a second order perceptron algorithm kernel based linear threshold algorithms such as support vector machines and perceptron like algorithms are among the best available techniques for solving pattern classification problems in this paper we describe an extension of the classical perceptron algorithm called second order perceptron and analyze its performance within the mistake bound model of on line learning the bound achieved by our algorithm depends on the sensitivity to second order data information and is the best known mistake bound for efficient kernel based linear threshold classifiers to date this mistake bound which strictly generalizes the well known perceptron bound is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues since the optimal setting of this parameter is not known a priori we also analyze two variants of the second order perceptron algorithm one that adaptively sets the value of the parameter in terms of the number of mistakes made so far and one that is parameterless based on pseudoinverses

an elementary proof of the strong law of large numbers in the following note we present a proof for the strong law of large numbers which is not only elementary in the sense that it does not use kolmogorov s inequality but it is also more applicable because we only require the random variables to be pairwise independent an extension to separable banach space valuedr dimensional arrays of random vectors is also discussed for the weak law of large numbers concerning pairwise independent random variables which follows from our result see theorem 5 2 2 in chung 1

universal randomized switching in this paper we consider a competitive approach to sequential decision problems suitable for a variety of signal processing applications where at each of a succession of times a selection must be made from among a fixed set of strategies or outcomes for each such decision and outcome pair loss is incurred and it is the time accumulation of these losses that is sought to be minimized rather than using a statistical performance measure our goal in this pursuit is to sequentially accumulate loss that is no larger than that of the best loss that could be obtained through a partitioning of the sequence of observations into an arbitrary fixed number of segments and independently selecting a different strategy for each segment for this purpose we introduce a randomized sequential algorithm built upon that of kozat and singer that asymptotically achieves the performance of a noncausal algorithm that would be able to choose the number of segments and the best algorithm for each segment based on observing the whole observation process a priori in addition to improving upon the bounds of kozat and singer as well as gyorgy the results we provide hold for more general loss functions than the square error loss studied therein

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

general entropy criteria for inverse problems with applications to data compression pattern classification and cluster analysis minimum distance approaches are considered for the reconstruction of a real function from finitely many linear functional values an optimal class of distances satisfying an orthogonality condition analogous to that enjoyed by linear projections in hilbert space is derived these optimal distances are related to measures of distances between probability distributions recently introduced by c r rao and t k nayak 1985 and possess the geometric properties of cross entropy useful in speech and image compression pattern classification and cluster analysis several examples from spectrum estimation and image processing are discussed

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

f purity and rational singularity in graded complete intersection rings a simple criterion is given for determining almost completely whether the positively graded complete intersection ring r k x1 xn t g of dimension n has an f pure type singularity at m x1 x specifically if deg x a 0 for 1 0 for 1 0 the characterization given by this theorem is particularly effective if the singularity of r at m is isolated in that case 8 0 so that only the condition t 1d y lta 0 is not solved by the above result in particular it follows from work of kei ichi watanabe that if r has an isolated rational singularity then r has f pure type the converse is also almost true with the only exception being the case where t 1d e i ta 0 in proving this criterion a weak but more stable form of f purity called f contractedness is defined and explored r is f contracted in characteristic p 0 if every system of parameters for m is contracted with respect to the frobenius map f r r just as for f purity the notion of f contracted type is defined in characteristic 0 by reduction to characteristic p the two notions of f pure type and f contracted type coincide when r is gorenstein whence in particular when r is a complete intersection ring 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m r m is exact a notion of f pure type can then be defined in characteristic 0 by reduction to characteristic p the concept of f purity was suggested originally by the hochster roberts proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 4 the same two authors later demonstrated that f purity measures nice singularities in the sense that it implies much simplification and vanishing in the computation of local cohomology 5 the problem of identifying those rings which are f pure type is quite difficult the following special cases are known 1 the invariant ring of a reductive linear group acting on a regular ring is f pure type 5 received by the editors october 8 1985 1980 mathematics subject classification 1985 revision primary 13h99 the author wishes to thank the research council of the university of missouri for a research grant which partially supported this project 1987 american mathematical society 0002 9947 87 1 00 25 per page

leading strategies in competitive on line prediction we start from a simple asymptotic result for the problem of on line regression with the quadratic loss function the class of continuous limited memory prediction strategies admits a leading prediction strategy which not only asymptotically performs at least as well as any continuous limited memory strategy but also satisfies the property that the excess loss of any continuous limited memory strategy is determined by how closely it imitates the leading strategy more specifically for any class of prediction strategies constituting a reproducing kernel hilbert space we construct a leading strategy in the sense that the loss of any prediction strategy whose norm is not too large is determined by how closely it imitates the leading strategy this result is extended to the loss functions given by bregman divergences and by strictly proper scoring rules

predicting a switching sequence of graph labelings we study the problem of predicting online the labeling of a graph we consider a novel setting for this problem in which in addition to observing vertices and labels on the graph we also observe a sequence of just vertices on a second graph a latent labeling of the second graph selects one of k labelings to be active on the first graph we propose a polynomial time algorithm for online prediction in this setting and derive a mistake bound for the algorithm the bound is controlled by the geometric cut of the observed and latent labelings as well as the resistance diameters of the graphs when specialized to multitask prediction and online switching problems the bound gives new and sharper results under certain conditions

kernelization of matrix updates when and how abstract we define what it means for a learning algorithm to be kernelizable in the case when the instances are vectors asymmetric matrices and symmetric matrices respectively we can characterize kernelizability in terms of an invariance of the algorithm to certain orthogonal transformations if we assume that the algorithm s action relies on a linear prediction then we can show that in each case the linear parameter vector must be a certain linear combination of the instances we give a number of examples of how to apply our methods in particular we show how to kernelize multiplicative updates for symmetric instance matrices

jensen s inequality for random elements in metric spaces and some applications abstract jensen s inequality is extended to metric spaces endowed with a convex combination operation applications include a dominated convergence theorem for both random elements and random sets a monotone convergence theorem for random sets and other results on set valued expectations in metric spaces and on random probability measures

an identity for kernel ridge regression this paper derives an identity connecting the square loss of ridge regression in on line mode with the loss of the retrospectively best regressor some corollaries about the properties of the cumulative loss of on line ridge regression are also obtained

the perceptron algorithm versus winnow linear versus logarithmic mistake bounds when few input variables are relevant abstract we give an adversary strategy that forces the perceptron algorithm to make kn mistakes in learning monotone disjunctions over n variables with at most k literals in contrast littlestone s algorithm winnow makes at most o k log n mistakes for the same problem both algorithms use thresholded linear functions as their hypotheses however winnow does multiplicative updates to its weight vector instead of the additive updates of the perceptron algorithm in general we call an algorithm additive if its weight vector is always a sum of a fixed initial weight vector and some linear combination of already seen instances thus the perceptron algorithm is an example of an additive algorithm we show that an adversary can force any additive algorithm to make n k 1 2 mistakes in learning a monotone disjunction of at most k literals simple experiments show that for k n winnow clearly outperforms the perceptron algorithm also on nonadversarial random data

online forecast combinations of distributions worst case bounds this paper considers forecasts with distribution functions that may vary through time the forecast is achieved by time varying combinations of individual forecasts we derive theoretical worst case bounds for general algorithms based on multiplicative updates of the combination weights the bounds are useful for studying properties of forecast combinations when data are non stationary and there is no unique best model

solving nonstationary classification problems with coupled support vector machines many learning problems may vary slowly over time in particular some critical real world applications when facing this problem it is desirable that the learning method could find the correct input output function and also detect the change in the concept and adapt to it we introduce the time adaptive support vector machine ta svm which is a new method for generating adaptive classifiers capable of learning concepts that change with time the basic idea of ta svm is to use a sequence of classifiers each one appropriate for a small time window but in contrast to other proposals learning all the hyperplanes in a global way we show that the addition of a new term in the cost function of the set of svms that penalizes the diversity between consecutive classifiers produces a coupling of the sequence that allows ta svm to learn as a single adaptive classifier we evaluate different aspects of the method using appropriate drifting problems in particular we analyze the regularizing effect of changing the number of classifiers in the sequence or adapting the strength of the coupling a comparison with other methods in several problems including the well known stagger dataset and the real world electricity pricing domain shows the good performance of ta svm in all tested situations

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

complete convergence for arrays of rowwise independent random variables and fuzzy random variables in convex combination spaces the aim of this study is to establish the complete convergence for arrays of rowwise independent random variables and fuzzy random variables in convex combination space our results not only generalize related previously reported results but also improve them some illustrative examples are provided

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

tracking the best linear predictor in most on line learning research the total on line loss of the algorithm is compared to the total loss of the best off line predictor u from a comparison class of predictors we call such bounds static bounds the interesting feature of these bounds is that they hold for an arbitrary sequence of examples recently some work has been done where the predictor ut at each trial t is allowed to change with time and the total on line loss of the algorithm is compared to the sum of the losses of ut at each trial plus the total cost for shifting to successive predictors this is to model situations in which the examples change over time and different predictors from the comparison class are best for different segments of the sequence of examples we call such bounds shifting bounds they hold for arbitrary sequences of examples and arbitrary sequences of predictors naturally shifting bounds are much harder to prove the only known bounds are for the case when the comparison class consists of a sequences of experts or boolean disjunctions in this paper we develop the methodology for lifting known static bounds to the shifting case in particular we obtain bounds when the comparison class consists of linear neurons linear combinations of experts our essential technique is to project the hypothesis of the static algorithm at the end of each trial into a suitably chosen convex region this keeps the hypothesis of the algorithm well behaved and the static bounds can be converted to shifting bounds

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

worst case quadratic loss bounds for prediction using linear functions and gradient descent studies the performance of gradient descent gd when applied to the problem of online linear prediction in arbitrary inner product spaces we prove worst case bounds on the sum of the squared prediction errors under various assumptions concerning the amount of a priori information about the sequence to predict the algorithms we use are variants and extensions of online gd whereas our algorithms always predict using linear functions as hypotheses none of our results requires the data to be linearly related in fact the bounds proved on the total prediction loss are typically expressed as a function of the total loss of the best fixed linear predictor with bounded norm all the upper bounds are tight to within constants matching lower bounds are provided in some cases finally we apply our results to the problem of online prediction for classes of smooth functions

a game theory approach to constrained minimax state estimation this paper presents a game theory approach to the constrained state estimation of linear discrete time dynamic systems in the application of state estimators there is often known model or signal information that is either ignored or dealt with heuristically for example constraints on the state values which may be based on physical considerations are often neglected because they do not easily fit into the structure of the state estimator this paper develops a method for incorporating state equality constraints into a minimax state estimator the algorithm is demonstrated on a simple vehicle tracking simulation

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

the generalization ability of online algorithms for dependent data we study the generalization performance of online learning algorithms trained on samples coming from a dependent source of data we show that the generalization error of any stable online algorithm concentrates around its regret an easily computable statistic of the online performance of the algorithm when the underlying ergodic process is or mixing we show high probability error bounds assuming the loss function is convex and we also establish sharp convergence rates and deviation bounds for strongly convex losses and several linear prediction problems such as linear and logistic regression least squares svm and boosting on dependent data in addition our results have straightforward applications to stochastic optimization with dependent data and our analysis requires only martingale convergence arguments we need not rely on more powerful statistical tools such as empirical process theory

analysis of the normalized lms algorithm with gaussian inputs the lms adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter for stability and convergence since the input power level is usually one of the statistical unknowns it is normally estimated from the data prior to beginning the adaptation process it is then assumed that the estimate is perfect in any subsequent analysis of the lms algorithm behavior in this paper the effects of the power level estimate are incorporated in a data dependent that appears explicitly within the algorithm the transient mean and second moment behavior of the modified lms nlms algorithm are evaluated taking into account the explicit statistical dependence of upon the input data the mean behavior of the algorithm is shown to converge to the wiener weight a constant coefficient matrix difference equation is derived for the weight fluctuations about the wiener weight the equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single frequency input in steady state for small expressions for the misadjustment error are also presented it is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the lms algorithm in the ale application the steady state weight fluctuations are shown to be mode dependent being largest at the frequency of the input

weighted last step min max algorithm with improved sub logarithmic regret abstract in online learning the performance of an algorithm is typically compared to the performance of a fixed function from some class with a quantity called regret forster 12 proposed a last step min max algorithm which was somewhat simpler than the algorithm of vovk 26 yet with the same regret in fact the algorithm he analyzed assumed that the choices of the adversary are bounded yielding artificially only the two extreme cases we fix this problem by weighing the examples in such a way that the min max problem will be well defined and provide analysis with logarithmic regret that may have better multiplicative factor than both bounds of forster 12 and vovk 26 we also derive a new bound that may be sub logarithmic as a recent bound of orabona et al 21 but may have better multiplicative factor finally we analyze the algorithm in a weak type of non stationary setting and show a bound that is sublinear if the non stationarity is sub linear as well

relative loss bounds for single neurons we analyze and compare the well known gradient descent algorithm and the more recent exponentiated gradient algorithm for training a single neuron with an arbitrary transfer function both algorithms are easily generalized to larger neural networks and the generalization of gradient descent is the standard backpropagation algorithm we prove worst case loss bounds for both algorithms in the single neuron case since local minima make it difficult to prove worst case bounds for gradient based algorithms we must use a loss function that prevents the formation of spurious local minima we define such a matching loss function for any strictly increasing differentiable transfer function and prove worst case loss bounds for any such transfer function and its corresponding matching loss the different forms of the two algorithms bounds indicates that exponentiated gradient outperforms gradient descent when the inputs contain a large number of irrelevant components simulations on synthetic data confirm these analytical results

the p norm generalization of the lms algorithm for adaptive filtering recently much work has been done analyzing online machine learning algorithms in a worst case setting where no probabilistic assumptions are made about the data this is analogous to the h sup spl infin setting used in adaptive linear filtering bregman divergences have become a standard tool for analyzing online machine learning algorithms using these divergences we motivate a generalization of the least mean squared lms algorithm the loss bounds for these so called p norm algorithms involve other norms than the standard 2 norm the bounds can be significantly better if a large proportion of the input variables are irrelevant i e if the weight vector we are trying to learn is sparse we also prove results for nonstationary targets we only know how to apply kernel methods to the standard lms algorithm i e p 2 however even in the general p norm case we can handle generalized linear models where the output of the system is a linear function combined with a nonlinear transfer function e g the logistic sigmoid

theorame ergodique ponctuel et lois fortes des grands nombres pour des points aleatoires d un espace metrique a courbure negative soit m un espace metrique separable complet a courbure negative suivant la definition de herer a l aide de la definition de herer de l esperance mathematique d un point aleatoire de m nous etendons a des suites de points aleatoires de m un theoreme ergodique ponctuel et plusieurs lois fortes des grands nombres ifgn connus dans le cas ou m est un espace de banach separable ifgn d etemadi de beck et giesy et de cuesta et matran dans les resultats obtenus la convergence a lieu au sens de hausdorff ou au sens de wijsman dans l espace des fermes de m

adaptive subgradient methods for online learning and stochastic optimization we present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient based learning metaphorically the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm we describe and analyze an apparatus for adaptively modifying the proximal function which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight we give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints we experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state of the art yet non adaptive subgradient algorithms

nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature we develop a nonlinear martingale theory for time discrete processes y n n n 0 these processes are defined on any filtered probability space f f n p n n 0 and have values in a metric space n d of nonpositive curvature in the sense of a d alexandrov the defining martingale property for such processes is e y n 1 f n y n p a s where the conditional expectation on the left hand side is defined as the minimizer of the functional z ed 2 z y n 1 within the space of f n measurable maps z n we give equivalent characterization of n valued martingales using merely the usual linear conditional expectations and derive fundamental properties of these martingales for example a martingale convergence theorem finally we exploit the relation with harmonic maps it turns out that a map f m n is harmonic w r t a given markov kernel p on m if and only if it maps markov chains x n n n with transition kernel p on m onto martingales f x n n n with values in n the nonlinear heat flow f n 0 x m n of a given initial map f 0 m n at time n is obtained as the filtered expectation f n x e x f x n f k k 0 of the random map f x n similarly the unique solution to the dirichlet problem for a given map g m n and a subset d m is obtained as f x e x g x d f k k 0 in both cases a crucial role is played by the notion of filtered expectation e x f k k 0 which will be discussed in detail moreover we prove jensen s inequality for expectations and filtered expectations and we prove weak and strong laws of large numbers for sequences of i i d random variables with values in n our theory is an extension of the classical linear martingale theory and of the nonlinear theory of martingales with values in manifolds as developed for example in emery 1989 and kendall 1990 the goal is to extend the previous framework towards processes with values in metric spaces this will lead to a stochastic approach to the theory of generalized harmonic maps with values in such singular spaces as developed by jost 1994 and korevaar and schoen 1993

a frobenius characterization of rational singularity in 2 dimensional graded rings a ring r is said to be f rational if for every prime p in r the local ring r p has the property that every system of parameters ideal is tightly closed as defined by hochster huneke it is proved that if r is a 2 dimensional graded ring with an isolated singularity at the irrelevant maximal ideal m then the following are equivalent 1 r has a rational singularity at m 2 r is f rational 3 a r 0 here a r as defined by goto watanabe denotes the least nonvanishing graded piece of the local cohomology module h m r the proof of this result relies heavily on the properties of derivations of r and suggests further questions in that direction paradigmatically if one knows that d a satisfies a certain property for every derivation d what can one conclude about the original ring element a

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

approach for a metric space with a convex combination operation and applications abstract in this study we embed a metric space endowed with a convex combination operation which is called a convex combination space into a banach space where the embedding preserves the structures of the metric and convex combination we also establish applications of this embedding for a random element that takes values in this type of space on the one hand we show some useful properties of mathematical expectation such as the representation of expectation through continuous affine mappings and the linearity of expectation on the other hand the notion of conditional expectation is also introduced and discussed using this embedding theorem we establish some basic properties of conditional expectation jensen s inequality the convergences of martingales and ergodic theorem

topological dynamical systems associated to ii1 factors abstract if n r is a separable ii1 factor the space h om n r of unitary equivalence classes of unital homomorphisms n r is shown to have a surprisingly rich structure if n is not hyperfinite h om n r is an infinite dimensional complete metrizable topological space with convex like structure and the outer automorphism group out n acts on it by affine homeomorphisms if n r then h om n r is just a point property t is reflected in the extreme points they re discrete in this case for certain free products n r every countable group acts nontrivially on h om n r and we show the extreme points are not discrete for these examples finally we prove that the dynamical systems associated to free group factors are isomorphic