least square completion and inconsistency repair methods for additively consistent fuzzy preference relations in this paper we explore the group decision making gdm problems with incomplete additively consistent fuzzy preference relations some properties of additively consistent fuzzy preference relations are also discussed a sufficient and necessary condition is proposed to keep the additive consistency of fuzzy preference relations methods for determining the priority weights of fuzzy preference relations are provided least square completion and inconsistency repair methods are developed to deal with incomplete and inconsistent fuzzy preference relations some numerical examples are also given to illustrate the proposed approaches

efficient solution of rational conics we present efficient algorithms for solving legendre equations over q equivalently for finding rational points on rational conics and parametrizing all solutions unlike existing algorithms no integer factorization is required provided that the prime factors of the discriminant are known

existence of primitive divisors of lucas and lehmer numbers we prove that for n 30 every n th lucas and lehmer number has a primitive divisor this allows us to list all lucas and lehmer numbers without a primitive divisor

a consistency improving method in the analytic hierarchy process in this paper we propose a method to modify a given comparison matrix by which the consistency ratio cr value of the modified matrix is less than that of the original one and give an algorithm to derive a positive reciprocal matrix with acceptable consistency i e cr 0 1 then the convergence theorem for the given algorithm is established and its practicality is shown by some examples

arithmetic multivariate descartes rule let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i be any number field or p adic field and consider f f 1 f k where f 1 f k inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i x 1 x n and no more than distinct exponent vectors occur in the monomial term expansions of the f i we prove that f has no more than 1 c n n 3 log n n geometrically isolated roots in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i n where c is an explicit and effectively computable constant depending only on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i this gives a significantly sharper arithmetic analogue of khovanski s theorem on real fewnomials and a higher dimensional generalization of an earlier result of hendrik w lenstra jr for the special case of a single univariate polynomial we also present some further refinements of our new bounds and a higher dimensional generalization of a bound of lipshitz on p adic complex roots connections to non archimedean amoebae and computational complexity including additive complexity and solving for the geometrically isolated rational roots are discussed along the way we thus provide the foundations for an effective arithmetic analogue of fewnomial theory

solving a system of linear diophantine equations with lower and upper bounds on the variables we develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables the algorithm is based on lattice basis reduction it first finds a short vector satisfying the system of diophantine equations and a set of vectors belonging to the nullspace of the constraint matrix due to basis reduction all these vectors are relatively short the next step is to branch on linear combinations of the null space vectors which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists the research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing our algorithm is tested with good results on real life data and on instances from the literature

a fast signature scheme based on congruential polynomial operations a novel digital signature scheme is proposed in which the computation time is much shorter than that of the rivest shamir adelman rsa scheme while the key length and signature length are comparable to those for the rsa scheme moreover the proposed scheme can be implemented easily and is therefore more practical for many digital signature applications the scheme is based on congruential polynomial operations whose degrees are more than three the secret key consists of two large prime numbers p and q and the public key is their product n p sup 2 q the security of this scheme depends on the difficulty of factorizing the number n variations using the number of zeros succeeding the significant bit are also proposed

multiplicative data envelopment analysis cross efficiency and stochastic weight space acceptability analysis for group decision making with interval multiplicative preference relations abstract to deal with group decision making gdm with interval multiplicative preference relations imprs this paper proposes a novel method based on multiplicative data envelopment analysis dea cross efficiency and stochastic weight space acceptability analysis we first develop a multiplicative dea model to evaluate the relative efficiency of all alternatives of a given multiplicative preference relation mpr then we present a method free from consistency adjustment to derive a priority vector using the multiplicative dea cross efficiency with respect to the given mpr for gdm with imprs we consider the decision makers weights as a uniform distribution for acceptability analysis a modified unacceptability index is further defined to measure the unlikeliness for a particular alternative in a particular rank finally we develop an assignment problem model to achieve an optimal ranking by minimizing the total rank unacceptability and to compute the expected priority vector of all alternatives numerical examples are provided to show the applicability and justifications of the proposed gdm method

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

robust estimation of priorities in the ahp abstract a pairwise comparison matrix of the analytic hierarchy process ahp is considered as a contingency table that helps to identify unusual or false data elicited from a judge special techniques are suggested for robust estimation of priority vectors they include transformation of a saaty matrix to matrix of shares of preferences and solving an eigenproblem designed for the transformed matrices we also introduce an optimizing objective that produces robust priority estimation numerical results are compared using the ahp with these differing approaches the comparison demonstrates that robust estimations yield priority vectors not prone to influence of possible errors among the elements of a pairwise comparison matrix

an efficient solution of the congruence x 2 ky 2 mpmod n the equation of the title arose in the proposed signature scheme of ong schnorr shamir the large integers n k and m are given and we are asked to find any solution x y it was believed that this task was of similar difficulty to that of factoring the modulus n we show that on the contrary a solution can easily be found if k and m are relatively prime to n under the assumption of the generalized riemann hypothesis a solution can be found by a probabilistic algorithm in o log n 2 log log k arithmetical steps on o log n bit integers the algorithm can be extended to solve the equation x 2 ky 2 m pmod n for quadratic integers k m in bf z sqrt d and to solve in integers the equation x 3 ky 3 k 2 z 3 3kxyz m pmod n

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

a note on a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making abstract in a recently published paper by liu et al liu f zhang w g wang z x 2012 a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making european journal of operational research 218 747 754 two equations are introduced to define consistency of incomplete interval multiplicative preference relations imprs and employed to develop a goal programming model for estimating missing values this note illustrates that such consistency definition and estimation model are technically incorrect new transitivity conditions are proposed to define consistent imprs and a two stage goal programming approach is devised to estimate missing values for incomplete imprs

integer forcing linear receivers linear receivers are often used to reduce the implementation complexity of multiple antenna systems in a traditional linear receiver architecture the receive antennas are used to separate out the codewords sent by each transmit antenna which can then be decoded individually although easy to implement this approach can be highly suboptimal when the channel matrix is near singular this paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer valued entries rather than attempting to recover transmitted codewords directly the decoder recovers integer combinations of the codewords according to the entries of the effective channel matrix the codewords are all generated using the same linear code which guarantees that these integer combinations are themselves codewords provided that the effective channel is full rank these integer combinations can then be digitally solved for the original codewords this paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter s which corresponds either to a multiuser uplink scenario or to single user v blast encoding in this setting the proposed integer forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean squared error receiver in the high signal to noise ratio regime the proposed receiver attains the optimal diversity multiplexing tradeoff for the standard multiple input multiple output mimo channel with no coding across transmit antennas it is further shown that in an extended mimo model with interference the integer forcing linear receiver achieves the optimal generalized degrees of freedom

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

implementing stochastic multicriteria acceptability analysis abstract stochastic multicriteria acceptability analysis smaa is a family of methods for aiding multicriteria group decision making in problems with inaccurate uncertain or missing information these methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one or that would give a certain rank for a specific alternative the main results of the analysis are rank acceptability indices central weight vectors and confidence factors for different alternatives the rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative the central weight vectors represent the typical preferences favouring each alternative and the confidence factors measure whether the criteria measurements are sufficiently accurate for making an informed decision the computations in smaa require the evaluation of multidimensional integrals that must in practice be computed numerically in this paper we present efficient methods for performing the computations through monte carlo simulation analyze the complexity and assess the accuracy of the presented algorithms we also test the efficiency of these methods empirically based on the tests the implementation is fast enough to analyze typical sized discrete problems interactively within seconds due to almost linear time complexity the method is also suitable for analysing very large decision problems for example discrete approximations of continuous decision problems

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

the optimal group continuous logarithm compatibility measure for interval multiplicative preference relations based on the cowga operator the calculation of compatibility measures is an important technique employed in group decision making with interval multiplicative preference relations in this paper a new compatibility measure called the continuous logarithm compatibility which considers risk attitudes in decision making based on the continuous ordered weighted geometric averaging cowga operator is introduced we also develop a group continuous compatibility model gcc model by minimizing the group continuous logarithm compatibility measure between the synthetic interval multiplicative preference relation and the continuous characteristic preference relation furthermore theoretical foundations are established for the proposed model such as the sufficient and necessary conditions for the existence of an optimal solution the conditions for the existence of a superior optimal solution and the conditions for the existence of redundant preference relations in addition we investigate certain conditions for which the optimal objective function of the gcc model guarantees its efficiency as the number of decision makers increases finally practical illustrative examples are examined to demonstrate the model and compare it with previous methods

the dea game cross efficiency model and its nash equilibrium in this paper we examine the cross efficiency concept in data envelopment analysis dea cross efficiency links one decision making unit s dmu performance with others and has the appeal that scores arise from peer evaluation however a number of the current cross efficiency approaches are flawed because they use scores that are arbitrary in that they depend on a particular set of optimal dea weights generated by the computer code in use at the time one set of optimal dea weights possibly out of many alternate optima may improve the cross efficiency of some dmus but at the expense of others while models have been developed that incorporate secondary goals aimed at being more selective in the choice of optimal multipliers the alternate optima issue remains in cases where there is competition among dmus this situation may be seen as undesirable and unfair to address this issue this paper generalizes the original dea cross efficiency concept to game cross efficiency specifically each dmu is v

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

smaa 2 stochastic multicriteria acceptability analysis for group decision making stochastic multicriteria acceptability analysis smaa is a multicriteria decision support method for multiple decision makers in discrete problems in smaa the decision makers need not express their preferences explicitly or implicitly instead the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one inaccurate or uncertain criteria values are represented by probability distributions from which the method computes confidence factors describing the reliability of the analysis in this paper we introduce the smaa 2 method which extends the original smaa by considering all ranks in the analysis in situations where the elitistic smaa may assess large acceptability only for extreme alternatives without sufficient majority support the more holistic smaa 2 analysis can be used to identify good compromise candidates the results are presented graphically we consider also situations where partial preference information is available we demonstrate the new method using a real life decision problem

on multilevel block modulation codes the multilevel technique for combining block coding and modulation is investigated a general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures a specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed given a multilevel block modulation code c with no interdependency among the binary component codes the proposed method gives a multilevel block modulation code c that has the same rate as c a minimum squared euclidean distance not less than that of c a trellis diagram with the same number of states as that of c and a smaller number of nearest neighbor codewords than that of c finally a technique is presented for analyzing the error performance of block modulation codes for an additive white gaussian noise awgn channel based on soft decision maximum likelihood decoding error probabilities of some specific codes are evaluated by simulation and upper bounds based on their euclidean weight distributions

interval estimation of priorities in the ahp this paper extends and modifies the analytic hierarchy process ahp and the synthetic hierarchy method shm of priority estimation to accommodate random data in the pairwise comparison matrices it employs a cauchy distribution to describe the pairwise comparison of alternatives in saaty matrices and shows how to modify these matrices in order to handle random data the use of random data yields saaty matrices that are not reciprocally symmetrical several variants of the ahp are then modified i to accommodate reciprocally asymmetric matrices and ii to allow each priority estimate to be expressed on an interval of possible values rather than as a single discrete point the merits of interval estimation are illustrated by an example

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

dual rsa and its security analysis we present new variants of an rsa whose key generation algorithms output two distinct rsa key pairs having the same public and private exponents this family of variants called dual rsa can be used in scenarios that require two instances of rsa with the advantage of reducing the storage requirements for the keys two applications for dual rsa blind signatures and authentication secrecy are proposed in addition we also provide the security analysis of dual rsa compared to normal rsa the security boundary should be raised when applying dual rsa to the types of small d small e and rebalanced rsa

lsb coded 8psk signals an 8psk scheme with the least significant bit lsb golay coded is proposed and its performance is analyzed with hard decision decoding the proposed scheme can achieve 1 db asymptotic coding gain and also save 26 bandwidth as compared with uncoded qpsk scheme

a multidimensional continued fraction based on a high order recurrence relation the paper describes and studies an iterative algorithm for finding of a set of linear forms over vectors of integers the algorithm small values of a set of linear forms over vectors of integers the algorithm uses a linear recurrence relation to generate a vector sequence the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values subject to the requirement that the integers should not become too large the problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one parameter family of lattices the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero the paper discusses two rounding procedures to solve the underlying inhomogeneous diophantine approximation problem the first which we call naive rounding leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties in particular when it is applied to the familiar problem of simultaneous rational approximation the algorithm reduces to the classical jacobi perron algorithm the second rounding procedure is babai s nearest plane procedure we compare the two rounding procedures numerically our experiments suggest that the multidimensional continued fraction corresponding to nearest plane rounding converges at an optimal asymptotic rate

axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations abstract this paper focuses on obtaining an interval extension of saaty s consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations imrprs as well as checking acceptability of imrprs four axiomatic properties are presented to characterize multiplicative consistency of imrprs the paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties a constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an imrpr based on the constructed interval matrix an ordinary interval multiplication based transitivity equation is developed to define consistency of imrprs a logarithmic least square model is established to find normalized interval weights from imrprs and its analytic solution is obtained by the lagrangian multiplier method the paper introduces an index to measure uniformity of row uncertainties in an imrpr and proposes a geometric consistency index to measure inconsistency of imrprs a novel method is put forward to determine acceptability of imrprs by checking acceptable consistency and acceptable uncertainty the proposed models are illustrated by eight numerical examples and a hierarchical multi criteria decision making problem dealing with recommending undergraduate students to graduate admission

some induced ordered weighted averaging operators and their use for solving group decision making problems based on fuzzy preference relations abstract in r r yager d p filev operations for granular computing mixing words and numbers in proceedings of the fuzz ieee world congress on computational intelligence anchorage 1998 pp 123 128 yager and filev introduced the induced ordered weighted averaging iowa operator in this paper we provide some iowa operators to aggregate fuzzy preference relations in group decision making gdm problems these iowa operators when guided by fuzzy linguistic quantifiers allow the introduction of some semantics or meaning in the aggregation and therefore allow for a better control over the aggregation stage developed in the resolution process of the gdm problems in particular we present the importance iowa i iowa operator which applies the ordering of the argument values based upon the importance of the information sources the consistency iowa c iowa operator which applies the ordering of the argument values based upon the consistency of the information sources and the preference iowa p iowa operator which applies the ordering of the argument values based upon the relative preference values associated to each one of them we provide a procedure to deal with ties in respect to the ordering induced by the application of one of these iowa operators it consists of a sequential application of the above iowa operators we also present a selection process for gdm problems based on the concept of fuzzy majority and the above three iowa operators finally we analyse the reciprocity and consistency properties of the collective fuzzy preference relations obtained using iowa operators

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

a primal all integer algorithm based on irreducible solutions this paper introduces an exact primal augmentation algorithm for solving general linear integer programs the algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities we prove that various versions of our algorithm are finite it is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively an implementation of the presented algorithms is given computational results for a number of hard 0 1 integer programs from the miplib demonstrate the practical power of the method

hard equality constrained integer knapsacks we consider the following integer feasibility problem given positive integer numbersa0 a1 a n with gcd a1 a n 1 anda a1 a n does there exist a vectorx z n 0satisfyinga x a0 we prove that if the coefficientsa1 a nhave a certain decomposable structure then the frobenius number associated witha1 a n i e the largest value ofa0 for whicha x a0 does not have a nonnegative integer solution is close to a known upper bound in the instances we consider we takea0 to be the frobenius number furthermore we show that the decomposable structure ofa1 a nmakes the solution of a lattice reformulation of our problem almost trivial since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small for branch and bound such instances are difficult to solve since they are infeasible and have large values ofa0 a i 1 i n we illustrate our results by some computational examples

an efficient low complexity detector for spatially multiplexed mc cdm we propose a novel suboptimum detection method for spatially multiplexed multicarrier code division multiplexing sm mc cdm communications compared to the spatially multiplexed ofdm sm ofdm the frequency domain spreading in sm mc cdm systems results in an additional diversity gain to take advantage of diversity and multiplexing while mitigating interference we design a low complexity efficient detector called unified successive interference cancellation sic detector for sm mc cdm communications further performance improvement is achieved by adopting in conjunction with the unified sic the iterative subcarrier reconstruction detection algorithm originally proposed for single antenna systems the results demonstrate significant performance improvement over other existing methods of comparable complexity a close approximation for the probability density function pdf of the proposed detector s output signal to interference plus noise ratio sinr is found and used to obtain error bounds for the bit error rate ber performance of the coded sm mc cdm transmission is also discussed in the paper

acceptable consistency analysis of interval reciprocal comparison matrices when a decision maker expresses his her opinions by means of an interval reciprocal comparison matrix the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution in the present paper an acceptably consistent interval reciprocal comparison matrix is defined which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers an interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency utilizing a convex combination method a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained whose weights are further found to exhibit a style of convex combination and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix a novel simple yet effective formula of possibility degree is presented to rank interval weights numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures