universal central extensions of twisted forms of split simple lie algebras over rings abstract we give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple lie algebra over a ring in particular the universal central extensions of twisted multiloop lie tori are obtained by the descent construction

demazure modules and weyl modules the twisted current case we study finite dimensional respresentations of twisted current algebras and show that any graded twisted weyl module is isomorphic to level one demazure modules for the twisted affine kac moody algebra using the tensor product property of demazure modules we obtain by analyzing the fundamental weyl modules dimension and character formulas moreover we prove that graded twisted weyl modules can be obtained by taking the associated graded modules of weyl modules for the loop algebra which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight these results were previously known for untwisted current algebras and are new for all twisted types

multi dimensional weyl modules and symmetric functions the weyl modules in the sense of v chari and a pressley cp over the current lie algebra on an affine variety are studied we show that local weyl modules are finite dimensional and generalize the tensor product decomposition theorem from cp more explicit results are stated for currents on a non singular affine variety of dimension d with coefficients in the lie algebra sl r the weyl modules with highest weights proportional to the vector representation one are related to the multi dimensional analogs of harmonic functions the dimensions of such local weyl modules are calculated in the following cases for d 1 we show that the dimensions are equal to powers of r for d 2 we show that the dimensions are given by products of the higher catalan numbers the usual catalan numbers for r 2

coordinate algebras of extended affine lie algebras of type a1 abstract the cores of extended affine lie algebras of reduced types were classified except for type a1 in this paper we determine the coordinate algebra of extended affine lie algebras of type a1 it turns out that such an algebra is a unital z n graded jordan algebra of a certain type called a jordan torus we classify jordan tori and get five types of jordan tori

towards a higher degree f transform the aim of this study is to show how the f transform technique can be generalized from the case of constant components to the case of polynomial components after a general presentation of an f m transform m 0 a detailed characterization of the f 1 transform is given we apply a numeric integration technique in order to simplify the computation of f 1 transform components the inverse f m transform m 0 is defined similarly to the ordinary inverse f transform the quality of approximation using the inverse f m transform increases with an increase in m

drinfel d jimbo coproduct of quantized gim lie algebras abstract we define a sequence of quotient algebras of a quantized gim lie algebra s tensor algebras modulo ideals generated by some imaginary root vectors so that there are algebra morphisms similar to the drinfel d jimbo coproduct of quantum groups and hence the quantized gim lie algebra has properties which are similar to those of hopf algebras weight modules and quantum loop modules of the corresponding quantum affine kac moody algebra are used to construct tensor modules of the quantized gim lie algebra via the coproduct

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

integrable representations of root graded lie algebras abstract in this paper we study the category of representations of a root graded lie algebra l which are integrable as representations of a finite dimensional semisimple subalgebra g and whose weights are bounded by some dominant weight of g we link this category to the module category of an associative algebra whose structure we determine for map algebras and sl n a our approach unifies recent work of chari and her collaborators on map algebras of fourier and savage and their collaborators on equivariant map algebras as well as the classical work of seligman on isotropic lie algebras

necessary and sufficient conditions for generalized uniform fuzzy partitions abstract the fundamental concept in the theory of fuzzy transform f transform is the fuzzy partition which is a generalization of the classical concept of partition the original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1 this condition can be generalized by relaxing the assumption of normality for fuzzy sets the result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse f transform a fuzzy partition with this property will be referred to as general the problem is how a general fuzzy partition can be effectively constructed if we use a generating function with special properties then it is not immediately clear whether it defines a general fuzzy partition in this paper we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily which is important in various practical applications of the f transform for example image processing time series analysis and solving differential equations with boundary conditions

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

central extension of quadratic lie algebras and its relation to dihedral homology if a is a k algebra where k is a commutative ring then kassel and w x w x loday in 5 study sl a s gl a gl a where gl a is the lie n n n n algebra of n n matrices over a they define a steinberg lie algebra st a which they prove is a universal central lie algebra extension of n sl a furthermore the kernel of this extension is isomorphic to a cyclic n homology group which in our current notation is designated as hc a 1 in this paper we prove a similar result for a a k algebra with involution a a a which acts trivially on k given a form parameter l and l g center a such that ll s 1 we have a form algebra a l l see w x 1 the lie algebra adaptation of the general l quadratic group gq a l will be denoted as gq l and will be a sub lie algebra of 2 n 2 n gl a we will show that the commutator subalgebra of gq l does not 2 n 2 n depend on the choice of parameter l this subalgebra will be denoted as sq a and will be called the special quadratic lie algebra of a there is 2 n another subalgebra sq l which does depend on the choice of l we 2 n

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

universal coverings of steinberg lie algebras of small characteristic abstract it is well known that the second homology group h 2 st n r of the steinberg lie algebra st n r is trivial when n 5 in this paper we will work out h 2 st n r explicitly for n 3 4 which are not necessarily trivial consequently we obtained h 2 sl n r for n 3 4

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

linguistic characterization of time series the goal of this paper is to provide an overview of applications of special soft computing theories the fuzzy transform and fuzzy natural logic to analysis forecasting and mining information from time series the focus is especially placed on the ability of methods of fuzzy natural logic to provide information in sentences of natural language our approach is based on the decomposition of time series into three components the trend cycle seasonal component and noise the trend cycle is extracted using the f transform and its course is characterized by automatically generated linguistic description the latter is then used to forecast the trend cycle the trend cycle can be furthermore decomposed into trend general tendency and cycle the former is computed again using the f transform moreover the f1 transform makes it possible to estimate the direction of the trend which can then be characterized by expressions of natural language stagnating slightly increasing sharply decreasing etc finally we focus on selected problems of mining information from time series first we suggest an algorithm for finding intervals of monotonous behavior and then show how the theory of intermediate quantifiers a constituent of fuzzy natural logic and generalized aristotle s syllogisms can be applied to automatic summarization of information on time series

weyl demazure and fusion modules for the current algebra of sl r 1 we construct a poincare birkhoff witt type basis for the weyl modules v chari a pressley weyl modules for classical and quantum affine algebras represent theory 5 2001 191 223 math qa 0004174 of the current algebra of as a corollary we prove the conjecture made in v chari a pressley weyl modules for classical and quantum affine algebras represent theory 5 2001 191 223 math qa 0004174 v chari a pressley integrable and weyl modules for quantum affine in quantum groups and lie theory durham 1999 in london math soc lecture note ser vol 290 cambridge univ press cambridge 2001 pp 48 62 math qa 0007123 on the dimension of the weyl modules in this case further we relate the weyl modules to the fusion modules defined in b feigin s loktev on generalized kostka polynomials and the quantum verlinde rule in differential topology infinite dimensional lie algebras and applications in amer math soc transl ser 2 vol 194 1999 pp 61 79 math qa 9812093 of the current algebra and the demazure modules in level one representations of the corresponding affine algebra in particular this allows us to establish substantial cases of the conjectures in b feigin s loktev on generalized kostka polynomials and the quantum verlinde rule in differential topology infinite dimensional lie algebras and applications in amer math soc transl ser 2 vol 194 1999 pp 61 79 math qa 9812093 on the structure and graded character of the fusion modules

spectral statistics of a system with sharply divided phase space we construct a family of non kolmogorov arnold moser non kam piecewise linear continuous 2d area preserving maps which have sharply divided phase space with regions of regular elliptic and chaotic hyperbolic motion for such systems the shape of the islands of regular motion is either a solid ellipse or a solid polygon depending on the ir rationality of the frequency and thus the total area of the regular region of phase space can be computed analytically or at least rigorously estimated from below we analyse the spectral statistics for a few examples of the quantization of our maps and show that they provide a convenient playground for testing and confirming the key assumptions required for the validity of berry robnik formulae

l fuzzy quantifiers of type determined by fuzzy measures the aim of this paper is first to introduce two new types of fuzzy integrals namely fuzzy integral and fuzzy integral the first integral is based on a fuzzy measure of l fuzzy sets and the second one on a complementary fuzzy measure of l fuzzy sets where l is a complete residuated lattice some of their properties and a relation to the fuzzy sugeno integral are investigated second using these integrals two classes of monadic l fuzzy quantifiers of type are defined these l fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like all some many none at most half etc several semantic properties of these l fuzzy quantifiers are studied

interpretation aware cognitive map construction for time series modeling abstract we raise a few issues regarding time series modeling using cognitive maps which is an example of a qualitative rather than a purely quantitative approach methods that operate at the level of concepts instead of numerical values are a worthy alternative for time series processing thanks to certain desirable properties such as abstraction and ease of interpretation those features are unavailable in classical time series modeling algorithms where the main concern is numerical accuracy of a prediction expressed as a series of numerical values in this work we argue that it is desirable for cognitive map based models to be simplified ideally to the point where their descriptiveness and numerical accuracy is not compromised in particular we show experimentally that both the size of the map and the size of the set of possible weight values can be minimized without increasing the error rate

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

irreducible finite dimensional representations of equivariant map algebras suppose a finite group acts on a scheme x and a finite dimensional lie algebra g the corresponding equivariant map algebra is the lie algebra m of equivariant regular maps from x to g we classify the irreducible finite dimensional representations of these algebras in particu lar we show that all such representations are tensor products of evaluation representations and one dimensional representations and we establish conditions ensuring that they are all evaluation representations for example this is always the case if m is perfect our results can be applied to multiloop algebras current algebras the onsager algebra and the tetrahedron algebra doing so we easily recover the known classifications of irreducible finite dimensional representations of these algebras moreover we obtain previously unknown classifica tions of irreducible finite dimensional representations of other types of equivariant map algebras such as the generalized onsager algebra

monadic l fuzzy quantifiers of the type the aim of this paper is to introduce monadic l fuzzy quantifiers of the type and investigate some of their semantic properties that are well known in generalized quantifier theory relationships between the properties of generalized quantifiers are also studied for monadic l fuzzy quantifiers finally a property to be extensional with respect to a given l permutation equivalence is proposed and some properties are shown

a comprehensive theory of trichotomous evaluative linguistic expressions in this paper a logical theory of the so called trichotomous evaluative linguistic expressions tev expressions is presented these are frequent expressions of natural language such as small very small roughly medium extremely big etc the theory is developed using the formal system of higher order fuzzy logic namely the fuzzy type theory generalization of classical type theory first we discuss informally what are properties of the meaning of tev expressions then we construct step by step axioms of a formal logical theory t e v of tev expressions and prove various properties of t e v all the proofs are syntactical and so our theory is very general we also outline construction of a canonical model of t e v the main elegancy of our theory consists in the fact that semantics of all kinds of evaluative expressions is modeled in a unified way we also prove theorems demonstrating that essential properties of the vagueness phenomenon can be captured within our theory

universal central extensions of elliptic affine lie algebras let g be a simple complex finite dimensional lie algebra and let r be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed lie algebras of the form g cr are considered these generalize kac moody loop algebras since for a curve of genus zero with two punctures r c t t 1 the universal central extension of g r is analogous to an untwisted affine kac moody algebra by kassel s theorem the kernel of the universal central extension is linearly isomorphic to the kahler differentials of r modulo exact differentials the dimension of the kernel for any r is determined first restricting to hyperelliptic curves with 2 3 or 4 special points removed a basis for the kernel is determined restricting further to an elliptic curve with punctures at two points of orders one and two in the group law we explicitly determine the cocycles which give the commutation relations for the universal central extension the results involve pollaczek polynomials which ar

simple lie subalgebras of locally finite associative algebras abstract we prove that any simple lie subalgebra of a locally finite associative algebra is either finite dimensional or isomorphic to the commutator algebra of the lie algebra of skew symmetric elements of some involution simple locally finite associative algebra the ground field is assumed to be algebraically closed of characteristic 0 this result can be viewed as a classification theorem for simple lie algebras that can be embedded in locally finite associative algebras we also establish a link between this class of lie algebras and that of lie algebras graded by finite root systems

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

a formal theory of generalized intermediate syllogisms in this paper we continue developing the formal theory of intermediate quantifiers expressions such as most few almost all a lot of many a great deal of a large part of a small part of the theory is a fuzzy logic formalization of the concept introduced by peterson in his book we will syntactically prove that 105 generalized aristotle s syllogisms introduced in this book are valid in our theory at the same time we will also prove that syllogisms listed there as invalid are invalid also in our theory therefore we believe that our theory provides a reasonable mathematical model of the generalized syllogistics

linguistic summarization of time series using a fuzzy quantifier driven aggregation we propose new types of linguistic summaries of time series data that extend those proposed in our previous papers the proposed summaries of time series refer to the summaries of trends identified here with straight line segments of a piecewise linear approximation of time series we first show how to construct such an approximation then we employ a set of features attributes to characterize the trends such as the slope of the line segment the goodness of approximation and the length of the trend the derivation of a linguistic summary of a time series is then related to a linguistic quantifier driven aggregation of trends for this purpose we employ the classic zadeh s calculus of linguistically quantified propositions but extending our previous works with different t norms in addition to the basic minimum we show an application to the analysis of time series data on daily quotations of an investment fund over an eight year period present some interesting linguistic summaries obtained and show results for different t norms the results are very promising

universal central extensions of direct limits of lie superalgebras abstract we show that the universal central extension of a direct limit of perfect lie superalgebras l i is isomorphic to the direct limit of the universal central extensions of l i as an application we describe the universal central extensions of some infinite rank lie superalgebras

twisted γ lie algebras and their vertex operator representations let be a generic subgroup of the multiplicative group c of nonzero complex numbers we define a class of lie algebras associated to called twisted lie algebras which are natural generalizations of the twisted affine lie algebras starting from an arbitrary even sublattice q of zn and an arbitrary finite order isometry of zn preserving q we construct a family of twisted vertex operators acting on generalized fock spaces which afford irreducible representations for certain twisted lie algebras as an application this recovers a number of known vertex operator realizations for infinite dimensional lie algebras such as twisted affine lie algebras extended affine lie algebras of type a trigonometric lie algebras of series a and b unitary lie algebras and bc graded lie algebras

classification and construction of quasisimple lie algebras abstract we study a class of possibly infinite dimensional lie algebras called the quasisimple lie algebras qsla s and generalizing semisimple and affine kac moody lie algebras they are characterized by the existence of a finite dimensional cartan subalgebra a non degenerate symmetric ad invariant killing form and nilpotent rootspaces attached to non isotropic roots we are then able to derive a classification theorem for the possible irreducible elliptic quasisimple root systems moreover we construct explicit realizations of some of them as untwisted and twisted current algebras generalizing the affine loop algebras

stickiness in hamiltonian systems from sharply divided to hierarchical phase space we investigate the dynamics of chaotic trajectories in simple yet physically important hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures we show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one parameter families of marginally unstable periodic orbits and is characterized by an exponent ensuremath gamma 2 for the asymptotic power law decay of the distribution of recurrence times generic perturbations lead to systems with hierarchical phase space where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and cantori in this case we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws depending on the relative contribution of the primary and secondary structures of the hierarchy numerical verification of our main results are provided for area preserving maps mushroom billiards and the newly defined magnetic mushroom billiards

the alternative torus and the structure of elliptic quasi simple lie algebras of type we present the complete classification of the tame irreducible elliptic quasi simple lie algebras of type a2 and in particular specialize on the case where the coordinates are not associative here the coordinates are cayley dickson algebras over laurent polynomial rings in v 3 variables which we call alternative tori in giving our classification we need to present much information on these alternative tori and the lie algebras coordinatized by them

tkk algebras and vertex operator representations let be the jordan algebra constructed from a semilattice s of v 1 see allison et al mem amer math soc 603 1997 let be the lie algebra obtained from the jordan algebra by the tits kantor koecher construction the tkk algebra is defined to be the universal central extension of the lie algebra in this paper we present a complete description of the tkk algebra where s is the smallest possible nonlattice semilattice which then allows us to give a representation to this lie algebra by vertex operators

imbedding of lie triple systems into lie algebras abstract we show that the category of lie triple systems is equivalent to a full subcategory of the category of z 2 graded lie algebras and give two intrinsic characterizations of this subcategory as a corollary we obtain an analogous result for imbedding of symmetric spaces into lie groups with involutions

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

central extensions of lie algebras graded by finite root systems lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by berman and moody benkart and zelmanov and neher in this paper we determine the central extensions of these lie algebras and hence describe them completely up to isomorphism

spectral properties of a piecewise linear intermittent map for a piecewise linear intermittent map the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the frobenius perron operator p are explicitly derived the evolution of the averages are shown to be a superposition of the contributions from two simple eigenvalues 1 and d 1 0 and a continuous spectrum on the unit interval 0 1 of p power law decay of correlations are controlled by the continuous spectrum also the non normalizable invariant measure in the non stationary regime is shown to determine the strength of the power law decay

fuzzy quantification a state of the art abstract quantified sentences are a very powerful notion for modelling statements in natural language nl but in practice they have been used to solve several problems this paper is intended to offer a global view of the development on this branch until now focusing in the different approaches dealing with quantification specially those involving imprecision called fuzzy quantification we put attention to the different mechanisms for defining them the evaluation methods for measuring their fulfilment as well as the properties they should satisfy

bcn graded lie algebras arising from fermionic representations abstract we use fermionic representations to obtain a class of bc n graded lie algebras coordinatized by quantum tori with nontrivial central extensions

vertex operators arising from the homogeneous realization for we use the underlying fock space for the homogeneous vertex operator representation of the affine lie algebra to construct a family of vertex operators as an application an irreducible module for an extended affine lie algebra of type a n 1 coordinatized by a quantum torus q of 2 variables or 3 variables is obtained moreover this module turns out to be a highest weight module which is an analog of the basic module for affine lie algebras

classical lie superalgebras over simple associative algebras over arbitrary fields of characteristic not equal to 2 we construct three families of simple lie algebras and six families of simple lie superalgebras of matrices with entries chosen from different one sided ideals of a simple associative algebra these families correspond to the classical lie algebras and superalgebras our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential compatible way many examples of simple associative algebras without an identity element arise as a by product the study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as weyl algebras and for that reason we illustrate our constructions by taking various one sided ideals from a weyl algebra or a quantum torus

analysis of generalized square of opposition with intermediate quantifiers abstract in this paper we continue the development of a formal theory of intermediate quantifiers these quantifiers are linguistic expressions such as most many few and almost all and they correspond to what are often called fuzzy quantifiers in the literature in a previous study we demonstrated that 105 generalized syllogisms with intermediate quantifiers are valid in our theory in this paper we turn our attention to another problem which is the analysis of the generalized aristotelian square of opposition which in addition to the classical quantifiers can be extended by several selected intermediate quantifiers we show that the expected relations can be well modeled in our theory our theory is developed within a special higher order fuzzy logic ukasiewicz fuzzy type theory and is very general with a high potential for applications

filtering out high frequencies in time series using f transform abstract in this paper we focus on application of fuzzy transform f transform to analysis of time series under the assumption that the latter can be additively decomposed into trend cycle seasonal component and noise we prove that when setting properly width of the basic functions the inverse f transform of the time series closely approximates its trend cycle this means that the f transform almost completely removes the seasonal component and noise the obtained theoretical results are demonstrated on two artificial time series whose trend cycle is precisely known and on three real time series at the same time comparison with three classical methods namely stl method ssa method and low pass butterworth filter is also provided

classification of division zn graded alternative algebras abstract the octonion torus or cayley torus appears as a coordinate algebra of extended affine lie algebras of type a 2 and f 4 a generalized version of the octonion torus called division z n graded alternative algebras is classified in this paper using the result we can complete the classification of division a 2 z n graded lie algebras up to central extensions which are a generalization of the cores of extended affine lie algebras of type a 2

integrable roots in split graded lie algebras abstract a lie algebra g is said to be split graded if it is graded by a torsion free abelian group q in such a way that the subalgebra g 0 is abelian and the operators ad g 0 are diagonalized by the grading the elements of q 0 with g 0 are called roots and a root is said to be integrable if there are root vectors x g which are ad nilpotent and generate an s l 2 subalgebra g in this paper we study subalgebras g g generated by the subalgebras g where is a set of integrable roots for 2 these subalgebras are essentially kac moody algebras which permits us to generalize several results on root strings from kac moody algebras to split graded algebras a central result is the local finiteness theorem saying that whenever all roots of a split graded lie algebra are integrable then is locally finite if differences of roots in are not roots then is called a simple system in this case we describe the structure of the subalgebras g in their relationship to the corresponding kac moody algebras

fuzzy transforms of higher order approximate derivatives a theorem in many practical applications it is useful to represent a function f x by its fuzzy transform i e by the average values f i f x a i x dx a i x dxover different elements of a fuzzy partitiona 1 x a n x for which a i x 0 and i 1 na i x 1 it is known that when we increase the number n of the partition elements a i x the resulting approximation gets closer and closer to the original function for each value x 0 the values f i corresponding to the function a i x for which a i x 0 1 tend to f x 0 in some applications if we approximate the function f x on each element a i x not by a constant but by a polynomial i e use a fuzzy transform of a higher order we get an even better approximation to f x in this paper we show that such fuzzy transforms of higher order and even sometimes the original fuzzy transforms not only approximate the function f x itself they also approximate its derivative s for example we have f i x 0 f x 0

fuzzy transforms theory and applications the technique of direct and inverse fuzzy f transforms of three different types is introduced and approximating properties of the inverse f transforms are described all three types of the direct f transform are transformations from a function space to a finite dimensional vector space the first ordinary f transform is constructed on the basis of the ordinary algebra of reals while the other two types of the f transform are constructed on the basis of residuated lattice the core idea of the technique of f transforms is a fuzzy partition of a universe into fuzzy subsets factors clusters granules etc we claim that for a sufficient representation of a function defined on this universe we may consider its average values over fuzzy subsets from the partition thus a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values a number of theorems establishing best approximation properties of the inverse f transforms are proved in fact three types of the inverse f transform are the best approximations in average from below and from above respectively as one of many possible applications we present a method of image compression and reconstruction on the basis of the f transform

reasoning about mathematical fuzzy logic and its future this paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author our goal is to think of the state of the art in mathematical fuzzy logic mfl and to outline some of the tasks on which in the author s opinion mfl should focus in the future in our discussion we will mention not only the basic theory but also its extension called fuzzy logic in broader sense flb the paradigm of the latter is to be the logic of natural human reasoning whose most essential characteristic is the use of natural language besides brief description of flb we will also mention some of its applications on the basis of that we will ponder on other possible directions for research namely the possibility of using fln as a metatheory of fuzzy mathematics as a proper tool for modeling of the main manifestations of the phenomenon of vagueness and as a reasonable tool for developing models of linguistic semantics

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

the structure of generalized intermediate syllogisms abstract in this paper we continue development of the formal theory of intermediate quantifiers which are expressions of the natural language most many few etc in the previous paper we demonstrated that 105 generalized syllogisms are valid in our theory we introduce the generalization of all the figures and we show that for the proof of validity of all the generalized syllogisms we need to prove the validity of only few of them so that the validity of the other ones immediately follows

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

escape time statistics for mushroom billiards chaotic orbits of the mushroom billiards display intermittent behaviors we investigate statistical properties of this system by constructing an infinite partition on the chaotic part of a poincare surface which illustrates details of chaotic dynamics each piece of the infinite partition has a unique escape time from the half disk region and from this result it is shown that for fixed values of the system parameters the escape time distribution obeys a power law 1 t esc 3