graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

compact operators into the spaces of strongly c1 summable and bounded sequences abstract we establish the necessary and sufficient conditions for the entries of an infinite matrix to map any of the classical sequence spaces l p 1 p c 0 and c into the spaces w 0 w and w of all sequences that are strongly summable to zero strongly summable and strongly bounded by the cesaro method of order 1 we also give the representations of the general bounded linear operators from c into any of the spaces w 0 w and w and compute or estimate the hausdorff measure of noncompactness in each case finally we apply our results to characterise the compact linear operators between the spaces mentioned above and generalise two classical results by steinhaus 1911 5 maddox 1967 31 and cohen and dunford 1937 24

covering problems with hard capacities we consider the classical vertex cover and set cover problems with hard capacity constraints this means that a set vertex can cover only a limited number of its elements adjacent edges and the number of available copies of each set vertex is bounded this is a natural generalization of the classical problems which also captures resource limitations in practical scenarios we obtain the following results for the unweighted vertex cover problem with hard capacities we give a 3 approximation algorithm that is based on randomized rounding with alterations we prove that the weighted version is at least as hard as the set cover problem yielding an interesting separation between the approximability of weighted and unweighted versions of a natural graph problem a logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of wolsey combinatorica 2 1982 pp 385 393 on submodular set cover we provide here a simple and intuitive proof for this bound

tight approximation for partial vertex cover with hard capacities abstract we consider the partial vertex cover problem with hard capacity constraints partial vc hc on hypergraphs in this problem we are given a hypergraph g v e with a maximum edge size f and a covering requirement r each edge is associated with a demand and each vertex is associated with a capacity and an integral available multiplicity the objective is to compute a minimum vertex multiset such that at least r units of demand from the edges are covered by the capacities of the vertices in the multiset and the multiplicity of each vertex does not exceed its available multiplicity in this paper we present an f approximation for this problem improving over a previous result of 2 f 2 1 by cheung et al to the tight extent possible our new ingredient of this work is a generalized analysis on the extreme points of the natural lp developed from previous works and a strengthened lp lower bound obtained for the optimal solutions

compactness by the hausdorff measure of noncompactness abstract in the present paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary b k space into the sequence spaces c 0 c l and l 1 and into the matrix domains of triangles in these spaces furthermore by using the hausdorff measure of noncompactness we apply our results to characterize some classes of compact operators on the b k spaces

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

approximation algorithms for the partition vertex cover problem we consider a natural generalization of the partial vertex cover problem here an instance consists of a graph g v e a cost function c v z a partition p 1 p r of the edge set e and a parameter k i for each partition p i the objective is to find a minimum cost set of vertices which cover at least k i edges from the partition p i we call this the partition vc problem in this paper we give matching upper and lower bound on the approximability of this problem our algorithm is based on a novel lp relaxation for this problem this lp relaxation is obtained by adding knapsack cover inequalities to a natural lp relaxation of the problem we show that this lp has integrality gap of o log r where r is the number of sets in the partition of the edge set we also extend our result to more general settings for example we consider a problem where additionally edges have profits and we would like to pick a minimum cost set of vertices which cover edges of total profit at least i for each partition p i we call this the knapsack partition vertex cover problem we give an o log r approximation algorithm for this problem as well

on compact operators and some euler b m difference sequence spaces abstract altay and basar 2005 1 and altay basar and mursaleen 2006 2 introduced the euler sequence spaces e 0 t e c t and e t basarir and kayikci 2009 3 defined the b m difference matrix and studied some topological and geometric properties of some generalized riesz b m difference sequence space in this paper we introduce the euler b m difference sequence spaces e 0 t b m e c t b m and e t b m consisting of all sequences whose b m transforms are in the euler spaces e 0 t e c t and e t respectively moreover we determine the and duals of these spaces and construct the schauder basis of the spaces e 0 t b m and e c t b m finally we characterize some matrix classes concerning the spaces e 0 t b m and e c t b m and give the characterization of some classes of compact operators on the sequence spaces e 0 t b m and e t b m

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

reduction techniques for strongly graded rings and finite representation type abstract we present reduction techniques for studying the category of lattices over strongly graded orders in particular we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

automorphisms of tiled orders abstract let be a tiled r order we give a description of autr as the semidirect product of inn and a certain subgroup of aut q where q is the link graph of additionally we give criteria for determining when an element of aut q belongs to this subgroup in terms of the exponent matrix for

compactness of matrix operators on some new difference sequence spaces abstract the difference sequence spaces c 0 and l have recently been introduced by mursaleen and noman 2010 m mursaleen a k noman on some new difference sequence spaces of non absolute type math comput modelling 52 3 4 2010 603 617 in the present paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain matrix operators on the spaces c 0 and l further by using the hausdorff measure of noncompactness we characterize some classes of compact operators on these spaces

crossed product algebras defined by separable extensions abstract we generalize the classical construction of crossed product algebras defined by finite galois field extensions to finite separable field extensions by studying properties of rings graded by groupoids we are able to calculate the jacobson radical of these algebras we use this to determine when the analogous construction of crossed product orders yield azumaya maximal or hereditary orders in a local situation thereby we generalize results by haile larson and sweedler

neat idempotents and tiled orders having large global dimension abstract we study neat primitive idempotents in a semiperfect noetherian ring and as an application we improve an example of a tiled order having large global dimension given by jansen and odenthal moreover another two tiled orders having large global dimension are added and two questions on tiled orders of finite global dimension are posed

improved approximation algorithms for the vertex cover problem in graphs and hypergraphs we obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs we use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations on graphs with maximum degree at most delta the algorithm achieves a performance ratio of 2 1 o 1 frac 2 ln ln delta ln delta for large delta which improves the previously known ratio of 2 frac log delta o 1 delta obtained by halld orsson and radhakrishnan using similar techniques we also present improved approximations for the vertex cover problem in hypergraphs for k uniform hypergraphs with n vertices we achieve a ratio of k 1 o 1 frac k ln ln n ln n for large n and for k uniform hypergraphs with maximum degree at most delta the algorithm achieves a ratio of k 1 o 1 frac k k 1 ln ln delta ln delta for large delta these results considerably improve the previous best ratio of k 1 c delta frac 1 k 1 for bounded degree k uniform hypergraphs and k 1 c n frac k 1 k for general k uniform hypergraphs both obtained by krivelevich using similar techniques we also obtain an approximation algorithm for the weighted independent set problem matching a recent result of halldorsson

on the euler sequence spaces which include the spaces lp and l ii abstract by e p r we mean the space of all sequences whose euler transforms of order r are in the sequence spaces l p and l see b altay f basar m mursaleen on the euler sequence spaces which include the spaces l p and l i inform sci 2005 in press where 1 p in the present paper we essentially characterize the classes e p r l e 1 r l p and e p r f of infinite matrices for 1 p and give the characterizations of some other matrix mappings from the space e p r to the euler riesz difference etc sequence spaces by means of a given basic lemma we devote the final section of the paper to examining some geometric properties of the space e p r

erratum to on the euler sequence spaces which include the spaces lp and l ii nonlinear anal tma 65 3 2006 707 717

on generalized means and some related sequence spaces in the present paper we introduce the notion of generalized means and define some related sequence spaces which include the most known sequence spaces as special cases further we study some topological properties of the spaces of generalized means and construct their bases finally we determine the b duals of these spaces and characterize some related matrix classes

approximation algorithms for data placement problems we develop approximation algorithms for the problem of placing replicated data in arbitrary networks where the nodes may both issue requests for data objects and have capacity for storing data objects so as to minimize the average data access cost we introduce the data placement problem to model this problem we have a set of caches mathcal f a set of clients mathcal d and a set of data objects mathcal o each cache i can store at most u i data objects each client j in mathcal d has demand d j for a specific data object o j in mathcal o and has to be assigned to a cache that stores that object storing an object o in cache i incurs a storage cost of f i o and assigning client j to cache i incurs an access cost of d jc ij the goal is to find a placement of the data objects to caches respecting the capacity constraints and an assignment of clients to caches so as to minimize the total storage and client access costs we present a 10 approximation algorithm for this problem our algorithm is based on rounding an optimal solution to a natural linear programming relaxation of the problem one of the main technical challenges encountered during rounding is to preserve the cache capacities while incurring only a constant factor increase in the solution cost we also introduce the connected data placement problem to capture settings where write requests are also issued for data objects so that one requires a mechanism to maintain consistency of data we model this by requiring that all caches containing a given object be connected by a steiner tree to a root for that object which issues a multicast message upon a write to any copy of that object the total cost now includes the cost of these steiner trees we devise a 14 approximation algorithm for this problem we show that our algorithms can be adapted to handle two variants of the problem a a k median variant where there is a specified bound on the number of caches that may contain a given object and b a generalization where objects have lengths and the total length of the objects stored in any cache must not exceed its capacity

matrix transformations between sequence spaces of generalized weighted means the main object of this paper is to introduce and study some sequence spaces which arise from concept of weighted means determine their b duals and characterize matrix transformations between them

a ptas for the cardinality constrained covering with unit balls in this paper we address the cardinality constrained covering with unit balls problem given a positive integer l and a set of n points in r d partition them into a minimum number of parts such that each part contains at most l points and it can be covered by a unit ball of the given p metric developing a constant factor approximation algorithm for this problem is an old open problem by proving a structural property in the problem and applying the shifting strategy and dynamic programming we derive the first email protected d approximation n o 1 e d time algorithm for this problem when d is a fixed constant

primal dual schema for capacitated covering problems primal dual algorithms have played an integral role in recent developments in approximation algorithms and yet there has been little work on these algorithms in the context of lp relaxations that have been strengthened by the addition of more sophisticated valid inequalities we introduce primal dual schema based on the lp relaxations devised by carr et al for the minimum knapsack problem as well as for the single demand capacitated facility location problem our primal dual algorithms achieve the same performance guarantees as the lp rounding algorithms of carr et al which rely on applying the ellipsoid algorithm to an exponentially sized lp furthermore we introduce new flow cover inequalities to strengthen the lp relaxation of the more general capacitated single item lot sizing problem using just these inequalities as the lp relaxation we obtain a primal dual algorithm that achieves a performance guarantee of 2 computational experiments demonstrate the effectiveness of this algorithm on generated problem instances

the group of automorphisms of certain subalgebras of matrix algebras the purpose of this note is to find and characterize the full group of automorphisms of certain prime algebras which are subalgebras of matrix algebras over commutative rings here we extend the methods which we used in 6 to characterize automorphisms of certain upper triangular matrix rings for a central simple k algebra a the group of k automorphisms of a is extensively treated in the literature 5 theorem 6 21 other cases where the automorphism group of a ring is known are the automorphism group of a polynomial ring in two indeterminates over a field and the free algebra in two variables over a field 4 chap 61 finally if a is a prime p z algebra with center c then every c automorphism of a is x inner 7 example 3 91

on the euler sequence spaces which include the spaces l p and l i in the present paper we introduce the euler sequence space e p r consisting of all sequences whose euler transforms of order r are in the space p of non absolute type which is the bk space including the space p and prove that the spaces e p r and p are linearly isomorphic for 1

on the fine spectrum of the generalized difference operator b r s over the sequence spaces lp and bvp 1 p abstract the main purpose of this paper is to determine the fine spectrum of the operator b r s defined by a band matrix over the sequence spaces l p and b v p with 1 p since the operator b r s is reduced to the right shift difference and zweier matrices in the special cases r s 0 1 r s 1 1 and r s r 1 r respectively our results are more general than the corresponding results in the existing literature

a distributed 2 e approximation for vertex cover in o log δ e log log δ rounds we present a simple deterministic distributed 2 e approximation algorithm for minimum weight vertex cover which completes in o log elog log rounds where is the maximum degree in the graph for any e 0 that is at most o 1 for a constant e this implies a constant approximation in o log log log rounds which contradicts the lower bound of kmw10

epsilon strongly graded rings separability and semisimplicity abstract we introduce the class of epsilon strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products we determine precisely when an epsilon strongly graded ring is separable over its principal component thereby we simultaneously generalize a result for strongly group graded rings by n st sescu van den bergh and van oystaeyen and a result for unital partial crossed products by bagio lazzarin and paques we also show that the class of unital partial crossed products appears in the class of epsilon strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings thereby we obtain in the special case of unital partial crossed products a short proof of a general result by dokuchaev exel and simon concerning when graded rings can be presented as partial crossed products we also provide some interesting classes of examples of separable epsilon strongly graded rings with finite as well as infinite grading groups in particular we obtain an answer to a question raised by le bruyn van den bergh and van oystaeyen in 1988

generalized spaces of difference sequences abstract let l c and c 0 be the banach spaces of bounded convergent and null sequences x x k 1 respectively write x x k x k 1 1 and 2 x x k x k 1 1 in canad math bull 24 1981 169 176 h kizmaz has introduced and studied the sequence spaces e x x e where e c 0 c l recently doga mat 17 1993 18 24 mikail et defined the sets e 2 x 2 x e he obtained duals of these sets and characterized the matrix class e f 2 where e f c 0 c l in this paper we generalize these sets and define e u 2 x u 2 x e where u u k is another sequence such that u k 0 k 1 2 we obtain and duals of these sets and further we characterize the matrix classes e u 2 f and e f u 2

a primal dual bicriteria distributed algorithm for capacitated vertex cover in this paper we consider the capacitated vertex cover problem which is the variant of vertex cover where each node is allowed to cover a limited number of edges we present an efficient deterministic distributed approximation algorithm for the problem our algorithm computes a 2 epsilon approximate solution which violates the capacity constraints by a factor of 4 epsilon in a polylogarithmic number of communication rounds on the other hand we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints our result is achieved in two steps we first develop a 2 approximate sequential primal dual algorithm that violates the capacity constraints by a factor of 2 subsequently we present a distributed version of this algorithm we demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds the challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

applications of measure of noncompactness in operators on the spaces sα s0α s c α lpα abstract in this paper we characterize some operators and matrix transformations in the sequence spaces s s 0 s c l p moreover using the hausdorff measure of noncompactness necessary and sufficient conditions are formulated for a linear operator between the mentioned spaces to be compact among other things some results of cohen and dunford are recovered

matrix transformations on some sequence spaces related to strong cesaro summability and boundedness the spaces a 0 r d a c r d and a r d introduced by aydin and basar c aydin f basar some new difference sequence spaces appl math comput 157 3 2004 677 693 can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero summable and bounded by the cesaro method of order 1 here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable summable to zero or bounded by the strong cesaro method of order 1 with index p 1 we determine the b duals of the new spaces and characterize matrix transformations on them into the sets of bounded convergent and null sequences

matrix transformations and compact operators on some new mth order difference sequences abstract we define some new sets of sequences the m th order differences of which are bounded convergent and convergent to zero and apply the general methods in e malkowsky v rakocevic on matrix domains of triangles appl math comput 189 2 2007 1146 1163 to give schauder bases for the latter two determine their duals and characterize matrix transformations on them our results generalize those in b de malafosse the banach algebra s and applications acta sci math szeged 70 1 2 2004 125 145 and improve those in e malkowsky s d parashar matrix transformations in spaces of bounded and convergent difference sequences of order m analysis 17 1997 87 97 we also establish identities and estimates for the hausdorff measure of non compactness of matrix operators from our spaces into the spaces of bounded convergent and null sequences and characterize the respective classes of compact operators some of these results generalize those in e malkowsky v rakocevic the measure of non compactness of linear operators between spaces of m th order difference sequences stud sci math hungar 33 1999 381 391

dynamic programming based algorithms for set multicover and multiset multicover problems given a universe n containing n elements and a collection of multisets or sets over n the multiset multicover msmc problem or the set multicover smc problem is to cover all elements at least a number of times as specified in their coverage requirements with the minimum number of multisets or sets in this paper we give various exact algorithms for these two problems with or without constraints on the number of times a multiset or set may be chosen first we show that the msmc without multiplicity constraints problem can be solved in o b 1 n f time and polynomial space where b is the maximum coverage requirement and f denotes the total number of given multisets over n the o notation suppresses a factor polynomial in n to our knowledge this is the first known exact algorithm for the msmc without multiplicity constraints problem second by combining dynamic programming and the inclusion exclusion principle we can exactly solve the smc without multiplicity constraints problem in o b 2 n time compared with two recent results in q s hua y wang d yu f c m lau set multi covering via inclusion exclusion theoretical computer science 410 38 40 2009 3882 3892 and j nederlof inclusion exclusion for hard problems master thesis utrecht university the netherlands 2008 respectively ours is the fastest exact algorithm for the smc without multiplicity constraints problem finally by directly using dynamic programming we give the first known exact algorithm for the msmc or the smc with multiplicity constraints problem in o b 1 n f time and o b 1 n space this algorithm can also be easily adapted as a constructive algorithm for the msmc without multiplicity constraints problem

hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means abstract in the present paper we derive some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain matrix operators on the sequence spaces of generalized means further by applying the hausdorff measure of noncompactness we obtain the necessary and sufficient conditions for such operators to be compact

measure of noncompactness of matrix operators on some difference sequence spaces of weighted means for a sequence x x k we denote the difference sequence by dx x k x k 1 let u u k k 0 and v v k k 0 be the sequences of real numbers such that u k 0 v k 0 for all email protected n the difference sequence spaces of weighted means l u v d are defined as l u v d x x k w x email protected where l c c 0 and and the matrix w w n k is defined by w n k u n v k v k 1 kn in this paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain matrix operators on l u v d further we characterize some classes of compact operators on these spaces by using the hausdorff measure of noncompactness

picard groups and infinite matrix rings we describe a connection between the picard group of a ring with local units t and the picard group of the unital overring end tt using this connection we show that the three groups pic r pic fm r and pic rfm r are isomorphic for any unital ring r furthermore eaclh ele ment of pic rfm r arises from an automorphism of rfm r which yields an isomorphsm between pic rfm r and out rfm r as one applica tion we extend a classical result of rosenberg and zelinsky by showing that the group outr rfm r is abelian for any commutative unital ring r

applications of measures of noncompactness to the infinite system of differential equations in lp spaces abstract in this paper we apply the technique of measures of noncompactness to the theory of infinite system of differential equations in the banach sequence spaces l p 1 p our aim is to present some existence results for infinite system of differential equations formulated with the help of measures of noncompactness

banach algebras of matrix transformations between some sequence spaces related to λ strong convergence and boundedness we study the spaces c 0 l c l and c l of sequences that are l strongly convergent to zero l strongly convergent and l strongly bounded and the related spaces v 0 l and v l in particular we give the characterizations of several classes of matrix transformations between those spaces furthermore we use our results to prove that the classes of matrix transformations from v l c l and c l into themselves are banach algebras as an application of our results we establish an estimate for the hausdorff measure of noncompactness of matrix operators that map c l into itself give a characterization of the subclass of compact operators and a sufficient condition for those operators to be fredholm

on matrix domains of triangles we prove some general results for the determination of the duals of and the characterisations of matrix transformations on matrix domains of arbitrary triangles in fk spaces our results contain almost all recently published ones as special cases we also study the measure of noncompactness of several matrix transformations in particular we obtain some known results of sember give the corrected version of a recent result by djolovic and study compact operators on some spaces of altay basar and mursaleen

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent

maximizing a monotone submodular function subject to a matroid constraint let f 2 x rightarrow cal r be a monotone submodular set function and let x cal i be a matroid we consider the problem rm max s in cal i f s it is known that the greedy algorithm yields a 1 2 approximation m l fisher g l nemhauser and l a wolsey math programming stud no 8 1978 pp 73 87 for this problem for certain special cases e g rm max s leq k f s the greedy algorithm yields a 1 1 e approximation it is known that this is optimal both in the value oracle model where the only access to f is through a black box returning f s for a given set s g l nemhauser and l a wolsey math oper res 3 1978 pp 177 188 and for explicitly posed instances assuming p neq np u feige j acm 45 1998 pp 634 652 in this paper we provide a randomized 1 1 e approximation for any monotone submodular function and an arbitrary matroid the algorithm works in the value oracle model our main tools are a variant of the pipage rounding technique of ageev and sviridenko j combin optim 8 2004 pp 307 328 and a continuous greedy process that may be of independent interest as a special case our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model j vondrak proceedings of the 38 th acm symposium on theory of computing 2008 pp 67 74 as a second application we show that the generalized assignment problem gap is also a special case although the reduction requires x to be exponential in the original problem size we are able to achieve a 1 1 e o 1 approximation for gap simplifying previously known algorithms additionally the reduction enables us to obtain approximation algorithms for variants of gap with more general constraints