on the geometry of sasakian einstein 5 manifolds on simply connected five manifolds sasakian einstein metrics coincide with riemannian metrics admitting real killing spinors which are of great interest as models of near horizon geometry for three brane solutions in superstring theory 24 we expand on the recent work of demailly and kollar 14 and johnson and kollar 20 who give methods for constructing kahler einstein metrics on log del pezzo surfaces by a previous result of the first two authors 9 circle v bundles over log del pezzo surfaces with kahler einstein metrics have sasakian einstein metrics on the total space of the bundle here these simply connected 5 manifolds arise as links of isolated hypersurface singularities which by the well known work of smale 36 together with 11 must be diffeomorphic to s 5 l s 2 s 3 more precisely using methods from mori theory in algebraic geometry we prove the existence of 14 inequivalent sasakian einstein structures on s 2 s 3 and infinite families of such structures on l s 2 s 3 with 2 l 7 we also discuss the moduli problem for these sasakian einstein structures

the algebra of complex 2 2 matrices and a general closed baker campbell hausdorff formula we derive a closed formula for the baker campbell hausdorff series expansion in the case of complex matrices for arbitrary matrices a and b and a matrix z such that our result expresses z as a linear combination of a and b their commutator and the identity matrix i the coefficients in this linear combination are functions of the traces and determinants of a and b and the trace of their product the derivation proceeds purely via algebraic manipulations of the given matrices and their products making use of relations developed here based on the cayley hamilton theorem as well as a characterization of the consequences of and or its determinant being zero or otherwise as a corollary of our main result we also derive a closed formula for the zassenhaus expansion we apply our results to several special cases most notably the parametrization of the product of two matrices and a verification of the recent result of van brunt and visser 2015 j phys a math theor 48 225207 for complex matrices in this latter case deriving also the related zassenhaus formula which turns out to be quite simple we then show that this simple formula should be valid for all matrices and operators

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics contact geometry has been applied to various mathematical sciences and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry here such a dually flat space is related to a legendre submanifold in a contact manifold in this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact hamiltonian vector fields based on these descriptions two ways of lifting vector fields on legendre submanifolds to contact manifolds are given for some classes of these lifted vector fields invariant measures in contact manifolds and stability analysis around legendre submanifolds are explicitly given throughout this paper legendre duality is explicitly stated in addition to show how to apply these general methodologies to applied mathematical disciplines electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed

characterizing the function space for bayesian kernel models kernel methods have been very popular in the machine learning literature in the last ten years mainly in the context of tikhonov regularization algorithms in this paper we study a coherent bayesian kernel model based on an integral operator defined as the convolution of a kernel with a signed measure priors on the random signed measures correspond to prior distributions on the functions mapped by the integral operator we study several classes of signed measures and their image mapped by the integral operator in particular we identify a general class of measures whose image is dense in the reproducing kernel hilbert space rkhs induced by the kernel a consequence of this result is a function theoretic foundation for using non parametric prior specifications in bayesian modeling such as gaussian process and dirichlet process prior distributions we discuss the construction of priors on spaces of signed measures using gaussian and levy processes with the dirichlet processes being a special case the latter computational issues involved with sampling from the posterior distribution are outlined for a univariate regression and a high dimensional classification problem

a note on exponential rosenbrock euler method for the finite element discretization of a semilinear parabolic partial differential equation abstract in this paper we consider the numerical approximation of a general second order semilinear parabolic partial differential equation equations of this type arise in many contexts such as transport in porous media using finite element method for space discretization and the exponential rosenbrock euler method for time discretization we provide a convergence proof in space and time under only the standard lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solutions this is in contrast to restrictive assumptions made in the literature where the authors have considered only approximation in time so far in their convergence proofs the main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data in particular the method achieves optimal convergence order o h 2 t 2 when the initial data belongs to the domain of the linear operator numerical simulations to sustain our theoretical result are provided

special case closed form of the baker campbell hausdorff formula the baker campbell hausdorff formula is a general result for the quantity where x and y are not necessarily commuting for completely general commutation relations between x and y the free lie algebra the general result is somewhat unwieldy however in specific physics applications the commutator while non zero might often be relatively simple which sometimes leads to explicit closed form results we consider the special case and show that in this case the general result reduces to furthermore we explicitly evaluate the symmetric function demonstrating that and relate this to previously known results for instance this result includes but is considerably more general than results obtained from either the heisenberg commutator or the creation destruction commutator

numerical values of goldberg s coefficients in the series for log e x e y calcul des coefficients de goldberg dans la serie infinie de log e x e y pour x et y non communitatifs avec des mots de longueur 20

a software package for lie algebraic computations the paper presents a computer algebra package that facilitates lie algebraic symbolic computations required in the solution of a variety of problems such as the solution of right invariant differential equations evolving on lie groups lie theory is a powerful tool helpful in the analysis and design of modern nonlinear control laws nonlinear filters and the study of particle dynamics the practical application of lie theory often results in highly complex symbolic expressions that are difficult to handle efficiently without the aid of a computer software tool the aim of the package is to facilitate and encourage further research relying on lie algebraic computations

multicategory support vector machines two category support vector machines svm have been very popular in the machine learning community for classification problems solving multicategory problems by a series of binary classifiers is quite common in the svm paradigm however this approach may fail under various circumstances we propose the multicategory support vector machine msvm which extends the binary svm to the multicategory case and has good theoretical properties the proposed method provides a unifying framework when there are either equal or unequal misclassification costs as a tuning criterion for the msvm an approximate leave one out cross validation function called generalized approximate cross validation is derived analogous to the binary case the effectiveness of the msvm is demonstrated through the applications to cancer classification using microarray data and cloud classification with satellite radiance profiles

bayesian classification of tumours by using gene expression data summary precise classification of tumours is critical for the diagnosis and treatment of cancer diagnostic pathology has traditionally relied on macroscopic and microscopic histology and tumour morphology as the basis for the classification of tumours current classification frameworks however cannot discriminate between tumours with similar histopathologic features which vary in clinical course and in response to treatment in recent years there has been a move towards the use of complementary deoxyribonucleic acid microarrays for the classification of tumours these high throughput assays provide relative messenger ribonucleic acid expression measurements simultaneously for thousands of genes a key statistical task is to perform classification via different expression patterns gene expression profiles may offer more information than classical morphology and may provide an alternative to classical tumour diagnosis schemes the paper considers several bayesian classification methods based on reproducing kernel hilbert spaces for the analysis of microarray data we consider the logistic likelihood as well as likelihoods related to support vector machine models it is shown through simulation and examples that support vector machine models with multiple shrinkage parameters produce fewer misclassification errors than several existing classical methods as well as bayesian methods based on the logistic likelihood or those involving only one shrinkage parameter

exponential polar factorization of the fundamental matrix of linear differential systems we propose a new constructive procedure to factorize the fundamental real matrix of a linear system of differential equations as the product of the exponentials of a symmetric and a skew symmetric matrix both matrices are explicitly constructed as series whose terms are computed recursively the procedure is shown to converge for sufficiently small times in this way explicit exponential representations for the factors in the analytic polar decomposition are found an additional advantage of the algorithm proposed here is that if the exact solution evolves in a certain lie group then it provides approximations that also belong to the same lie group thus preserving important qualitative properties

homotopy groups of k contact toric manifolds contact toric manifolds of reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps we compute their first and second homotopy group terms of the images of the moment map we also explain why they are k contact

combinatorics of rooted trees and hopf algebras we begin by considering the graded vector space with a basis consisting of rooted trees with grading given by the count of non root vertices we define two linear operators on this vector space the growth and pruning operators which respectively raise and lower grading their commutator is the operator that multiplies a rooted tree by its number of vertices and each operator naturally associates a multiplicity to each pair of rooted trees by using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint and obtain several results about the associated multiplicities now the symmetric algebra on the vector space of rooted trees after a degree shift can be endowed with a coproduct to make a hopf algebra this was defined by kreimer in connection with renormalization we extend the growth and pruning operators as well as the inner product mentioned above to kreimer s hopf algebra on the other hand the vector space of rooted trees itself can be given a noncommutative multiplication with an appropriate coproduct this leads to the hopf algebra of grossman and larson we show that the inner product on rooted trees leads to an isomorphism of the grossman larson hopf algebra with the graded dual of kreimer s hopf algebra correcting an earlier result of panaite

toric geometry sasaki einstein manifolds and a new infinite class of ads cft duals recently an infinite family of explicit sasaki einstein metrics yp q on s2 s3 has been discovered where p and q are two coprime positive integers with q p these give rise to a corresponding family of calabi yau cones which moreover are toric aided by several recent results in toric geometry we show that these are kahler quotients open image in new window namely the vacua of gauged linear sigma models with charges p p p q p q thereby generalising the conifold which is p 1 q 0 we present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold open image in new window for all q p with fixed p we hence find that the yp q manifolds are ads cft dual to an infinite class of open image in new window superconformal field theories arising as ir fixed points of toric quiver gauge theories with gauge group su n 2p as a non trivial example we show that y2 1 is an explicit irregular sasaki einstein metric on the horizon of the complex cone over the first del pezzo surface the dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its ir fixed point using the ads cft correspondence the value we obtain is a quadratic irrational number and remarkably agrees with a recent purely field theoretic calculation using a maximisation

semiparametric regression using student t processes in this paper we propose a latent factor regression model in which priors are assigned to both the latent regression vector and the error term by using reproducing kernels the resulting regression function follows a stochastic process known as a student process the model is attractive because its implementation is based on a tractable posterior predictive distribution and a simple expectation maximization em estimation algorithm in addition treating the transductive inference as a missing data problem we devise the em algorithm to deal with the parameter estimation as well as the response prediction in a single paradigm the model is also elaborated for multivariate response regression problems for this purpose we present a generalization of multivariate models and some of its properties experimental results show our approaches to be effective

thermostat algorithm for generating target ensembles we present a deterministic algorithm called contact density dynamics that generates any prescribed target distribution in the physical phase space akin to the famous model of nose and hoover our algorithm is based on a non hamiltonian system in an extended phase space however the equations of motion in our case follow from contact geometry and we show that in general they have a similar form to those of the so called density dynamics algorithm as a prototypical example we apply our algorithm to produce a gibbs canonical distribution for a one dimensional harmonic oscillator

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

some inclusion properties of a certain family of integral operators abstract the authors introduce several new subclasses of analytic functions which are defined by means of a general integral operator i and investigate various inclusion properties of these subclasses many interesting applications involving these and other families of integral operators are also considered

5d super yang mills on y p q sasaki einstein manifolds on any simply connected sasaki einstein five dimensional manifold one can construct a super yang mills theory which preserves at least two supersymmetries we study the special case of toric sasaki einstein manifolds known as y p q manifolds we use the localisation technique to compute the full perturbative part of the partition function the full equivariant result is expressed in terms of a certain special function which appears to be a curious generalisation of the triple sine function as an application of our general result we study the large n behaviour for the case of single hypermultiplet in adjoint representation and we derive the n 3 behaviour in this case

a linear operator and associated families of meromorphically multivalent functions making use of a linear operator which is defined here by means of the hadamard product or convolution the authors introduce and investigate the various properties and characteristics of two novel families of meromorphically multivalent functions they also extend the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

liouville s theorem and the canonical measure for nonconservative systems from contact geometry standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space this is exploited to derive hamilton s equations liouville s theorem and to find the canonical invariant measure in this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry in particular we find out the generalized hamilton s equations liouville s theorem and the microcanonical and canonical measures invariant under the contact flow remarkably the latter measure has a power law density distribution with respect to the standard contact volume form finally we argue on the several possible applications of our results

convergence proof for goldberg s exponential series abstract the goldberg presentation of the formal series for log e x e y in noncommutative symbols x and y is shown to converge when x and y are replaced by normed elements in the unit sphere about zero provided the norm defining the unit sphere is compatible with multiplication

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

differential subordination and superordination results for cho kwon srivastava operator making use of the concepts of differential subordination and superordination many subordination and superordination results are obtained for analytic functions in the open unit disk using the cho kwon srivastava operator by investigating appropriate classes of admissible functions sandwich type results are also obtained

maximal reductions in the baker hausdorff formula a preliminary expression for the baker hausdorff formula is found up to ninth order i e a series expansion of z in terms of multiple commutators where ez exey with x and y noncommuting up to ninth degree in x y by means of complete sets of linear relations between multiple commutators maximal reduction of the number of different multiple commutators in the series are obtained

sasakian einstein structures on 9 we show that 9 s 2 s 3 admits an 8 dimensional complex family of inequivalent non regular sasakian einstein structures these are the first known einstein metrics on this 5 manifold in particular the bound b 2 m 8 which holds for any regular sasakian einstein m does not apply to the non regular case we also discuss the failure of the hitchin thorpe inequality in the case of 4 orbifolds and describe the orbifold version

an application of certain integral operators the integral operator i n f z was introduced by salagean for func tions of the form f z z 1 p k 2 a kz k which are analytic in the unit disc

inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators abstract by making use of a general linear operator i p a c the authors introduce several new subclasses of multivalent functions and investigate various inclusion relationships and argument properties associated with these subclasses some interesting applications involving such and other families of linear operators are also considered the results presented here include a number of known results as their special cases

variational bayesian multinomial probit regression with gaussian process priors it is well known in the statistics literature that augmenting binary and polychotomous response models with gaussian latent variables enables exact bayesian analysis via gibbs sampling from the parameter posterior by adopting such a data augmentation strategy dispensing with priors over regression coefficients in favor of gaussian process gp priors over functions and employing variational approximations to the full posterior we obtain efficient computational methods for gp classification in the multiclass setting the model augmentation with additional latent variables ensures full a posteriori class coupling while retaining the simple a priori independent gp covariance structure from which sparse approximations such as multiclass informative vector machines ivm emerge in a natural and straightforward manner this is the first time that a fully variational bayesian treatment for multiclass gp classification has been developed without having to resort to additional explicit approximations to the nongaussian likelihood term empirical comparisons with exact analysis use markov chain monte carlo mcmc and laplace approximations illustrate the utility of the variational approximation as a computationally economic alternative to full mcmc and it is shown to be more accurate than the laplace approximation

an effective version of the lazard correspondence the lazard correspondence establishes an equivalence of categories between p groups of nilpotency class less than p and nilpotent lie rings of the same class and order the main tools used to achieve this are the baker campbell hausdorff formula and its inverse formulae here we describe methods to compute the inverse baker campbell hausdorff formulae using these we get an algorithm to compute the lie ring structure of a p group of class p furthermore the baker campbell hausdorff formula yields an algorithm to construct a p group from a nilpotent lie ring of order pn and class less than p at the end of the paper we discuss some applications of and practical experiences with the algorithms

on the convergence and optimization of the baker campbell hausdorff formula abstract in this paper the problem of the convergence of the baker campbell hausdorff series for z log e x e y is revisited we collect some previous results about the convergence domain and present a new estimate which improves all of them we also provide a new expression of the truncated lie presentation of the series up to sixth degree in x and y requiring the minimum number of commutators numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

sufficient conditions for the convergence of the magnus expansion two different sufficient conditions are given for the convergence of the magnus expansion arising in the study of the linear differential equation y a t y the first one provides a bound on the convergence domain based on the norm of the operator a t the second condition links the convergence of the expansion with the structure of the spectrum of y t thus yielding a more precise characterization several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions

on certain classes of analytic functions defined by noor integral operator abstract integral operator introduced by noor is defined by using convolution let fn z z 1 z n 1 n n0 and let f be analytic in the unit disc e then inf f 1 n f where fn f 1 n z 1 z using this operator certain classes of analytic functions related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation are defined and studied in detail some basic properties rate of growth of coefficients and a radius problem are investigated it is shown that these classes are closed under convolution with convex functions most of the results are best possible in some sense

bayesian classification with gaussian processes we consider the problem of assigning an input vector to one of m classes by predicting p c x for c 1 m for a two class problem the probability of class one given x is estimated by spl sigma y x where spl sigma y 1 1 e sup y a gaussian process prior is placed on y x and is combined with the training data to obtain predictions for new x points we provide a bayesian treatment integrating over uncertainty in y and in the parameters that control the gaussian process prior the necessary integration over y is carried out using laplace s approximation the method is generalized to multiclass problems m 2 using the softmax function we demonstrate the effectiveness of the method on a number of datasets

some subclasses of multivalent functions involving a certain linear operator the authors investigate various inclusion and other properties of several subclasses of the class ap of normalized p valent analytic functions in the open unit disk which are defined here by means of a certain linear operator problems involving generalized neighborhoods of analytic functions in the class ap are investigated finally some applications of fractional calculus operators are considered

bayesian lasso regression summary the lasso estimate for linear regression corresponds to a posterior mode when independent double exponential prior distributions are placed on the regression coefficients this paper intro duces new aspects of the broader bayesian treatment of lasso regression a direct characterization of the regression coefficients posterior distribution is provided and computation and inference under this characterization is shown to be straightforward emphasis is placed on point estima tion using the posterior mean which facilitates prediction of future observations via the posterior predictive distribution it is shown that the standard lasso prediction method does not neces sarily agree with model based bayesian predictions a new gibbs sampler for bayesian lasso regression is introduced

inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure making use of the familiar convolution structure of analytic functions in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order among the various results obtained here for each of these function classes we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here

subordinations for analytic functions defined by the dziok srivastava linear operator in the present investigation we obtain certain sufficient conditions for a normalized analytic function f z defined by the dziok srivastava linear operator h l a1 to satisfy the certain subordination our results extend corresponding previously known results on starlikeness convexity and close to convexity

inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations making use of a multiplier transformation which is defined here by means of the hadamard product or convolution the authors introduce some new subclasses of meromorphic functions and investigate their inclusion relationships and argument properties some integral preserving properties in a given sector are also considered

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

some applications of fractional integral operators and ruscheweyh derivatives abstract recently owa saigo and srivastava introduced and studied a certain generalized fractional integral operatorj0 z involving the gaussian hypergeometric function the object of this paper is to investigate various properties and relationships involvingj0 z the carlson shaffer operator l a c and the ruscheweyh derivatived a number of interesting subclasses of analytic and univalent functions are also considered in our investigation

simplifying the reinsch algorithm for the baker campbell hausdorff series the goldberg version of the baker campbell hausdorff series computes the quantity z x y lnexey wg w w x y where x and y are not necessarily commuting in terms of words constructed from the x y alphabet the so called goldberg coefficients g w are the central topic of this article this baker campbell hausdorff series is a general purpose tool of very wide applicability in mathematical physics quantum physics and many other fields the reinsch algorithm for the truncated series permits one to calculate the goldberg coefficients up to some fixed word length w by using nilpotent w 1 w 1 matrices we shall show how to further simplify the reinsch algorithm making its implementation in principle utterly straightforward using off the shelf symbolic manipulation software specific computations provide examples which help to provide a deeper understanding of the goldberg coefficients and their properties for instance we shall establish some strict bounds and some equalities on the number of non zero goldberg coefficients unfortunately we shall see that the number of nonzero goldberg coefficients often grows very rapidly in fact exponentially with the word length w furthermore the simplified reinsch algorithm readily generalizes to many closely related but still quite distinct problems we shall also present closely related results for the symmetric product s x y lnex 2eyex 2 wgs w w x y variations on such themes are straightforward for instance one can just as easily consider the loop product l x y lnexeye xe y wgl w w x y this loop type of series is of interest for instance when considering either differential geometric parallel transport around a closed curve non abelian versions of stokes theorem or even wigner rotation thomas precession in special relativity several other closely related series are also briefly investigated

on the numerical integration of fpu like systems abstract this paper concerns the numerical integration of systems of harmonic oscillators coupled by nonlinear terms like the common fpu models we show that the most used integration algorithm namely leap frog behaves very gently with such models preserving in a beautiful way some peculiar features which are known to be very important in the dynamics in particular the selection rules which regulate the interaction among normal modes this explains why leap frog in spite of being a low order algorithm behaves so well as numerical experimentalists always observed at the same time we show how the algorithm can be improved by introducing at a low cost a counterterm which eliminates the dominant numerical error

spacetime singularity resolution by m theory fivebranes calibrated geometry anti de sitter solutions and special holonomy metrics the supergravity description of various configurations of supersymmetric m fivebranes wrapped on calibrated cycles of special holonomy manifolds is studied the description is provided by solutions of eleven dimensional supergravity which interpolate smoothly between a special holonomy manifold and an event horizon with anti de sitter geometry for known examples of anti de sitter solutions the associated special holonomy metric is derived one explicit anti de sitter solution of m theory is so treated for fivebranes wrapping each of the following cycles kahler cycles in calabi yau two three and four folds special lagrangian cycles in three and four folds associative three and co associative four cycles in g 2 manifolds complex lagrangian four cycles in sp 2 manifolds and cayley four cycles in spin 7 manifolds in each case the associated special holonomy metric is singular and is a hyperbolic analogue of a known metric the analogous known metrics are respectively eguchi hanson the resolved conifold and the four fold resolved conifold the deformed conifold and the stenzel four fold metric the bryant salamon gibbons page pope g 2 metrics on an mathbb r 4 bundle over s 3 and an mathbb r 3 bundle over s 4 or mathbb cp 2 the calabi hyper kahler metric on t mathbb cp 2 and the bryant salamon gibbons page pope spin 7 metric on an mathbb r 4 bundle over s 4 by the ads cft correspondence a conformal field theory is associated to each of the new singular special holonomy metrics and defines the quantum gravitational physics of the resolution of their singularities

magnus and fer expansions for matrix differential equations the convergence problem approximate solutions of matrix linear differential equations by matrix exponentials are considered in particular the convergence issue of magnus and fer expansions is treated upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained the very few previously published bounds are improved bounds to the error of approximate solutions are also reported all results are based just on algebraic manipulations of the recursive relation of the expansion generators

an efficient algorithm for computing the baker campbell hausdorff series and some of its applications we provide a new algorithm for generating the baker campbell hausdorff bch series z log exey in an arbitrary generalized hall basis of the free lie algebra l x y generated by x and y it is based on the close relationship of l x y with a lie algebraic structure of labeled rooted trees with this algorithm the computation of the bch series up to degree of 20 111 013 independent elements in l x y takes less than 15min on a personal computer and requires 1 5gbytes of memory we also address the issue of the convergence of the series providing an optimal convergence domain when x and y are real or complex matrices

some classes of p valent analytic functions defined by certain integral operator let a p p n be the class of functions f z z p a p 1z p 1 analytic in the open unit disc e for n 0 1 2 n p a certain integral operator i n p 1 a p a p is defined as i n p 1f f 1 n p 1 f such that f 1 n p 1 f n p 1 z z p1 z where f n p 1 z z p 1 z n p and denotes convolution or hadamard product using this integral operator a new subclass s n p a of a p 0 a

special cases of a matrix exponential formula abstract some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations counterexamples to certain natural conjectures involving a product of matrix exponentials are also given and comments relating to certain recently discovered exponential identities are included

a note of sasakian metrics with constant scalar curvature in this paper we consider compact sasakian manifolds with constant scalar curvature under some positive curvature assumption we prove that such sasakian metrics must be einstein metrics

bayesian approximate kernel regression with variable selection abstractnonlinear kernel regression models are often used in statistics and machine learning because they are more accurate than linear models variable selection for kernel regression models is a challenge partly because unlike the linear regression setting there is no clear concept of an effect size for regression coefficients in this paper we propose a novel framework that provides an effect size analog for each explanatory variable in bayesian kernel regression models when the kernel is shift invariant for example the gaussian kernel we use function analytic properties of shift invariant reproducing kernel hilbert spaces rkhs to define a linear vector space that i captures nonlinear structure and ii can be projected onto the original explanatory variables this projection onto the original explanatory variables serves as an analog of effect sizes the specific function analytic property we use is that shift invariant kernel functions can be approximated via random fourier bases based

bayesian generalized kernel mixed models we propose a fully bayesian methodology for generalized kernel mixed models gkmms which are extensions of generalized linear mixed models in the feature space induced by a reproducing kernel we place a mixture of a point mass distribution and silverman s g prior on the regression vector of a generalized kernel model gkm this mixture prior allows a fraction of the components of the regression vector to be zero thus it serves for sparse modeling and is useful for bayesian computation in particular we exploit data augmentation methodology to develop a markov chain monte carlo mcmc algorithm in which the reversible jump method is used for model selection and a bayesian model averaging method is used for posterior prediction when the feature basis expansion in the reproducing kernel hilbert space is treated as a stochastic process this approach can be related to the karhunen loeve expansion of a gaussian process gp thus our sparse modeling framework leads to a flexible approximation method for gps

contact variational integrators we present geometric numerical integrators for contact flows that stem from a discretization of herglotz variational principle first we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle then we discuss the backward error analysis of our variational integrators including the construction of a modified lagrangian surprisingly this construction presents some interesting simplifications compared to the corresponding construction for symplectic systems throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues

the optimal force gradient symplectic finite difference time domain scheme for electromagnetic wave propagation in this communication we design an optimal force gradient symplectic finite difference time domain sfdtd scheme for numerically solving maxwell s equations based on the additional constraint of the minimized norm of fourth order truncation errors the numerical dissipation can be minimized and the optimal time coefficients for the third order force gradient symplectic method can be obtained due to the second order accurate adopted in space domain in general the proposed optimal sfdtd method is still second order accurate the numerical stability and numerical dispersion analysis indicate that the optimal force gradient sfdtd method has higher accuracy and stability compared with the nonoptimal force gradient sfdtd method moreover in order to verify the validity of the presented method the scattering parameters of a line fed rectangular microstrip antenna are calculated

hopf algebraic structure of families of trees in this paper we describe hopf algebras which are associated with certain families of trees these hopf algebras originally arose in a natural fashion one of the authors 5 was investigating data structures based on trees which could be used to efficiently compute certain differential operators given data structures such as trees which can be multiplied and which act as higherorder derivations on an algebra one expects to find a hopf algebra of some sort we were pleased to find that not only was there a hopf algebra associated with these data structures but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees previous work applying hopf algebras to combinatorial objects such as 10 13 or 14 has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets rather than on trees themselves

exact baker campbell hausdorff formula for the contact heisenberg algebra in this work we introduce the contact heisenberg algebra which is the restriction of the jacobi algebra on contact manifolds to the linear and constant functions we give the exact expression of its corresponding baker campbell hausdorff formula we argue that this result is relevant to the quantization of contact systems

dynkin s method of computing the terms of the baker campbell hausdorff series the infinite series for log exp x exp y for noncommuting x and y is expressible in terms of iterated commutators of x and y except for the linear term x y dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences this paper presents a practical algorithm for applying dynkin s formula and gives several illustrative examples

on the noncommutative geometry of square superpotential algebras a superpotential algebra is square if its quiver admits an embedding into a two torus such that the image of its underlying graph is a square grid possibly with diagonal edges in the unit squares examples are provided by dimer models in physics such an embedding reveals much of the algebras representation theory through a device we introduce called an impression let a be a square superpotential algebra z its center and m the maximal ideal at the origin of spec z using an impression we give a classification of all simple a modules up to isomorphism and give algebraic and homological characterizations of the simple a modules of maximal k dimension show that z is a 3 dimensional normal toric domain and zm is gorenstein by determining transcendence bases and z regular sequences and show that am is a noncommutative crepant resolution of zm and thus a local calabi yau algebra a particular class of square superpotential algebras the yp q algebras is considered in detail we show that the azumaya and smooth loci of the centers coincide and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

canonical sasakian metrics let m be a closed manifold of sasaki type a polarization of m is defined by a reeb vector field and for any such polarization we consider the set of all sasakian metrics compatible with it on this space we study the functional given by the square of the l2 norm of the scalar curvature we prove that its critical points or canonical representatives of the polarization are sasakian metrics that are transversally extremal we define a sasaki futaki invariant of the polarization and show that it obstructs the existence of constant scalar curvature representatives for a fixed cr structure of sasaki type we define the sasaki cone of structures compatible with this underlying cr structure and prove that the set of polarizations in it that admit a canonical representative is open we use our results to describe fully the case of the sphere with its standard cr structure showing that each element of its sasaki cone can be represented by a canonical metric we compute their sasaki futaki invariant and use it to describe the canonical metrics that have constant scalar curvature and to prove that only the standard polarization can be represented by a sasaki einstein metric

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

a simple expression for the terms in the baker campbell hausdorff series a simple expression is derived for the terms in the baker campbell hausdorff series one formulation of the result involves a finite number of operations with matrices of rational numbers generalizations are discussed

einstein manifolds of dimension five with small first eigenvalue of the dirac operator soit m n une variete de spin compacte d einstein a courbure scalaire positive r 0 et on note par d t s t s l operateur de dirac agissant sur des sections du fibre spinoriel on etudie le probleme de la classification des varietes pour le cas a 5 dimensions

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

mixtures of g priors for bayesian variable selection zellner s g prior remains a popular conventional prior for use in bayesian variable selection despite several undesirable consistency issues in this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular we present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors empirical bayes approaches and other default procedures please see arnold zellner s letter and the author s response

a unifying view of sparse approximate gaussian process regression we provide a new unifying view including all existing proper probabilistic sparse approximations for gaussian process regression our approach relies on expressing the effective prior which the methods are using this allows new insights to be gained and highlights the relationship between existing methods it also allows for a clear theoretically justified ranking of the closeness of the known approximations to the corresponding full gps finally we point directly to designs of new better sparse approximations combining the best of the existing strategies within attractive computational constraints

efficient methods for linear schrodinger equation in the semiclassical regime with time dependent potential we build efficient and unitary hence stable methods for the solution of the linear time dependent schrodinger equation with explicitly time dependent potentials in a semiclassical regime the magnus zassenhaus schemes presented here are based on a combination of the zassenhaus decomposition bader et al 2014 found comput math 14 689 720 doi 10 1007 s10208 013 9182 8 1 with the magnus expansion of the time dependent hamiltonian we conclude with numerical experiments 1 http dx doi org 10 1007 s10208 013 9182 8