learning prototypes and distances a prototype reduction technique based on nearest neighbor error minimization a prototype reduction algorithm is proposed which simultaneously trains both a reduced set of prototypes and a suitable local metric for these prototypes starting with an initial selection of a small number of prototypes it iteratively adjusts both the position features of these prototypes and the corresponding local metric weights the resulting prototypes metric combination minimizes a suitable estimation of the classification error probability good performance of this algorithm is assessed through experiments with a number of benchmark data sets and with a real task consisting in the verification of images of human faces

multidimensional unrepetitive configurations note abstract we show that one can associate a letter of a finite alphabet with each lattice point of the n dimensional space so that an r th power free word corresponds to each straight line of the space this result is obtained as a consequence of the existence of bi infinite words on finite alphabets which are r th power free together with a certain class of their subwords an analogous property is obtained for some avoidable sets of patterns

two non holonomic lattice walks in the quarter plane we present two classes of random walks restricted to the quarter plane with non holonomic generating functions the non holonomicity is established using the iterated kernel method a variant of the kernel method this adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions m mishna classifying lattice walks in the quarter plane j combin theory ser a 116 2009 460 477 the method also yields an asymptotic expression for the number of walks of length n

words avoiding repetitions in arithmetic progressions carpi constructed an infinite word over a 4 letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference we show a connection between carpi s construction and the paperfolding words we extend carpi s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference we also construct infinite words avoiding overlaps and infinite words avoiding all sufficiently large squares in arithmetic progressions of odd difference we use these words to construct labelings of the 2 dimensional integer lattice such that any line through the lattice encounters a squarefree resp overlapfree sequence of labels

linear reconstruction measure steered nearest neighbor classification framework the linear reconstruction measure lrm which determines the nearest neighbors of the query sample in all known training samples by sorting the minimum l 2 norm error linear reconstruction coefficients is introduced in this paper the intuitive interpretation and mathematical proofs are presented to reveal the efficient working mechanism of lrm through analyzing the physical meaning of coefficients and regularization items we find that lrm provides more useful information and advantages than the conventional similarity measure model which calculates the distance between two entities i e conventional point to point c ptp inspired by the advantages of lrm the linear reconstruction measure steered nearest neighbor classification framework lrm nncf is designed with eight classifiers according to different decision rules and models of lrm evaluation on several face databases and the experimental results demonstrate that these proposed classifiers can achieve greater performance than the c ptp based 1 nns and competitive recognition accuracy and robustness compared with the state of the art classifiers

an elementary solution of gessel s walks in the quadrant abstract around 2000 ira gessel conjectured that the number of lattice walks in the quadrant n 2 starting and ending at the origin 0 0 and taking their steps in had a simple hypergeometric form in the following decade this problem was recast in the systematic study of walks with small steps that is steps in 1 0 1 2 confined to the quadrant the generating functions of such walks are archetypal solutions of partial discrete differential equations a complete classification of quadrant walks according to the nature of their generating function algebraic d finite or not is now available but gessel s walks remained mysterious because they were the only model among the 23 d finite ones that had not been given an elementary solution instead gessel s conjecture was first proved using an inventive computer algebra approach in 2008 a year later the associated three variate generating function was proved to be algebraic by a computer algebra tour de force this was re proved recently using elaborate complex analysis machinery we give here an elementary and constructive proof our approach also solves other quadrant models with multiple steps recently proved to be algebraic via computer algebra

irahc instance reduction algorithm using hyperrectangle clustering abstract in instance based classifiers there is a need for storing a large number of samples as training set in this work we propose an instance reduction method based on hyperrectangle clustering called instance reduction algorithm using hyperrectangle clustering irahc irahc removes non border interior instances and keeps border and near border ones this paper presents an instance reduction process based on hyperrectangle clustering a hyperrectangle is an n dimensional rectangle with axes aligned sides which is defined by min and max points and a corresponding distance function the min max points are determined by using the hyperrectangle clustering algorithm instance based learning algorithms are often confronted with the problem of deciding which instances must be stored to be used during an actual test storing too many instances can result in a large memory requirements and a slow execution speed in irahc core of instance reduction process is based on set of hyperrectangles the performance has been evaluated on real world data sets from uci repository by the 10 fold cross validation method the results of the experiments have been compared with state of the art methods which show superiority of the proposed method in terms of classification accuracy and reduction percentage

a mating of trees approach for graph distances in random planar maps we introduce a general technique for proving estimates for certain random planar maps which belong to the gamma liouville quantum gravity lqg universality class for gamma in 0 2 the family of random planar maps we consider are those which can be encoded by a two dimensional random walk with i i d increments via a mating of trees bijection and includes the uniform infinite planar triangulation uipt gamma sqrt 8 3 and planar maps weighted by the number of different spanning trees gamma sqrt 2 bipolar orientations gamma sqrt 4 3 or schnyder woods gamma 1 that can be put on the map using our technique we prove estimates for graph distances in the above family of random planar maps in particular we obtain non trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the watabiki prog theor phys suppl 114 1 17 1993 prediction for the hausdorff dimension of gamma lqg and we establish the existence of an exponent for certain distances in the map the basic idea of our approach is to compare a given random planar map m to a mated crt map a random planar map constructed from a correlated two dimensional brownian motion using a strong coupling zaitsev in esaim probab stat 2 41 108 1998 of the encoding walk for m and the brownian motion used to construct the mated crt map this allows us to deduce estimates for graph distances in m from the estimates for graph distances in the mated crt map which we proved using continuum theory in a previous work in the special case when gamma sqrt 8 3 we instead deduce estimates for the sqrt 8 3 mated crt map from known results for the uipt the arguments of this paper do not directly use sle lqg and can be read without any knowledge of these objects

a novel approach for designing adaptive fuzzy classifiers based on the combination of an artificial immune network and a memetic algorithm in this study we propose a novel approach for designing fuzzy classifiers the first part of our approach is a new preprocess algorithm called spp silhouette cluster validity index aided pre process via k means the spp algorithm has been performed on the data set to determine the numbers of the membership functions and their initial boundaries then the mopt ainetls algorithm modified version of opt ainet combined with local search strategy of memetic algorithm the second part of the approach examines search space to find the optimal values of fuzzy rules and membership functions for the system the mopt ainetls is the combination of the memetic algorithm and a modified version of the opt ainet algorithm in which some changes were made in the suppression and hypermutation mechanisms of the original opt ainet algorithm these two new mechanisms are called the intelligent suppression mechanism and the adaptive hypermutation operator combining the modified version of opt ainet with the local search strategy of the memetic algorithm improves the accuracy of the classification rate an effective search process has been realized using the mopt ainetls because the global search capability of opt ainet is complemented by the local search strategy of the memetic algorithm to test the performance of this new approach twenty different well known classification dataset benchmark problems from the uci dataset were used the average 3x10 cross fold validation results obtained from these datasets are presented and compared with the results of certain classification algorithms reported in the literature the wilcoxon signed rank test was also used for statistical comparisons the obtained results demonstrate the effectiveness of the proposed approach

proof of ira gessel s lattice path conjecture abstract we present a computer aided yet fully rigorous proof of ira gessel s tantalizingly simply stated conjecture that the number of ways of walking 2n steps in the region x y 0 y 0 of the square lattice with unit steps in the east west north and south directions that start and end at the origin equals

a memetic algorithm for evolutionary prototype selection a scaling up approach prototype selection problem consists of reducing the size of databases by removing samples that are considered noisy or not influential on nearest neighbour classification tasks evolutionary algorithms have been used recently for prototype selection showing good results however due to the complexity of this problem when the size of the databases increases the behaviour of evolutionary algorithms could deteriorate considerably because of a lack of convergence this additional problem is known as the scaling up problem memetic algorithms are approaches for heuristic searches in optimization problems that combine a population based algorithm with a local search in this paper we propose a model of memetic algorithm that incorporates an ad hoc local search specifically designed for optimizing the properties of prototype selection problem with the aim of tackling the scaling up problem in order to check its performance we have carried out an empirical study including a comparison between our proposal and previous evolutionary and non evolutionary approaches studied in the literature the results have been contrasted with the use of non parametric statistical procedures and show that our approach outperforms previously studied methods especially when the database scales up

walks confined in a quadrant are not always d finite we consider planar lattice walks that start from a prescribed position take their steps in a given finite subset of z2 and always stay in the quadrant x 0 y 0 we first give a criterion which guarantees that the length generating function of these walks is d finite that is satisfies a linear differential equation with polynomial coefficients this criterion applies among others to the ordinary square lattice walks then we prove that walks that start from 1 1 take their steps in 2 1 1 2 and stay in the first quadrant have a non d finite generating function our proof relies on a functional equation satisfied by this generating function and on elementary complex analysis

ifs coco instance and feature selection based on cooperative coevolution with nearest neighbor rule feature and instance selection are two effective data reduction processes which can be applied to classification tasks obtaining promising results although both processes are defined separately it is possible to apply them simultaneously this paper proposes an evolutionary model to perform feature and instance selection in nearest neighbor classification it is based on cooperative coevolution which has been applied to many computational problems with great success the proposed approach is compared with a wide range of evolutionary feature and instance selection methods for classification the results contrasted through non parametric statistical tests show that our model outperforms previously proposed evolutionary approaches for performing data reduction processes in combination with the nearest neighbor rule

experimental study on prototype optimisation algorithms for prototype based classification in vector spaces prototype based classification relies on the distances between the examples to be classified and carefully chosen prototypes a small set of prototypes is of interest to keep the computational complexity low while maintaining high classification accuracy an experimental study of some old and new prototype optimisation techniques is presented in which the prototypes are either selected or generated from the given data these condensing techniques are evaluated on real data represented in vector spaces by comparing their resulting reduction rates and classification performance usually the determination of prototypes is studied in relation with the nearest neighbour rule we will show that the use of more general dissimilarity based classifiers can be more beneficial an important point in our study is that the adaptive condensing schemes here discussed allow the user to choose the number of prototypes freely according to the needs if such techniques are combined with linear dissimilarity based classifiers they provide the best trade off of small condensed sets and high classification accuracy

a human proof of gessel s lattice path conjecture gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either west east south west or north east in 2001 ira gessel conjectured a nice closed form expression for the number of gessel walks ending at the origin in 2008 kauers koutschan and zeilberger gave a computer aided proof of this conjecture the same year bostan and kauers showed again using computer algebra tools that the complete generating function of gessel walks is algebraic in this article we propose the first human proofs of these results they are derived from a new expression for the generating function of gessel walks in terms of weierstrass zeta functions

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

on using prototype reduction schemes to optimize dissimilarity based classification the aim of this paper is to present a strategy by which a new philosophy for pattern classification namely that pertaining to dissimilarity based classifiers dbcs can be efficiently implemented this methodology proposed by duin and his co authors see refs experiments with a featureless approach to pattern recognition pattern recognition lett 18 1997 1159 1166 relational discriminant analysis pattern recognition lett 20 1999 1175 1181 dissimilarity representations allow for buillding good classifiers pattern recognition lett 23 2002 943 956 dissimilarity representations in pattern recognition concepts theory and applications ph d thesis delft university of technology delft the netherlands 2005 prototype selection for dissimilarity based classifiers pattern recognition 39 2006 189 208 is a way of defining classifiers between the classes and is not based on the feature measurements of the individual patterns but rather on a suitable dissimilarity measure between them the advantage of this methodology is that since it does not operate on the class conditional distributions the accuracy can exceed the bayes error bound the problem with this strategy is however the need to compute store and process the inter pattern dissimilarities for all the training samples and thus the accuracy of the classifier designed in the dissimilarity space is dependent on the methods used to achieve this in this paper we suggest a novel strategy to enhance the computation for all families of dbcs rather than compute store and process the dbc based on the entire data set we advocate that the training set be first reduced into a smaller representative subset also rather than determine this subset on the basis of random selection or clustering etc we advocate the use of a prototype reduction scheme prs whose output yields the points to be utilized by the dbc the rationale for this is explained in the paper apart from utilizing prss in the paper we also propose simultaneously employing the mahalanobis distance as the dissimilarity measurement criterion to increase the dbcs classification accuracy our experimental results demonstrate that the proposed mechanism increases the classification accuracy when compared with the conventional approaches for samples involving real life as well as artificial data sets even though the resulting dissimilarity criterion is not symmetric

local zero estimates and effective division in rings of algebraic power series we give a necessary condition for algebraicity of finite modules over the ring of formal power series this condition is given in terms of local zero estimates in fact we show that this condition is also sufficient when the module is a ring with some additional properties to prove this result we show an effective weierstrass division theorem and an effective solution to the ideal membership problem in rings of algebraic power series finally we apply these results to prove a gap theorem for power series which are remainders of the grauert hironaka galligo division theorem

exit times from cones in ℝ n of brownian motion for a fairly general class of cones inn dimensions n 3 we determine the corresponding distributions of brownian first exit times asymptotic results may then be read off

some approaches to improve tree based nearest neighbour search algorithms nearest neighbour search is a widely used technique in pattern recognition during the last three decades a large number of fast algorithms have been proposed in this work we are interested in algorithms that can be used with any dissimilarity function provided that it fits the mathematical notion of distance some of such algorithms organize in preprocessing time the data in a tree structure that is traversed in search time to find the nearest neighbour the speedup is obtained using some pruning rules that avoid the traversal of some parts of the tree in this work two new decomposition methods to build the tree and three new pruning rules are explored the behaviour of our proposal is studied through experiments with synthetic and real data

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

uniformly growing k th power free homomorphisms a string is called kth power free if it does not have xk as a nonempty substring for all nonnegative rational numbers k kth power free strings and kth power free homomorphisms are investigated and the shortest uniformly growing square free k 2 and cube free k 3 homomorphisms mapping into least alphabets with three and two letters are introduced it is shown that there exist exponentially many square free and cube free strings of each length over these alphabets sharpening the kth power freeness to the repetitive threshold rt n of n letter alphabets we provide arguments for the nonexistence of various rt n th power free homomorphisms

remark on exit times from cones in mathbb r n of brownian motion our purpose is to show how the asymptotics in corollary 1 3 of 2 can be obtained under much weaker hypotheses it turns out the problem essentially reduces to showing that if r s is a bessel process u 0 and 0 then p left mathop f limits 0 1 r s 2 ds leqq u right o t alpha as t we provide a simple proof of this fact