mutations and the derived picard group of the brauer star algebra abstract in this paper we show that the derived picard group trpic a of the brauer star algebra of type n t is generated by shift pic a and equivalences h i i 1 n in the case t 1 where h i were shown to satisfy the relations of the braid group on the affine diagram a n 1 by schaps and zakay illouz in the multiplicity free case we show that trpic a is generated by a slightly bigger set

maps between tree and band modules the indecomposable modules over special biserial algebras are known to be of a specially simple form so called tree and band modules cf 6 2 crawley boevey has shown how to describe the homomorphisms between tree modules cf 3 and our objective in this paper is to extend this result to band modules tree and band modules occur as indecomposable representations of arbitrary algebras nevertheless we follow crawley boevey and consider zero relation algebras as the appropriate context given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k linear maps which are associated to these quiver homomorphisms it is interesting to note that in general the classification of maps between representations of a special biserial algebra a is a wild problem in fact the maps are equivalent to the representations of t a for instance take for a the path algebra of an a n 2 6 then t a is wild although the maps between indecomposable representations of a are easily to describe throughout k will be a fixed algebraically closed field maps are written on the left this paper results from a stay at the university of liverpool supported by the deutscher akademischer austauschdienst i am very grateful to sheila brenner for her suggestions

aggregation via empirical risk minimization given a finite set f of estimators the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in f it was conjectured that empirical minimization performed in the convex hull of f is an optimal aggregation method but we show that this conjecture is false despite that we prove that empirical minimization in the convex hull of a well chosen empirically determined subset of f is an optimal aggregation method

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

adaptive regression by mixing adaptation over different procedures is of practical importance different procedures perform well under different conditions in many practical situations it is rather hard to assess which conditions are approximately satisfied so as to identify the best procedure for the data at hand thus automatic adaptation over various scenarios is desirable a practically feasible method named adaptive regression by mixing arm is proposed to convexly combine general candidate regression procedures under mild conditions the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best simulations are conducted in several settings including comparing a parametric model with nonparametric alternatives comparing a neural network with a projection pursuit in multidimensional regression and combining bandwidths in kernel regression the results clearly support the theoretical property of arm the arm algorithm assigns we

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

on uniserial modules in the auslander reiten quiver abstract this article begins the study of irreducible maps involving finite dimensional uniserial modules over finite dimensional associative algebras we work on the classification of irreducible maps between two uniserials over triangular algebras and give estimates for the number of middle terms of an almost split sequence with a uniserial end term

resolutions over symmetric algebras with radical cube zero abstract let be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k satisfying j 3 0 let s 1 s r be representatives of the isomorphism classes of simple modules and let e be the r r matrix whose i j entry is dim k ext 1 s i s j if there exists an eigenvalue of e satisfying 2 then the minimal resolution of each non projective finitely generated module has exponential growth with radius of convergence 1 2 2 4 on the other hand if all eigenvalues of e satisfy 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated module are either bounded or grow linearly in this case we classify the possibilities for the matrix e the proof is an application of the perron frobenius theorem

special multiserial algebras are quotients of symmetric special multiserial algebras in this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles as a consequence we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra

a simple randomized algorithm for sequential prediction of ergodic time series we present a simple randomized procedure for the prediction of a binary sequence the algorithm uses ideas from previous developments of the theory of the prediction of individual sequences we show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges almost surely to that of the optimum given by the bayes predictor the desirable finite sample properties of the predictor are illustrated by its performance for markov processes in such cases the predictor exhibits near optimal behavior even without knowing the order of the markov process prediction with side information is also considered

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

the derivative of minkowski s x function abstract minkowski s x function can be seen as the confrontation of two number systems regular continued fractions and the alternated dyadic system this way of looking at it enables us to prove that its derivative when it exists in a wide sense can only attain two values zero and infinity it is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k 5 31972 and x exists then x 0 in the same way if the same average is less than 2 log 2 where is the golden ratio then x finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions

on auslander reiten components of blocks and self injective biserial algebras we investigate the existence of auslander reiten components of euclidean type for special biserial self injective algebras and for blocks of group algebras in particular we obtain a complete description of stable auslander reiten quivers for the tame self injective algebras considered here

on the duality of aggregation operators and k negations in this paper we study a class of duality functions given by the solution of a system of functional equations related to the de rham system with the aid of a generalized dyadic representation system in the unit interval we study a negation n which is a duality function for pairs of operators satisfying certain boundary conditions new properties of n are investigated including its singularity and fractal dimensions for several related sets as an application we obtain an explicit expression for k negations

universal linear prediction by model order weighting a common problem that arises in adaptive filtering autoregressive modeling or linear prediction is the selection of an appropriate order for the underlying linear parametric model we address this problem for linear prediction but instead of fixing a specific model order we develop a sequential prediction algorithm whose sequentially accumulated average squared prediction error for any bounded individual sequence is as good as the performance attainable by the best sequential linear predictor of order less than some m this predictor is found by transforming linear prediction into a problem analogous to the sequential probability assignment problem from universal coding theory the resulting universal predictor uses essentially a performance weighted average of all predictors for model orders less than m efficient lattice filters are used to generate the predictions of all the models recursively resulting in a complexity of the universal algorithm that is no larger than that of the largest model order examples of prediction performance are provided for autoregressive and speech data as well as an example of adaptive data equalization

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

lower bounds and aggregation in density estimation in this paper we prove the optimality of an aggregation procedure we prove lower bounds for aggregation of model selection type of m density estimators for the kullback leibler divergence kl the hellinger s distance and the l1 distance the lower bound with respect to the kl distance can be achieved by the on line type estimate suggested among others by yang 2000a combining these results we state that log m n is an optimal rate of aggregation in the sense of tsybakov 2003 where n is the sample size

a singular function with a non zero finite derivative on a dense set with hausdorff dimension one abstract this article closes a trilogy on the existence of singular functions with non zero finite derivatives in two previous papers the authors had exhibited a continuous strictly increasing singular function from 0 1 into 0 1 with a derivative that takes non zero finite values at two different zero measure sets first at the points of an uncountable set then at the points of a dense set in 0 1 in the present paper the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval a b has hausdorff dimension one another feature of this third article is the construction of the required function using the most paradigmatic of the singular functions the cantor lebesgue one

model selection in nonparametric regression model selection using a penalized data splitting device is studied in the context of nonparametric regression finite sample bounds under mild conditions are obtained the resulting estimates are adaptive for large classes of functions

local rademacher complexities and oracle inequalities in risk minimization let be a class of measurable functions f s 0 1 defined on a probability space s mathcal a p given a sample x1 xn of i i d random variables taking values in s with common distribution p let pn denote the empirical measure based on x1 xn we study an empirical risk minimization problem pnf min f given a solution fn of this problem the goal is to obtain very general upper bounds on its excess risk mathcal e p hat f n p hat f n inf f in mathcal f pf expressed in terms of relevant geometric parameters of the class using concentration inequalities and other empirical processes tools we obtain both distribution dependent and data dependent upper bounds on the excess risk that are of asymptotically correct order in many examples the bounds involve localized sup norms of empirical and rademacher processes indexed by functions from the class we use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification

wiener system modelling by exponentially weighted aggregation abstractin the paper the aggregation algorithm for wiener system modelling is proposed the method uses exponential weights to combine aggregate models from a given collection of d linear dynamic and dm nonlinear static models of the genuine system blocks that can be a priori hypotheses other system identification outcomes etc the resulting model has a noteworthy property it is almost as accurate as the best hypothetical wiener model built from a given collection this is caused by the fact that the error introduced by the aggregation routine decreases in the mean squared sense with growing number of measurements n as fast as cqn 1ln dmd where c and q are known constants simulations confirm theoretical findings and demonstrate applicability of the algorithm in engineering problems for small and moderate measurement data sets

multiserial and special multiserial algebras and their representations abstract in this paper we study multiserial and special multiserial algebras these algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type we define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module we show that all finitely generated modules over a special multiserial algebra are multiserial in particular this implies that in analogy to special biserial algebras being biserial special multiserial algebras are multiserial we then show that the class of symmetric special multiserial algebras coincides with the class of brauer configuration algebras where the latter are a generalization of brauer graph algebras we end by showing that any symmetric algebra with radical cube zero is special multiserial and so in particular it is a brauer configuration algebra

riesz nagy singular functions revisited abstract in 1952 f riesz and sz nagy published an example of a monotonic continuous function whose derivative is zero almost everywhere that is to say a singular function besides the function was strictly increasing their example was built as the limit of a sequence of deformations of the identity function as an easy consequence of the definition the derivative when it existed and was finite was found to be zero in this paper we revisit the riesz nagy family of functions and we relate it to a system for real number representation which we call 1 expansions with the help of these real number expansions we generalize the family the singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity

a singular function with a non zero finite derivative abstract this paper exhibits for the first time in the literature a continuous strictly increasing singular function with a derivative that takes non zero finite values at some points for all the known classic singular functions cantor s hellinger s minkowski s and the riesz nagy one including its generalizations and variants the derivative when it existed and was finite had to be zero as a result there arose a strong suspicion almost a conjecture that this had to be the case for any singular function we present here a singular function constructed as a patchwork of known classic singular functions with derivative 1 on a subset of the cantor set

a singular function with a non zero finite derivative on a dense set abstract the authors had exhibited in a previous paper a continuous strictly increasing singular function from 0 1 into 0 1 with a derivative that takes non zero finite values at the points of an uncountable set in this article the construction is improved to encompass a dense set

differentiability of the minkowski question mark function abstract we get necessary and sufficient conditions for the derivative of the minkowski question mark function x to be equal to zero or infinity these conditions are formulated in terms of sums s x t a 1 a t of partial quotients of continued fraction expansion to x 0 a 1 a t in particular we prove that if there exists c such that s x t 1 t log t log 2 c with 1 2 log 1 5 2 log 2 1 38 8 then x exists and x another result is as follows assume that there exists a constant c such that s x t 2 t c with 2 4 log 5 29 2 5 log 2 5 log 5 29 2 log 2 5 log 2 4 40 1 then x exists and x 0 we show that our conditions on the sum s x t are the best possible our results improve upon earlier theorems by paradis viader bibiloni and dushistova moshchevitin

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

trivial extensions of gentle algebras and brauer graph algebras abstract we show that two well studied classes of tame algebras coincide namely the class of symmetric special biserial algebras coincides with the class of brauer graph algebras we then explore the connection between gentle algebras and symmetric special biserial algebras by explicitly determining the trivial extension of a gentle algebra by its minimal injective co generator this is a symmetric special biserial algebra and hence a brauer graph algebra of which we explicitly give the brauer graph we further show that a brauer graph algebra gives rise via admissible cuts to many gentle algebras and that the trivial extension of a gentle algebra obtained via an admissible cut is the original brauer graph algebra as a consequence we prove that the trivial extension of a jacobian algebra of an ideal triangulation of a riemann surface with marked points in the boundary is isomorphic to the brauer graph algebra with brauer graph given by the arcs of the triangulation

information theory and mixing least squares regressions for gaussian regression we develop and analyze methods for combining estimators from various models for squared error loss an unbiased estimator of the risk of the mixture of general estimators is developed special attention is given to the case that the component estimators are least squares projections into arbitrary linear subspaces such as those spanned by subsets of explanatory variables in a given design we relate the unbiased estimate of the risk of the mixture estimator to estimates of the risks achieved by the components this results in simple and accurate bounds on the risk and its estimate in the form of sharp and exact oracle inequalities that is without advance knowledge of which model is best the resulting performance is comparable to or perhaps even superior to what is achieved by the best of the individual models furthermore in the case that the unknown parameter has a sparse representation our mixture estimator adapts to the underlying sparsity simulations show that the performance of these mixture estimators is better than that of a related model selection estimator which picks a model with the highest weight also the connection between our mixtures with bayes procedures is discussed

on the invariant theory for acyclic gentle algebras in this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions along the way we exhibit for such fields of rational invariants a transcendence basis in terms of schofield s determinantal semi invariants we also show that the moduli spaces of modules over regular irreducible component are just a products of projective spaces contents

sequential procedures for aggregating arbitrary estimators of a conditional mean in this correspondence a sequential procedure for aggregating linear combinations of a finite family of regression estimates is described and analyzed particular attention is given to linear combinations having coefficients in the generalized simplex the procedure is based on exponential weighting and has a computationally tractable approximation analysis of the procedure is based in part on techniques from the sequential prediction of nonrandom sequences here these techniques are applied in a stochastic setting to obtain cumulative loss bounds for the aggregation procedure from the cumulative loss bounds we derive an oracle inequality for the aggregate estimator for an unbounded response having a suitable moment generating function the inequality shows that the risk of the aggregate estimator is less than the risk of the best candidate linear combination in the generalized simplex plus a complexity term that depends on the size of the coefficient set the inequality readily yields convergence rates for aggregation over the unit simplex that are within logarithmic factors of known minimax bounds some preliminary results on model selection are also presented

simultaneous adaptation to the margin and to complexity in classification we consider the problem of adaptation to the margin and to complexity in binary classification we suggest a learning method with a numerically easy aggregation step adaptivity both to the margin and complexity in classification usually involves empirical risk minimization or rademacher complexities which lead to numerical difficulties on the other hand there exist classifiers that are easy to compute and that converge with fast rates but are not adaptive combining these classifiers by our aggregation procedure we get numerically realizable adaptive classifiers that converge with fast rates

the hausdorff dimension of the level sets of takagi s function abstract recently maddock 2006 12 has conjectured that the hausdorff dimension of each level set of takagi s function is at most 1 2 we prove this conjecture using the self affinity of the function of takagi and the existing relationship between the hausdorff and box counting dimensions

tame biserial algebras a similar result is suspected if we drop the property representationfinite in classes 1 and 2 and we will start an investigation of biserial algebras of infinite representation type the key to the result for representation finite algebras was to observe that each representation finite biserial algebra is special 13 see also the basic notations below which means that it can be presented by a quiver with nice relations special algebras play an important role in the modular representation theory of finite groups namely each representation finite block of a group algebra and some of the tame blocks they occur only in characteristic 2 are special 7 10 4 moreover in the complex representation theory of the lorentz group the so called harish chandra modules are defined over tame special algebras 6 in this paper we show that l a 62 for any special algebra a and that special algebras of infinite representation type are tame our paper consists of four parts first we prove that each special algebra is a factor of a special symmetric algebra theorem 1 5 for the latter the methods of gelfand and ponomarev used in the classification of the indecomposable harish chandra modules of the lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras proposition 2 3 presented in section 2 from this list we 480 0021 8693 85 3 00