complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

application of fuzzy soft set in decision making problems based on grey theory there are many uncertain problems in practical production and life which need decisions made with soft sets and fuzzy soft sets however the basis of evaluation of the decision method is single and simple the same decision problem can obtain different results from using a different evaluation basis in this paper in order to obtain the right result we discuss fuzzy soft set decision problems a new algorithm based on grey relational analysis is presented the evaluation bases of the new algorithm are multiple there is more information in a decision result based on multiple evaluation bases which is more easily accepted and logical to one s thinking for the two cases examined the results show that the new algorithm is efficient for solving decision problems

applications of soft sets in ideal theory of bck bci algebras the notion of soft ideals and idealistic soft bck bci algebras is introduced and several examples are given relations between soft bck bci algebras and idealistic soft bck bci algebras are provided the intersection union and operation and or operation of soft ideals and idealistic soft bck bci algebras are established

soft sets and soft groups molodtsov introduced the concept of soft set theory which can be used as a generic mathematical tool for dealing with uncertainty in this paper we introduce the basic properties of soft sets and compare soft sets to the related concepts of fuzzy sets and rough sets we then give a definition of soft groups and derive their basic properties using molodtsov s definition of the soft sets

on the moment map for the variety of lie algebras we consider the moment map m pvn iu n for the action of gl n on vn 2 cn cn the critical points of the functional fn m 2 pvn r are studied in order to understand the stratification of ln pvn defined by the negative gradient flow of fn where ln is the algebraic subset of all lie algebras we obtain a description of the critical points which lie in ln in terms of those which are nilpotent as well as the minima and maxima of fn ln r a characterization of the critical points modulo isomorphism as the union of categorical quotients of suitable actions is considered and some applications to the study of ln are given

finding einstein solvmanifolds by a variational method we use a variational approach to prove that any nilpotent lie algebra having a codimension one abelian ideal and anyone of dimension leq 5 admits a rank one solvable extension which can be endowed with an einstein left invariant riemannian metric a curve of 8 dimensional einstein solvmanifolds is also given

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

on optimal sequential prediction for general processes this paper considers several aspects of the sequential prediction problem for unbounded nonstationary processes under pth power loss spl lscr sub p u v u v sup p 1 p spl infin in the first part it is shown that bayes prediction schemes are cesaro optimal under general conditions that cesaro optimal prediction schemes are unique in a natural sense and that cesaro optimality is equivalent to a form of weak calibration connections between calibration and stronger forms of optimality are considered extensions of the existence and uniqueness results to generalized prediction and prediction from observations with additive noise are established for binary processes it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the hamming loss in the second part of the paper it is shown how to construct from a countable family of prediction schemes a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family the construction is based on aggregating methods for individual binary sequences using the construction some results of algoet 1994 1995 on the existence of cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes

einstein solvmanifolds existence and non existence questions the aim of this paper is to study the problem of which solvable lie groups admit an einstein left invariant metric the space mathcal n of all nilpotent lie brackets on mathbb r n parametrizes a set of n 1 dimensional rank one solvmanifolds s mu mu in mathcal n containing the set of all those which are einstein in that dimension the moment map for the natural gl n action on mathcal n evaluated at mu in mathcal n encodes geometric information on s and suggests the use of strong results from geometric invariant theory for instance the functional on mathcal n whose critical points are precisely the einstein s s is the square norm of this moment map we use a gl n invariant stratification for the space mathcal n and show that there is a strong interplay between the strata and the einstein condition on the solvmanifolds s as an application we obtain criteria to decide whether a given nilpotent lie algebra can be the nilradical of a rank one einstein solvmanifold or not we find several examples of mathbb n graded even 2 step nilpotent lie algebras which are not a classification in the 7 dimensional 6 step case and an existence result for certain 2 step algebras associated to graphs are also given

the strong law of large numbers for sequential decisions under uncertainty combines optimization and ergodic theory to characterize the optimum long run average performance that can be asymptotically attained by nonanticipating sequential decisions let x sub t be a stationary ergodic process and suppose an action b sub t must be selected in a space spl bscr with knowledge of the t past x sub 0 spl middot spl middot spl middot x sub t 1 at the beginning of every period t spl ges 0 action b sub t will incur a loss l b sub t x sub t at the end of period t when the random variable x sub t is revealed the author proves under mild integrability conditions that the optimum strategy is to select actions that minimize the conditional expected loss given the currently available information at each step the minimum long run average loss per decision can be approached arbitrarily closely by strategies that are finite order markov and under certain continuity conditions it is equal to the minimum expected loss given the infinite past if the loss l b x is bounded and continuous and if the space spl bscr is compact then the minimum can be asymptotically attained even if the distribution of the process x sub t is unknown a priori and must be learned from experience

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

the game of normal numbers we introduce a two player game where at each period one player say player 2 chooses a distribution and the other player player 1 chooses a realization player 1 wins the game if the sequence of realized outcomes is normal with respect to the sequence of distributions we present a pure winning strategy of player 1 and thereby provide a universal algorithm that generates a normal sequence for any discrete stochastic process it turns out that to select thenth digit the algorithm conductso n2 calculations the proof uses approachability in infinite dimensional spaces lehrer 2002

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

generalised fuzzy soft sets in this paper we define generalised fuzzy soft sets and study some of their properties application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown

some stochastic properties of memoryless individual sequences an individual sequence of real numbers is memoryless if no continuous markov prediction scheme of finite order can outperform the best constant predictor under the squared loss it is established that memoryless sequences satisfy an elementary law of large numbers and sliding block versions of hoeffding s inequality and the central limit theorem it is further established that memoryless binary sequences have convergent sample averages of every order and that their limiting distributions are bernoulli several examples and sources of memoryless sequences are given and it is shown how memoryless binary sequences may be constructed from aggregating methods for sequential prediction

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

on the classification of degree two algebras the classification of algebras of generic degree two and of nonzero generic discriminant is presented in the case of zero discriminant a structural theorem is proven

noncompact homogeneous einstein manifolds attached to graded lie algebras in this paper we study the nilradicals of parabolic subalgebras of semisimple lie algebras and the natural one dimensional solvable extensions of them we investigate the structures curvatures and einstein conditions of the associated nilmanifolds and solvmanifolds we show that our solvmanifold is einstein if the nilradical is two step new examples of einstein solvmanifolds with three step and four step nilradicals are also given

on efficient probability forecasting systems we study the asymptotic behaviour of probability forecasting systems and discuss their usefulness as inferential tools for statistical problems such as model verification and selection our theoretical setting is the prequential or predictive sequential framework proposed by dawid 1984 we study especially the notion of prequential efficiency of a forecasting system and present some new results we focus on plug in or estimative forecasting systems where the forecast distribution is generated by replacing the unknown parameter with an estimate

the normal parameter reduction of soft sets and its algorithm this paper is concerned with the reduction of soft sets and fuzzy soft sets firstly the problems of suboptimal choice and added parameter set of soft sets are analyzed then we introduce the definition of normal parameter reduction in soft sets to overcome these problems in addition a heuristic algorithm of normal parameter reduction is presented two new definitions parameter important degree and decision partition are proposed for analyzing the algorithm of normal parameter reduction furthermore the normal parameter reduction is also investigated in fuzzy soft sets

on efficient point prediction systems summary assume that a forecaster observes a sequence of random variables and issues predictions according to a point prediction system i e a rule which at every time t issues a point prediction for the next observation at time t 1 we introduce the concept of efficiency of a point prediction system for the case that the joint distribution of the sequence of observations is known to belong to a parametric family and performance is assessed by the long run sum of squared prediction errors independence is not a requirement under weak conditions the class of efficient point prediction systems is non empty and any two efficient point prediction systems will in a certain strong sense make asymptotically identical predictions for the infinite future we discuss the efficiency of point prediction systems based on bayesian predictive means and on plugging in parameter estimates the results are applied to probability forecasting and stochastic regression

soft matrix theory and its decision making in this work we define soft matrices and their operations which are more functional to make theoretical studies in the soft set theory we then define products of soft matrices and their properties we finally construct a soft max min decision making method which can be successfully applied to the problems that contain uncertainties

kernels and closures of soft set relations and soft set relation mappings in this paper the notions of anti reflexive kernel symmetric kernel reflexive closure and symmetric closure of a soft set relation are first introduced respectively then their accurate calculation formulae and some properties are obtained finally soft set relation mappings and inverse soft set relation mappings are proposed and some related properties are discussed

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

prediction in the worst case a predictor is a method of estimating the probability of future events over an infinite data sequence one predictor is as strong as another if for all data sequences the former has at most the mean square error mse of the latter given any countable set mathscr d of predictors we explicitly construct a predictor s that is at least as strong as every element of mathscr d finite sample bounds are also given which hold uniformly on the space of all possible data

exclusive disjunctive soft sets soft sets theory initiated by molodtsov is an emerging tool to deal with uncertain problems and has been studied by scholars in both theory and practice this paper proposes the notion of exclusive disjunctive soft sets and studies some of its operations such as restricted relaxed and operations dependency between exclusive disjunctive soft sets and bijective soft sets exclusive disjunctive soft decision systems reduction of exclusive disjunctive soft decision systems core of exclusive disjunctive soft decision systems decision rules in exclusive disjunctive decision soft sets moreover this study gives an application of exclusive disjunctive soft sets which shows that it can be applied to attribute reduction of incomplete information system

universal schemes for learning the best nonlinear predictor given the infinite past and side information let x sub t be a real valued time series the best nonlinear predictor of x sub 0 given the infinite past x sub spl infin sup 1 in the least squares sense is equal to the conditional mean e x sub 0 x sub spl infin sup 1 previously it has been shown that certain predictors based on growing segments of past observations converge to the best predictor given the infinite past whenever x sub t is a stationary process with values in a bounded interval the present paper deals with universal prediction schemes for stationary processes with finite mean we also discuss universal schemes for learning the conditional mean e x sub 0 x sub spl infin sup 1 y sub spl infin sup 1 y sub 0 from past observations of a stationary pair process x sub t y sub t and schemes for learning the repression function m y e x y y from independent samples of x y

calibration based empirical probability probability forecasts for a sequence of uncertain events may be compared with the outcomes of those events by means of a natural criterion of empirical validity calibration it is shown that any two sequences of forecasts which both meet this criterion must be in asymptotic agreement these agreed values can then be considered as correct objective probability forecasts for the particular sequence of outcome results obtained however the objective forecasts vary with the extent of the information taken into account when they are formulated we thus obtain a general theory of empirical probability relative to an information base this theory does not require that such probabilities be interpreted in terms of repeated trials of the same event some implications of this theory are discussed

the problems of classifying pairs of forms and local algebras with zero cube radical are wild we prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is hermitian pairs of bilinear forms in which the second is symmetric skew symmetric and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n by n matrices up to simultaneous similarity

soft bck bci algebras molodtsov d molodtsov soft set theory first results comput math appl 37 1999 19 31 introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches in this paper we apply the notion of soft sets by molodtsov to the theory of bck bci algebras the notion of soft bck bci algebras and soft subalgebras are introduced and their basic properties are derived

attribute and size reduction mechanisms in multi adjoint concept lattices attribute reduction and size reduction in concept lattices are key research topics in formal concept analysis fca this paper combines both strategies in the multi adjoint concept lattice framework in order to simplify the information provided by the original context specifically we present three procedures which merge the attribute reduction and the size reduction by means of an irreducible cut concept lattice analyzing the obtained properties

sequential prediction of individual sequences under general loss functions we consider adaptive sequential prediction of arbitrary binary sequences when the performance is evaluated using a general loss function the goal is to predict on each individual sequence nearly as well as the best prediction strategy in a given comparison class of possibly adaptive prediction strategies called experts by using a general loss function we generalize previous work on universal prediction forecasting and data compression however here we restrict ourselves to the case when the comparison class is finite for a given sequence we define the regret as the total loss on the entire sequence suffered by the adaptive sequential predictor minus the total loss suffered by the predictor in the comparison class that performs best on that particular sequence we show that for a large class of loss functions the minimax regret is either spl theta log n or spl omega spl radic spl lscr log n depending on the loss function where n is the number of predictors in the comparison class and spl lscr is the length of the sequence to be predicted the former case was shown previously by vovk 1990 we give a simplified analysis with an explicit closed form for the constant in the minimax regret formula and give a probabilistic argument that shows this constant is the best possible some weak regularity conditions are imposed on the loss function in obtaining these results we also extend our analysis to the case of predicting arbitrary sequences that take real values in the interval 0 1

wildness of the problems of classifying two dimensional spaces of commuting linear operators and certain lie algebras abstract for each two dimensional vector space v of commuting n n matrices over a field f with at least 3 elements we denote by v the vector space of all n 1 n 1 matrices of the form a 0 0 with a v we prove the wildness of the problem of classifying lie algebras v with the bracket operation u v u v v u we also prove the wildness of the problem of classifying two dimensional vector spaces consisting of commuting linear operators on a vector space over a field

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

problems of classifying associative or lie algebras and triples of symmetric or skew symmetric matrices are wild we prove that the problems of classifying triples of symmetric or skew symmetric matrices up to congruence local commutative associative algebras with zero cube radical and square radical of dimension 3 and lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n by n matrices up to simultaneous similarity

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

a combined forecasting approach based on fuzzy soft sets forecasting the export and import volume in international trade is the prerequisite of a government s policy making and guidance for a healthier international trade development however an individual forecast may not always perform satisfactorily while combination of forecasts may result in a better forecast than component forecasts we believe the component forecasts employed in combined forecasts are a description of the actual time series which is fuzzy this paper attempts to use forecasting accuracy as the criterion of fuzzy membership function and proposes a combined forecasting approach based on fuzzy soft sets this paper also examines the method with data of international trade from 1993 to 2006 in the chongqing municipality of china and compares it with a combined forecasting approach based on rough sets and each individual forecast the experimental results show that the combined approach provided in this paper improves the forecasting performance of each individual forecast and is free from a rough sets approach s restrictions as well it is a promising forecasting approach and a new application of soft sets theory

on some new operations in soft set theory molodtsov introduced the theory of soft sets which can be seen as a new mathematical approach to vagueness in this paper we first point out that several assertions proposition 2 3 iv vi proposition 2 4 and proposition 2 6 iii iv in a previous paper by maji et al p k maji r biswas a r roy soft set theory comput math appl 45 2003 555 562 are not true in general by counterexamples furthermore based on the analysis of several operations on soft sets introduced in the same paper we give some new notions such as the restricted intersection the restricted union the restricted difference and the extended intersection of two soft sets moreover we improve the notion of complement of a soft set and prove that certain de morgan s laws hold in soft set theory with respect to these new definitions

combination of interval valued fuzzy set and soft set the soft set theory proposed by molodtsov can be used as a general mathematical tool for dealing with uncertainty by combining the interval valued fuzzy set and soft set models the purpose of this paper is to introduce the concept of the interval valued fuzzy soft set the complement and and or operations are defined on the interval valued fuzzy soft sets the demorgan s associative and distribution laws of the interval valued fuzzy soft sets are then proved finally a decision problem is analyzed by the interval valued fuzzy soft set some numerical examples are employed to substantiate the conceptual arguments

soft semirings molodtsov introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainty in this paper we initiate the study of soft semirings by using the soft set theory the notions of soft semirings soft subsemirings soft ideals idealistic soft semirings and soft semiring homomorphisms are introduced and several related properties are investigated

soft bl algebras based on fuzzy sets by means of soft sets and q soft sets some characterizations of implicative positive implicative and fantastic filteristic soft bl algebras are investigated finally we prove that a soft set is an implicative filteristic soft bl algebra if and only if it is both a positive implicative filteristic soft bl algebra and a fantastic filteristic soft bl algebra

parabolic subgroups of semisimple lie groups and einstein solvmanifolds in this paper we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple lie algebras these solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type we show that the ricci curvatures of our solvmanifolds coincide with the restrictions of the ricci curvatures of the ambient symmetric spaces consequently all of our solvmanifolds are einstein which provide a large number of new examples of noncompact homogeneous einstein manifolds we also show that our solvmanifolds are minimal but not totally geodesic submanifolds of symmetric spaces

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

stabilizers of subspaces under similitudes of the klein quadric and automorphisms of heisenberg algebras we determine the groups of automorphisms and their orbits for nilpotent lie algebras of class 2 and small dimension over arbitrary fields including the characteristic 2 case

soft sets and soft rings molodtsov 1999 introduced the concept of soft sets in 1 then maji et al 2003 defined some operations on soft sets in 2 aktas and ca man 2007 defined the notion of soft groups in 3 finally soft semirings are defined by feng et al 2008 in 5 in this paper we have introduced initial concepts of soft rings

nilpotent lie algebras with 2 dimensional commutator ideals abstract we classify all finitely dimensional nilpotent lie k algebras h with 2 dimensional commutator ideals h extending a known result to the case where h is non central and k is an arbitrary field it turns out that while the structure of h depends on the field k if h is central it is independent of k if h is non central and is uniquely determined by the dimension of h in the case where k is algebraically or real closed we also list all nilpotent lie k algebras h with 2 dimensional central commutator ideals h and dim k h 11

application of level soft sets in decision making based on interval valued fuzzy soft sets molodtsov s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty research on fuzzy soft set based decision making has received much attention in recent years this paper aims to give deeper insights into decision making involving interval valued fuzzy soft sets a hybrid model combining soft sets with interval valued fuzzy sets the concept called reduct fuzzy soft sets of interval valued fuzzy soft sets is introduced using reduct fuzzy soft sets and level soft sets flexible schemes for decision making based on weighted interval valued fuzzy soft sets are proposed and some illustrative examples are employed to show that the proposals presented here are not only more reasonable but more efficient in practical applications

gambling using a finite state machine sequential gambling schemes in which the amount wagered on the future outcome is determined by a finite state fs machine are defined and analyzed it is assumed that the fs machine determines the fraction of the capital wagered at each time instance i on the outcome at the next time instance i 1 and that wagers are paid at even odds the maximal capital achieved by any fs machine is found and its dependence on an empirical entropy measure h sup fs x defined as the finite state complexity of x is shown a specific gambling scheme is then proposed based on the lempel ziv method for universal compression the capital gained by this method is found and it is observed that asymptotically its exponential growth rate dominates the experimental growth rate achieved by gambling using any fs machine furthermore this specific scheme suggests a class of gambling methods based on a class of variable to variable length lossless compression methods in which the capital is doubled for every bit compressed these results emphasize the relation between gambling and data compression

soft set relations and functions the traditional soft set is a mapping from a parameter to the crisp subset of universe molodtsov introduced the theory of soft sets as a generalized tool for modeling complex systems involving uncertain or not clearly defined objects in this paper the concepts of soft set relations are introduced as a sub soft set of the cartesian product of the soft sets and many related concepts such as equivalent soft set relation partition composition function etc are discussed

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

fuzzy soft set theory applied to bck bci algebras abstract in the paper p k maji r biswas a r roy fuzzy soft sets j fuzzy math 9 3 2001 589 602 maji et al introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem the aim of this manuscript is to apply fuzzy soft set for dealing with several kinds of theories in bck bci algebras the notions of fuzzy soft bck bci algebras closed fuzzy soft ideals and fuzzy soft p ideals are introduced and related properties are investigated

the bijective soft set with its operations soft set theory is a newly emerging tool to deal with uncertain problems and has been studied by scholars in theory and practice this paper proposes the concept of bijective soft set and some of its operations such as the restricted and and the relaxed and operations on a bijective soft set dependency between two bijective soft sets bijective soft decision system significance of bijective soft set with respect to bijective soft decision system reduction of bijective soft set with respect to bijective soft decision system and decision rules in bijective soft decision system with these notions and operations an application of bijective soft set in decision making problems is also discussed

soft set theory applied to ideals in d algebras the notions of transitive soft d algebras soft edge d algebras soft d algebras soft d ideals soft d ideals soft d ideals and d idealistic d idealistic or d idealistic soft d algebras are introduced also their related properties are surveyed

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains

letter to the editor comment on a fuzzy soft set theoretic approach to decision making problems the algorithm for identification of an object in a previous paper of a r roy et al a r roy p k maji a fuzzy soft set theoretic approach to decision making problems j comput appl math 203 2007 412 418 is incorrect using the algorithm the right choice cannot be obtained in general the problem is illustrated by a counter example

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence