allowing agents to be imprecise a proposal using multiple linguistic terms in this paper we propose a decision making procedure where the agents judge the alternatives through linguistic terms such as very good good acceptable etc if the agents are not confident about their opinions they can use a linguistic expression formed by several consecutive linguistic terms to obtain a ranking on the set of alternatives the method consists of three different stages the first stage looks for the alternatives in which the overall opinion is closer to the ideal assessment the overall opinion is developed by a distance based process among the individual assessments the next two stages form a tie breaking process firstly by using a dispersion index based on the gini coefficient and secondly by taking into account the number of best assessments the main characteristics of the proposed decision making procedure are analyzed

the solution to the conjecture on properness of weakly relaxed delayed controls in 1986 warga proposed a weak relaxation procedure applicable to fully nonlinear problems with delays in the control variables and showed that the resulting relaxed problem has a solution however in the event of commensurate delays several examples were found for which weakly relaxed controls cannot be approximated with original controls so that this extension fails to be proper although the case of commensurate delays was solved by the introduction of a strong model the question of how to properly relax noncommensurately delayed controls has remained open and a natural candidate has been precisely that of weakly relaxed controls in this paper a general counterexample is constructed which rules out the weak relaxation as a proper relaxation when there are two or more delays it is hoped that this result will provide some insight into the problem of finding a general representation of properly relaxed controls

coordinatization of triangulated jordan systems we obtain partial coordinatizations of any quadratic jordan triple system containing a triangle e u e i e a split jo ke ku 0 ke z hz k such that e e ez is invertible and u is faithful in the sense that l x e u 0 x 0 this is automatic if j is nondegenerate or t e k we identify a hermitian matrix subsystem jh j e du j2 e2 e h d d 7t and a clifford subsystem j k no k e j q s c we show that a simple j must be all hermitian or all clifford coincides with j or j yielding a shorter proof osborn s capacity 2 theorem for jordan algebras we also can coordinatize most unital bimodules for hermitian matrix systems the impetus for this study came from two sources in presenting a selfcontained treatment of jordan structure theory n jacobson 4 sought a shorter proof of the osborn capacity 2 theorem the proof is not short for linear jordan algebras 2 and is quite involved for quadratic jordan algebras 3 in coordinatizing jordan triple systems the second author needed a version of the capacity 2 theorem making no semisimplicity restriction on the ambient system and without assuming that the tripotents e are division tripotents lo by avoiding semisimplicity assumptions such a formulation also coordinatizes unital bimodules in the triangulated case one cannot expect as precise a coordinatization as for rank n 3 in the latter case if j is a jordan algebra then jz

ranking multi attribute alternatives on the basis of linguistic labels in group decisions this paper presents a new approach based upon qualitative reasoning techniques for representing and synthesising the information given by a group of evaluators a mathematical formulation is developed that contributes to decision making analysis in the context of multi attribute and group decision making this method is applied in choice and ranking problems and can work at different precision levels to represent non trivial domain knowledge the patterns or alternatives to be ranked are characterised by a set of features which are evaluated by each member of the group through linguistic labels corresponding to ordinal values different levels of precision are considered to draw the distinctions required by evaluators reasoning processes the method used for ranking alternatives is based on comparing distances against an optimal reference point a theorem on the consistency of the proposed method is presented and proved three real life applications are presented to outline areas of management where the proposed method has been implemented and achieved interesting results

local orders in jordan algebras abstract making use of results on general algebras of quotients of jordan algebras we study a notion of local order based on the version for linear jordan algebras of the ideas of fountain and gould 14 as adapted to the jordan context by fernandez lopez and garcia rus in 7 in particular we characterize the set of lesieur croisot elements of a nondegenerate jordan algebra as those elements of the jordan algebra lying in the socle of its maximal algebra of quotients and apply this relationship to extend to quadratic jordan algebras the results of fernandez lopez and garcia rus on local orders in nondegenerate jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors

higher order conditions for conical controllability let q be a convex subset of a linear space mathcal u subset q mathcal z a topological vector space c subset mathcal z convex with a nonempty interior bar q in q varphi varphi 1 varphi 2 q to mathbb r m times mathcal z and varphi 2 bar q in c also let mathcal a u in mathcal u varphi 2 u in c this framework can be used to model many optimization problems including control problems with unilateral state inequality constraints sufficient conditions are derived for varphi 1 mathcal a to cover the interior of a convex set containing varphi 1 bar q and an open neighborhood of a point varphi 1 bar q w for w in mathbb r m these sufficient conditions can be used to provide information about the attainable sets of control problems and in some cases to show that varphi 1 mathcal a contains a neighborhood of varphi 1 bar q several illustrative examples are also discussed

a general class of simple majority decision rules based on linguistic opinions in this paper we have introduced a class of decision rules related to simple majority by considering individual intensities of preference these intensities will be shown by means of linguistic labels in order to compare the amount of opinion obtained by each alternative we have considered the total ordered monoid generated by the sums of the original labels according to an addition and an ordering in this general framework different sets of linguistic labels can be employed and these sets can be represented by means of diverse mathematical objects moreover on these mathematical representations of linguistic labels several orderings can be considered thus flexibility is an important feature of this new class of group decision making procedures some examples of putting in practice the simple majority decision rules based on linguistic labels are provided and the main properties of these voting systems are analyzed it is worth emphasizing that these properties are satisfied for any total ordered monoid regardless of the mathematical representation of linguistic labels or the ordering used to compare collective opinions

derivations and automorphisms of jordan algebras in characteristic two a jordan algebra j over a field k of characteristic 2 becomes a 2 lie algebra l j with lie product x y x y and squaring x 2 x 2 we determine the precise ideal structure of l j in case j is simple finite dimensional and k is algebraically closed we also decide which of these algebras have smooth automorphism groups finally we study the derivation algebra of a reduced albert algebra j h3 o k and show that der j has a unique proper nonzero ideal vj isomorphic to l j k 1j with quotient der j v j independent of o on the group level this gives rise to a special isogeny between the automorphism group of j and that of the split albert algebra whose kernel is the infinitesimal group determined by vj

fine gradings on simple exceptional jordan pairs and triple systems abstract we give a classification up to equivalence of the fine group gradings by abelian groups on the jordan pairs and triple systems of types bi cayley and albert under the assumption that the base field is algebraically closed of characteristic different from 2 the associated weyl groups are computed we also determine for each fine grading on the bi cayley and albert pairs the induced grading on the exceptional simple lie algebra given by the tits kantor koecher construction

racines orthogonales formes quadratiques et orbites d algebres de lie simples graduees abstract we give a unified orbital decomposition for irreducible regular prehomogeneous vector spaces of parabolic type graded by strongly orthogonal roots of same length with one dimensional corresponding root space on a local or global field of characteristic 0 in terms of a class of quadratic forms related to a root system weyl action and the invariants of the prehomogeneous vector spaces

strongly and weakly relaxed controls for time delay systems in recent articles two different relaxation procedures for time delay problems in optimal control have been proposed a weak procedure due to warga nonadditively coupled delayed controls privately circulated 1986 for which the existence of minimizers is assured applies to fully nonlinear problems with delays in the state and control variables nevertheless an example is found for which the effect of weak relaxation reduces the infimum cost the example involves two delays one being twice the value of the other and it belongs to a class of problems that is called commensurate where the quotient of any two delays is rational for these problems a strong procedure is proposed and it is proved that the extended problem has a solution and the extension is proper i e the infimum costs coincide in this paper it is shown through several examples why in general the weak relaxation technique may fail to provide a proper extension and for noncommensurate delay problems an approximation r

local pi theory of jordan systems ii abstract in this paper we study jordan systems having nonzero local algebras that satisfy a polynomial identity we prove that in nondegenerate jordan systems the set of elements at which the local algebra is pi is an ideal and that if it is nonzero it coincides with the socle when the system is primitive

a topological index condition for conical controllability covers the interior of a convex set containing e q and a neighborhood of a y w we say that p1 is controllable at 4 if it is w conically controllable at w 0 these concepts are motivated by problems of the optimal control theory and are closely related to the investigation of attainable sets certain higher order sufficient conditions for controllability and conical controllability were derived in l 21 these conditions also ensure that controllability respectively conical controllability is preserved when 9 p2 is replaced by v e pz ez for all higher order perturbations et ej in the present paper we generalize the basic result of 2 by replacing a homeomorphism condition with a more general nonzero index condition let v be an open subset of ky h i r continuous and w e r h av we refer to the elements of g w as w points of g let 71 be a bounded connected component of h w admitting an open neighborhood u whose closure is contained in i and is disjoint from other components of h w we refer to such a component rc as a bounded isolated w patch of h and denote by i h rrw w the topological degree d h u w for any choice of u as described above 3 we refer to i h rr w as the index of h at w with respect to rc we also denote by jh x the jacobian determinant det h x

higher order conditions with and without lagrange multipliers let q be a convex subset of a vector space mathcal u subset q mathcal s a topological vector space c a convex subset of mathcal z with a nonempty interior phi phi 1 phi 2 q to mathbb r m times mathcal z bar q in q and phi 2 bar q in c we assume that phi has a pth order taylor approximation at bar q when it is restricted to an arbitrary finite dimensional simplex in q with a vertex at bar q in the case when mathcal u is a proper subset of q we also assume that qis a uniform space phi continuous and mathcal u abundant we establish a number of higher order sufficient conditions not involving any lagrange multipliers for the existence of neighborhoods g 1 and g 2 of the origins such that phi 1 bar q g 1 subset phi 1 u u in mathcal u phi 2 u g 2 subset c these sufficient conditions involving nonconvex sets of variations are shown by an example to be stronger than those in the literature we

a fuzzy linguistic methodology to deal with unbalanced linguistic term sets many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects in most of these problems a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed however there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets i e using term sets that are not uniformly and symmetrically distributed the use of linguistic variables implies processes of computing with words cw different computational approaches can be found in the literature to accomplish those processes the 2 tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed in this paper we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets to do so we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2 tuple fuzzy linguistic computational model to accomplish processes of cw with unbalanced term sets in a precise way and without loss of information

goldie theory for jordan algebras it is shown that zelmanov s version of goldie s conditions still characterizes quadratic jordan algebras having an artinian algebra of quotients which is nondegenerate at the same time jordan versions of the main notions of the associative theory such as those of the uniform ideal uniform element singular ideal and uniform dimension are studied moreover it is proved that the nondegenerate unital jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element 2002 elsevier science usa

defining the borda count in a linguistic decision making context different kinds of decision rules have been successfully implemented under a linguistic approach this paper aims the same goal for the borda count a well known procedure with some interesting features in order to this two ways of extension from the borda rule to a linguistic framework are proposed taking into account all the agents opinions or only the favorable ones for each alternative when compared with each other in the two cases both individual and collective borda counts are analyzed asking for properties as good as those of the original patterns

jordan canonical form for finite rank elements in jordan algebras the usual jordan canonical form for matrices is extended first to nilpotent elements of the socle of a nondegenerate jordan algebra and then to elements of a nondegenerate jordan algebra which is reduced over an algebraically closed field 0 elsevier science inc 1997

a proper relaxation of shifted and delayed controls abstract we consider optimal control problems defined by functional integral equations with variable shifts in the controls and with the shifted controls nonadditively coupled as a special case we obtain differential equations with delays and or advances in both the state and the control functions we prove that if the original controls are measurable functions u with values in a compact metric space then the k 1 tuples u u u h 1 u h k of shifted controls have relaxed limits that form a compact and convex set this enables us to prove an existence theorem to approximate any relaxed solution with original solutions and to derive necessary conditions generalizeing pontryagin s maximum principle in integral form even for nonsmooth problems we also discuss the dependence on the shifts hj of the set of original controls u of the set of their relaxed limits and of the set of the corresponding solutions of the functional integral equation our results include the proof that the d model of rosenblueth and vinter is proper

relaxation procedures for time delay systems abstract despite the attention that optimal control problems involving time delays have received over the last two decades little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers recently warga has proposed a relaxation procedure for fully nonlinear problems with delays in the state and control variables and showed that the resulting relaxed problem has a solution nevertheless the effect of this relaxation may reduce the infimum cost as we show through an example which belongs to a class of problems with commensurate delays i e the quotient of any two delays is rational for problems of this nature we provide a new relaxation procedure for which the extension is proper i e the infimum costs coincide we also present an abstract relaxation procedure applicable to problems with possibly noncommensurate delays

diagonalization in jordan pairs as every student of linear algebra knows a rectangular matrix over a division ring can be diagonalized by elementary row and column operations similarly there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations since matrices of this kind form the main examples of jordan pairs it is natural to ask whether similar results hold in the jordan setting we show that this is in fact the case study the obstruction to diagonalizability the defect and also prove that nondegenerate jordan pairs admit a rank function sharing many properties with classical matrix rank let v v v be a jordan pair and s e e a set of orthogonal idempotents the s diagonal elements are those in xi 1 v e now suppose that v is nondegenerate and satisfies dcc on principal inner ideals an element x e v a i is called diagonalizabfe if x is s diagonal for some s consisting of division idempotents to see that for matrices this is the same as the usual notion of diagonalizability suppose that v has in addition act on principal inner ideals then v contains a frame that is a finite set f of orthogonal division idempotents such that v f 0 for rectangular or hermitian matrices over division rings f can be taken to consist of the diagonal matrix units e eii whereas for alternating matrices over a field e eli1 2ie zi the jordan analogues of elementary row and column operations are the inner automorphisms b v e v e and v e v e where e e f which generate the group of f elementary automorphisms of v 6 any set of orthogonal division idempotents can be transformed up to association into f by an f elementary automorphism 6 th 21 it follows that x is diagonalizable if and only if cp x is f diagonal for some elementary automorphism cp of v this shows the equivalence with the usual definition

global directional controllability this paper derives sufficient conditions for the image of a function varphi 1 q to mathbb r to cover an open segment originating at varphi 1 bar q and with direction w or even a neighborhood of such an interval subject to restrictions of the form varphi 2 q in c or varphi 2 q g subset c here varphi 2 q to mathcal z q is an arbitrary set c a convex subset of the topological vector space mathcal z and g a neighborhood of 0 in mathcal z also derived are similar directional controllability conditions subject to the additional restriction q in mathcal u subset q for an abundant subset mathcal u of q these conditions are applicable to unilateral problems of control theory and of infinite dimensional mathematical programming these results are global and apply to problems defined by functions whose restrictions to certain finite dimensional sets are differentiable but not necessarily c 1 or are locally uniform limits of ifferentiable fu

minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making in some real world decision processes decision makers may prefer to provide their opinions using linguistic expressions instead of a single linguistic term particularly they may hesitate between several linguistic terms in this paper we deal with the consensus issue in the hesitant linguistic group decision making gdm problem firstly a novel distance based consensus measure is proposed then using this consensus measure we develop an optimization based consensus model in the hesitant linguistic gdm which minimizes the number of adjusted simple terms in the consensus building furthermore a two stage model is displayed to further optimize the solutions to the proposed consensus model through which we obtain the unique optimal adjustment suggestion to support the consensus reaching process in the hesitant linguistic gdm finally several desirable properties are proposed to justify the proposal and two examples are used to demonstrate the validity of the models