restricted one dimensional central extensions of restricted simple lie algebras abstract we study the restricted one dimensional central extensions of an arbitrary finite dimensional restricted simple lie algebra for p 5 for h 2 g 0 we explicitly describe the cocycles spanning h 2 g and in the case h 2 g 0 we give a procedure to describe a basis for h 2 g

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

necessary and sufficient conditions for the output feedback regularization of descriptor systems the problem of output feedback regularization of descriptor systems is studied necessary and sufficient conditions are given to regulate the descriptor systems by derivative and proportional output feedback into a closed loop regular system which is of index at most one

central extensions and h1 l l of the graded lie algebras of cartan type let f be an algebraically closed field of characteristic p 0 l i rsl i a graded lie algebra of cartan type over f we give a description of the dual adjoint module l by means of the mixed product and the coinduced module such that the computation of h1 l l can be reduced to that of the first cohomology of l 0 then we determine the central extensions and h2 l f by computing h1 l l

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

training a support vector machine in the primal most literature on support vector machines svms concentrates on the dual optimization problem in this letter we point out that the primal problem can also be solved efficiently for both linear and nonlinear svms and that there is no reason for ignoring this possibility on the contrary from the primal point of view new families of algorithms for large scale svm training can be investigated

on regularizing singular systems by decentralized output feedback this paper considers linear time invariant decentralized singular systems which are either nonregular or if they are regular they have impulsive modes it derives algebraic necessary and sufficient conditions for making a singular system both regular and impulse free by decentralized output feedback control laws and decentralized proportional plus derivative output feedback control laws

classification with nonmetric distances image retrieval and class representation a key problem in appearance based vision is understanding how to use a set of labeled images to classify new images systems that model human performance or that use robust image matching methods often use nonmetric similarity judgments but when the triangle inequality is not obeyed most pattern recognition techniques are not applicable exemplar based nearest neighbor methods can be applied to a wide class of nonmetric similarity functions the key issue however is to find methods for choosing good representatives of a class that accurately characterize it we show that existing condensing techniques are ill suited to deal with nonmetric dataspaces we develop techniques for solving this problem emphasizing two points first we show that the distance between images is not a good measure of how well one image can represent another in nonmetric spaces instead we use the vector correlation between the distances from each image to other previously seen images second we show that in nonmetric spaces boundary points are less significant for capturing the structure of a class than in euclidean spaces we suggest that atypical points may be more important in describing classes we demonstrate the importance of these ideas to learning that generalizes from experience by improving performance we also suggest ways of applying parametric techniques to supervised learning problems that involve a specific nonmetric distance functions showing how to generalize the idea of linear discriminant functions in a way that may be more useful in nonmetric spaces

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

minimum norm regularization of descriptor systems by mixed output feedback we study the regularization problem for linear constant coefficient descriptor systems ex ax bu y1 cx y2 x by proportional and derivative mixed output feedback necessary and sufficient conditions are given which guarantee that there exist output feedbacks such that the closed loop system is regular has index at most one and e bg has a desired rank i e there is a desired number of differential and algebraic equations to resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of e is decreased numerical procedures are derived to construct the desired feedback gains these numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way

combining multiple classifiers by averaging or by multiplying abstract in classification tasks it may be wise to combine observations from different sources not only it decreases the training time but it can also increase the robustness and the performance of the classification combining is often done by just weighted averaging of the outputs of the different classifiers using equal weights for all classifiers then results in the mean combination rule this works very well in practice but the combination strategy lacks a fundamental basis as it cannot readily be derived from the joint probabilities this contrasts with the product combination rule which can be obtained from the joint probability under the assumption of independency in this paper we will show differences and similarities between this mean combination rule and the product combination rule in theory and in practice

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

a note on decomposition numbers for reductive algebraic groups in a recent paper 7 james proves two theorems on the decomposition numbers for the general linear groups and working within the context of green s monograph 5 derives properties of the decomposition numbers for the symmetric groups this note grew out of an attempt to formulate for arbitrary reductive groups these theorems of james we prove a natural result on decomposition numbers for reductive groups from which james s theorems are deduced the result the theorem of section 2 is the analogue of jantzen s satz 1 18 of 8 on decomposition numbers for representations of complex semisimple lie algebras also proved independently by deodhar very little direct use of 5 is made though some of the ideas are cribbed from that source as one may see by comparing the properties of the functor fa in section 2 with green s treatment see section 6 2 of s of ese theory s a ring e an idempotent

on regularizing descriptor systems by output feedback in this paper linear time invariant descriptor variable systems are considered which are either nonregular or if they are regular they have impulsive modes it is shown that if a certain necessary and sufficient condition is satisfied then both regularity can be established and impulsive modes can be eliminated by output feedback we have also considered the problem of achieving structural properties of the regularized system

dissimilarity based ensembles for multiple instance learning in multiple instance learning objects are sets bags of feature vectors instances rather than individual feature vectors in this paper we address the problem of how these bags can best be represented two standard approaches are to use dis similarities between bags and prototype bags or between bags and prototype instances the first approach results in a relatively low dimensional representation determined by the number of training bags whereas the second approach results in a relatively high dimensional representation determined by the total number of instances in the training set however an advantage of the latter representation is that the informativeness of the prototype instances can be inferred in this paper a third intermediate approach is proposed which links the two approaches and combines their strengths our classifier is inspired by a random subspace ensemble and considers subspaces of the dissimilarity space defined by subsets of instances as prototypes we provide insight into the structure of some popular multiple instance problems and show state of the art performances on these data sets

algebraic properties of singular systems subject to decentralized output feedback this paper investigates algebraic properties of decentralized singular systems dsss under local static output feedback new concepts of the geometric multiplicity gm of the finite decentralized fixed mode dfm the decentralized output feedback variable polynomial dvp and the finite decentralized output feedback cycle index dci of the dss are defined the formulas for determining the gm and the dci are given in terms of the dfm and the system matrices it is shown that almost any decentralized output feedback can make the zeros of the dvp i e the closed loop poles that are variable distinct and away from any given finite set in the complex plane it is also shown that the finite dfm of the dss are those uncontrollable and or observable finite modes of the closed loop dss through any single channel finally the minimal number of the inputs and outputs that guarantee the finite modes of the closed loop dss controllable and observable is shown to be the finite dci of the dss an illustrative example is provided

solving the multiple instance problem with axis parallel rectangles the multiple instance problem arises in tasks where the training examples are ambiguous a single example object may have many alternative feature vectors instances that describe it and yet only one of those feature vectors may be responsible for the observed classification of the object this paper describes and compares three kinds of algorithms that learn axis parallel rectangles to solve the multiple instance problem algorithms that ignore the multiple instance problem perform very poorly an algorithm that directly confronts the multiple instance problem by attempting to identify which feature vectors are responsible for the observed classifications performs best giving 89 correct predictions on a musk odor prediction task the paper also illustrates the use of artificial data to debug and compare these algorithms

on the regularizability of singular systems the property of regularity of singular systems is not a feedback invariant to correct this deficiency a new property of regularizability is defined and geometric tests are given for it in terms of system matrices regularizability is shown to be the natural extension of regularity which is a condition on the homogeneous system to controlled singular systems definitions of controllability and reachability are modified depending on regularizability rather than regularity a brief comparison of proportional and proportional plus derivative feedback laws in the context of making the closed loop system regular regular and reachable and regular and controllable is also given dynamical interpretations of these properties are also presented

prototype selection for dissimilarity based classifiers a conventional way to discriminate between objects represented by dissimilarities is the nearest neighbor method a more efficient and sometimes a more accurate solution is offered by other dissimilarity based classifiers they construct a decision rule based on the entire training set but they need just a small set of prototypes the so called representation set as a reference for classifying new objects such alternative approaches may be especially advantageous for non euclidean or even non metric dissimilarities the choice of a proper representation set for dissimilarity based classifiers is not yet fully investigated it appears that a random selection may work well in this paper a number of experiments has been conducted on various metric and non metric dissimilarity representations and prototype selection methods several procedures like traditional feature selection methods here effectively searching for prototypes mode seeking and linear programming are compared to the random selection in general we find out that systematic approaches lead to better results than the random selection especially for a small number of prototypes although there is no single winner as it depends on data characteristics the k centres works well in general for two class problems an important observation is that our dissimilarity based discrimination functions relying on significantly reduced prototype sets 3 10 of the training objects offer a similar or much better classification accuracy than the best k nn rule on the entire training set this may be reached for multi class data as well however such problems are more difficult

derivations and central extensions of finitely generated graded lie algebras abstract some general results concerning derivations of finitely generated lie algebras are established these are employed in order to determine the derivations and central extensions of kac moody lie algebras

the associative forms of the graded cartan type lie algebras this paper determines the cartan type lie algebras that possess a nonsingular associative form o introduction the structure of cartan type lie algebras has been extensively studied for several years derivation algebras and isomorphism classes of cartan subalgebras are well known and it is the objective of the present paper to determine those graded cartan type lie algebras which possess a nonsingular associative form in contrast with trace forms general nonsingular associative forms do not play an eminent role in the classification theory of simple lie algebras of prime characteristic their significance primarily rests on cohomology theory nonsingular associative forms allow the study of central extensions of lie algebras by means of derivations we will employ a slight generalization of this known interrelation in a forthcoming paper the introductory remarks of the first section are followed by a brief investigation of cartan subalgebras of graded lie algebras our results generalize those of m frank cf 8 3 establishes general properties of associative forms of graded simple lie algebras and provides the basic tools for the study of the graded cartan type lie algebras the author would like to express his gratitude to professor helmut strade for many helpful suggestions as well as to the referee for proposing proposition 3 4 and the resulting simplifications in 4 1 graded representations of graded lie algebras in the sequel l ik r l r k 1 denotes a graded lie algebra over an arbitrary field f a representation t l gl v is said to be graded if v 13 vj and t x vj c vl j for every x e li and s 1 we consider the subalgebras l eik l li and l d3 li using the p b w theorem it can be readily verified that we have u l u l u lo u l u l u lo u l for the respective universal enveloping algebras received by the editors may 17 1984 and in revised form april 10 1985 1980 mathematics subject classification primary 17b20 t1986 american mathematical society 0002 9947 86 1 00 25 per page

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy