on the uniqueness problems of entire functions and their derivatives in this paper we obtain some uniqueness theorems for entire functions and their derivatives sharing the same fixed points with the same multiplicities

explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations abstract we find an explicit combinatorial interpretation of the coefficients of kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups s n in terms of free cumulants r 2 r 3 of the corresponding young diagram our interpretation is based on counting certain factorizations of a given permutation

posterior probability support vector machines for unbalanced data this paper proposes a complete framework of posterior probability support vector machines ppsvms for weighted training samples using modified concepts of risks linear separability margin and optimal hyperplane within this framework a new optimization problem for unbalanced classification problems is formulated and a new concept of support vectors established furthermore a soft ppsvm with an interpretable parameter spl nu is obtained which is similar to the spl nu svm developed by scho spl uml lkopf et al and an empirical method for determining the posterior probability is proposed as a new approach to determine spl nu the main advantage of an ppsvm classifier lies in that fact that it is closer to the bayes optimal without knowing the distributions to validate the proposed method two synthetic classification examples are used to illustrate the logical correctness of ppsvms and their relationship to regular svms and bayesian methods several other classification experiments are conducted to demonstrate that the performance of ppsvms is better than regular svms in some cases compared with fuzzy support vector machines fsvms the proposed ppsvm is a natural and an analytical extension of regular svms based on the statistical learning theory

an entire function and its derivatives sharing a polynomial in this paper we study the growth of solutions of a first order linear differential equation and that of a second order linear differential equation from this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities the results in this paper also improve some known results some examples show that the results in this paper are best possible

weighted sharing three values and brosch s theorem this paper studies the idea of weighted sharing and proves two main theorems which generalize some results given by g brosch q zhang and other authors

weighted sharing three values and uniqueness of meromorphic functions using the idea of weighted sharing we prove some results on uniqueness of meromorphic functions sharing three values which improve some results given by h x yi i lahiri x hua and other authors

meromorphic functions sharing three weighted values in this paper we deal with the relation between the characteristics of two meromorphic functions that share three weighted values which improve the results of q c zhang and through the results two conjectures which proposed by mues and osgood yang are solved in a weaker condition

gaussian fluctuations of jack deformed random young diagrams we introduce a large class of random young diagrams which can be regarded as a natural one parameter deformation of some classical young diagram ensembles a deformation which is related to jack polynomials and jack characters we show that each such a random young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically gaussian

multiclass posterior probability support vector machines tao et al have recently proposed the posterior probability support vector machine ppsvm which uses soft labels derived from estimated posterior probabilities to be more robust to noise and outliers tao et al s model uses a window based density estimator to calculate the posterior probabilities and is a binary classifier we propose a neighbor based density estimator and also extend the model to the multiclass case our bias variance analysis shows that the decrease in error by ppsvm is due to a decrease in bias on 20 benchmark data sets we observe that ppsvm obtains accuracy results that are higher or comparable to those of canonical svm using significantly fewer support vectors

on meromorphic functions that share three values of finite weights abstract a uniqueness theorem for two distinct non constant meromorphic functions that share three values of finite weights is proved which generalizes two previous results by h x yi and x m li and h x yi as applications of it many known results by h x yi and p li etc could be improved furthermore with the concept of finite weight sharing extensions on osgood yang s conjecture and mues conjecture and a generalization of some prevenient results by m ozawa and h ueda ect could be obtained

spectra of random linear combinations of matrices defined via representations and coxeter generators of the symmetric group we consider the asymptotic behavior as n of the spectra of random matrices of the form k where for each n the random variables z nk are i i d standard gaussian and the matrices n k k 1 are obtained by applying an irreducible unitary representation n of the symmetric group on 1 2 n to the transposition k k 1 that interchanges k and k i thus n k k 1 is both unitary and self adjoint with all eigenvalues either 1 or 1 irreducible representations of the symmetric group on 1 2 n are indexed by partitions n of n a consequence of the results we establish is that if n 1 n 2 0 0 is the partition of n corresponding to n n 1 n 2 0 is the corresponding conjugate partition of n i e the young diagram of n is the transpose of the young diagram of n lim n n i n pi for each i 1 and lim n n j n q j for each j 1 then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is gaussian with random mean z and variance 1 2 where is the constant i p 2 i j q 2 j and z is a standard gaussian random variable

asymptotics of q plancherel measures in this paper we are interested in the asymptotic size of the rows and columns of a random young diagram under a natural deformation of the plancherel measure coming from hecke algebras the first lines of such diagrams are typically of order n so it does not fit in the context of the work of biane int math res notices 4 179 192 2001 and niady probab theory relat fields 136 263 297 2006 using the theory of polynomial functions on young diagrams of kerov and olshanski we are able to compute explicitly the first and second order asymptotics of the length of the first rows our method also works for other measures for example those coming from schur weyl representations

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

second order freeness and fluctuations of random matrices i gaussian and wishart matrices and cyclic fock spaces abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we introduce the concept of second order freeness and interpret the global fluctuations of gaussian and wishart random matrices by a general limit theorem for second order freeness by introducing cyclic fock space we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation annihilation and preservation operators we show that orthogonal families of gaussian and wishart random matrices are asymptotically free of second order

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

representations of symmetric groups and free probability abstract we consider representations of symmetric groupssqfor largeq we give the asymptotic behaviour of the characters when the corresponding young diagrams rescaled by a factorq 1 2 converge to some prescribed shape this behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram we also show that the basic operations of representation theory like taking tensor products restriction or induction have a limiting behavior which can be described using the free probability theory of d voiculescu

stein s method and random character ratios stein s method is used to prove limit theorems for random character ratios tools are developed for four types of structures finite groups gelfand pairs twisted gelfand pairs and association schemes as one example an error term is obtained for a central limit theorem of kerov on the spectrum of the cayley graph of the symmetric group generated by i cycles other main examples include an error term for a central limit theorem of ivanov on character ratios of random projective representations of the symmetric group and a new central limit theorem for the spectrum of certain random walks on perfect matchings the results are obtained with very little information a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk the limit theorems stated in this paper are for normal approximation but many of the tools developed are applicable for arbitrary distributional approximation

on the characteristics of meromorphic functions that share three values cm in this paper by studying the counting functions of the common 1 points of meromorphic functions a more precise relation between the characteristics of meromorphic functions that share three values cm has been obtained as applications of this many known results can be improved

cumulants in noncommutative probability theory i noncommutative exchangeability systems cumulants linearize convolution of measures we use a formula of good to define noncommutative cumulants in a very general setting it turns out that the essential property needed is exchangeability of random variables roughly speaking the formula says that cumulants are moments of a certain discrete fourier transform of a random variable this provides a simple unified method to understand the known examples of cumulants like classical free and various q cumulants

second order freeness and fluctuations of random matrices ii unitary random matrices abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we show how the concept of second order freeness which was introduced in part i allows one to understand global fluctuations of haar distributed unitary random matrices in particular independence between the unitary ensemble and another ensemble goes in the large n limit over into asymptotic second order freeness two important consequences of our general theory are i we obtain a natural generalization of a theorem of diaconis and shahshahani to the case of several independent unitary matrices ii we can show that global fluctuations in unitarily invariant multi matrix models are not universal

zonal polynomials via stanleyʼs coordinates and free cumulants abstract we study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power sum symmetric functions we show that the zonal characters just like the characters of the symmetric groups admit a nice combinatorial description in terms of stanley s multirectangular coordinates of young diagrams we also study the analogue of kerov polynomials namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients in this way we prove two recent conjectures of lassalle for jack polynomials in the special case of zonal polynomials

kernel matching pursuits with arbitrary loss functions the purpose of this research is to develop a classifier capable of state of the art performance in both computational efficiency and generalization ability while allowing the algorithm designer to choose arbitrary loss functions as appropriate for a give problem domain this is critical in applications involving heavily imbalanced noisy or non gaussian distributed data to achieve this goal a kernel matching pursuit kmp framework is formulated where the objective is margin maximization rather than the standard error minimization this approach enables excellent performance and computational savings in the presence of large imbalanced training data sets and facilitates the development of two general algorithms these algorithms support the use of arbitrary loss functions allowing the algorithm designer to control the degree to which outliers are penalized and the manner in which non gaussian distributed data is handled example loss functions are provided and algorithm performance is illustrated in two groups of experimental results the first group demonstrates that the proposed algorithms perform equivalent to several state of the art machine learning algorithms on well published balanced data the second group of results illustrates superior performance by the proposed algorithms on imbalanced non gaussian data achieved by employing loss functions appropriate for the data characteristics and problem domain

weighted mahalanobis distance kernels for support vector machines the support vector machine svm has been demonstrated to be a very effective classifier in many applications but its performance is still limited as the data distribution information is underutilized in determining the decision hyperplane most of the existing kernels employed in nonlinear svms measure the similarity between a pair of pattern images based on the euclidean inner product or the euclidean distance of corresponding input patterns which ignores data distribution tendency and makes the svm essentially a ldquolocalrdquo classifier in this paper we provide a step toward a paradigm of kernels by incorporating data specific knowledge into existing kernels we first find the data structure for each class adaptively in the input space via agglomerative hierarchical clustering ahc and then construct the weighted mahalanobis distance wmd kernels using the detected data distribution information in wmd kernels the similarity between two pattern images is determined not only by the mahalanobis distance md between their corresponding input patterns but also by the sizes of the clusters they reside in although wmd kernels are not guaranteed to be positive definite pd or conditionally positive definite cpd satisfactory classification results can still be achieved because regularizers in svms with wmd kernels are empirically positive in pseudo euclidean pe spaces experimental results on both synthetic and real world data sets show the effectiveness of ldquopluggingrdquo data structure into existing kernels

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

representations of lie groups and random matrices we study the asymptotics of representations of a fixed compact lie group we prove that the limit behavior of a sequence of such representations can be described in terms of certain random matrices in particular operations on representations for example tensor product restriction to a subgroup correspond to some natural operations on random matrices respectively sum of independent random matrices taking the corners of a random matrix our method of proof is to treat the canonical block matrix associated to a representation as a random matrix with non commutative entries

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

small functions and uniqueness of meromorphic functions we discuss the uniqueness of meromorphic functions sharing three weighted values and provide a complete answer to a question of t c alzahary

meromorphic functions that share four small functions in this paper we prove that if two nonconstant meromorphic functions f and g share two small functions cm and share other two small functions im then f must be a quasi mobius transformation of g this result is a generalization of several results obtained by g g gundersen and li yang

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

a kernel based two class classifier for imbalanced data sets many kernel classifier construction algorithms adopt classification accuracy as performance metrics in model evaluation moreover equal weighting is often applied to each data sample in parameter estimation these modeling practices often become problematic if the data sets are imbalanced we present a kernel classifier construction algorithm using orthogonal forward selection ofs in order to optimize the model generalization for imbalanced two class data sets this kernel classifier identification algorithm is based on a new regularized orthogonal weighted least squares rowls estimator and the model selection criterion of maximal leave one out area under curve loo auc of the receiver operating characteristics rocs it is shown that owing to the orthogonalization procedure the loo auc can be calculated via an analytic formula based on the new regularized orthogonal weighted least squares parameter estimator without actually splitting the estimation data set the proposed algorithm can achieve minimal computational expense via a set of forward recursive updating formula in searching model terms with maximal incremental loo auc value numerical examples are used to demonstrate the efficacy of the algorithm