weyl manifolds and connections a study is made of weyl connections and their associated holonomy groups the extent to which a weyl connection determines its associated metric and 1 form structures is also discussed finally the related concept of a locally metric connection is introduced and some relations between metric locally metric and weyl connections are established

discrete j spectral factorization of possibly singular polynomial matrices we develop two state space algorithms for discrete time j spectral factorization of possibly singular possibly non column reduced and with possible zeros at the origin para hermitian matrices by assignment of matrices in optimal lq return difference equality the column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete time algebraic riccati system equals the sum of the column degrees necessary and sufficient condition for the least order property of our state space realizations is column reduction of the given matrix

conditions for a symmetric connection to be a metric connection a recent result in general circumstances for a four dimensional space time giving algebraic conditions on a curvature tensor of a symmetric connection so that the connection be metric is shown to be a special case of a more general result both of these results are shown formally to be of a generic nature in this new result the conditions are imposed on a tensor of a more general character than the curvature tensor in addition it is shown that once the symmetric connection is known to be metric the metric is uniquely defined up to a constant conformal factor for those special curvature tensors which are excluded from the original result supplementary conditions are suggested which alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also riemann tensors

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

algorithm of j spectral factorization of polynomial matrices numerical algorithms for the canonical j factorization of polynomial matrices with respect to the imaginary axis are given the factorization problems for the non regular polynomial matrices and for polynomials whose determinants have zero roots are considered the constructed algorithms are not connected to the finding of roots of the polynomial the basic calculation procedure is the construction of the stabilizing solution of the matrix algebraic riccati equation

on the number of factors of sturmian words abstract we prove that for m 1 card a m 1 m i 1 m i 1 i where a m is the set of factors of length m of all the sturmian words and is the euler function this result was conjectured by dulucq and gouyou beauchamps 1987 who proved that this result implies that the language m 0 a m c is inherently ambiguous we also give a combinatorial version of the riemann hypothesis

purely magnetic space times purely magnetic space times in which the riemann tensor satisfies rabcdubud 0 for some unit timelike vector ua are studied the algebraic consequences for the weyl and ricci tensors are examined in detail and consideration given to the uniqueness of ua some remarks concerning the nature of the congruence associated with ua are made

on the algebra of parallel endomorphisms of a pseudo riemannian metric on a pseudo riemannian manifold m g some fields of endomorphisms i e sections of end tm may be parallel for g they form an associative algebra e which is also the commutant of the holonomy group of g as any associative algebra e is the sum of its radical and of a semi simple algebra s this s may be of eight different types see 8 then for any self adjoint nilpotent element n of the commutant of such an s in end tm the set of germs of metrics such that e s n is non empty we parametrize it generically the holonomy algebra of those metrics is the full commutant o g s n and then apart from some degenerate cases e s n where n is the ideal spanned by n to prove it we introduce an analogy with complex differential calculus the ring r x x n replacing the field c this treats the case where the radical of e is principal and consists of self adjoint elements we add a glimpse in the case where this radical is not principal we build germs of pseudo riemannian metric admitting a parallel nilpotent endomorphism n end tm together with a semisimple associative algebra of parallel endomorphisms s 6 n possibly s rid motivation let m g be a pseudo riemannian manifold h its

normal form of a metric admitting a parallel field of planes local coordinate normal forms for two classes of metrics are obtained the first class is comprised of those metrics that have a field of degenerate but not necessarily null planes thereby extending a result of a g walker the second class of metrics are artinian metrics that possess a pair of parallel complementary null distributions this latter class is shown to be equivalent to a certain class of symplectic forms

spectral factorization of non symmetric polynomial matrices the topic of the paper is spectral factorization of rectangular and possibly non full rank polynomial matrices to each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients the associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix the matrix dimensions of the pencil we obtain by solving an integer linear programming ilp minimization problem then by extracting a deflating subspace of the pencil we come to the required spectral factorization we apply the algorithm to most general case of inner outer factorization regardless continuous or discrete time case and to finding the greatest common divisor of polynomial matrices

double warped space times an invariant characterization of double warped space times is given in terms of newman penrose formalism and a classification scheme is proposed a detailed study of the conformal algebra of these space times is also carried out and some remarks are made on certain classes of exact solutions

the holonomy lie algebras of neutral metrics in dimension four the lie algebra isomorphism between su 1 1 su 1 1 and o 2 2 is used to obtain a list of subalgebras of the latter the resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the lie algebra of the holonomy group of a neutral metric in four dimensions the conclusions taken in conjunction with previously known results furnish a classification of such lie subalgebras of o 2 2 with only one case remaining unresolved

paper efficient algorithm for matrix spectral factorization an algorithm is presented for the spectral factorization of polynomial or rational matrices arising in optimal control and filtering theory as well as in network theory there are two versions of the algorithm one applicable to continuous time problems the other to discrete time ones both versions are based on newton s method feature quadratic convergence and provide a significant improvement in efficiency over the existing methods

a spectral factorization approach for h spl infin estimation problems in discrete time this note deals with the h filtering prediction and smoothing problems for discrete time linear systems the j spectral factorization approach is pursued and for each of the three problems a computationally efficient procedure to derive a j spectral factor is presented the construction of filter smoother and predictor hinges on the stabilizing solution of the same algebraic riccati equation this fact allows for the avoidance of any fictitious augmentation of the state space dimensions

brief polynomial j spectral factorization in minimal state space by a specific choice of matrices in optimal lq return difference equality we develop a simple state space algorithm for polynomial j spectral factorization for this purpose we solve a minimal order algebraic riccati equation the order obtained by solving an integer linear programming ilp problem an example that cannot be solved by the existing algorithms illustrates our algorithm

a spinor approach to walker geometry a four dimensional walker geometry is a four dimensional manifold m with a neutral metric g and a parallel distribution of totally null two planes this distribution has a natural characterization as a projective spinor field subject to a certain constraint spinors therefore provide a natural tool for studying walker geometry which we exploit to draw together several themes in recent explicit studies of walker geometry and in other work of dunajski 11 and plebanski 30 in which walker geometry is implicit in addition to studying local walker geometry we address a global question raised by the use of spinors

holonomy groups in general relativity the infinitesimal holonomy group structure of space time is discussed and related to the petrov type of the weyl tensor and the algebraic segre type of the energy momentum tensor the number of covariant derivatives of the curvature tensor required to determine the infinitesimal holonomy group is determined in each case and the complete classification scheme is tabulated some special cases of physical interest are investigated in more detail a geometrical approach is followed throughout

de rham wu decomposition of holomorphic riemannian manifolds it is shown that a de rham wu decomposition for holomorphic riemannian manifolds is obtained easily from the underlying real geometry of holomorphic riemannian manifolds by appealing to h wu s extension to pseudo riemannian manifolds of de rham s decomposition theorem

curvature structure in general relativity this paper investigates the extent to which the curvature structure of space time determines the metric stucture it continues the work of earlier papers by prescribing the curvature structure and the curvature covariant derivatives up to certain orders it is shown that with the exception of the so called generalized pp waves the curvature and its first and second covariant derivatives essentially determine the metric up to coordinate transformations

on the fragmentary complexity of symbolic sequences a measure of the ability of a symbolic sequence to be covered by initial fragments of another symbolic sequence is introduced and its basic properties are investigated applications to the characterization of symbolic sequences associated with shift mappings on a torus corresponding to a special partitioning of the torus and to multirate systems of coprocessors are considered

affine collineations in general relativity and their fixed point structure a general discussion of affine vector fields collineations in space time is given following a previous article by hall and da costa some general remarks are presented on the zeros of affine vector fields and then the general theory of such zeros is established some preliminary results regarding homothetic symmetries in two and three dimensional space times are also given the impossibility of reducing an affine algebra containing proper affine vector fields to homothetic or killing vector fields by means of a change of compatible metric is demonstrated

a coprime factorisation and its application in ℋ2 control given a plant transfer matrix as a fraction of two polynomial matrices we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix and

spacetimes admitting a vector field whose inner product with the riemann tensor is zero the equation v r 0 arises in various places in general relativity in particular as the integrability conditions of the equations lvg 2 g where is a constant these are the equations of a homothetic vector field v with a zero homothetic bivector dv 0 in some space time with metric g and r are the components of the riemann tensor of that metric in some frame in this paper the equation v r 0 is examined and the components of the riemann tensor for the spacetimes which admit nonzero solutions v of this equation are given the petrov types of the weyl tensors of these spacetimes are listed and as a result a correction is then made to a theorem in a paper by collinson and fugere about the petrov types of spacetimes which admit the type of separation that they require of the hamilton jacobi equation for these spacetimes

walker s theorem without coordinates we provide a coordinate free version of the local classification due to walker q j math 1 69 1950 of null parallel distributions on pseudo riemannian manifolds the underlying manifold is realized locally as the total space of a fiber bundle each fiber of which is an affine principal bundle over a pseudo riemannian manifold all structures just named are naturally determined by the distribution and the metric in contrast with the noncanonical choice of coordinates in the usual formulation of walker s theorem

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

a toeplitz algorithm for polynomial j spectral factorization a block toeplitz algorithm is proposed to perform the j spectral factorization of a para hermitian polynomial matrix the input matrix can be singular or indefinite and it can have zeros along the imaginary axis the key assumption is that the finite zeros of the input polynomial matrix are given as input data the algorithm is based on numerically reliable operations only namely computation of the null spaces of related block toeplitz matrices polynomial matrix factor extraction and linear polynomial matrix equations solving

curvature collineations in general relativity ii this paper is the second of a set of two papers on curvature collineations in general relativity the first paper presented the mathematical basis of curvature collineations and a possible approach to their study this paper continues from the first one by investigating in detail many of the cases where curvature collineations can occur in space time it is based on a classification of the curvature tensor which is discussed in the first paper and reviewed briefly here this together with the geometrical approach favored in this paper leads to a rather general discussion of the problem which it is hoped does not obscure those physical aspects of the situation that are important in einstein s theory

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

spectral and inner outer factorizations for discrete time systems the spectral and inner outer factorization problems are solved for discrete time systems with transfer matrices of arbitrary rank simple and explicit state space formulas expressed in terms of the stabilizing solution to the discrete time algebraic riccati system are derived the results are numerically effective since a reliable algorithm for computing such a stabilizing solution is available