ordinal regression revisited multiple criteria ranking using a set of additive value functions abstract we present a new method called utagms for multiple criteria ranking of alternatives from set a using a set of additive value functions which result from an ordinal regression the preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives ar a called reference alternatives the preference model built via ordinal regression is the set of all additive value functions compatible with the preference information using this model one can define two relations in the set a the necessary weak preference relation which holds for any two alternatives a b from set a if and only if for all compatible value functions a is preferred to b and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b these relations establish a necessary and a possible ranking of alternatives from a being respectively a partial preorder and a strongly complete relation the utagms method is intended to be used interactively with an increasing subset ar and a progressive statement of pairwise comparisons when no preference information is provided the necessary weak preference relation is a weak dominance relation and the possible weak preference relation is a complete relation every new pairwise comparison of reference alternatives for which the dominance relation does not hold is enriching the necessary relation and it is impoverishing the possible relation so that they converge with the growth of the preference information distinguishing necessary and possible consequences of preference information on the complete set of actions utagms answers questions of robustness analysis moreover the method can support the decision maker when his her preference statements cannot be represented in terms of an additive value function the method is illustrated by an example solved using the utagms software some extensions of the method are also presented

a generalization of coxeter groups root systems and matsumoto s theorem the root systems appearing in the theory of lie superalgebras and nichols algebras admit a large symmetry extending properly the one coming from the weyl group based on this observation we set up a general framework in which the symmetry object is a groupoid we prove that in our context the groupoid is generated by simple reflections and coxeter relations in a broad sense this answers a question of serganova our weak version of the exchange condition allows us to prove matsumoto s theorem therefore the word problem is solved for the groupoid

a borcherds kac moody superalgebra with conway symmetry we construct a borcherds kac moody bkm superalgebra on which the conway group hbox co 0 acts faithfully we show that the bkm algebra is generated by the physical states brst cohomology classes in a chiral superstring theory we use this construction to produce denominator identities for the chiral partition functions of the conway module v s natural a supersymmetric c 12 chiral conformal field theory whose twisted partition functions enjoy moonshine properties and which has automorphism group isomorphic to hbox co 0 in particular these functions satisfy a genus zero property analogous to that of monstrous moonshine finally we suggest how one may promote the denominators to spacetime bps indices in type ii string theory which might thus furnish a physical explanation of the genus zero property of conway moonshine

isolated singularities of graphs in warped products and monge ampere equations we study graphs of positive extrinsic curvature with a non removable isolated singularity in 3 dimensional warped product spaces and describe their behavior at the singularity in several natural situations we use monge ampere equations to give a classification of the surfaces in 3 dimensional space forms which are embedded around a non removable isolated singularity and have a prescribed real analytic positive extrinsic curvature function at every point specifically we prove that this space is in one to one correspondence with the space of regular analytic strictly convex jordan curves in the 2 dimensional sphere s2

missing modules the gnome lie algebra and e10 we study the embedding of kac moody algebras into borcherds or gener alized kac moody algebras which can be explicitly realized as lie algebras of physical states of some completely compactified bosonic string the extra missing states can be decomposed into irreducible highest or lowest weight missing modules w r t the relevant kac moody subalgebra the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of cer tain polarization states of the associated string we analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular lorentzian latticeii 1 1 orii 9 1 respectively the former leads to the borcherds algebra gii 1 1 which we call gnome lie algebra with maximal kac moody subalgebraa1 by the use of the denominator formula a complete set of imaginary simple roots can be exhibited while the ddf construction provides an explicit lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions the second example is the borcherds algebra gii 9 1 whose maximal kac moody subalgebra is the hyperbolic algebra e10 the imaginary simple roots at level 1 which give rise to irreducible low est weight modules fore10 can be completely characterized furthermore our explicit analysis of two non trivial level 2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for gii 9 1

automorphisms of borcherds superalgebras and fixed point subalgebras abstract let g be a borcherds superalgebra and let be a finite order automorphism of g satisfying certain conditions we prove that the fixed point subalgebra g is equal to a borcherds superalgebra we also compute the trace of on g from the twisted denominator identity of g and derive closed form formula for root multiplicities of g

dimension formula for graded lie algebras and its applications abstract in this paper we investigate the structure of infinite dimensional lie algebras l l l graded by a countable abelian semigroup satisfying a certain finiteness condition the euler poincare principle yields the denominator identities for the graded lie algebras from which we derive a dimension formula for the homogeneous subspaces l our dimension formula enables us to study the structure of the graded lie algebras in a unified way we will discuss some interesting applications of our dimension formula to the various classes of graded lie algebras such as free lie algebras kac moody algebras and generalized kac moody algebras we will also discuss the relation of graded lie algebras and the product identities for formal power series

liftings of nichols algebras of diagonal type iii cartan type g2 abstract we complete the classification of hopf algebras whose infinitesimal braiding is a principal yetter drinfeld realization of a braided vector space of cartan type g 2 over a cosemisimple hopf algebra we develop a general formula for a class of liftings in which the quantum serre relations hold we give a detailed explanation of the procedure for finding the relations based on the recent work of andruskiewitsch angiono and rossi bertone

meta goal programming abstract in most of the goal programming gp applications reported in the literature the election of the gp variant is made in a mechanistic way by the analyst however each variant implies a different preference structure and thus it should be chosen by the decision maker moreover the final solution of the problem depends critically on the variant chosen in this paper it is assumed that the use of a combination of gp variants may reflect in many cases more precisely the actual preferences of the decision maker in this way a sensitivity analysis of the solution obtained as well as of the own model structure can be very helpful to let the decision maker clarify his her knowledge about the reality analysed this type of problem requires a more general approach that leads to a gp formulation of the initial gp model this type of meta gp approach is presented in this paper

bi matching a new preference assessment method to reduce compatibility effects preference models and utility functions are often assessed by eliciting value trade offs among attributes prior research has shown that trade off judgments can be biased in systematic ways for example the attribute which is used as response receives more relative subjective weight i e the so called scale compatibility effects tversky et al tversky a s sattath p slovic 1988 contingent weighting in judgment and choice psych rev 95 371 384 this paper proposes a new procedure to elicit value trade offs called bidimensional matching or bi matching designed to alleviate this effect bi matching differs from traditional trade off judgments in that both attributes are adjusted simultaneously to reach indifference judgments bi matching is compared with simple matching and choice in four experimental studies to measure preferences for lotteries and riskless multiattribute alternatives the main results are 1 bi matching produces trade offs intermediate between those derived from matching on the more important attribute and matching on the less important attribute although closer to the former 2 the trade offs derived from choice reflect more relative weight on the more important dimension than those from bi matching 3 bi matching appears to reduce response error compared to standard matching these results are generally consistent with theoretical predictions we discuss the normative question of which preference assessment method is preferable the current results as a whole and the built in features of the bi matching procedure already position this elicitation method as a worthwhile alternative to traditional methods for helping decision makers introspect and construct their value trade offs

when conflict induces the expression of incomplete preferences multicriteria conflict arises in pairwise comparisons where each alternative outperforms the other one on some criterion which imposes a trade off comparing two alternatives can be difficult if their respective advantages are of high magnitude the attribute spread is large in this paper we investigate to which extent conflict in a comparison situation can lead decision makers to express incomplete preferences that is to refuse to compare the two alternatives or to be unable to compare them with confidence we report on an experiment in which subjects expressed preferences on pairs of alternatives involving varying conflicts results show that depending on whether the participants are allowed to express incomplete preferences or not attribute spread has a different effect a large attribute spread increases the frequency of incomparability statements when available while it increases the use of indifference statements when only indifference and preference answers are permitted these results lead us to derive some implications for preference elicitation methods involving comparison tasks

grothendieck rings of basic classical lie superalgebras the grothendieck rings of nite dimensional representations of the basic classical lie superalgebras are explicitly described in terms of the corresponding generalized root systems we show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super weyl groupoids

a classification model based on goal programming with non standard preference functions with application to the prediction of cinema going behaviour this paper presents a goal programming model that allows for the flexible handling of the two group classification problem the goal programming model is based around the concepts of non standard preference functions and penalty function modelling an extension to a generalised distance metric case is given the inclusion of multiple levels of classification based upon different levels of certainty is incorporated into the model the model is tested on a real life data set pertaining to cinema going attendance and conclusions are drawn both in the context of the methodology and of the application

monstrous moonshine from orbifolds we consider the monster module of frenkel lepowsky and meurman as az2 orbifold of a bosonic string compactified by the leech lattice we show that the main conway and norton monstrous moonshine properties stating that the thompson series for each monster group conjugacy class has a modular invariance group of genus zero follow from an orbifold construction based on an orbifold group composed of monster group elements it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each thompson series it is also shown that the power map formula of conway and norton follows from the same vacuum structure finally we demonstrate the validity of the vacuum conjectures for sectors twisted by leech lattice automorphisms in many cases

rank 4 finite dimensional nichols algebras of diagonal type in positive characteristic abstract nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed hopf algebras by lifting method of andruskiewitsch and schneider arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property in the present paper all rank 4 nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified our proof uses the classification of the finite arithmetic root systems of rank 4

standard and nonstandard extensions of lie algebras we study the problem of quadruple extensions of simple lie algebras we find that adding a new simple root 4 it is not possible to have an extended kac moody algebra described by a dynkin kac diagram with simple links and no loops between the dots while it is possible if 4 is a borcherds imaginary simple root we also comment on the root lattices of these new algebras the folding procedure is applied to the simply laced triple extended lie algebras obtaining all the nonsimply laced ones nonstandard extension procedures for a class of lie algebras are proposed it is shown that the two extensions of e8 with a dot simply linked to the dynkin kac diagram of e9 are rank 10 subalgebras of e10 finally the simple root systems of a set of rank 11 subalgebras of e11 containing as sub algebra e10 are explicitly written

logarithmic widehat s ell 2 cft models from nichols algebras i we construct chiral algebras that centralize rank 2 nichols algebras with at least one fermionic generator this gives logarithmic w algebra extensions of a fractional level algebra we discuss crucial aspects of the emerging general relation between nichols algebras and logarithmic conformal field theory cft models i the extra input beyond the nichols algebra proper needed to uniquely specify a conformal model ii a relation between the cft counterparts of nichols algebras connected by weyl groupoid maps and iii the common double bosonization u x of such nichols algebras for an extended chiral algebra candidates for its simple modules that are counterparts of the u x simple modules are proposed as a first step toward a functorial relation between u x and w algebra representation categories

involutive automorphisms and iwasawa decomposition of some hyperbolic kac moody algebras the involutive automorphisms of hyperbolic kac moody algebras are computed from satake diagrams of these algebras which are then used to furnish a general procedure of their iwasawa decomposition in particular 1 1 and 2 1 are considered as representative examples to illustrate the efficacy of the underlying treatment

quantized enveloping algebras for borcherds superalgebras the rst author was supported in part by nsf grant dms 9300523 the second author was supported in part by the nondirected research fund korea research foundation 1996 the third author was supported in part by a faculty research grant from st lawrence university

on nichols algebras over basic hopf algebras this is a contribution to the classification of finite dimensional hopf algebras over an algebraically closed field mathbb k of characteristic 0 concretely we show that a finite dimensional hopf algebra whose hopf coradical is basic is a lifting of a nichols algebra of a semisimple yetter drinfeld module and we explain how to classify nichols algebras of this kind we provide along the way new examples of nichols algebras and hopf algebras with finite gelfand kirillov dimension

the effect of bi criteria conflict on matching elicited preferences we focus on multicriteria preference elicitation by matching in this widely employed task the decision maker dm is presented with two multicriteria options a and b and must assess the performance value on one criterion for b left blank so that she is indifferent between the two options a reverse matching which is normatively equivalent can be created by integrating the answer to the description of b and letting the dm adjust a performance value on the previously totally specified option a such a procedure is called a bi matching consistency requires that isopreferences resulting from the forward and backward matchings be identical but they empirically differ in a systematic direction in a matching task multicriteria conflict refers to the magnitude of the advantage or disadvantage to be compensated we investigate the effect of the multicriteria conflict or trade off size on the difference of judgement between forward and backward matchings we observed that the difference of judgement is increased both by multicriteria conflict and by asking deteriorating rather than improving judgements at both steps of the bi matching we derive some implications for the practice of preference elicitation

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

nichols algebras over groups with finite root system of rank two iii abstract we study nichols algebras of direct sums of two absolutely simple yetter drinfeld modules over non abelian groups and arbitrary fields we construct families of new examples where the nichols algebra is finite dimensional and not of diagonal type

can a linear value function explain choices an experimental study abstract we investigate in a simple bi criteria experimental study whether subjects are consistent with a linear value function while making binary choices many inconsistencies appeared in our experiment however the impact of inconsistencies on the linearity vs non linearity of the value function was minor moreover a linear value function seems to predict choices for bi criteria problems quite well this ability to predict is independent of whether the value function is diagnosed linear or not inconsistencies in responses did not necessarily change the original diagnosis of the form of the value function our findings have implications for the design and development of decision support tools for multiple criteria decision making problems

vertex algebras lie algebras and superstrings abstract we construct lie algebras from vertex superalgebras and study their structure they are sometimes generalized kac moody algebras in some special cases we can calculate the multiplicities of the roots

minimal generators of ringel hall algebras of affine quivers abstract let h kq be the ringel hall algebra of affine quiver q all indecomposable representations of q which can be generated inside h kq by smaller representations of q are classified systems of minimal homogeneous generators of h kq are explicitly written out

multi brace cotensor hopf algebras and quantum groups we give a construction of the whole quantum group associated to a symmetrizable kac moody lie algebra as a quantum quasi symmetric algebra a particular case of multi brace cotensor hopf algebras in the spirit of the quantum shuffle construction of the plus part all highest weight irreducible representations are constructed using this machinery it also provides a systematic way to construct simple modules over the quantum double of a quantum group

the weak hopf algebras related to generalized kac moody algebra we define a kind of quantized enveloping algebra of a generalized kac moody algebra g by adding a generator j satisfying jm jm 1 for some integer m we denote this algebra by wuq g this algebra is a weak hopf algebra if and only if m 2 in general it is a bialgebra and contains a hopf subalgebra this hopf subalgebra is isomorphic to the usually quantum envelope algebra uq g of a generalized kac moody algebra g

multi attribute sequential decision problem with optimizing and satisficing attributes abstract we deal with the multi attribute decision problem with sequentially presented decision alternatives our decision model is based on the assumption that the decision maker has a major attribute that must be optimized and minor attributes that must be satisficed in the vendor selection problem for example the product price could be the major factor that should be optimized while the product quality and delivery time could be the minor factors that should satisfy certain aspiration levels we first derive the optimal selection strategy for the discrete time case in which one alternative is presented at each time period the discrete time model is then extended to the continuous time case in which alternatives are presented sequentially at random times a numerical example is used to analyze the effects of the satisficing condition and the uncertainty on the optimal selection strategy

skew differential operator algebras of twisted hopf algebras abstract by introducing a twisted hopf algebra we unify several important objects of study skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied this generalizes results in the weyl algebra applying this investigation to the twisted ringel hall algebra we get in particular a natural realization of the non positive part of a quantized generalized kac moody algebra by identifying the canonical generators with some linear skew differential operators this also induces some algebras which are quantum group like

graded lie superalgebras and the superdimension formula in this paper we investigate the structure of graded lie superalgebras l a a l a where is a countable abelian semigroup and a is a countable abelian group with a coloring map satisfying a certain finiteness condition given a denominator identity for the graded lie superalgebra l we derive a superdimension formula for the homogeneous subspaces l a a a which enables us to study the structure of graded lie superalgebras in a unified way we discuss the applications of our superdimension formula to free lie superalgebras generalized kac moody superalgebras and monstrous lie superalgebras in particular the product identities for normalized formal power series are interpreted as the denominator identities for free lie superalgebras we also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of monstrous lie superalgebras

on defining ideals and differential algebras of nichols algebras abstract this paper is devoted to understanding the defining ideal of a nichols algebra from the decomposition of specific elements in the group algebra of braid groups a family of primitive elements is found and algorithms are proposed to prove the main result the differential algebra of a nichols algebra is constructed moreover another point of view on serre relations is provided

generators and relations for lie superalgebras of cartan type we give an analog of a chevalley serre presentation for the lie superalgebras w n and s n of cartan type these are part of a wider class of lie superalgebras the so called tensor hierarchy algebras denoted w g and s g where g denotes the kac moody algebra a r d r or e r then w a n 1 and s a n 1 are the lie superalgebras w n and s n the algebras w g and s g are constructed from the dynkin diagram of the borcherds kac moody superalgebras b g obtained by adding a single grey node representing an odd null root to the dynkin diagram of g we redefine the algebras w a r and s a r in terms of chevalley generators and defining relations we prove that all relations follow from the defining ones at level 2 and higher the analogous definitions of the algebras in the d and e series are given in the latter case the full set of defining relations is conjectured

a practical weight sensitivity algorithm for goal and multiple objective programming this paper presents a weight sensitivity algorithm that can be used to investigate a portion of weight space of interest to the decision maker in a goal or multiple objective programme the preferential information required from the decision maker is an initial estimate of their starting solution with an equal weights solution being used as a default if this is not available and preference information that will define the portion of weight space on which the sensitivity analysis is to be conducted the different types of preferential information and how they are incorporated by the algorithm are discussed the output of the algorithm is a set of distinct solutions that characterise the portion of weight space searched the possible different output requirements of decision makers are detailed in the context of the algorithm the methodology is demonstrated on two examples one hypothetical and the other relating to predicting cinema going behaviour conclusions and avenues for future research are given

a generalization of borcherds algebra and denominator formula abstract we study generalized lie superalgebras an extension of kac s generalized lie superalgebras and show that they are transformed forms ofl graded lie superalgebras for some abelian groupl we then introduce a generalized lie superalgebra version of the generalized kac moody algebra borcherds algebra since it is a transformed borcherds superalgebra it has several properties similar to those of borcherds superalgebra for example it is defined by similar relations and has a similar denominator formula and character formulas

a classification result on prime hopf algebras of gk dimension one abstract in this paper we classify all prime hopf algebras h of gk dimension one satisfying the following two conditions 1 h has a 1 dimensional representation of order pi deg h and 2 the invariant components of h with respect to this 1 dimensional representation are domains see section 2 for related definitions as consequences 1 a number of new hopf algebras of gk dimension one are found and some of them are not pointed 2 we give a partial answer to a question posed in 12 and 3 the question 1 1 is solved

on cartan subalgebras abstract in this paper we study the concept of a cartan subalgebra in the context of locally finite and borcherds kac moody lie algebras

the cauchy problem for improper affine spheres and the hessian one equation we give a conformal representation for improper affine spheres which is used to solve the cauchy problem for the hessian one equation with this representation we characterize the geodesics of an improper affine sphere study its symmetries and classify the helicoidal ones finally we obtain the complete classification of the isolated singularities of the hessian one monge ampere equation

kazhdan lusztig conjecture for generalized kac moody algebras ii proof of the conjecture generalized kac moody algebras were introduced by borcherds in the study of conway and norton s moonshine conjectures for the monster sporadic simple group in this paper we prove the kazhdan lusztig conjecture for generalized kac moody algebras under a certain mild condition by using a generalization to the case of generalized kac moody algebras of jantzen s character sum formula our main formula generalizes the celebrated result for the case of kac moody algebras and describes the characters of irreducible highest weight modules over generalized kac moody algebras in terms of the extended kazhdan lusztig polynomials

on the algebras of bps states we define an algebra on the space of bps states in theories with extended supersymmetry we show that the algebra of perturbative bps states in toroidal compactification of the heterotic string is closely related to a generalized kac moody algebra we use d brane theory to compare the formulation of rr charged bps algebras in type ii compactification with the requirements of string string duality and find that the rr charged bps states should be regarded as cohomology classes on moduli spaces of coherent sheaves the equivalence of the algebra of bps states in heterotic iia dual pairs elucidates certain results and conjectures of nakajima and gritsenko nikulin on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras in particular to any calabi yau 3 fold there are two canonically associated algebras exchanged by mirror symmetry

ringel hall algebras and lusztig s symmetries abstract let d be the double ringel hall algebra of a finite dimensional hereditary algebra the present paper shows that the bgp reflection operators of d coincide with lusztig s symmetries up to canonical isomorphisms and satisfy the braid group relations on the whole double ringel hall algebra d this answers a question of sevenhant and van den bergh j algebra 221 1999 135 160

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

representations of u qsl 2 1 at even roots of unity we construct all projective modules of the restricted quantum group u qsl 2 1 at an even 2p th root of unity this 64p4 dimensional hopf algebra is a common double bosonization of two rank 2 nichols algebras x1d505 x with fermionic generator s we show that the category of u qsl 2 1 modules is equivalent to the category of yetter drinfeld x1d505 x modules in c hhy kern 1ptd for h 2p 2p where the coaction is defined by a universal r matrix h h as an application of the projective module construction we study the basic algebra of u qsl 2 1 and find the associative algebra structure and the dimension 5p2 p 4 of its center

on the classification of finite dimensional pointed hopf algebras we classify finite dimensional complex hopf algebras a which are pointed that is all of whose irreducible comodules are one dimensional and whose group of group like elementsg a is abelian such that all prime divisors of the order of g a are 7 since these hopf algebras turn out to be deformations of a natural class of generalized small quantum groups our result can be read as an axiomatic description of generalized small quantum groups

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

bi goal evolution for many objective optimization problems this paper presents a meta objective optimization approach called bi goal evolution bige to deal with multi objective optimization problems with many objectives in multi objective optimization it is generally observed that 1 the conflict between the proximity and diversity requirements is aggravated with the increase of the number of objectives and 2 the pareto dominance loses its effectiveness for a high dimensional space but works well on a low dimensional space inspired by these two observations bige converts a given multi objective optimization problem into a bi goal objective optimization problem regarding proximity and diversity and then handles it using the pareto dominance relation in this bi goal domain implemented with estimation methods of individuals performance and the classic pareto nondominated sorting procedure bige divides individuals into different nondominated layers and attempts to put well converged and well distributed individuals into the first few layers from a series of extensive experiments on four groups of well defined continuous and combinatorial optimization problems with 5 10 and 15 objectives bige has been found to be very competitive against five state of the art algorithms in balancing proximity and diversity the proposed approach is the first step towards a new way of addressing many objective problems as well as indicating several important issues for future development of this type of algorithms

probability versus certainty equivalence methods in utility measurement are they equivalent certainty equivalence ce and probability equivalence pe methods are the two most frequently used procedures for constructing von neumann morgenstern utility functions in this paper we compare these methods experimentally using a two stage within subject design by asking subjects first for a ce judgment and later for a related pe judgment or vice versa a consistency test is devised which any deterministic expectation model including those allowing probability transformations should meet using four related experiments this consistency test is applied separately to gain and loss questions and to the two sequences of linked equivalence judgments namely ce pe and pe ce the empirical results reveal serious inconsistencies between the ce and pe responses for each of the four experiments the extent of discrepancy depends strongly on the subject s initial risk attitude and whether the gain or loss domain is examined to explain the complex pattern of results the second part of the paper explores several plausible hypotheses the first of these concerns the role of random error in either the responses or the utility function itself it is shown that both can lead to bias although not of a type that could explain our results thereafter shifts in reference points are examined a particular reframing of the pe response mode is postulated in which a pure gamble is psychologically translated into a mixed one leading to increased risk aversion this hypothesis which is also supported by other evidence offers a complete and simple explanation of the results finally several other behavioral hypotheses are examined after developing a weighted average model to simulate them they concern anchoring effects differences in salience between the probability and outcome dimensions strategic misrepresentation regret or rejoice influences and endowment effects although each hypothesis predicts some type of bias none of these five could singly explain the particular pattern of bias observed in general the study demonstrates 1 that serious discrepancies exist between the ce and pe methods of utility measurement 2 that the particular results are incompatible with traditional deterministic choice models 3 how random response errors through propagation can induce systematic biases in the utility function 4 that reframing of the pe mode offers a simple reference shift explanation of the complex findings and 5 how various heuristics and biases can be operationalized and simulated to assess their effects on utility measurement as such this study represents a further step toward a systematic investigation of response mode biases in utility measurement

a general structure of achievement function for a goal programming model abstract a key element of a goal programming model is the achievement function that is the function that measures the degree of minimisation of the unwanted deviation variables of the goals considered in the model the three oldest and still most widely used forms of achievement functions are the following archimedean lexicographic and minmax while they are the most common other newer versions can offer theoretical and practical advantages these newer forms will be analytically described and interpreted in utility terms throughout this paper the presentation will end with a very general structure which encompasses as particular cases basically all the achievement functions proposed in the literature

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group

surfaces of constant curvature in r3 with isolated singularities abstract we prove that finite area isolated singularities of surfaces with constant positive curvature k 0 in r 3 are removable singularities branch points or immersed conical singularities we describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in s 2 with admissible cusp singularities characterizing when the singularity is actually embedded in the global setting we describe the space of peaked spheres in r 3 i e compact convex surfaces of constant curvature k 0 with a finite number of singularities and give applications to harmonic maps and constant mean curvature surfaces

the weyl groupoid of a nichols algebra of diagonal type the theory of nichols algebras of diagonal type is known to be closely related to that of semi simple lie algebras in this paper the connection between both theories is made closer for any nichols algebra of diagonal type invertible transformations are introduced which remind one of the action of the weyl group on the root system associated to a semi simple lie algebra they give rise to the definition of a groupoid as an application an alternative proof of classification results of rosso andruskiewitsch and schneider is obtained without using any technical assumptions on the braiding

on subsequences of quiddity cycles and nichols algebras abstract we provide a tool to obtain local descriptions of quiddity cycles as an application we classify rank two affine nichols algebras of diagonal type

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

the r matrix of quantum doubles of nichols algebras of diagonal type let h be the quantum double of a nichols algebra of diagonal type we compute the r matrix of 3 tuples of modules for general finite dimensional highest weight modules over h we also calculate a multiplicative formula for the universal r matrix when h is finite dimensional we show the unicity of a pbw basis or a lusztig type poincare birkhoff witt basis with a given convex order

extended affine root supersystems abstract the interaction of a lie algebra l having a weight space decomposition with respect to a nonzero toral subalgebra with its corresponding root system forms a powerful tool in the study of the structure of l this in particular suggests a systematic study of the root system apart from its connection with the lie algebra although there have been a lot of researches in this regard on lie algebra level such an approach has not been considered on lie superalgebra level in this work we introduce and study extended affine root supersystems which are a generalization of both affine reflection systems and locally finite root supersystems extended affine root supersystems appear as the root systems of the super version of extended affine lie algebras and invariant affine reflection algebras including affine lie superalgebras this work provides a framework to study the structure of this kind of lie superalgebras refereed to as extended affine lie superalgebras

on pointed hopf algebras associated with unmixed conjugacy classes in sm let sm be a product of disjoint cycles of the same length c the conjugacy class of and an irreducible representation of the centralizer of we prove that either the nichols algebra b c is infinite dimensional or the braiding of the yetter drinfeld module is negative

the arithmetic mirror symmetry and calabi yau manifolds we extend our variant of mirror symmetry for k3 surfaces gn3 and clarify its relation with mirror symmetry for calabi yau manifolds we introduce two classes for the models a and b of calabi yau manifolds fibrated by k3 surfaces with some special picard lattices these two classes are related with automorphic forms on iv type domains which we studied in our papers gn1 gn6 conjecturally these automorphic forms take part in the quantum intersection pairing for model a yukawa coupling for model b and mirror symmetry between these two classes of calabi yau manifolds recently there were several papers by physicists where it was shown on some examples we propose a problem of classification of introduced calabi yau manifolds our papers gn1 gn6 and n3 n14 give hope that this is possible they describe possible picard or transcendental lattices of general k3 fibers of the calabi yau manifolds

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

nichols algebras of group type with many quadratic relations abstract we classify nichols algebras of irreducible yetter drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two all such nichols algebras are finite dimensional and all known finite dimensional nichols algebras of nonabelian group type appear in the result of our classification we find a new finite dimensional nichols algebra over fields of characteristic two

incorporating additional meta objectives into the extended lexicographic goal programming framework this paper introduces two new meta objectives into the extended goal programming framework the first meta objective is the number of unmet goals and the second is a measure of closeness to the pairwise comparisons given by the decision maker these complement the original two meta objectives of the weighted sum of deviations and the maximal weighted deviation to provide a flexible four meta objective framework lexicographic and non lexicographic representations of the framework are developed an example relating to transportation is solved in both the lexicographic and non lexicographic forms weight sensitivity analysis is applied to the meta weight parameters for the non lexicographic case in order to find a range of available distinct solutions

root systems and weyl groupoids for nichols algebras motivated by work of kac and lusztig we define a root system and a weyl groupoid for a large class of semisimple yetter drinfeld modules over an arbitrary hopf algebra the ob tained combinatorial structure fits perfectly into an existing frame work of generalized root systems associated to a family of cartan matrices and provides novel insight into nichols algebras we demonstrate the power of our construction with new results on nichols algebras over finite non abelian simple groups and sym metric groups

a simplicial complex of nichols algebras we translate the concept of restriction of an arrangement in terms of hopf algebras in consequence every nichols algebra gives rise to a simplicial complex decorated by nichols algebras with restricted root systems as applications some of these nichols algebras provide weyl groupoids which do not arise for diagonal nichols algebras and in fact we realize all root systems of finite weyl groupoids of rank greater than three further our result explains the root systems of the folded nichols algebras over nonabelian groups and of generalized satake diagrams

on a factorization of graded hopf algebras using lyndon words abstract we find a generalization of the restricted pbw basis for pointed hopf algebras over abelian groups constructed by kharchenko we obtain a factorization of the hilbert series for a wide class of graded hopf algebras these factors are parametrized by lyndon words and they are the hilbert series of certain graded hopf algebras