polynomial approximation of certain biharmonic mappings abstract in this paper we first determine conditions on the analytic polynomials p and q such that the family of univalent sense preserving biharmonic polynomials of the form p z 2 p q is dense in the family of univalent sense preserving biharmonic mappings f defined on the unit disk d of the form f z 2 g next we consider the univalency of harmonic function g whenever f z 2 g is univalent in d finally we give a partial answer to the problem raised by muhanna about the radius of univalence of a family of sense preserving biharmonic mappings with the form f z 2 g h where g is univalent in d

local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

a direct formulation for sparse pca using semidefinite programming given a covariance matrix we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination this problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis pca and has wide applications ranging from biology to finance we use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix where cardinality is constrained and derive a semidefinite programming based relaxation for our problem we also discuss nesterov s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse pca problem the method has complexity o n 4 sqrt log n epsilon where n is the size of the underlying covariance matrix and epsilon is the desired absolute accuracy on the optimal value of the problem

on lengths areas and lipschitz continuity of polyharmonic mappings abstract in this paper we continue our investigation of polyharmonic mappings in the complex plane first we establish two landau type theorems we also show a three circles type theorem and an area version of the schwarz lemma finally we study lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric

an interface capturing method for the simulation of multi phase compressible flows a novel finite volume interface contact capturing method is presented for simulation of multi component compressible flows with high density ratios and strong shocks in addition the materials on the two sides of interfaces can have significantly different equations of state material boundaries are identified through an interface function which is solved in concert with the governing equations on the same mesh for long simulations the method relies on an interface compression technique that constrains the thickness of the diffused interface to a few grid cells throughout the simulation this is done in the spirit of shock capturing schemes for which numerical dissipation effectively preserves a sharp but mesh representable shock profile for contact capturing the formulation is modified so that interface representations remain sharp like captured shocks countering their tendency to diffuse via the same numerical diffusion needed for shock capturing special techniques for accurate and robust computation of interface normals and derivatives of the interface function are developed the interface compression method is coupled to a shock capturing compressible flow solver in a way that avoids the spurious oscillations that typically develop at material boundaries convergence to weak solutions of the governing equations is proved for the new contact capturing approach comparisons with exact riemann problems for model one dimensional multi material flows show that the interface compression technique is accurate the method employs cartesian product stencils and therefore there is no inherent obstacles in multiple dimensions examples of two and three dimensional flows are also presented including a demonstration with significantly disparate equations of state a shock induced collapse of three dimensional van der waal s bubbles air in a stiffened equation of state liquid water adjacent to a mie gruneisen equation of state wall copper

covariance eigenvector sparsity for compression and denoising sparsity in the eigenvectors of signal covariance matrices is exploited in this paper for compression and denoising dimensionality reduction dr and quantization modules present in many practical compression schemes such as transform codecs are designed to capitalize on this form of sparsity and achieve improved reconstruction performance compared to existing sparsity agnostic codecs using training data that may be noisy a novel sparsity aware linear dr scheme is developed to fully exploit sparsity in the covariance eigenvectors and form noise resilient estimates of the principal covariance eigenbasis sparsity is effected via norm one regularization and the associated minimization problems are solved using computationally efficient coordinate descent iterations the resulting eigenspace estimator is shown capable of identifying a subset of the unknown support of the eigenspace basis vectors even when the observation noise covariance matrix is unknown as long as the noise power is sufficiently low it is proved that the sparsity aware estimator is asymptotically normal and the probability to correctly identify the signal subspace basis support approaches one as the number of training data grows large simulations using synthetic data and images corroborate that the proposed algorithms achieve improved reconstruction quality relative to alternatives

estimates on bloch constants for planar harmonic mappings the schwarz lemma and bloch constants for planar bounded harmonic mappings are considered sharper form and better estimates are obtained our results improve the one made by dorff and nowak as well as by chen gauthier and hengartner

a discontinuous galerkin method for the simulation of compressible gas gas and gas water two medium flows abstract in this paper we develop a new discontinuous galerkin method for the simulation of shocks and interfaces in compressible gas gas and gas water two medium flows by solving the based model the spatial discretization is carefully designed to possess the following features discrete conservation in terms of the total mass total momentum and total energy high order accuracy and consistency for smooth flows free of oscillations at an isolated material interface in order to handle potential discontinuities arising in the simulation a nonlinear limiter based on the weighted essentially non oscillatory weno strategy is employed to suppress numerical oscillations and to preserve high order of accuracy in regions of smooth flows the weno reconstruction is imposed on suitably selected quantities rather than the conserved ones in case for discontinuities with large pressure ratio low density and dramatic change of material property where unphysical variables may be encountered a posteriori solution correction on the subcell level is locally adopted to enhance the robustness a series of typical test cases for both one and two dimensional problems are provided to demonstrate the performance of the proposed method

quasiconformal and harmonic mappings between jordan domains let and 1 be jordan domains let 0 1 and let f omega mapsto omega 1 be a harmonic homeomorphism the object of the paper is to prove the following results a if f is q c and 1 c 1 then f is lipschitz b if f is q c 1 c 1 and 1 is convex then f is bi lipschitz and c if is the unit disk 1 is convex and 1 c 1 then f is quasiconformal if and only if its boundary function is bi lipschitz and the hilbert transform of its derivative is in l these extend the results of pavlovic ann acad sci fenn 27 365 372 2002

on certain geometric properties of polyharmonic mappings abstract the main purpose of this paper is twofold first the three quarter theorem is established for a large class of polyharmonic mappings second a class of polyharmonic mappings with positive jacobian and being univalent in the unit disk d are constructed

a robust and accurate approach to computing compressible multiphase flow stratified flow model and ausm up scheme in this paper we propose a new approach to compute compressible multifluid equations firstly a single pressure compressible multifluid model based on the stratified flow model is proposed the stratified flow model which defines different fluids in separated regions is shown to be amenable to the finite volume method we can apply the conservation law to each subregion and obtain a set of balance equations secondly the ausm scheme which is originally designed for the compressible gas flow is extended to solve compressible liquid flows by introducing additional dissipation terms into the numerical flux the new scheme called ausm up can be applied to both liquid and gas flows thirdly the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact riemann solver we will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities such as shock waves and fluid interfaces several one dimensional test problems are used to demonstrate the capability of our method including the ransom s water faucet problem and the air water shock tube problem finally several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities such as interactions between water shock wave and air bubble between air shock wave and water column s and underwater explosion

numerical prediction of interfacial instabilities sharp interface method sim we introduce a sharp interface method sim for the direct numerical simulation of unstable fluid fluid interfaces the method is based on the level set approach and the structured adaptive mesh refinement technology endowed with a corridor of irregular cut cell grids that resolve the interfacial region to third order spatial accuracy key in that regard are avoidance of numerical mixing and a least squares interpolation method that is supported by irregular datasets distinctly on each side of the interface results on test problems show our method to be free of the spurious current problem of the continuous surface force method and to converge on grid refinement at near theoretical rates simulations of unstable rayleigh taylor and viscous kelvin helmholtz flows are found to converge at near theoretical rates to the exact results over a wide range of conditions further we show predictions of neutral stability maps of the viscous kelvin helmholtz flows yih instability as well as self selection of the most unstable wave number in multimode simulations of rayleigh taylor instability all these results were obtained with a simple seeding of random infinitesimal disturbances of interface shape as opposed to seeding by a complete eigenmode for other than elementary flows the latter would normally not be available and extremely difficult to obtain if at all sample comparisons with our code adapted to mimic typical diffuse interface treatments were not satisfactory for shear dominated flows on the other hand the sharp dynamics of our method would appear to be compatible and possibly advantageous to any interfacial flow algorithm in which the interface is represented as a discrete heaviside function

on the univalence of the log biharmonic mappings in this paper we establish the univalence and starlikeness connection between log biharmonic mappings and logharmonic mappings

simulation of compressible two phase flows with topology change of fluid fluid interface by a robust cut cell method we develop a robust cut cell method for numerical simulation of compressible two phase flows with topology change of the fluid fluid interface in cut cell methods the flows can be solved in the finite volume framework and the jump conditions at the interface are resolved by solving a local riemann problem therefore cut cell methods can obtain interface evolution with high resolution and at the same time satisfactorily maintain the conservation of flow quantities however it remains a challenge for the cut cell methods to handle interfaces with topology change or very high curvature where the mesh is not sufficiently fine to resolve the interface inappropriate treatment could give rise to either distorted interface advection or unphysical oscillation of flow variables especially when the regularization process e g reinitialization in the level set methods is implemented a robust cut cell method is proposed here with the interface being tracked by a level set function the local unphysical oscillation of flow variables in the presence of topology change is shown to be greatly suppressed by using a delayed reinitialization the method can achieve second order accuracy with respect to the interface position in the absence of topology changes of interface while locally degrading to first order at the interface region where topology change occurs its performance is examined through a variety of numerical tests such as rayleigh collapse shock bubble interaction and shock induced bubble collapse in water numerical results are compared against either benchmark solutions or experimental observations and good agreement has been achieved qualitatively and or quantitatively finally we apply the method to investigating the collapse process of two tandem bubbles in water

sparse pca optimal rates and adaptive estimation principal component analysis pca is one of the most commonly used statistical procedures with a wide range of applications this paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting under mild technical conditions we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate the lower bound is obtained by calculating the local metric entropy and an application of fano s lemma the rate optimal estimator is constructed using aggregation which however might not be computationally feasible we then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently it is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces a key idea in our construction is a reduction scheme which reduces the sparse pca problem to a high dimensional multivariate regression problem this method is potentially also useful for other related problems

on some properties of solutions of the biharmonic equation in the present paper the properties of the linear complex operator l f zfz z fz which is defined on the class of complex valued c1 functions in the plane are investigated it is shown that harmonicity and biharmonicity are invariant under the linear operatorl results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered the operator l can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way

bloch constant and landau s theorem for planar p harmonic mappings abstract in this paper our main aim is to introduce the concept of planar p harmonic mappings and investigate the properties of these mappings first we discuss the p harmonic bloch mappings two estimates on the bloch constant are obtained which are generalizations of the main results in colonna 1989 9 as a consequence of these investigations we establish a bloch and landau s theorem for p harmonic mappings

accuracies and conservation errors of various ghost fluid methods for multi medium riemann problem since the original ghost fluid method ogfm was proposed by fedkiw et al in 1999 5 a series of other gfm based methods such as the gas water version gfm gwgfm the modified gfm mgfm and the real gfm rgfm have been developed subsequently systematic analysis however has yet to be carried out for the various gfms on their accuracies and conservation errors in this paper we develop a technique to rigorously analyze the accuracies and conservation errors of these different gfms when applied to the multi medium riemann problem with a general equation of state eos by analyzing and comparing the interfacial state provided by each gfm to the exact one of the original multi medium riemann problem we show that the accuracy of interfacial treatment can achieve third order accuracy in the sense of comparing to the exact solution of the original mutli medium riemann problem for the mgfm and the rgfm while it is of at most first order accuracy for the ogfm and the gwgfm when the interface approach is actually near in balance similar conclusions are also obtained in association with the local conservation errors a special test method is exploited to validate these theoretical conclusions from the numerical viewpoint

a theory for learning by weight flow on stiefel grassman manifold recently we introduced the concept of neural network learning on stiefel grassman manifold for multilayer perceptron like networks contributions of other authors have also appeared in the scientific literature about this topic this article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here

optimization algorithms exploiting unitary constraints this paper presents novel algorithms that iteratively converge to a local minimum of a real valued function f x subject to the constraint that the columns of the complex valued matrix x are mutually orthogonal and have unit norm the algorithms are derived by reformulating the constrained optimization problem as an unconstrained one on a suitable manifold this significantly reduces the dimensionality of the optimization problem pertinent features of the proposed framework are illustrated by using the framework to derive an algorithm for computing the eigenvector associated with either the largest or the smallest eigenvalue of a hermitian matrix

stokes flow in curved channels in this paper the problem of stokes flow in curved channels is solved by applying the matched eigenfunction expansion method the flow region is partitioned into a union of sectorial subregions then within each subregion the stream function is represented by an expansion of papkovich fadle eigenfunctions the coefficients in these expansions are obtained by requiring weak c 3 continuity of the stream function across common interfaces biorthogonality conditions in cylindrical coordinates are utilized

landau s theorem for log p harmonic mappings abstract the main aim of this paper is to prove the existence of landau bloch constant for log p harmonic mappings

on discontinuity adaptive smoothness priors in computer vision a variety of analytic and probabilistic models in connection with markov random fields mrfs have been proposed in the last decade for solving low level vision problems involving discontinuities this paper presents a systematic study of these models and defines a general discontinuity adaptive da mrf model by analyzing the euler equation associated with the energy minimization it shows that the fundamental difference between different models lies in the behavior of interaction between neighboring points which is determined by the a priori smoothness constraint encoded into the energy function an important necessary condition is derived for the interaction to be adaptive to discontinuities to avoid oversmoothing this forms the basis on which a class of adaptive interaction functions aifs is defined the da model is defined in terms of the euler equation constrained by this class of aifs its solution is c sup 1 continuous and allows arbitrarily large but bounded slopes in dealing with discontinuities because of the continuous nature it is stable to changes in parameters and data a good property for regularizing ill posed problems experimental results are shown

norm of the pre schwarzian derivative bloch s constant and coefficient bounds in some classes of harmonic mappings abstract the aim of the paper is a study of a family of sense preserving complex valued harmonic functions f that are normalized in the open unit disk and such that its analytic part is convex in one direction we prove the univalence and establish estimates on pre schwarzian derivative and the bloch constant for co analytic part of harmonic mapping the bounds of the coefficients of co analytic part are also given

representation formula and bi lipschitz continuity of solutions to inhomogeneous biharmonic dirichlet problems in the unit disk abstract the aim of this paper is twofold first we establish the representation formula and the uniqueness of the solutions to a class of inhomogeneous biharmonic dirichlet problems and then prove the bi lipschitz continuity of the solutions

lengths areas and lipschitz type spaces of planar harmonic mappings abstract in this paper we give bounds for length and area distortion for harmonic k quasiconformal mappings and investigate certain lipschitz type spaces on harmonic mappings

runge kutta discontinuous galerkin methods for compressible two medium flow simulations one dimensional case the runge kutta discontinuous galerkin rkdg method for solving hyperbolic conservation laws is a high order finite element method which utilizes the useful features from high resolution finite volume schemes such as the exact or approximate riemann solvers tvd runge kutta time discretizations and limiters in this paper we investigate using the rkdg finite element method for compressible two medium flow simulation with conservative treatment of the moving material interfaces numerical results for both gas gas and gas water flows in one dimension are provided to demonstrate the characteristic behavior of this approach

vector l_0 sparse variable pca principal component analysis pca achieves dimension reduction by replacing the original measured variables with a smaller set of derived variables called the principal components sparse pca improves this with sparsity there are two kinds of sparse pca sparse loading pca slpca which keeps all the measured variables but zeroes out some of their loadings and sparse variable pca svpca which removes some measured variables completely by simultaneously zeroing out all their loadings because it zeroes out some measured variables completely svpca is capable of huge additional dimension reduction beyond pca while slpca keeps all measured variables and does not have this capability here we consider a vector l0 penalized likelihood approach to svpca and develop a penalized expectation maximization pem algorithm which remarkably in an l0 setting leads to a closed form m step and we provide a convergence analysis

sparse variable pca using geodesic steepest descent principal component analysis pca is a dimensionality reduction technique used in most fields of science and engineering it aims to find linear combinations of the input variables that maximize variance a problem with pca is that it typically assigns nonzero loadings to all the variables which in high dimensional problems can require a very large number of coefficients but in many applications the aim is to obtain a massive reduction in the number of coefficients there are two very different types of sparse pca problems sparse loadings pca slpca which zeros out loadings while generally keeping all of the variables and sparse variable pca which zeros out whole variables typically leaving less than half of them in this paper we propose a new svpca which we call sparse variable noisy pca svnpca it is based on a statistical model and this gives access to a range of modeling and inferential tools estimation is based on optimizing a novel penalized log likelihood able to zero out whole variables rather than just some loadings the estimation algorithm is based on the geodesic steepest descent algorithm finally we develop a novel form of bayesian information criterion bic for tuning parameter selection the svnpca algorithm is applied to both simulated data and real functional magnetic resonance imaging fmri data

a generalized approach for sub and super critical flows using the local discontinuous galerkin method abstract we present a generalized approach to solve sub and super critical flows using the local discontinuous galerkin ldg method the 5 equation model is used to capture multiple phases this model converges to the euler equation when only one phase is present the formulation is independent of the equation of state eos in each phase and thus allows for a more accurate representation of the different fluids by using their appropriate eos the multi phase model includes the convection of the volume fraction which is modeled by a hamilton jacobi equation this appears to be the first study that demonstrates the solution of the volume fraction in a two phase model using the ldg method a modified moment limiter is used to prevent oscillations close to discontinuities and shows that limiting using primitive instead of conservative variables is significantly bigger in two phase flows when compared to a single phase flow an additional limiting technique to minimize the numerical diffusion is introduced which is shown to be especially important at the interface canonical test cases are used to verify and validate the formulation in one two and three dimensions finally the new two phase ldg approach is applied to shock bubble interaction studies and compared to available data

landau s theorem for planar harmonic mappings chen gauthier and hengartner obtained two versions of landau s theorem for bounded planar harmonic mappings later dorff and nowak improved their estimates furthermore grigoryan and huang independently improved their results in this note we improve these last results by obtaining better coefficient estimates for bounded and normalized planar harmonic mappings in particular our theorems are sharp when m 1 which are consistent with landau s theorem when m 1

landau s theorem for certain biharmonic mappings abstract in this paper we show the existence of landau and bloch constants for biharmonic mappings of the form l f here l represents the linear complex operator l z z z z defined on the class of complex valued c 1 functions in the plane and f belongs to the class of biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

biorthogonality condition for creeping flow in wedge shaped trenches in the present paper we derive the biorthogonality condition for stokes flow in wedge shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge this condition is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding stokes flow problem

the ghost fluid method for compressible gas water simulation an analysis is carried out for the ghost fluid method gfm based algorithm as applied to the gas water riemann problems which can be construed as two single medium gfm riemann problems it is found that the inability to provide correct and consistent riemann waves in the respective real fluids by these two gfm riemann problems may lead to inaccurate numerical results based on this finding two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity using these two conditions to analyse some of the existing gfm based algorithms such as the original gfm j comput phys 152 1999 457 the new version gfm j comput phys 166 2001 1 j comput phys 175 2002 200 and the modified gfm mgfm j comput phys 190 2003 651 it is found that there are ranges of conditions for each type of solution where either the original gfm or the new version gfm or both are unable to provide correct or consistent riemann waves in one of the real fluids within these ranges examples can be found such that either the original gfm or the new version gfm or both are unable to provide accurate results the mgfm is also found to encounter difficulties when applied to nearly cavitating flow various examples are presented to demonstrate the conclusions obtained the mgfm with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results

a five equation reduced model for compressible two phase flow problems this paper studies an eulerian diffuse interface model for the simulation of compressible multifluid and two phase flow problems we first show how to derive this model from a seven equation two pressure two velocity model of baer nunziato type using an asymptotic analysis in the limit of zero relaxation time we then study the mathematical properties of the system the structure of the waves the expression of the riemann s invariants and the existence of a mathematical entropy we also describe two different numerical approximation schemes for this system the first one relies on a linearized riemann solver while the second uses more heavily the mathematical structure of the system and relies on a linearization of the characteristic relations finally we present some numerical experiments and comparisons with the results obtained by the two pressure two velocity model as well as some test cases and comparisons with another five equation model recently proposed for interface computations between compressible fluids

a cell centered lagrangian scheme for two dimensional compressible flow problems we present a new lagrangian cell centered scheme for two dimensional compressible flows the primary variables in this new scheme are cell centered i e density momentum and total energy are defined by their mean values in the cells the vertex velocities and the numerical fluxes through the cell interfaces are not computed independently contrary to standard approaches but are evaluated in a consistent manner due to an original solver located at the nodes the main new feature of the algorithm is the introduction of four pressures on each edge two for each node on each side of the edge this extra degree of freedom allows us to construct a nodal solver which fulfills two properties first the conservation of momentum and total energy is ensured second a semidiscrete entropy inequality is provided in the case of a one dimensional flow the solver reduces to the classical godunov acoustic solver it can be considered as its two dimensional generalization many numerical tests are presented they are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

bohr s inequalities for the analytic functions with lacunary series and harmonic functions abstract we determine the bohr radius for the class of all functions f of the form f z z m k 0 a k p z k p analytic in the unit disk z 1 and satisfy the condition f z 1 for all z 1 in particular our result also contains a solution to a recent conjecture of ali et al 9 for the bohr radius for odd analytic functions solved by the authors in 17 we consider a more flexible approach by introducing the p bohr radius for harmonic functions which in turn contains the classical bohr radius as special case also we prove several other new results and discuss p bohr radius for the class of odd harmonic bounded functions

one pointed univalent logharmonic mappings abstract a complete characterization of univalent logharmonic mappings from the exterior of the unit disku onto c 1 is given

on gradient adaptation with unit norm constraints in this correspondence we describe gradient based adaptive algorithms within parameter spaces that are specified by w 1 where spl middot is any vector norm we provide several algorithm forms and relate them to true gradient procedures via their geometric structures we also give algorithms that mitigate an inherent numerical instability for l sub 2 norm constrained optimization tasks simulations showing the performance of the techniques for independent component analysis are provided

on the univalence of polyharmonic mappings abstract a 2p times continuously differentiable complex valued function f u i v in a simply connected domain is polyharmonic or p harmonic if it satisfies the polyharmonic equation p f 0 every polyharmonic mapping f can be written as f z k 1 p z 2 p 1 g p k 1 z where each g p k 1 is harmonic in this paper we investigate the univalence of polyharmonic mappings on linearly connected domains and the relation between univalence of f z and that of g p z the notion of stable univalence and logpolyharminc mappings are also considered

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

landau s theorem for biharmonic mappings abstract in this paper we show the existence of landau constant for biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

some properties of a class of elliptic partial differential operators abstract we prove schwarz pick type estimates and coefficient estimates for a class of functions induced by the elliptic partial differential operators then we apply these results to obtain a landau type theorem

landau s theorem for polyharmonic mappings abstract in this paper we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk d then we obtain two versions of landau s theorem for polyharmonic mappings f and for the mappings of the type l f where l is the differential operator of abdulhadi abu muhanna and khuri examples and numerical estimates are given

short note a comment on the computation of non conservative products we are interested in the solution of non conservative hyperbolic systems and consider in particular the so called path conservative schemes see e g 2 3 which rely on the theoretical work in 1 the example of the standard euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach

a gradient algorithm locally equivalent to the em algorithm in many problems of maximum likelihood estimation it is impossible to carry out either the e step or the m step of the em algorithm the present paper introduces a gradient algorithm that is closely related to the em algorithm this em gradient algorithm approximately solves the m step of the em algorithm by one iteration of newton s method since newton s method converges quickly the local properties of the em gradient algorithm are almost identical with those of the em algorithm any strict local maximum point of the observed likelihood locally attracts the em and em gradient algorithm at the same rate of convergence and near the maximum point the em gradient algorithm always produces an increase in the likelihood with proper modification the em gradient algorithm also exhibits global convergence properties that are similar to those of the em algorithm our proof of global convergence applies and improves existing theory for the em algorithm these theoretical points are reinforced by a discussion of three realistic examples illustrating how the em gradient algorithm can succeed where the em algorithm is intractable

simultaneous variable selection we propose a new method for selecting a common subset of explanatory variables where the aim is to model several response variables the idea is a natural extension of the lasso technique proposed by tibshirani 1996 and is based on the joint residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region the properties of the resulting convex programming problem are analyzed for the special case of an orthonormal design for the general case we develop an efficient interior point algorithm the method is illustrated on a dataset with infrared spectrometry measurements on 14 qualitatively different but correlated responses using 770 wavelengths the aim is to select a subset of the wavelengths suitable for use as predictors for as many of the responses as possible

runge kutta discontinuous galerkin method for the approximation of baer and nunziato type multiphase models a high order numerical method is developed for the computation of compressible multiphase flows the model we use is based on the baer and nunziato type systems 4 among all the other available models in the literature these systems present the advantage to be able to simulate either interface or mixture problems nevertheless they still raise some issues mainly based on their non conservative feature the numerical method we propose is a discontinuous galerkin type in this work the interior side integrals are computed thanks to 2 robustness and high order of accuracy of the method are proved on classical interface problems but also on suitably derived analytical solutions

an interface treating technique for compressible multi medium flow with runge kutta discontinuous galerkin method the high order accurate runge kutta discontinuous galerkin rkdg method is applied to the simulation of compressible multi medium flow generalizing the interface treating method given in chertock et al 2008 9 in mixed cells where the interface is located riemann problems are solved to define the states on both sides of the interface the input states to the riemann problem are obtained by extrapolation to the cell boundary from solution polynomials in the neighbors of the mixed cell the level set equation is solved by using a high order accurate rkdg method for hamilton jacobi equations resulting in a unified dg solver for the coupled problem the method is conservative if we include the states in the mixed cells which are however not used in the updating of the numerical solution in other cells the states in the mixed cells are plotted to better evaluate the conservation errors manifested by overshoots undershoots when compared with states in neighboring cells these overshoots undershoots in mixed cells are problem dependent and change with time numerical examples show that the results of our scheme compare well with other methods for one and two dimensional problems in particular the algorithm can capture well complex flow features of the one dimensional shock entropy wave interaction problem and two dimensional shock bubble interaction problem

biorthogonality condition for axisymmetric stokes flow in a toroidal region in the present paper the biorthogonality condition for axisymmetric stokes flow in a toroidal geometry is derived this condition is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding stokes flow problem

discontinuous galerkin finite element methods for hyperbolic nonconservative partial differential equations we present space and space time discontinuous galerkin finite element dgfem formulations for systems containing nonconservative products such as occur in dispersed multiphase flow equations the main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form then the formulation must reduce to that for conservative systems standard dgfem formulations cannot be applied to nonconservative systems of partial differential equations we therefore introduce the theory of weak solutions for nonconservative products into the dgfem formulation leading to the new question how to define the path connecting left and right states across a discontinuity the effect of different paths on the numerical solution is investigated and found to be small we also introduce a new numerical flux that is able to deal with nonconservative products our scheme is applied to two different systems of partial differential equations first we consider the shallow water equations where topography leads to nonconservative products in which the known possibly discontinuous topography is formally taken as an unknown in the system second we consider a simplification of a depth averaged two phase flow model which contains more intrinsic nonconservative products

distortion theorems for bloch functions we establish various distortion theorems for both normalized locally schlicht bloch functions and normalized bloch function with branch points these distortion theorems give lower bounds on either f z or re f z most of our distortion theorems are sharp and all extremal functions identified the main tools used in establishing these distortion theorems are the classical form of julia s lemma and a new version of julia s lemma that applies to certain multiple valued analytic functions

linear dimensionality reduction survey insights and generalizations linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data due to their simple geometric interpretations and typically attractive computational properties these methods capture many data features of interest such as covariance dynamical structure correlation between data sets input output relationships and margin between data classes methods have been developed with a variety of names and motivations in many fields and perhaps as a result the connections between all these methods have not been highlighted here we survey methods from this disparate literature as optimization programs over matrix manifolds we discuss principal component analysis factor analysis linear multidimensional scaling fisher s linear discriminant analysis canonical correlations analysis maximum autocorrelation factors slow feature analysis sufficient dimensionality reduction undercomplete independent component analysis linear regression distance metric learning and more this optimization framework gives insight to some rarely discussed shortcomings of well known methods such as the suboptimality of certain eigenvector solutions modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver which accepts as input data and an objective to be optimized and returns as output an optimal low dimensional projection of the data this simple optimization framework further allows straightforward generalizations and novel variants of classical methods which we demonstrate here by creating an orthogonal projection canonical correlations analysis more broadly this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox objective agnostic numerical technology

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

sparse principal component analysis and iterative thresholding principal component analysis pca is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix however it behaves poorly when the number of features p is comparable to or even much larger than the sample size n in this paper we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse under a spiked covariance model we find that the new approach recovers the principal subspace and leading eigenvectors consistently and even optimally in a range of high dimensional sparse settings simulated examples also demonstrate its competitive performance

a reconstructed discontinuous galerkin method for the compressible navier stokes equations on arbitrary grids a reconstruction based discontinuous galerkin rdg method is presented for the solution of the compressible navier stokes equations on arbitrary grids the rdg method originally developed for the compressible euler equations is extended to discretize viscous and heat fluxes in the navier stokes equations using a so called inter cell reconstruction where a smooth solution is locally reconstructed using a least squares method from the underlying discontinuous dg solution similar to the recovery based dg rdg methods this reconstructed dg method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional dg methods unlike rdg methods this rdg method does not need to judiciously choose a proper form of a recovered polynomial thus is simple flexible and robust and can be used on arbitrary grids the developed rdg method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy efficiency robustness and versatility the numerical results indicate that this rdg method is able to deliver the same accuracy as the well known bassi rebay ii scheme at a half of its computing costs for the discretization of the viscous fluxes in the navier stokes equations clearly demonstrating its superior performance over the existing dg methods for solving the compressible navier stokes equations

on the linear combinations of harmonic univalent mappings abstract in this paper we derive several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the direction of the real axis furthermore some illustrative examples and imagine domains of the linear combinations satisfying the desired conditions are enumerated

schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings abstract in this paper we discuss the properties of the schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings first we introduce the schwarzian derivative s f for biharmonic mappings f z 2g h and obtain several necessary and sufficient conditions for s f to be analytic second we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings finally we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the jacobian of functions in these invariant families

convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh canonical edge problems for the biharmonic equation can be solved by separating variables the eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of r c smith 1952 we study the reduced system which is governed by a vector ordinary differential equation a solution of the biharmonic problem governed by a partial differential equation can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem the theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f y with values in 2 can be represented in terms of eigenvectors of the reduced problem this paper adds new theorems and conjectures to the theory we extend smith s generalization to fourth order problems of the methods introduced by titchmarsh 1946 to study eigenfunction expansions associated with second order problems we use this method to prove that if f y f1 y f2y 1 y 1 f y e c1 1 1 f e l2 1 1 then the series expressing f y converges uniformly to f y in the open interval 1 1 uniformly in 1 1 if f1 1 0 and in any case to 0 f2 1 f1 1 at y 1 this is unlike fourier series which converge to the mean value of the periodic extension of a function the series exhibits a gibbs phenomenon near the end points of discontinuity when f1 1 0

a procrustes problem on the stiefel manifold an orthogonal procrustes problem on the stiefel manifold is studied where a matrix q with orthonormal columns is to be found that minimizes aq b rm f for an l times m matrix a and an l times n matrix b with l geq m and m n based on the normal and secular equations and the properties of the stiefel manifold necessary conditions for a global minimum as well as necessary and sufficient conditions for a local minimum are derived

a discontinuous galerkin method for viscous compressible multifluids abstract we present a generalized discontinuous galerkin method for a multicomponent compressible barotropic navier stokes system of equations the system presented has a functional viscosity which depends on the pressure p p i of the flow with the density and the local concentration i high order runge kutta time discretization techniques are employed and different methods of dealing with arbitrary coupled boundary conditions are discussed analysis of the energy consistency of the scheme is performed in addition to inspection of the relative error of the solution compared to exact analytic test cases finally several examples comparisons generalizations and physical applications are presented

a note on the eigenfunction expansion for the elastic strip the papkovitch fadle expansion of the biharmonic stress function for a semi infinite strip x leqq 1 y geqq 0 with free edges x pm 1 is obtained as the residue sum of a fourier integral for the case of end data corresponding to prescribed normal stress and transverse displacement which implies knowledge of phi xx x 0 equiv f 2 x and delta phi x 0 equiv f 4 x it is shown that if f f 2 f 4 is twice differentiable with bf f in l 2 0 1 and the data are compatible with the edge conditions at the corners x pm 1 y 0 the derived expansions for phi xx x y and delta phi x y converge to f 2 x f 4 x as y downarrow 0 for x on the closed interval 0 1 even when the compatibility conditions are not met the expansions converge to the data on the open interval 0 1

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt