an h likelihood method for spatial mixed linear models based on intrinsic auto regressions summary we consider sparse spatial mixed linear models particularly those described by besag and higdon and develop an h likelihood method for their statistical inference the method proposed allows for singular precision matrices as it produces estimates that coincide with those from the residual maximum likelihood based on appropriate differencing of the data and has a novel approach to estimating precision parameters by a gamma linear model furthermore we generalize the h likelihood method to include continuum spatial variations by making explicit use of scaling limit connections between gaussian intrinsic markov random fields on regular arrays and the de wijs process keeping various applications of spatial mixed linear models in mind we devise a novel sparse conjugate gradient algorithm that allows us to achieve fast matrix free statistical computations we provide two applications the first is an extensive analysis of an agricultural variety trial that brings forward various new aspects of nearest neighbour adjustment such as effects on statistical analyses to changes of scale and use of implicit continuum spatial formulation the second application concerns an analysis of a large cotton field which gives a focus to matrix free computations the paper closes with some further considerations such as applications to irregularly spaced data use of the parametric bootstrap and some generalizations to the gaussian matern mixed effect models

extension and optimization of the find algorithm computing green s and less than green s functions the find algorithm is a fast algorithm designed to calculate certain entries of the inverse of a sparse matrix such calculation is critical in many applications e g quantum transport in nano devices we extended the algorithm to other matrix inverse related calculations those are required for example to calculate the less than green s function and the current density through the device for a 2d device discretized as an n xxn y mesh the best known algorithms have a running time of o n x 3n y whereas find only requires o n x 2n y even though this complexity has been reduced by an order of magnitude the matrix inverse calculation is still the most time consuming part in the simulation of transport problems we could not reduce the order of complexity but we were able to significantly reduce the constant factor involved in the computation cost by exploiting the sparsity and symmetry the size of the problem beyond which find is faster than other methods typically decreases from a 130x130 2d mesh down to a 40x40 mesh these improvements make the optimized find algorithm even more competitive for real life applications

complete real kahler submanifolds in codimension two minimal isometric immersions f m 2n rightarrow mathbb r 2n 2 in codimension two from a complete kahler manifold into euclidean space had been classified in dajczer and gromoll invent math 119 235 242 1995 for n 3 in this note we describe the non minimal situation showing that if f is real analytic but not everywhere minimal then f is a cylinder over a real kahler surface g n 4 rightarrow mathbb r 6 that is m 2n n 4 times mathbb c n 2 and f g id split where id mathbb c n 2 cong mathbb r 2n 4 is the identity map moreover g can be further described

an approximate framework for quantum transport calculation with model order reduction a new approximate computational framework is proposed for computing the non equilibrium charge density in the context of the non equilibrium green s function negf method for quantum mechanical transport problems the framework consists of a new formulation called the x formulation for single energy density calculation based on the solution of sparse linear systems and a projection based nonlinear model order reduction mor approach to address the large number of energy points required for large applied biases the advantages of the new methods are confirmed by numerical experiments

computing entries of the inverse of a sparse matrix using the find algorithm an accurate and efficient algorithm called fast inverse using nested dissection find for computing non equilibrium green s functions negf for nanoscale transistors has been developed and applied in the simulation of a novel dual gate metal oxide semiconductor field effect transistor mosfet device structure the method is based on the algorithm of nested dissection a graph of the matrix is constructed and decomposed using a tree structure an upward and downward traversal of the tree yields significant performance improvements for both the speed and memory requirements compared to the current state of the art recursive methods for negf this algorithm is quite general and can be applied to any problem where certain entries of the inverse of a sparse matrix e g its diagonal entries the first row or column etc need to be computed as such it is applicable to the calculation of the green s function of partial differential equations find is applicable even when complex boundary conditions are used for example non reflecting boundary conditions

rigidity of minimal submanifolds in space forms 1 1 let c be a real number represent by a4 c a n dimensional space form of curvature c let m n be a n dimensional connected riemannian manifold the question that served as the starting point for this paper was to find simple conditions on the metric of m so that if f m ffl p c p 1 is an isometric minimal immersion then f is rigid in the following sense given another minimal immersion g m a4 q c q p then there exists a rigid motion t of m q c such that g to f l p c being considered as a totally geodesic submanifold of

compact kahler manifolds with nonpositive bisectional curvature in this article we prove that for any compact kahler manifold mn with real analytic metric and nonpositive bisectional curvature there exists a finite cover m of m such that m is a holomorphic and metric fiber bundle over a compact kahler manifold n with nonpositive bisectional curvature and c1 n 0 and the fiber is a flat complex torus this partially confirms a conjecture of yau

conformal hypersurfaces with the same gauss map in this paper we provide a complete classification of all hypersurfaces of euclidean space which admit conformal deformations other than the ones obtained through conformal diffeomorphisms of the ambient space preserving the gauss map elie cartan in one of his earlier papers in differential geometry classified all euclidean hypersurfaces of dimension at least five which admit conformal deformations other than the trivial ones obtained through compositions with conformal diffeomorphisms of the ambient space when all possible conformal deformations are trivial a hypersurface is called conformally rigid first cartan proved that a hypersurface is conformally rigid if at no point does there exist a principal curvature of multiplicity at least n 2 see ca cd2 or da using this he concluded that a conformally deformable hypersurface is either conformally flat or a 2 parameter envelope of spheres or planes of some very special type moreover the set of all conformal deformations is either a 1parameter family or there is just one other deformation in this paper we classify all euclidean hypersurfaces which admit nontrivial conformal deformations preserving the gauss map in the isometric case a complete classification for any codimension has been obtained by dajczer and gromoll dg2 while the conformal case but only for surfaces is due to vergasta ve2 theorem let f g mn rn 1 n 3 be conformal immersions of an ndimensional connected riemannian manifold with the same gauss map assume that on no open subset do they differ by a conformal diffeomorphism of rn l then f mn is part of one of the following examples while g mn is of the same type i a minimal real kaehler hypersurface ii a rotation hypersurface over a plane curve iii a rotation hypersurface over a minimal surface in r3 minimal real kaehler hypersurfaces have been completely classified by dajczer and gromoll in dg2 the assumption that f and g do not differ locally by a conformal deformation of the ambient space has been introduced in order to produce a global result without that assumption our proof shows that the only other possibility is to have an open subset where up to homothety received by the editors december 10 1993 1991 mathematics subject classification primary 53c42 1995 american mathematical society

splittings and cr structures for manifolds with nonpositive sectional curvature let m n denote the universal covering space of a compact riemannian manifold m n with sectional curvature 1 k mn 0 we show that a collection of deck transformations of m n satisfying certain metric dependent conditions determines an open dense subset of m n at every point of which there exists a local isometric splitting with nontrivial flat factor such a collection which we call an abelian structure also gives rise to an essentially canonical cr structure in the sense of buyalo i e an atlas for an injective f structure for which additional conditions hold it follows in particular that the minimal volume of m n vanishes we show that an abelian structure exists if the injectivity radius at all points of m n is less than e n 0 this yields a conjecture of buyalo as well as a strengthened version of the conclusion of gromov s gap conjecture in our special situation in addition we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric

exact inference on gaussian graphical models of arbitrary topology using path sums we present the path sum formulation for exact statistical inference of marginals on gaussian graphical models of arbitrary topology the path sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth our method originates from the closed form resummation of infinite families of terms of the walk sum representation of the covariance matrix we prove that the path sum formulation always exists for models whose covariance matrix is positive definite i e it is valid for both walk summable and non walk summable graphical models of arbitrary topology we show that for graphical models on trees the path sum formulation is equivalent to gaussian belief propagation we also recover as a corollary an existing result that uses determinants to calculate the covariance matrix we show that the path sum formulation formulation is valid for arbitrary partitions of the inverse covariance matrix we give detailed examples demonstrating our results

on nonpositively curved compact riemannian manifolds with degenerate ricci tensor in this article we prove that if the ricci tensor of a compact nonpositively curved manifold m n is nowhere negative definite then it admits local flat factors to be more precise let u c m be the open subset where the ricci tensor has maximum rank r then for any connected component u a of u its universal cover u a is isometric to r n r n r a where n r a is a nonpositively curved manifold with negative ricci curvature in particular if m n is compact nonpositively curved without euclidean de rham factor and both the manifold and the metric are real analytic then its ricci tensor must be negative definite somewhere

the nagata automorphism is wild it is proved that the well known nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild that is it can not be decomposed into a product of elementary automorphisms

a fast parallel algorithm for selected inversion of structured sparse matrices with application to 2d electronic structure calculations an efficient parallel algorithm is presented for computing selected components of a 1 where a is a structured symmetric sparse matrix calculations of this type are useful for several applications including electronic structure analysis of materials in which the diagonal elements of the green s functions are needed the algorithm proposed here is a direct method based on a block ldl t factorization the selected elements of a 1 we compute lie in the nonzero positions of l l t we use the elimination tree associated with the block ldl t factorization to organize the parallel algorithm and reduce the synchronization overhead by passing the data level by level along this tree using the technique of local buffers and relative indices we demonstrate the efficiency of our parallel implementation by applying it to a discretized two dimensional hamiltonian matrix we analyze the performance of the parallel algorithm by examining its load balance and communication overhead and show that our parallel implementation exhibits an excellent weak scaling on a large scale high performance distributed memory parallel machine

domain decomposition type methods for computing the diagonal of a matrix inverse this paper presents two methods based on domain decomposition concepts for determining the diagonal of the inverse of specific matrices the first uses a divide and conquer principle and the sherman morrison woodbury formula and assumes that the matrix can be decomposed into a 2 times 2 block diagonal matrix and a low rank matrix the second method is a standard domain decomposition approach in which local solves are combined with a global correction both methods can be successfully combined with iterative solvers and sparse approximation techniques the efficiency of the methods usually depends on the specific implementation which should be fine tuned for different test problems preliminary results for some two dimensional 2d problems are reported to illustrate the proposed methods

a hybrid method for the parallel computation of green s functions quantum transport models for nanodevices using the non equilibrium green s function method require the repeated calculation of the block tridiagonal part of the green s and lesser green s function matrices this problem is related to the calculation of the inverse of a sparse matrix because of the large number of times this calculation needs to be performed this is computationally very expensive even on supercomputers the classical approach is based on recurrence formulas which cannot be efficiently parallelized this practically prevents the solution of large problems with hundreds of thousands of atoms we propose new recurrences for a general class of sparse matrices to calculate green s and lesser green s function matrices which extend formulas derived by takahashi and others we show that these recurrences may lead to a dramatically reduced computational cost because they only require computing a small number of entries of the inverse matrix then we propose a parallelization strategy for block tridiagonal matrices which involves a combination of schur complement calculations and cyclic reduction it achieves good scalability even on problems of modest size

harmonically immersed surfaces of r n some generalizations of classical results in the theory of minimal surfaces f m rn are shown to hold in the more general case of harmonically immersed surfaces introduction let m g be a connected riemann surface with a prescribed metric g in its conformal class and let f m rn be an immersion it is well known that f realizes m with the induced metric from rn as a minimal surface if and only if f is a conformal with respect to g harmonic map cf for example 3 or 8 that is the theory of minimal surfaces is substantially the theory of conformally immersed harmonic surfaces our purpose is to analyze the case when f is simply a harmonic immersion to introduce an appropriate gauss map and as the main achievement to establish in this new setting the analogue of three fundamental results cf theorems 1 1 and 2 1 and 3 in the theory of minimal surfaces the harmonicity of the gauss map ruh and vilms 13 equidistribution properties of the gauss map in cpn chern and osserman 1 and 11 and the enneper weierstrass representation formulas recently due for arbitrary codimension to hoffman and osserman 7 the last two topics the third for n 3 have alrealy been treated by t k milnor see for instance her survey article 9 but as we remark in 2 our results complement hers in an interesting way the method of the moving frame as well as the einstein summation convention are used throughout this paper 1 the gauss map and first properties let m g be as in the introduction we fix the index ranges 1 rn be an immersion a darboux frame al g f is a map e u c m e n u ope in m such that received by the editors april 6 1987 selected results of this paper were presented by the first named author november 1 1987 at the 837th meeting of the american mathematical society held at lincoln nebraska 1980 mathematics subject classification 1985 revision primary 53a07 53a10

derivations of tensor products of nonassociative algebras abstract let r and s be nonassociative unital algebras assuming that either one of them is finite dimensional or both are finitely generated we show that every derivation of r s is the sum of derivations of the following three types a ad u where u belongs to the nucleus of r s b l z f where f is a derivation of s and z lies in the center of r and c g l w where g is a derivation of r and w lies in the center of s

complete minimal hypersurfaces in the hyperbolic space mathbb h 4 with vanishing gauss kronecker curvature we investigate 3 dimensional complete minimal hypersurfaces in the hyperbolic space h 4 with gauss kronecker curvature identically zero more precisely we give a classification of complete minimal hypersurfaces with gauss kronecker curvature identically zero a nowhere vanishing second fundamental form and a scalar curvature bounded from below

complete minimal submanifolds with nullity in euclidean space in this paper we investigate minimal submanifolds in euclidean space with positive index of relative nullity let m m be a complete riemannian manifold and let f m m rightarrow mathbb r n be a minimal isometric immersion with index of relative nullity at least m 2 at any point we show that if the omori yau maximum principle for the laplacian holds on m m for instance if the scalar curvature of m m does not decrease to infty too fast or if the immersion f is proper then the submanifold must be a cylinder over a minimal surface

rigidity for complete weingarten hypersurfaces we classify locally and globally the ruled weingarten hypersurfaces of the euclidean space as a consequence of the local classification and a rigidity theorem of dajczer and gromoll it follows that a complete weingarten hypersurface which does not contain an open subset of the form l3 x rn 3 where l3 is unbounded and n 3 is rigid

skolem noether algebras abstract an algebra s is called a skolem noether algebra sn algebra for short if for every central simple algebra r every homomorphism r r s extends to an inner automorphism of r s one of the important properties of such an algebra is that each automorphism of a matrix algebra over s is the composition of an inner automorphism with an automorphism of s the bulk of the paper is devoted to finding properties and examples of sn algebras the classical skolem noether theorem implies that every central simple algebra is sn in this article it is shown that actually so is every semilocal and hence every finite dimensional algebra not every domain is sn but for instance unique factorization domains polynomial algebras and free algebras are further an algebra s is sn if and only if the power series algebra s is sn

the gauss map for kahlerian submanifolds of ⁿ we introduce a gauss map for kahler submanifolds of euclidean space and study its geometry in relation to that of the given immersion in particular we generalize a number of results of the classical theory of minimal surfaces in euclidean space

a fast algorithm for sparse matrix computations related to inversion we have developed a fast algorithm for computing certain entries of the inverse of a sparse matrix such computations are critical to many applications such as the calculation of non equilibrium green s functions g r and g for nano devices the find fast inverse using nested dissection algorithm is optimal in the big o sense however in practice find suffers from two problems due to the width 2 separators used by its partitioning scheme one problem is the presence of a large constant factor in the computational cost of find the other problem is that the partitioning scheme used by find is incompatible with most existing partitioning methods and libraries for nested dissection which all use width 1 separators our new algorithm resolves these problems by thoroughly decomposing the computation process such that width 1 separators can be used resulting in a significant speedup over find for realistic devices up to twelve fold in simulation the new algorithm also has the added advantage that desired off diagonal entries can be computed for free consequently our algorithm is faster than the current state of the art recursive methods for meshes of any size furthermore the framework used in the analysis of our algorithm is the first attempt to explicitly apply the widely used relationship between mesh nodes and matrix computations to the problem of multiple eliminations with reuse of intermediate results this framework makes our algorithm easier to generalize and also easier to compare against other methods related to elimination trees finally our accuracy analysis shows that the algorithms that require back substitution are subject to significant extra round off errors which become extremely large even for some well conditioned matrices or matrices with only moderately large condition numbers when compared to these back substitution algorithms our algorithm is generally a few orders of magnitude more accurate and our produced round off errors stay at a reasonable level

symmetry diffeomorphism group of a manifold of nonpositive curvature let m denote a complete simply connected manifold of nonpos itive sectional curvature for each point p s m let sp denote the diffeomor phism of m that fixes p and reverses all geodesies through p the symmetry diffeomorphism group g generated by all diffeomorphisms sp p m ex tends naturally to group of homeomorphisms of the boundary sphere m oo a subset x of m oo is called involutive if it is invariant under g theorem let x c m co be a proper closed involutive subset for each point p s m let n p denote the linear span in tpm of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7 00 belongs to x if n p is a proper subspace of tpm for some point p m then m splits as a riemannian product my x mi such that n is the distribution of m induced by my this result has several applications that include new results as well as great simplifications in the proofs of some known results in a sequel to this paper it is shown that if m is irreducible and m oo admits a proper closed involutive subset x then m is isometric to a symmetric space of noncompact type and rank k 2