms of pro coherent sheaves

towards robustness in residue number systems the problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli namely the robust remaindering problem may occur in many applications including phase unwrapping frequency detection from several undersampled waveforms wireless sensor networks etc assuming that the dynamic range of the large number is the maximal possible one i e the least common multiple lcm of all the moduli a method called robust chinese remainder theorem for solving the robust remaindering problem has been recently proposed in this paper by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli a tradeoff between the dynamic range and the robustness bound for two modular systems is studied it basically says that a decrease in the dynamic range may lead to an increase of the robustness bound we first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed form formula for the robustness to hold we then propose simple closed form reconstruction algorithms furthermore the newly obtained two modular results are applied to the robust reconstruction for multimodular systems and generalized to real numbers finally some simulations are carried out to verify our proposed theoretical results

fast fault tolerant digital convolution using a polynomial residue number system a fault tolerant convolution algorithm that is an extension of residue number system fault tolerance schemes applied to polynomial rings is described the algorithm is suitable for implementation on multiprocessor systems and is able to concurrently mask processor failures a fast algorithm based on long division for detecting and correcting multiple processor failures is presented moduli polynomials that yield an efficient and robust fast fourier transform fft based algorithm are selected for this implementation a single fault detection and correction is studied and a generalized likelihood ratio test is applied to optimally detect system failures in the presence of computational noise the coding scheme is capable of protecting over 90 of the computation involved in convolution parts not covered by the scheme are assumed to be protected via triple modular redundancy this hybrid approach can detect and correct any single system failure with as little as 70 overhead compared with 200 needed for a system fully protected via modular redundancy