on the eisenstein cohomology of arithmetic groups the cohomology h gamma e of an arithmetic subgroup gamma of a connected reductive algebraic group g defined over some algebraic number field f can be interpreted in terms of the automorphic spectrum of gamma with this framework in place there is a sum decomposition of the cohomology into the cuspidal cohomology i e classes represented by cuspidal automorphic forms for g and the so called eisenstein cohomology constructed as the span of appropriate residues or derivatives of eisenstein series attached to cuspidal automorphic forms on the levi components of proper parabolic f subgroups of g the main objective of this paper is to isolate a specific structural part in the eisenstein cohomology this pertains to regular eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered it is shown that the cohomological degree of these classes is bounded from below by the constant q 0 g mathbb r 1 2 dim x g mathbb r rk g mathbb r rk k mathbb r where k mathbb r denotes a maximal compact subgroup of the real lie group g mathbb r where x g mathbb r is the associated symmetric space this investigation has various applications one of these is a vanishing result for the cohomology in the generic case i e where the representation determining the coefficient system e has regular highest weight in degrees below q 0 g mathbb r this is a sharp bound depending only on the underlying real lie group g mathbb r corollary 5 6 proposition 5 8 this result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular eisenstein cohomology classes

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

enhancing an algorithm for set covering problems abstract in this note we present enhancements to a previously published algorithm for the optimal solution of set covering problems these enhancements relate to the use of a langrangean heuristic feasible solution exclusion constraints gomory f cuts and an improved branching strategy computational results for problems involving up to 400 rows and 4000 columns indicate that the enhanced algorithm gives better computational results than the original algorithm

surrogate constraint normalization for the set covering problem the set covering problem scp is central in a wide variety of practical applications for which finding good feasible solutions quickly often in real time is crucial surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming this framework remains the core of the most effective constructive heuristics for the solution of the scp chiefly represented by the widely used chvatal method this paper introduces a number of normalization rules and demonstrates their superiority to the classical chvatal rule especially when solving large scale and real world instances directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided

on the set covering polytope i all the facets with coefficients in 0 1 2 while the set packing polytope through its connection with vertex packing has lent itself to fruitful investigations little is known about the set covering polytope we characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0 1 or 2 and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining we show that all inequalities in the above class are contained in the elementary closure of the constraint set and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure we point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix finally we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

a heuristic method for the set covering problem we present a lagrangian based heuristic for the well known set covering problem scp the algorithm was initially designed for solving very large scale scp instances involving up to 5 000 rows an

the eigenstructure of finite field trigonometric transforms in this paper we discuss the eigenstructure of finite field trigonometric transforms fftt this transform family includes different types of cosine and sine transforms that correspond in the finite field case to the well known discrete cosine and sine transforms defined over the real numbers more specifically we determine the eigenvalues of the fftt matrices and study their multiplicities we propose procedures for constructing the associated eigenvectors applications of the developed theory to multiuser communication systems and error correcting codes are also suggested

graph theoretic relaxations of set covering and set partitioning problems in this paper we review the graph theoretic relaxations of the set covering problem scp and the set partitioning problem ssp these are a network relaxation which can be solved by the greedy method a matching relaxation and a graph covering relaxation other relaxations such as assignment shortest route and minimum spanning tree are also presented

a lagrangian based heuristic for large scale set covering problems we present a new lagrangian based heuristic for solving large scale set covering problems arising from crew scheduling at the italian railways ferrovie dello stato our heuristic obtained impressive results when compared to state of the art codes on a test bed provided by the company which includes instances with sizes ranging from 50 000 variables and 500 constraints to 1 000 000 variables and 5000 constraints 1998 the mathematical programming society inc published by elsevier science b v

multiple parameter discrete fractional transform and its applications in recent years several special multiple parameter discrete fractional transforms mpdfrts have been proposed and their advantages have been demonstrated in the fields of communication systems and information security however the general theoretical framework of mpdfrts has not yet been established in this paper we propose two separate theoretical frameworks called the type i and ii mpdfrt that can include existing multiple parameter transforms as special cases the properties of the type i and ii mpdfrt have been analyzed in detail and their high dimensional operators have been defined under the theoretical frameworks we can construct new types of transforms that may be useful in signal processing and information security finally we perform two applications about image encryption and image feature extraction in the type i and ii mpdfrt domain the simulation results demonstrate that the typical transforms constructed under the proposed theoretical frameworks yield promising results in these applications

the discrete fractional cosine and sine transforms this paper is concerned with the definitions of the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst the definitions of dfrct and dfrst are based on the eigen decomposition of dct and dst kernels this is the same idea as that of the discrete fractional fourier transform dfrft the eigenvalue and eigenvector relationships between the dfrct dfrst and dfrft can be established the computations of dfrft for even or odd signals can be planted into the half size dfrct and dfrst calculations this will reduce the computational load of the dfrft by about one half

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

a genetic algorithm for the set covering problem abstract in this paper we present a genetic algorithm based heuristic for non unicost set covering problems we propose several modifications to the basic genetic procedures including a new fitness based crossover operator fusion a variable mutation rate and a heuristic feasibility operator tailored specifically for the set covering problem the performance of our algorithm was evaluated on a large set of randomly generated problems computational results showed that the genetic algorithm based heuristic is capable of producing high quality solutions

a block cipher cryptosystem using wavelet transforms over finite fields we present a new private key cryptosystem based on the finite field wavelet the encryption and decryption are performed by the synthesis and analysis banks of the nonlinear finite field wavelet transform whose filter coefficients are determined by the keys of the users we exploit the polyphase representation of the wavelets to introduce a shared key mechanism for the wavelet cryptosystem we propose to use wavelets that operate over gf 256 and a nonlinear device that performs a mapping on the field elements to their inverse in the field the introduced cryptographic system can operate in either stream cipher or block cipher modes depending on whether the filterbanks perform linear or circular convolution the block cipher system has a key length of 16 symbols 128 bits and an input block size of 30 symbols 240 bits to evaluate the efficiency of the developed two round wavelet cryptographic scheme we compare it with des and aes our results suggest that the wavelet cryptosystem has comparable computational complexity to aes and approximately half the complexity of des the security is tied to the length of the wavelet basis function and to the nonlinearity within the wavelet transform we study the security of the block cipher wavelet cryptosystem in response to classical attacks and those specific to this algorithm particularly those which use variations of the divide and conquer interpolation attack and discrete fourier transform techniques we show that chosen ciphertext attacks of the wavelet encryption system can reduce to the problem of solving a set of nonlinear equations over finite fields by considering existing classical and structure specific attacks we conclude that the lowest complexity of any of these attacks is greater than an exhaustive key search

a multi product loading problem a model and solution method abstract an important operational problem arises during the transportation and delivery of several products which cannot be mixed in the same vehicle at regular intervals the vehicle has compartments to keep the products separately therefore a scheme of allocation of compartments which we call vehicle loading problem to maximize the efficiency of the system while the demands for the products at the destination s are satisfied a mixed binary model is developed for this multi product loading problem the solution method is based on simultaneously exploring the primal and dual structures derived from the lagrangian relaxation subset sum problems are obtained as subproblems to the partial lagrangian an algorithm is developed and its convergence is proved the efficiency of the method is demonstrated by running randomly chosen test problems an initial solution finding method is also developed

simple lagrangian heuristic for the set covering problem abstract in this paper a simple lagrangian heuristic is proposed for the set covering problem it is based on simple and classical ideas lagrangian duality greedy heuristic for the set covering problem subgradient optimization and redundant covers the main interesting point of the paper lies in the fact that the proposed heuristic turns out to be efficient for low density set covering problems with a large number of variables

optimal solution of set covering partitioning problems using dual heuristics we present an algorithm for a mixed set covering partitioning model that includes as special cases the well known set covering problem and set partitioning problem the novel feature of our algorithm is the use of continuous heuristics applied to the dual of the linear programming relaxation to provide lower bounds for a branch and bound algorithm the heuristics are continuous adaptations of the well known greedy and 3 opt methods that have been applied to a variety of combinatorial optimization problems our algorithm has outperformed the current best set covering algorithm of balas and ho 1980 by about a factor of 3 and appears to improve on the best existing set partitioning algorithm by more than an order of magnitude

the fractional discrete cosine transform the extension of the fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications the paper introduces and develops the fractional discrete cosine transform dct on the same lines discussing multiplicity and computational aspects similarities and differences with respect to the fractional fourier transform are pointed out

an algorithm for the planar three index assignment problem abstract a branch and bound algorithm for the planar three index assignment problem is presented the branching operation is performed on sets of variables belonging to a common constraint rather than on a single variable the algorithm comprises an upper and lower bound procedure the upper bound procedure consists of two parts one for finding an initial solution and one for improving it the local improvement rule takes advantage of the latin square structure of the solution the lower bound procedure combines a new dual heuristic and a lagrangean relaxation scheme solved by subgradient optimization the use of reduced costs in the evaluation of bounds is another feature of the algorithm computational experience both on the exact algorithm and on the primal heuristics as standalone procedures is reported

algorithms for railway crew management crew management is concerned with building the work schedules of crews needed to cover a planned timetable this is a well known problem in operations research and has been historically associated with airlines and mass transit companies more recently railway applications have also come on the scene especially in europe in practice the overall crew management problem is decomposed into two subproblems called crew scheduling and crew rostering in this paper we give an outline of different ways of modeling the two subproblems and possible solution methods two main solution approaches are illustrated for real world applications in particular we discuss in some detail the solution techniques currently adopted at the italian railway company ferrovie dello stato spa for solving crew scheduling and rostering problems

an application oriented guide for designing lagrangean dual ascent algorithms abstract lagrangean relaxation is an effective method for providing bounds in integer programming an efficient solution of the lagrangean dual often requires a specialized algorithm that must be tailored for each application model lagrangean dual ascent is a generic name that describes a broad class of algorithms lagrangean dual ascent is often preferred to subgradient optimization in solving a lagrangean dual since an ascent procedure guarantees monotonic bound improvement also if embedded within a branch and bound scheme lagrangean dual ascent may adjust only a few multipliers at each node thus conserving computer storage and enhancing processing efficiency this paper describes a framework to construct a lagrangean dual ascent procedure and adapts it for the generalized assignment problem and the constrained arborescence problem

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

the residual eisenstein cohomology of sp_ 4 over a totally real number field let g sp4 k be the k split symplectic group of k rank 2 where k is a totally real numbereld in this paper we compute the eisenstein cohomology of g with respect to anynite dimensional irreducible k rational representation e of g1 rk qg r where rk q denotes the restriction of scalars from k to q the approach is based on the work of schwermer regarding the eisenstein cohomology for sp4 q kim s description of the residual spectrum of sp4 and the frankeltration of the space of automorphic forms in fact taking the representation theoretic point of view we write for the group g the frankeltration with respect to the cuspidal support and give a precise description of theltration quotients in terms of induced representations this is then used as a prerequisite for the explicit computation of the eisenstein cohomology the special focus is on the residual eisenstein cohomology under a certain compatibility condition for the coefficient system e and the cuspidal support we prove the existence of non trivial residual eisenstein cohomology classes which are not square integrabl e that is represented by a non square integrabl residue of an eisenstein series

on limit multiplicites of discrete series representations in spaces of automorphic forms of course the groups f are then arithmetic subgroups of g in this situation it seems natural to assume that as n the spectral decomposition of l2 f g will approximate that of g in particular let 3 be a discrete series representation of g if f is a subgroup of g let re f 6 be the multiplicity of 3 in the representation of g by right translations on l2 f g choose a haar measure dg on g if f is arithmetic let v f g be the finite volume of f g for the measure induced by dg let d 3 be the formal degree of 3 for the measure dg

fractional cosine and sine transforms over finite fields abstract in this paper we introduce fractional cosine and sine transforms over finite fields the basis for the definition of such transforms which are respectively referred by the acronyms gfrct and gfrst is the recently proposed fractional fourier transform over finite fields gfrft 1 more specifically we use the eigenvectors of the finite field fourier transform to construct eigenvectors of the finite field cosine and sine transforms such eigenvectors are then used in spectral expansions that allow to compute fractional powers of the finite field cosine and sine transform matrices

transform order division multiplexing multiplexing in the fractional fourier domain frfd only transmits the signals in a single direction in the time frequency plane and the transform order of the fractional fourier transform frft is not used sufficiently in this paper we apply the perfect reconstruction transmultiplexer in the frfd to develop the transform order division multiplexing todm which transmits the original information in the multidirections in the time frequency plane simultaneously the proposed todm systems can be divided into two types one needs the multitransmission of the original information while the other needs the modifications of the original information by the interpolation and the inverse discrete frft the performance analysis of the todm systems is also presented including the system capacity the reconstruction accuracy the security and the limitation compared with the multiplexing in the frfd the todm is more flexible and the security of the system is highly strengthened finally simulation results confirm the obtained analytical results

dual decomposition of a single machine scheduling problem we design a fast ascent direction algorithm for the lagrangian dual problem of the single machine scheduling problem of minimizing total weighted completion time subject to precedence constraints we show that designing such an algorithm is relatively simple if a scheduling problem is formulated in terms of the job completion times rather than as an 0 1 linear program also we show that upon termination of such an ascent direction algorithm we get a dual decomposition of the original problem which can be exploited to develop approximative and enumerative approaches for it computational results exhibit that in our application the ascent direction leads to good lagrangian lower and upper bounds

on the cuspidal cohomology of arithmetic subgroups of mathrm sl 2n and the first betti number of arithmetic 3 manifolds

a 3 flip neighborhood local search for the set covering problem the set covering problem scp calls for a minimum cost family of subsets from n given subsets which together covers the entire ground set in this paper we propose a local search algorithm for scp which has the following three characteristics 1 the use of 3 flip neighborhood which is the set of solutions obtainable from the current solution by exchanging at most three subsets as the size of 3 flip neighborhood is o n3 the neighborhood search becomes expensive if implemented naively to overcome this we propose an efficient implementation that reduces the number of candidates in the neighborhood without sacrificing the solution quality 2 we allow the search to visit the infeasible region and incorporate the strategic oscillation technique realized by adaptive control of penalty weights 3 the size reduction of the problem by using the information from the lagrangian relaxation is incorporated which is indispensable for solving very large instances according to computational comparisons on benchmark instances with other existing heuristic algorithms for scp our algorithm performs quite effectively for various types of problems especially for very large scale instances

closed form orthogonal number theoretic transform eigenvectors and the fast fractional ntt in this paper we propose a new method to find the closed form solution of number theoretic transform ntt eigenvectors we construct the complete generalized legendre sequence over the finite field cglsf and use it to solve the ntt eigenvector problem we derive the cglsf like ntt eigenvectors successfully including the case where the operation field is defined over the fermat and mersenne numbers the derived ntt eigenvector set is orthogonal and has a closed form it is suitable for constructing sub ntt building blocks for ntt implementation in addition with different eigenvalue assignment rule we can construct the fractional number theoretic transform fntt including the fractional fermat number transform ffnt the fractional complex mersenne number transform fcmnt and the fractional new mersenne number transform fnmnt they are the generalizations of the original transforms and all have the complexities of o nlog2n

a surrogate heuristic for set covering problems abstract the purpose of this paper is to present a new heuristic for set covering problems based upon continuous surrogate relaxations and subgradient optimization the algorithm combines problem reduction tests an adequate step size control and avoid preliminary sorting in solving the continuous surrogate relaxations computational tests for large scale set covering problems up to 1 000 rows and 12 000 columns indicate better quality results than algorithms based on lagrangian relaxations in terms of final solutions and mainly in computer times although the solving of a single surrogate optimization problem is slower than a corresponding lagrangian optimization the overall performance is almost twice as fast this is due to the smaller number of iterations which is a result of faster convergence and less oscillation

an algorithm for set covering problem in this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent subgradient optimisation and linear programming computational results are presented for problems involving up to 400 rows and 4000 columns

a dynamic subgradient based branch and bound procedure for set covering we discuss a branch and bound algorithm for set covering whose centerpiece is a new integrated upper bounding lower bounding procedure called dynamic subgradient optimization dynsgrad this new procedure applied to a lagrangean dual at every node of the search tree combines the standard subgradient method with primal and dual heuristics that interact to change the lagrange multipliers and tighten the upper and lower bounds fix variables and periodically restate the lagrangean itself extensive computational testing is reported as a stand alone heuristic dynsgrad performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort when incorporated into a branch and bound algorithm dynsgrad considerably advances the state of the art in solving set covering problems

tautological classes on moduli spaces of hyper kahler manifolds we study algebraic cycles on moduli spaces f h of h polarized hyperkahler manifolds following previous work of marian oprea and pandharipande on the tautological conjecture on moduli spaces of k3 surfaces we first define the tautological ring on f h we then study the images of these tautological classes in the cohomology groups of f h and prove that most of them are linear combinations of noether lefschetz cycle classes in particular we prove the cohomological version of the tautological conjecture on moduli space of k3 n type hyperkahler man ifolds with n 2 secondly we prove the cohomological generalized franchetta conjecture on universal family of these hyperkahler mani folds

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

the discrete fractional fourier transform we propose and consolidate a definition of the discrete fractional fourier transform that generalizes the discrete fourier transform dft in the same sense that the continuous fractional fourier transform generalizes the continuous ordinary fourier transform this definition is based on a particular set of eigenvectors of the dft matrix which constitutes the discrete counterpart of the set of hermite gaussian functions the definition is exactly unitary index additive and reduces to the dft for unit order the fact that this definition satisfies all the desirable properties expected of the discrete fractional fourier transform supports our confidence that it will be accepted as the definitive definition of this transform

an adaptation of sh heuristic to the location set covering problem abstract in a recent paper a new surrogate heuristic sh has been proposed for the set covering problem here we present an adaptation of it in order to solve more efficiently the location set covering problem we will show that our new version not only outperforms algorithm sh but that it is more accurate than the pair cma fmc its power is experimentally tested over a set of 65 randomly generated problems

an efficient local search heuristic with row weighting for the unicost set covering problem abstract the set covering problem scp is np hard we propose a new row weighting local search rwls algorithm for solving the unicost variant of the scp i e uscps where the costs of all sets are identical rwls is a heuristic algorithm that has three major components united in its local search framework 1 a weighting scheme which updates the weights of uncovered elements to prevent convergence to local optima 2 tabu strategies to avoid possible cycles during the search and 3 a timestamp method to break ties when prioritizing sets rwls has been evaluated on a large number of problem instances from the or library and compared with other approaches it is able to find all the best known solutions bks and improve 14 of them although requiring a higher computational effort on several instances rwls is especially effective on the combinatorial or library instances and can improve the best known solution to the hardest instance cyc11 considerably rwls is conceptually simple and has no instance dependent parameters which makes it a practical and easy to use uscp solver

an effective and simple heuristic for the set covering problem abstract this paper investigates the development of an effective heuristic to solve the set covering problem scp by applying the meta heuristic meta raps meta heuristic for randomized priority search in meta raps a feasible solution is generated by introducing random factors into a construction method then the feasible solutions can be improved by an improvement heuristic in addition to applying the basic meta raps the heuristic developed herein integrates the elements of randomizing the selection of priority rules penalizing the worst columns when the searching space is highly condensed and defining the core problem to speedup the algorithm this heuristic has been tested on 80 scp instances from the or library the sizes of the problems are up to 1000 rows 10 000 columns for non unicost scp and 28 160 rows 11 264 columns for the unicost scp this heuristic is only one of two known scp heuristics to find all optimal best known solutions for those non unicost instances in addition this heuristic is the best for unicost problems among the heuristics in terms of solution quality furthermore evolving from a simple greedy heuristic it is simple and easy to code this heuristic enriches the options of practitioners in the optimization area

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is

a multiplier adjustment method for the generalized assignment problem we describe a branch and bound algorithm for the generalized assignment problem in which bounds are obtained from a lagrangian relaxation with the multipliers set by a heuristic adjustment method the algorithm was tested on a large sample of small random problems and a number of large problems derived from a vehicle routing application computation times were reasonable in all cases and the branch and bound trees generated had nearly two orders of magnitude fewer nodes than for competing algorithms although comparison of running times on different machines is difficult the multiplier adjustment method appears to be about one order of magnitude faster than the best previously existing algorithms for this problem

the l2 lefschetz numbers of hecke operators introduction 5 1 hecke operators and l2 cohomology 52 the spectral decomposition of cohomology 6 3 application of the trace formula 54 the functions pa t 5 5 proof of theorem 5 1 6 the main formula appendix bibliography

generalized eigenvectors and fractionalization of offset dfts and dcts the offset discrete fourier transform dft is a discrete transform with kernel exp j2 spl pi m a n b n it is more generalized and flexible than the original dft and has very close relations with the discrete cosine transform dct of type 4 dct iv dct viii discrete sine transform dst iv dst viii and discrete hartley transform dht iv in this paper we derive the eigenvectors eigenvalues of the offset dft especially for the case where a b is an integer by convolution theorem we can derive the close form eigenvector sets of the offset dft when a b is an integer we also show the general form of the eigenvectors in this case then we use the eigenvectors eigenvalues of the offset dft to derive the eigenvectors eigenvalues of the dct iv dct viii dst iv dst viii and dht iv after the eigenvectors eigenvalues are derived we can use the eigenvectors decomposition method to derive the fractional operations of the offset dft dct iv dct viii dst iv dst viii and dht iv these fractional operations are more flexible than the original ones and can be used for filter design data compression encryption and watermarking etc