codimensions of algebras and growth functions abstract let a be an algebra over a field f of characteristic zero and let c n a n 1 2 be its sequence of codimensions we prove that if c n a is exponentially bounded its exponential growth can be any real number 1 this is achieved by constructing for any real number 1 an f algebra a such that lim n c n a n exists and equals the methods are based on the representation theory of the symmetric group and on properties of infinite sturmian and periodic words

preference modeling experiments with surrogate weighting procedures for the promethee method abstract one of the main tasks in a multi criteria decision making process is to define weights for the evaluation criteria however in many situations the decision maker dm may not be confident about defining specific values for these weights and may prefer to use partial information to represent the values of such weights with surrogate weights although for the additive model the use of surrogate weighting procedures has been already explored in the literature there is a gap with regard to experimenting with such kind of preference modeling in outranking based methods such as promethee for which there already are applications with surrogate weights in the literature thus this paper presents an experimental study on preference modeling based on simulation so as to increase understanding and acceptance of a recommendation obtained when using surrogate weights within the promethee method the main approaches to surrogate weights in the literature ew rs rr and roc have been evaluated for choice and ranking problematics throughout statistical procedures including kendall s tau coefficient the surrogate weighting procedure that most faithfully represents a dm s value system according to this analysis is the roc procedure

on some minimal supervarieties of exponential growth abstract in the present paper we investigate minimal supervarieties of given superexponent over fields of characteristic zero we show that any minimal supervariety of finite basic rank is generated by one of the minimal superalgebras introduced by giambruno and zaicev in 2003 furthermore it is proved that any minimal superalgebra whose graded simple components of the semisimple part are simple generates a minimal supervariety finally we state that the same conclusion holds when the semisimple part of a minimal superalgebra has exactly two arbitrary graded simple components

exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

robustness and information levels in case based multiple criteria sorting abstract case based preference elicitation methods for multiple criteria sorting problems have the advantage of posing rather small cognitive demands on a decision maker but they may lead to ambiguous results when preference parameters are not uniquely determined we use a simulation approach to determine the extent of this problem and to study the impact of additional case information on the quality of results our experiments compare two decision analysis tools case based distance sorting and the simple additive weighting method in terms of the effects of additional case information on sorting performance depending on problem dimension number of groups number of criteria etc our results confirm the expected benefit of additional case information on the precision of estimates of the decision maker s preferences problem dimension however has some unexpected effects

tools for interactive multiattribute decisionmaking with incompletely identified information abstract the aim of this paper is to present tools or techniques for an interactive multiattribute decisionmaking with the decision maker s incomplete information that is 1 we formally describe mathematical models under certainty and uncertainty for establishing pairwise dominance where the incomplete information becomes the set of constraints in the model 2 we present a description on styles of incomplete parameter value information 3 the characteristic of a weak dominance is described when the decision maker is not willing or able to provide more information on parameters the weak dominance technique is useful for a final choicemaking 4 we propose a dominance graph and also present an algorithm of generating the dominance graph based on the information of pairwise dominance the dominance graph compactly displays the dominance structure of all alternatives thus it can be used for aiding the selection of preferable alternatives

ranking with partial information a method and an application a method is presented for ranking multiattributed alternatives using a weighted additive evaluation function with partial information about the weighting scaling constants the method is applied to evaluate materials for use in nuclear waste containment the paper derives conditions to determine whether a pair of alternatives can be ranked given the partial information about weighting constants and presents an algorithm that partially rank orders the complete set of alternatives based on the pairwise ranking information

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

heuristics for selecting pair wise elicitation questions in multiple criteria choice problems abstract we present a set of heuristic approaches for selecting pair wise elicitation questions in an interactive process for multiple criteria choice problems our heuristics aim at minimizing the number of question answer iterations leading to the univocal recommendation of the decision maker s dm s most preferred alternative to identify the myopically best question at a given stage of interaction the proposed approaches ask the dm to compare a pair of alternatives that contributes to the greatest reduction of uncertainty with respect to the indication of the best alternative by all compatible value functions this uncertainty is measured either in terms of the number of potentially optimal alternatives or the entropy of first rank acceptabilities while assuming different a priori unknown probabilities of the dm s answers we discuss results from the extensive experiments on artificially generated and real world decision problems depending on the complexity of the considered problem instances we either perform a comprehensive analysis of the question answer interaction trees constructed by the heuristics or traverse their paths randomly by simulating a numerous set of decision policies we demonstrate that the greatest benefits from using our questioning procedures can be observed for problems involving numerous alternatives and few criteria and when the applied piece wise linear value functions consist of a small number of characteristic points the study allows us to identify two approaches that perform well in the average or the least advantageous elicitation scenario

polynomial identities on superalgebras and exponential growth abstract let a be a finitely generated superalgebra over a field f of characteristic 0 to the graded polynomial identities of a one associates a numerical sequence cnsup a n 1 called the sequence of graded codimensions of a in case a satisfies an ordinary polynomial identity such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra lim n c n sup a n exists and is a non negative integer we denote such integer by supexp a and we give an effective way for computing it as an application we construct eight superalgebras ai i 1 8 characterizing the identities of any finitely generated superalgebra a with supexp a 2 in the following way supexp a 2 if and only if idsup a idsup ai for some i 1 8 where idsup b is the ideal of graded identities of the algebra b we also compare the superexponent and the exponent see a giambruno m zaicev adv math 140 1998 145 155 of any finitely generated superalgebra

compatible weighting method with rank order centroid maximum entropy ordered weighted averaging approach in a situation where imprecise attribute weights such as a rank order are captured various approximate weighting methods have been proposed to aid multiattribute decision analysis among others it is well known that the rank order centroid roc weights result in the highest performance in terms of the identification of the best alternative under the ranked attribute weights in this paper we aim to reinterpret the meaning of the roc weights and to develop a compatible weighting method that is based on other well established academic disciplines the ordered weighted averaging owa method is a nonlinear aggregation method in that the weights are associated with the objects reordered according to their magnitudes in the aggregation process some interesting semantics can be attached to the approximate weights in view of the measure developed in the owa method furthermore the weights generated by the maximum entropy method show equally compatible performance with the roc weights under some condition which is demonstrated by theoretical and simulation analysis

functional dependencies in horn theories abstract this paper studies functional dependencies in horn theories both when the theory is represented by its clausal form and when it is defined as the horn envelope of a set of models we provide polynomial algorithms for the recognition of whether a given functional dependency holds in a given horn theory as well as polynomial algorithms for the generation of some representative sets of functional dependencies we show that some problems of inferring functional dependencies e g constructing an irredundant fd cover are computationally difficult we also study the structure of functional dependencies that hold in a horn theory showing that every such functional dependency is in fact a single positive term boolean function and prove that for any horn theory the set of its minimal functional dependencies is quasi acyclic finally we consider the problem of condensing a horn theory prove that any horn theory has a unique condensation and develop an efficient polynomial algorithm for condensing horn theories

lattices closures systems and implication bases a survey of structural aspects and algorithms abstract concept lattices and closed set lattices are graphs with the lattice property they have been increasingly used this last decade in various domains of computer science such as data mining knowledge representation databases or information retrieval a fundamental result of lattice theory establishes that any lattice is the concept lattice of its binary table a consequence is the existence of a bijective link between lattices contexts via the table and a set of implicational rules via the canonical direct basis the possible transformations between these objects give rise to relevant tools for data analysis in this paper we present a survey of lattice theory from the algebraic definition of a lattice to that of a concept lattice through closure systems and implicational rules including the exploration of fundamental bijective links between lattices reduced contexts and bases of implicational rules and concluding with the presentation of the main generation algorithms of these objects

a note on scale transformations in the promethee v method abstract as already noted by mavrotas et al 2006 the promethee v method for multi attribute analysis of portfolio problems will fail to include an item in a portfolio if it has a negative net flow with respect to other items although adding such an item might actually improve the entire portfolio however the model formulated by mavrotas et al introduces a bias in favor of large portfolios because the promethee v method is sensitive to scale transformations the present paper analyzes this effect and proposes a method to correct it

fundamental study the multiple facets of the canonical direct unit implicational basis the notion of dependencies between attributes arises in many areas such as relational databases data analysis data mining formal concept analysis knowledge structures formalization of dependencies leads to the notion of so called full implicational systems or full family of functional dependencies which is in one to one correspondence with the other significant notions of closure operator and of closure system an efficient generation of a full implicational system or a closure system can be performed from equivalent implicational systems and in particular from the bases for such systems for example the so called canonical basis this paper shows the equality between five other bases originating from different works and satisfying various properties in particular they are unit implicational systems the three main properties of this unique basis are the directness canonical and minimal properties whence the name canonical direct unit implicational basis given to this unit implicational system the paper also gives a nice characterization of this canonical basis and makes precise its link with the prime implicants of the horn function associated to a closure operator it concludes that it is necessary to compare more closely related works made independently and with a different terminology in order to take advantage of the really new results in these works

note a preference ranking organisation method the promethee method for multiple criteria decision making principles for a new family of outranking methods are given the main aim of the proposed promethee approach is to be as easily understood as possible by the decision maker it is based on extensions of the notion of criterion six possible extensions are considered these extensions can easily be identified by the decision maker because the parameters to be defined at most 2 have an economic significance a valued outranking graph is constructed by using a preference index two possibilities are considered to solve the ranking problem by using this valued graph promethee i provides a partial preorder and promethee ii a total preorder on the set of the possible actions

a method for determining the weights of criteria the centralized weights abstract in multicriteria analysis many methods use weights to represent the relative importance of criteria the noncompensatory aggregation procedure used in outranking methods enables comparisons of subsets of criteria from this point of view an interactive method is presented which requests only ordinal comparisons from the decision maker the relation more important than is assumed to be a semiorder therefore the indifference part of this relation is not necessarily transitive the judgements can be formulated as linear inequalities which constrain the set of feasible weights and threshold values all vertices of this polyhedron are determined the threshold and the unknown real weights are estimated by the centroid some characteristics a possible refinement in the estimation and an illustrative example are also given

p i algebras with hopf algebra actions abstract ifais a p i algebra in characteristic zero with action from a finite dimensional semisimple hopf algebrah thenahas a nilpotenth idealnsuch thata nwill beh verbally semiprime everyh verbally semiprime algebra ish p i equivalent to a direct sum ofh verbally prime algebras in the case of a finite group action or a grading by an abelian group we show that the sum can be taken to be finite in the case of an action by a finite cyclic groupg we classify allg p i algebras up to equivalence this paper generalizes the work of a r kerner 1985 math ussr izv 25

aggregating preferences rankings with variable weights abstract one of the most important issues for aggregating preferences rankings is the determination of the weights associated with the different ranking places to avoid the subjectivity in determining the weights cook and kress 1990 5 suggested evaluating each candidate with the most favorable scoring vector for him her with this purpose various models based on data envelopment analysis have appeared in the literature although these methods do not require predetermine the weights subjectively some of them have a serious drawback the relative order between two candidates may be altered when the number of first second k th ranks obtained by other candidates changes although there is not any variation in the number of first second k th ranks obtained by both candidates in this paper we propose a model that allows each candidate to be evaluated with the most favorable weighting vector for him her and avoids the previous drawback moreover in some cases we give a closed expression for the score assigned with our model to each candidate

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

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

identities of graded algebras and codimension growth let a g g a g be a g graded associative algebra over a field of characteristic zero in this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component a e to that of the whole of a in the case where the support of the grading is finite we prove the conjecture in several natural cases one of them being the case where a is finite dimensional and a e has polynomial growth

a conjecture of regev about the capelli polynomial abstract let x1 xn2 y1 yn2 be generic n n matrices over a field k of characteristic zero if x 1 x n is a multilinear invariant of x1 xn then s z 2 sign f x 1 y 1 x 2 y 2 y 3 y 4 x 3 y 5 y 9 x n y n 2 2n 2 y n 2 f x 1 x n y where sn2 is the symmetric group of degree n2 y is the discriminant of y1 yn2 and tf x 1 x n is a uniquely defined multilinear invariant of x1 xn thus tf defines a function from the vector space of multilinear invariants of x1 xn to itself an analysis of this function is used to prove regev s conjecture that s z 2 sign x n 1 y 1 x 2 x 3 x 4 y 2 y 3 y 4 x 5 x n 2 2 2 x n 2 y n 2 2n 2 y n 2 is nonzero in addition a variant of the above function is used to evaluate the capelli polynomial

ordinal judgments in multiattribute decision analysis abstract the article discusses the contradiction between the ambiguity of human judgment in a multicriterion environment and the exactness of the assessments required in the majority of the decision making methods preferential information from the decision makers in the ordinal form e g more preferable less preferable etc is argued to be more stable and more reliable than cardinal input ways of obtaining and using ordinal judgments for rank ordering of multiattribute alternatives are discussed the effectiveness of the step wise procedure of using ordinal tradeoffs for comparison of alternatives is evaluated we introduce the notion of ordinal tradeoffs presentation of ordinal tradeoffs as a flexible three stage process a paired joint ordinal scale pjos and evaluation of the effectiveness of the three stage process simulation results examine the sensitivity of the number of pairwise comparisons required for given numbers of criteria and categories within criteria as well as the number of alternatives analyzed this simulation shows that ordinal pairwise comparisons provide sufficient power to discriminate between 75 and 80 of the alternatives compared while the proportional number of pairwise comparisons relative to the maximum possible decreases with the number of criteria and categories the method is relatively insensitive to the number of alternatives considered

using assignment examples to infer weights for electre tri method some experimental results given a finite set of alternatives a the sorting or assignment problem consists in the assignment of each alternative to one of the pre defined categories in this paper we are interested in multiple criteria sorting problems and more precisely in the existing method electre tri this method requires the elicitation of preferential parameters weights thresholds category limits in order to construct a preference model which the decision maker dm accepts as a working hypothesis in the decision aid study a direct elicitation of these parameters requiring a high cognitive effort from the dm v mosseau r slowinski journal of global optimization 12 2 1998 174 proposed an interactive aggregation disaggregation approach that infers electre tri parameters indirectly from holistic information i e assignment examples in this approach the determination of electre tri parameters that best restore the assignment examples is formulated through a nonlinear optimization program in this paper we consider the subproblem of the determination of the weights only the thresholds and category limits being fixed this subproblem leads to solve a linear program rather than nonlinear in the global inference model numerical experiments were conducted so as to check the behaviour of this disaggregation tool results showed that this tool is able to infer weights that restores in a stable way the assignment examples and that it is able to identify inconsistencies in the assignment examples

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

use of piecewise linear value functions in interactive multicriteria decision support a monte carlo study this paper describes a monte carlo study conducted to evaluate the effects of modelling assumptions and design parameters on the behaviour of interactive methods for the discrete choice mcdm problem based on explicit value function models the purpose of the study is to identify those assumptions and parameters which lead to the most efficient use of preference judgements made by the decision maker and to the greatest robustness to judgmental errors it is concluded that nonlinearities in the value function need to be modelled achieved here by use of a piecewise linear form it was also found that search for indifference points rather than using simple preference judgements alone is of great advantage best realized by expressing judgements in terms of pairwise trade offs methods incorporating these features are highly robust to judgmental errors interactive methods of this class are compared with a priori fitting of similar value functions and found to give a very similar quality of solution

generating random vectors uniformly distributed inside and on the surface of different regions abstract an acceptance rejection algorithm for generating random vectors uniformly distributed over inside or on the surface of a complex region inserted in a minimal multidimensional rectangle is considered for regions having simple forms simplex hypersphere hyperellipsoid several algorithms are presented as well

on codimension growth of finitely generated associative algebras abstract letabe a pi algebra over a fieldf we study the asymptotic behavior of the sequence of codimensionscn a ofa we show that ifais finitely generated overftheninv a limn c n a always exists and is an integer we also obtain the following characterization of simple algebras ais finite dimensional central simple overfif and only ifinv a dim a

decision quality using ranked attribute weights three published approximation formulae for selecting the best multiattribute alternative based on rank ordered weights are evaluated all formulae are surprisingly efficacious in determining the best multiattribute alternative rank order centroid roc weights are more accurate than the other rank based formulae furthermore the roc formula generalizes to incorporate both other forms of partial information about attribute weights and partial rank order information as well because a roc based analysis is so straightforward and efficacious it provides an appropriate implementation tool

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

the joy of implications aka pure horn formulas pure horn clauses have also been called among others functional dependencies strong association rules or simply implications we survey the mathematical theory of implications with an emphasis on the progress made in the last 30 years

polynomial identities and noncommutative versal torsors abstract to any cleft hopf galois object i e any algebra h obtained from a hopf algebra h by twisting its multiplication with a two cocycle we attach two universal algebras a h and u h the algebra a h is obtained by twisting the multiplication of h with the most general two cocycle formally cohomologous to the cocycle takes values in the field of rational functions on h by construction a h is a cleft h galois extension of a big commutative algebra b h any form of h can be obtained from a h by a specialization of b h and vice versa if the algebra h is simple then a h is an azumaya algebra with center b h the algebra u h is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras it is the universal comodule algebra in which all comodule algebra identities of h are satisfied we construct an embedding of u h into a h this embedding maps the center z h of u h into b h when the algebra h is simple in this case under an additional assumption a h b h z h u h thus turning a h into a central localization of u h we completely work out these constructions in the case of the four dimensional sweedler algebra