noncommutative differential geometry and new models of gauge theory the noncommutative differential geometry of the algebra c v mn c of smooth mn c valued functions on a manifold v is investigated for n 2 the analog of maxwell s theory is constructed and interpreted as a field theory on v it describes a u n yang mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential the euclidean action which is positive vanishes on exactly two distinct gauge orbits which are interpreted as two vacua of the theory in one of the corresponding vacuum sectors the su n part of the yang mills field is massive for the case n 2 analogies with the standard model of electroweak theory are pointed out finally a brief description is provided of what happens if one starts from the analog of a general yang mills theory instead of maxwell s theory which is a particular case

numerical methods for the solution of special and general sixth order boundary value problems with applications to benard layer eigenvalue problems a family of numerical methods is developed for the solution of special nonlinear sixth order boundary value problems methods with second fourth sixth and eighth order convergence are contained in the family the problem is also solved by writing the sixth order differential equation as a system of three second order differential equations a family of second and fourth order convergent methods is then used to obtain the solution a second order convergent method is discussed for the numerical solution of general nonlinear sixth order boundary value problems this method with modifications where necessary is applied to the sixth order eigenvalue problems associated with the onset of instability in a benard layer numerical results are compared with asymptotic estimates appearing in the literature

the bifurcating autoregression model in cell lineage studies a model for cell lineage data is presented and analysed the model is an extension of the classical first order autoregression used in time series studies to bifurcating data trees of general size and shape maximum likelihood theory is developed and compared with an extensive simulation study some properties of moment estimators are also presented

numerical solution of fourth order problems with separated boundary conditions in this paper the fourth order linear and nonlinear differential equations with separated boundary conditions is converted into an optimal control problem then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied the analysis is accompanied by numerical examples the obtained results demonstrate reliability and efficiency of the proposed scheme

a natural basis of states for the noncommutative sphere and its moyal bracket an infinite dimensional algebra which is a nondecomposable reducible representation of su 2 is given this algebra is defined with respect to two real parameters if one of these parameters is zero the algebra is the commutative algebra of functions on the sphere otherwise it is a noncommutative analog this is an extension of the algebra normally referred to as the berezin quantum sphere or fuzzy sphere a natural indefinite inner product and a basis of the algebra orthogonal with respect to it are given the basis elements are homogeneous polynomials eigenvectors of a laplacian and related to the hahn polynomials it is shown that these elements tend to the spherical harmonics for the sphere a moyal bracket is constructed and shown to be the standard moyal bracket for the sphere

gl and geometric quantization an axiomatic approach to the approximation of infinite dimensional algebras is presented examples illustrating the need for a rigorous treatment of this subject geometric quantization is employed to construct systematicallysu n approximations of diffeomorphism algebras which first appeared in the theory of relativistic membranes

numerical solution of sixth order boundary value problems with sixth degree b spline functions abstract siddiqi and twizell using sextic splines to approximate solution of sixth order linear boundary value problems in the present paper a sixth degree b spline functions is used to construct an approximate solution for sixth order linear and nonlinear boundary value problems in fact we will show that for every e 0 there exist an approximate solution ve such that the least square error of this ve is less than e 0 and also ve satisfies the exact boundary conditions

topologically nontrivial field configurations in noncommutative geometry in the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated fuzzy sphere the corresponding field theory actions conserve all basic symmetries of the standard commutative version space isometries and global chiral symmetry but due to the noncommutativity of the space the fields are regularized and they contain only a finite number of modes

anti periodic oscillations of some second order differential equations and optimal control problems abstract this paper deals with periodic and anti periodic boundary conditions for wave and heat equations as well as for some second order ordinary differential equations optimal control problems associated with such equations are studied and some numerical experiments to determine the solution of the problem y t 3 sin t y t cos t 0 t y i 0 y i 0 i 0 1 are given some open problems related to the invertibility of i e ta are pointed out

application of least square method to arbitrary order problems with separated boundary conditions in this paper differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied in fact it will be shown that for every e 0 there exists an approximate solution v e of b spline functions such that the corresponding least square error is less than e 0 and also v e satisfies the exact boundary conditions some examples are given and the optimal errors are obtained for the sake of comparison

variance components models for dependent cell populations cells grown in culture can be tracked for several generations and measurements taken on size or age at division and other cell characteristics the observations for the offspring of each cell form a family tree of dependent data such cell lineage data are here modeled as repeated measurements on different family trees arising from individual ancestor cells selected at random from a population of cultured cells the bifurcating autoregression model is embedded in a process that allows for measurement error and variation from tree to tree robust methods are presented that accommodate outliers in this time dependent and branching environment while allowing the statistician to interactively build a variance components model for the process the methodology is illustrated on a substantial data set of 41 trees of emt6 cells with the surprising conclusion that after removing measurement error sister cell lifetimes are nearly identical

adomian decomposition method with green s function for sixth order boundary value problems in this paper the adomian decomposition method with green s function is applied to solve linear and nonlinear sixth order boundary value problems the numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the decomposition method is of high accuracy more convenient and efficient for solving high order boundary value problems

a natural basis for spinor and vector fields on the noncommutative sphere the product of two heisenberg weil algebras contains the jordan schwinger representation of su 2 this algebra is quotiented by the square root of the casimir to produce a nonassociative algebra denoted by psi this algebra may be viewed as the right module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere it is now possible to interpret other subspaces as the space of spinor or vector fields on the noncommutative sphere a natural basis of psi is given which may be interpreted as the deformed entries in the rotation matrices of su 2

α and β curves sister sister and mother daughter correlations in cell population dynamics abstract the and curves and sister sister and mother daughter cycle length correlations are analyzed for a general model of age and size structured cell population dynamics some basic facts concerning these concepts are established a criterion is given for log a and log a to have asymptotically equal slopes the significance of size dependence upon sister sister and mother daughter correlations is demonstrated some numerical examples are given to illustrate the concepts

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

cell lineage analysis variance components models for dependent cell populations abstract cells grown in culture can be tracked for several generations and measurements taken on size or age at division and other cell characteristics the observations for the offspring of each cell form a family tree of dependent data such cell lineage data are here modeled as repeated measurements on different family trees arising from individual ancestor cells selected at random from a population of cultured cells the bifurcating autoregression model is embedded in a process that allows for measurement error and variation from tree to tree robust methods are presented that accommodate outliers in this time dependent and branching environment while allowing the statistician to interactively build a variance components model for the process the methodology is illustrated on a substantial data set of 41 trees of emt6 cells with the surprising conclusion that after removing measurement error sister cell lifetimes are nearly identical

numerical methods for special nonlinear boundary value problems of order 2 m abstract general algorithms based on transforming the problem into a system of m second order problems and on solving the problem as a 2m th order problem are proposed for obtaining the solution of the special nonlinear boundary value problem y 2m x f x y x0 x x1 y 2i x0 a2i y 2i x1 b2i i 0 1 m 1 and m 1 the former is to be preferred to the latter as it does not suffer from rounding errors and also it is easy and inexpensive to implement the latter is found to be vulnerable to word length problems when it is used to solve high order boundary value problems beyond using finite difference methods an alternative method based on the assumption that the desired solution can be adequately represented by a finite polynomial is also outlined for solving the special nonlinear boundary value problem comparisons with the proposed equivalent low order form show that the results obtained are less accurate and apparently at more computational expense a simple example is carried out on the special nonlinear eighth order boundary value problem to illustrate the results given by the methods moreover numerical results for the special nonlinear tenth order boundary value problem using the equivalent low order method are also reported

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

chirality and dirac operator on noncommutative sphere we give a derivation of the dirac operator on the noncommutative 2 sphere within the framework of the bosonic fuzzy sphere and define connes triple it turns out that there are two different types of spectra of the dirac operator and correspondingly there are two classes of quantized algebras as a result we obtain a new restriction on the planck constant in berezin s quantization the map to the local frame in noncommutative geometry is also discussed

deformation estimates for the berezin toeplitz quantization deformation estimates for the berezin toeplitz quantization ofcn are obtained these estimates justify the description of ccr k as a first order quantum deformation of ap c0 where ccr is the usualc algebra of boson canonical commutation relations k is the full algebra of compact operators ap is the algebra of almost periodic functions andc0 is the algebra of continuous functions which vanish at infinity

finite difference methods for twelfth order boundary value problems the effect of rotation acting on a layer of fluid heated from below is similar to the effect of a magnetic field acting under the same conditions they both inhibit the onset of instability and they both elongate the cells which appear at marginal stability however it must not be assumed that acting together rotation and magnetic field reinforce each other in fact they have conflicting tendencies when acting together the effect of rotation and magnetic field together leads to a twelfh order eigenvalue problem chandrasekhar 1961 which will be addressed in a future paper experience in solving high order boundary value problems has shown that considerable insight may be obtained by solving the special problem first moreover beside the mathematical interest the computational aspects of the special problem need to be considered to this end finite difference methods of order two and global extrapolation on two grids are proposed to solve the following special nonlinear twelfth order boundary value problem w xii x f x w a x b w 2i a a2i w 2i b b2i i 0 1 5 a simple example is carried out to illustrate the results given by the methods

robust inference for variance components models for single trees of cell lineage data previously huggins and staudte examined robust estimators for a variance components formulation of the bifurcating autoregressive model for cell lineage data they gave asymptotic properties of the estimators if a large number of trees were observed however for single trees the derivation of these asymptotic properties is more complex here the asymptotic distributions of robust estimators of parameters associated with the stationary bifurcating autoregressive process as a single tree becomes large are obtained these results follow from the formulation of the estimating functions as the product of a nonrandom matrix and the sum of vectors of functions of an infinite sequence of exchangeable random variables

the use of quartic splines in the numerical solution of a fourth order boundary value problem abstract this paper derives via uniform quartic polynomial splines a few new consistency relations connecting the quartic spline function values at midknots and the corresponding values of the second and fourth derivatives respectively it is shown how these relations may be used in an algorithm for computing approximations to the solution and its higher derivatives for a linear fourth order boundary value problem two numerical illustrations are also included to demonstrate the practical usefulness of our algorithm

the commutative limit of a matrix geometry the noncommutative geometry of complex n n matrices is considered as an approximation to the geometry of the two sphere s2 and the differential forms defined over the matrices are mapped into the ordinary differential forms on su2 considered as a fiber bundle over s2 it is found that the extra gauge invariant vacuum studied in previous publications has as image a constant multiple of the connection on su2 that defines the fibration the dirac monopole with unit magnetic charge