new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

stabilized nearest neighbor classifier and its statistical properties abstractthe stability of statistical analysis is an important indicator for reproducibility which is one main principle of the scientific method it entails that similar statistical conclusions can be reached based on independent samples from the same underlying population in this article we introduce a general measure of classification instability cis to quantify the sampling variability of the prediction made by a classification method interestingly the asymptotic cis of any weighted nearest neighbor classifier turns out to be proportional to the euclidean norm of its weight vector based on this concise form we propose a stabilized nearest neighbor snn classifier which distinguishes itself from other nearest neighbor classifiers by taking the stability into consideration in theory we prove that snn attains the minimax optimal convergence rate in risk and a sharp convergence rate in cis the latter rate result is established for general plug in classifiers under a low noise condition ext

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

asymptotics and optimal bandwidth selection for highest density region estimation we study kernel estimation of highest density regions hdr our main contributions are twofold firstly we derive a uniform in bandwidth asymptotic approximation to a risk that is appropriate for hdr estimation this approximation is then used to derive a bandwidth selection rule for hdr estimation possessing attractive asymptotic properties we also present the results of numerical studies that illustrate the benefits of our theory and methodology

pairwise local fisher and naive bayes improving two standard discriminants abstract the fisher discriminant is probably the best known likelihood discriminant for continuous data another benchmark discriminant is the naive bayes which is based on marginals only in this paper we extend both discriminants by modeling dependence between pairs of variables in the continuous case this is done by local gaussian versions of the fisher discriminant in the discrete case the naive bayes is extended by taking geometric averages of pairwise joint probabilities we also indicate how the two approaches can be combined for mixed continuous and discrete data the new discriminants show promising results in a number of simulation experiments and real data illustrations

generalization of the von staudt clausen theorem abstract the localization ls x of log 1 x at a set of primes s is defined by taking those powers of x in the logarithmic series for log 1 x which lie in the span of s the functional inverse ls 1 x of ls x also localizes the functional inverse ex 1 of log 1 x and a generalization of the von staudt clausen theorem is proved for the even coefficients in the power series expansion for x l s 1 x this reduces to the von staudt clausen theorem when s is the set of all primes and to a weaker version of theorem 3 9 of i dibag j algebra87 1984 332 341 when s consists of a single prime

an l stability analysis for the finite difference solution of one dimensional linear convection diffusion equations on moving meshes the stability of three moving mesh finite difference schemes is studied in the l norm for one dimensional linear convection diffusion equations these schemes use central finite differences for spatial discretization and the q method for temporal discretization and they are based on conservative and non conservative forms of transformed partial differential equations the stability conditions obtained consist of the cfl condition and the mesh speed related conditions the cfl condition is independent of the mesh speed and has the same form as that for fixed meshes the mesh speed related conditions restrict how fast the mesh can move the conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine illustrative numerical results are presented

smooth discrimination analysis ties f and g can be based on the estimation of the set g x fx gx we consider applications where it is appropriate to assume that the region g has a smooth boundary or belongs to another nonparametric class of sets in particular this assumption makes sense if discrimination is used as a data analytic tool decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered their rates of convergence are obtained we show that these rules achieve optimal rates for estimation of g and optimal rates of convergence for bayes risks an interesting conclusion is that the optimal rates for bayes risks can be very fast in particular faster than the parametric root n rate these fast rates cannot be guaranteed for plug in rules

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

on balanced moving mesh methods moving mesh methods are a widely used approach in the numerical solution of pdes where the original pdes are transformed from a physical domain to a computational domain the objective is to utilize a uniform mesh in the computational domain to get a non uniform physical mesh that better captures the behavior of the solution the movement of the physical mesh points can be governed by a moving mesh pde associated with a corresponding monitor function and both the original pdes and the moving mesh pdes must be solved simultaneously the motivation for this paper is to study a balanced moving mesh method where the aim is to strike a balance between the properties of the solution of the original pde and that of the moving mesh pde we focus on particular choices of the monitor function that give both a well behaved mesh transformation and a well behaved solution in the computational domain both theoretical analysis and numerical experiments are presented as illustrations

bandwidth choice for nonparametric classification it is shown that for kernel based classification with univariate distributions and two populations optimal bandwidth choice has a dichotomous character if the two densities cross at just one point where their curvatures have the same signs then minimum bayes risk is achieved using bandwidths which are an order of magnitude larger than those which minimize pointwise estimation error on the other hand if the curvature signs are different or if there are multiple crossing points then bandwidths of conventional size are generally appropriate the range of different modes of behavior is narrower in multivariate settings there the optimal size of bandwidth is generally the same as that which is appropriate for pointwise density estimation these properties motivate empirical rules for bandwidth choice

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

variational mesh adaptation methods for axisymmetrical problems we study variational mesh adaptation for axially symmetric solutions to two dimensional problems the study is focused on the relationship between the mesh density distribution and the monitor function and is carried out for a traditional functional that includes several widely used variational methods as special cases and a recently proposed functional that allows for a weighting between mesh isotropy or regularity and global equidistribution of the monitor function the main results are stated in theorems ref thm4 1 and ref thm4 2 for axially symmetric problems it is natural to choose axially symmetric mesh adaptation to this end it is reasonable to use the monitor function in the form g lambda 1 r mbox boldmath e r mbox boldmath e r t lambda 2 r mbox boldmath e theta mbox boldmath e theta t where mbox boldmath e r and mbox boldmath e theta are the radial and angular unit vectors it is shown that when higher mesh concentration at the origin is desired a choice of lambda 1 and lambda 2 satisfying lambda 1 0 0 by choosing lambda 1 and lambda 2 in contrast numerical results show that the new functional provides better control of the mesh concentration through the monitor function two dimensional numerical results are presented to support the analysis

optimal weighted nearest neighbour classifiers we derive an asymptotic expansion for the excess risk regret of a weighted nearest neighbour classifier this allows us to find the asymptotically optimal vector of nonnegative weights which has a rather simple form we show that the ratio of the regret of this classifier to that of an unweighted k nearest neighbour classifier depends asymptotically only on the dimension d of the feature vectors and not on the underlying populations the improvement is greatest when d 4 but thereafter decreases as d rightarrow infty the popular bagged nearest neighbour classifier can also be regarded as a weighted nearest neighbour classifier and we show that its corresponding weights are somewhat suboptimal when d is small in particular worse than those of the unweighted k nearest neighbour classifier when d 1 but are close to optimal when d is large finally we argue that improvements in the rate of convergence are possible under stronger smoothness assumptions provided we allow negative weights our findings are supported by an empirical performance comparison on both simulated and real data sets

stability of isotropic singularities for the nonlinear schro dinger equation abstract we describe a method by which the fully two and three dimensional cubic schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities in both cases anisotropic initial data collapse very rapidly towards isotropic singularities in three dimensions the solutions become self similar with a blowup rate t t 1 2 in two dimensions the self similarity is weakly broken and the blowup rate is t t ln ln 1 t t 1 2 the stability of the singular isotropic solutions is very firmly backed by the numerical results

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

estimation and inference of heterogeneous treatment effects using random forests many scientific and engineering challenges ranging from personalized medicine to customized marketing recommendations require an understanding of treatment effect heterogeneity in this article we

the weighted nearest neighbor estimate for hammerstein system identification this paper concerns the nonparametric identification problem for a class of nonlinear discrete time dynamical systems that is characterized by its cascade structure this is a hammerstein system being a series connection of a nonlinear memoryless element followed by a linear dynamic system the input output training data generated from the system are dependent and they do not reveal the strong mixing property the nonlinear part of the system is recovered with the weighted k nearest neighbor regression estimate the a priori information is nonparametric both the nonlinear characteristic and the impulse response of the linear part are completely unknown and can be of any form local and global properties of the estimate are examined whatever the probability density of the input signal the estimate converges at every continuity point of the characteristic as well as in the global sense we derive the formulas for asymptotic bias and the variance and evaluate the corresponding rate of convergence the convergence rate is independent of the shape of the input density and is proved to be optimal these results allow us to find a set of optimal nonnegative weights that further improve the accuracy of our estimation algorithm our findings are supported by simulation experiments

an adaptive grid method for two dimensional viscous flows this paper extends the gas kinetic bgk ns scheme to an adaptive grid for the viscous flow simulations the grid movement and adaptation is controlled by a monitor function which may depend on velocity gradient or other flow variables such as density or pressure for the viscous flow computation the use of adaptive mesh much improves the efficiency and accuracy of the method in comparison with the methods with static mesh points the current method is an accurate and efficient method for the viscous flow computation where the grid points can be easily moved and concentrated on the regions with large velocity and density gradients such as the boundary layer and multi material interface many numerical examples validate the current approach for the viscous flow simulations

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

efficient k nn based hep 2 cells classifier human epithelial hep 2 cells are commonly used in the indirect immunofluorescence iif tests to detect autoimmune diseases the diagnosis consists of searching and classification to specific patterns created by anti nuclear antibodies anas in the patient serum evaluation of the iif test is mostly done by humans which means that it is highly dependent on the experience and expertise of the physician therefore a significant amount of research has been focused on the development of computer aided diagnostic systems which could help with the analysis of images from microscopes this work deals with the design and development of hep 2 cells classifier the classifier is able to categorize pre segmented images of hep 2 cells into 6 classes the core of this engine consists of the following image descriptors haralick features local binary patterns sift surface description and a granulometry based descriptor these descriptors produce vectors that form metric spaces k nn classification is based on aggregated distance function which combines several features together an extensive set of evaluations was performed on the publicly available mivia hep 2 images dataset which allows a direct comparison of our approach with other solutions the results show that our approach is one of the leading classifiers when comparing with other participants in the hep 2 cells classification contest

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

numerical study of blow up to the purely elliptic generalized daveystewartson system blow up solutions for the purely elliptic generalized daveystewartson system are studied by using a relaxation numerical method the numerical method is based on an implicit finite difference scheme with a second order accuracy in both time and space the stability of the numerical method is analyzed by investigating the linear stability of plane wave solutions to evaluate the ability of the relaxation method to detect blow up numerical simulations are conducted for several test problems a particular attention is paid to the gap interval neither a global existence nor a blow up result is established the monotonicity properties of blow up time on the coupling parameter are also investigated numerically

out of bag estimation of the optimal sample size in bagging the performance of m out of n bagging with and without replacement in terms of the sampling ratio m n is analyzed standard bagging uses resampling with replacement to generate bootstrap samples of equal size as the original training set m w o r n without replacement methods typically use half samples m w r n 2 these choices of sampling sizes are arbitrary and need not be optimal in terms of the classification performance of the ensemble we propose to use the out of bag estimates of the generalization accuracy to select a near optimal value for the sampling ratio ensembles of classifiers trained on independent samples whose size is such that the out of bag error of the ensemble is as low as possible generally improve the performance of standard bagging and can be efficiently built

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

grid evolution in time asymptotic problems the selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration a new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh the technique has been used to solve the simple viscous burgers equation in one and two dimensions substantial error reductions are demonstrated the treatment of boundary points and the effect of using different error criteria in generating grids are discussed

projective and coarse projective integration for problems with continuous symmetries temporal integration of equations possessing continuous symmetries e g systems with translational invariance associated with traveling solutions and scale invariance associated with self similar solutions in a co evolving frame i e a frame which is co traveling co collapsing or co exploding with the evolving solution leads to improved accuracy because of the smaller time derivative in the new spatial frame the slower time behavior permits the use of projective and coarse projective integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems respectively these methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries the ideas of projective integration in a co evolving frame are illustrated on the one dimensional translationally invariant nagumo partial differential equation pde a corresponding kinetic monte carlo model motivated from the nagumo kinetics is used to illustrate the coarse grained method a simple one dimensional diffusion problem is used to illustrate the scale invariant case the efficiency of projective integration in the co evolving frame for both the macroscopic diffusion pde and for a random walker particle based model is again demonstrated

fast learning rates for plug in classifiers it has been recently shown that under the margin or low noise assumption there exist classifiers attaining fast rates of convergence of the excess bayes risk that is rates faster than n 1 2 the work on this subject has suggested the following two conjectures i the best achievable fast rate is of the order n 1 and ii the plug in classifiers generally converge more slowly than the classifiers based on empirical risk minimization we show that both conjectures are not correct in particular we construct plug in classifiers that can achieve not only fast but also super fast rates that is rates faster than n 1 we establish minimax lower bounds showing that the obtained rates cannot be improved

classification with imperfect training labels summary we study the effect of imperfect training data labels on the performance of classification methods in a general setting where the probability that an observation in the training dataset is mislabelled may depend on both the feature vector and the true label we bound the excess risk of an arbitrary classifier trained with imperfect labels in terms of its excess risk for predicting a noisy label this reveals conditions under which a classifier trained with imperfect labels remains consistent for classifying uncorrupted test data points furthermore under stronger conditions we derive detailed asymptotic properties for the popular k nearest neighbour support vector machine and linear discriminant analysis classifiers one consequence of these results is that the k nearest neighbour and support vector machine classifiers are robust to imperfect training labels in the sense that the rate of convergence of the excess risk of these classifiers remains unchanged in fact our theoretical and empirical results even show that in some cases imperfect labels may improve the performance of these methods the linear discriminant analysis classifier is shown to be typically inconsistent in the presence of label noise unless the prior probabilities of the classes are equal our theoretical results are supported by a simulation study

singular solutions of the biharmonic nonlinear schrodinger equation we consider singular solutions of the l 2 critical biharmonic nonlinear schrodinger equation we prove that the blowup rate is bounded by a quartic root the solution approaches a quasi self similar profile and a finite amount of l 2 norm which is no less than the critical power concentrates into the singularity we also prove the existence of a ground state solution we use asymptotic analysis to show that the blowup rate of peak type singular solutions is slightly faster than that of a quartic root and the self similar profile is given by the ground state standing wave these findings are verified numerically up to focusing levels of 10 8 using an adaptive grid method we also use the spectral renormalization method to compute the ground state of the standing wave equation and the critical power for collapse in one two and three dimensions

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

an analogue of the von staudt clausen theorem the integrality theorem e 1 e z 1 1 proved in 2 j and subsequently in 131 enables us to interpret l x as the localization of l x log 1 x at the prime p this motivates us to replace in 1 above ex 1 which is the functional inverse of l x by the functional inverse of l x the purpose of this paper is to write down a power series expansion for the function x l x and determine the congruences mod p satisfied by the numerators of its coeffkients the main result in this direction is theorem 3 9 which states that x a x rl l ii 7 l x n u p i n ln pi 1

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients

blow up properties for a degenerate parabolic system with nonlinear localized sources in this paper we investigate the initial boundary problem of a degenerate parabolic system with nonlinear localized sources we classify the blow up solutions into global blow up cases and single point blow up cases according to the values of m n p i q i furthermore we obtain the uniform blow up profiles of solutions for the global blow up case finally we give some numerical examples to verify the results these extend and generalize a recent work of one of the authors l du blow up for a degenerate reaction diffusion systems with nonlinear localized sources j math anal appl 324 2006 304 320 which only considered uniform blow up profiles under the special case p 1 p 2 0

on dibag s generalisation of von staudt s theorem abstract we show how a stronger version of a generalisation of von staudt s theorem due to i dibag an analogue of the von staudt clausen theorem j algebra 125 1989 519 523 follows from the universal von staudt theorems of f clarke the universal von staudt theorems trans amer math soc 315 1989 591 603