on positive semidefinite modification schemes for incomplete cholesky factorization incomplete cholesky factorizations have long been important as preconditioners for use in solving large scale symmetric positive definite linear systems in this paper we focus on the relationship between two important positive semidefinite modification schemes that were introduced to avoid factorization breakdown namely the approach of jennings and malik and that of tismenetsky we present a novel view of the relationship between the two schemes and implement them in combination with a limited memory approach we explore their effectiveness using extensive numerical experiments involving a large set of test problems arising from a wide range of practical applications

enhanced block fsai preconditioning using domain decomposition techniques adaptive block factorized sparse approximate inverse fsai abf is a novel algebraic preconditioner for the cost effective parallel solution of symmetric positive definite linear systems however a main drawback stems from its reduced scalability as the iteration count to convergence tends to grow when the number of processors increases a domain decomposition schur complement approach can enhance both the abf performance and scalability it is demonstrated that the enhanced abf preconditioner is superior to the native block fsai reducing at the same time the construction and communication computational burden numerical results from different large size applications show that the proposed algorithm can improve significantly the preconditioner allowing for its efficient use in massively parallel simulations as well

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

compressed graphs and the minimum degree algorithm the minimum degree algorithm is well known and widely used to find a permutation matrix p such that a sparse symmetric positive definite matrix a can be factored as pap t ll t to yield a lower triangular matrix l with relatively few nonzero entries in this paper we show that by compressing the graph of a when possible the run time of the minimum degree algorithm can be significantly reduced some empirical results for a collection of test matrices are presented

discrete integrable systems and deformations of associative algebras interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed theory of deformations for associative algebras is presented closed left ideal generated by the elements representing the multiplication table plays a central role in this theory deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations it is demonstrated that many discrete equations such as discrete boussinesq equation discrete wdvv equation discrete schwarzian kp and bkp equations discrete hirota miwa equations for kp and bkp hierarchies are particular realizations of the central system an interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful an interpretation of the menelaus relation discrete schwarzian kp equation discrete hirota miwa equation for kp hierarchy consistency around the cube as the associativity conditions and the concept of gauge equivalence for instance between the menelaus and kp configurations are particular examples

a reduced order model based preconditioner for the efficient solution of transient diffusion equations this paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive definite linear systems arising from the numerical discretization of transient parabolic and selfadjoint partial differential equations the preconditioners are obtained by nesting appropriate projections of reduced order models into the classical iteration of the preconditioned conjugate gradient pcg the main idea is to employ the reduced order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases this approach is particularly appealing for transient systems where the full model solution has to be computed at each time step in these cases the natural reduced space is the one generated by full model solutions at previous time steps when increasing the size of the projection space the proposed methodology highly reduces the system conditioning number and the number of pcg iterations at every time step the cost of the application of the preconditioner linearly increases with the size of the projection basis and a trade off must be found to effectively reduce the pcg computational cost the quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models c 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

integrable discrete differential geometry of plated membranes in equilibrium recently it has been established that a well known system of classical shell theory descriptive of membranes in equilibrium is in fact amenable to the techniques of soliton theory here it is d

a scalable parallel algorithm for incomplete factor preconditioning we describe a parallel algorithm for computing incomplete factor ilu preconditioners the algorithm attains a high degree of parallelism through graph partitioning and a two level ordering strategy both the subdomains and the nodes within each subdomain are ordered to preserve concurrency we show through an algorithmic analysis and through computational results that this algorithm is scalable experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors the resulting preconditioned krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors

quantization of three wave equations the subject of this paper is the revision of discretization and quantization of two similar classical integrable hamiltonian systems in three dimensional spacetime the standard three wave equations and the modified three wave equations the darboux manakov zakharov system in two distinguished gauges the quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories integrability of the theories and in particular the quantum tetrahedron equations for vertex operators follow from the quantum auxiliary linear problems the principal object of the lattice field theories is the heisenberg discrete time evolution operator constructed with the help of vertex operators

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

a block fsai ilu parallel preconditioner for symmetric positive definite linear systems a novel parallel preconditioner for symmetric positive definite matrices is developed coupling a generalized factored sparse approximate inverse fsai with an incomplete lu ilu factorization the generalized fsai called block fsai is derived by requiring the preconditioned matrix to resemble a block diagonal matrix in the sense of the minimal frobenius norm an incomplete block jacobi algorithm is then effectively used to accelerate the convergence of a krylov subspace method the block fsai ilu preconditioner proves superior to both fsai and the incomplete block jacobi by themselves in a number of realistic finite element test cases and is fully scalable for a given number of blocks

malliavin calculus geometric mixing and expansion of diffusion functionals under geometric mixing condition we presented asymptotic expansion of the distribution of an additive functional of a markov or an e markov process with finite autoregression including markov type semimartingales and time series models with discrete time parameter the emphasis is put on the use of the malliavin calculus in place of the conditional type cramer condition whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach in the second part by means of the perturbation method and the operational calculus we proved the geometric mixing property for non symmetric diffusion processes and presented a sufficient condition which is easily checked in practice accordingly we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions e g without the strong contractivity condition

from integrable nets to integrable lattices inspired by the results of jonas einsenhart demoulin and bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions of pseudo orthogonal pseudo symmetric and pseudo egorov types dressing transformations of the n component kp hierarchy described within the grassmannian are used to generate quadrilateral lattices and its corresponding reductions as a byproduct we get the corresponding discrete dressing transformations in particular we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice

fundamental transformations for quadrilateral lattices first potentials and τ functions symmetric and pseudo egorov reductions we find the behaviour of first potentials and functions for quadrilateral lattices under vectorial fundamental transformations we also give those transformations which preserve the symmetric and pseudo egorov reductions

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

regular incomplete factorizations of real positive definite matrices abstract we study the classes of matrices which admit a regular incomplete factorization with respect to any graph set we extend the construction to the class of real positive definite matrices the convergence of the basic iteration associated with the splittings we have is discussed

adaptive techniques for improving the performance of incomplete factorization preconditioning three techniques for improving the robustness and performance of incomplete factorization preconditioners for sparse systems with symmetric positive definite or mildly indefinite coefficient matrices are introduced the primary contribution is two new block algorithms for incomplete factorization that result in an improvement in the performance of both the preconditioner generation and the iterative solution phases one of the algorithms applies to matrices that have a natural block structure in their original form and the other applies to matrices without natural dense blocks additionally two relatively simple but highly effective heuristic strategies are introduced these include selecting the solver based on the definiteness properties of the preconditioner and automatic selection and tuning of incomplete factorization parameters all three techniques have adaptive components i e the preconditioner solver combination chooses parameters or algorithmic components based on the properties of the coefficient matrix and its incomplete factors two of the three techniques are applicable to incomplete lu factorization for unsymmetric systems as well

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

enhancing performance and robustness of ilu preconditioners by blocking and selective transposition incomplete factorization is one of the most effective general purpose preconditioning methods for krylov subspace solvers for large sparse systems of linear equations two techniques for enhancing the robustness and performance of incomplete lu factorization for sparse unsymmetric systems are described a block incomplete factorization algorithm based on the crout variation of lu factorization is presented the algorithm is suitable for incorporating threshold based dropping as well as unrestricted partial pivoting and it overcomes several limitations of existing incomplete lu factorization algorithms with and without blocking it is shown that blocking has a three pronged impact it speeds up the computation of incomplete factors and the solution of the associated triangular systems it permits denser and more robust factors to be computed economically and it permits a trade off with the restart parameter of gmres to further improve the overall speed and robustness a highly effective heuristic for imp

solving hermitian positive definite systems using indefinite incomplete factorizations incomplete ldl factorizations sometimes produce an indefinite preconditioner even when the input matrix is hermitian positive definite the two most popular iterative solvers for symmetric systems cg and minres cannot use such preconditioners they require a positive definite preconditioner one approach that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process we explore a different approach use the incomplete factorization with a krylov method that can accept an indefinite preconditioner the conventional wisdom has been that long recurrence methods like gmres or alternatively non optimal short recurrence methods like symmetric qmr and bicgstab must be used if the preconditioner is indefinite we explore the performance of these methods when used with an incomplete factorization but also explore a less known krylov method called pcg odir that is both optimal and uses a short recurrence and can use an indefinite preconditioner furthermore we propose another optimal short recurrence method called ip minres that can use an indefinite preconditioner and a variant of pcg odir which we call ip cg that is more numerically stable and usually requires fewer iterations

asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of malliavin watanabe the asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the malliavin calculus from this the second order efficiency of the maximum likelihood estimator is proved

malliavin calculus and asymptotic expansion for martingales we present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale the emphasis is put on the use of the malliavin calculus the uniform nondegeneracy of the malliavin covariance under certain truncation plays an essential role as the cramer condition did in the case of independent observations applications to statistics are presented

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

charged free fermions vertex operators and the classical theory of conjugate nets we show that the quantum field theoretical formulation of the function theory has a geometrical interpretation within the classical transformation theory of conjugate nets in particular we prove that i the partial charge transformations preserving the neutral sector are laplace transformations ii the basic vertex operators are levy and adjoint levy transformations and iii the diagonal soliton vertex operators generate fundamental transformations we also show that the bilinear identity for the multicomponent kadomtsev petviashvili hierarchy becomes through a generalized miwa map a bilinear identity for the multidimensional quadrilateral lattice equations

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

three dimensional integrable lattices in euclidean spaces conjugacy and orthogonality it is shown that the discrete darboux system descriptive of conjugate lattices in euclidean spaces admits constraints on the adjoint eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices thus it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic orthogonal lattices on lines planes and spheres are discussed and the underlying integrable systems in one two and three dimensions are derived explicitly a discrete analogue of bianchi s ribaucour transformation is set down and particular orthogonal lattices are given as a by product discrete dini surfaces are obtained

asymptotic lattices and their integrable reductions i the bianchi ernst and the fubini ragazzi lattices we review recent results on asymptotic lattices and their integrable reductions we present the theory of general asymptotic lattices in 3 together with the corresponding theory of their darboux type transformations then we find a novel permutability theorem for bianchi surfaces which can be reinterpreted as a discrete version of the bianchi ernst system and coincides with an equation recently introduced by schief schief w k 2001 stud appl math 106 85 137 using the well known connection between the bianchi and ernst systems we also propose the discrete analogue of the ernst system finally we present the theory of the discrete analogues of isothermally asymptotic fubini ragazzi nets together with their transformations

asymptotic expansions and bootstrapping distributions for dependent variables a martingale approach the paper develops a one step triangular array asymptotic expansion for continuous martingales which are asymptotically normal mixing conditions are not required but the quadratic variations of the martingales must satisfy a law of large numbers and a central limit type condition from this result we derive expansions for the distributions of estimators in asymptotically ergodic differential equation models and also for the bootstrapping estimators of these distributions

adaptive pattern research for block fsai preconditioning an adaptive algorithm is presented to generate automatically the nonzero pattern of the block factored sparse approximate inverse bfsai preconditioner it is demonstrated that in symmetric positive definite spd problems bfsai minimizes an upper bound to the kaporin number of the preconditioned matrix the mathematical structure of this bound suggests an efficient and easily parallelizable strategy for improving the given nonzero pattern of bfsai thus providing a novel adaptive bfsai abf preconditioner numerical experiments performed on large sized finite element problems show that abf coupled with a block incomplete cholesky ic outperforms bfsai ic even by a factor of 4 preserving the same preconditioner density and exhibiting an excellent parallelization degree

partial mixing and edgeworth expansion introducing a conditional mixing property gotze and hipp s theory is generalized to a continuous time conditional markov process satisfying this property the malliavin calculus for jump processes applies to random coefficient stochastic differential equations with jumps with the aid of the support theorem to verify the non degeneracy condition i e a conditional type cramer condition

on the rate of convergence in the central limit theorem for martingales with discrete and continuous time heyde and brown 1970 established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order 2 28 with 0 0 moreover an example is constructed demonstrating that this bound is asymptotically exact for all 8 0 the result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time

reductions of the multidimensional quadrilateral lattice the multidimensional circular lattice we apply a recently introduced 21 15 reduction method based on the dressing to construct a large class of integrable reductions of the equations characterizing the multidimensional quadrilateral lattice an n dimensional lattice in m n m whose elementary quadrilaterals are planar and whose continuous limit describes submanifolds parametrized by conjugate lines 11 we also show that generically in the limit of the small lattice parameter half of these reductions lead to the darboux equations for symmetric fields and the second half lead to the generalized lame equations describing n dimensional submanifolds of m parametrized by conjugate orthogonal systems of coordinates we finally show that a distinguished example of the second class of reductions corresponds to the multidimensional circular lattice an n dimensional lattice in m n m whose elementary quadrilaterals are inscribed in circles 7

generalized isothermic lattices we study multi dimensional quadrilateral lattices satisfying simultaneously two integrable constraints a quadratic constraint and the projective moutard constraint when the lattice is two dimensional and the quadric under consideration is the mobius sphere one obtains after the stereographic projection the discrete isothermic surfaces defined by bobenko and pinkall by an algebraic constraint imposed on the complex cross ratio of the circular lattice we derive the analogous condition for our generalized isothermic lattices using steiner s projective structure of conics and we present basic geometric constructions which encode integrability of the lattice in particular we introduce the darboux transformation of the generalized isothermic lattice and we derive the corresponding bianchi permutability principle finally we study two dimensional generalized isothermic lattices in particular geometry of their initial boundary value problem

stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics the solution of linear systems arising in the finite element analysis of shells and solids by the preconditioned conjugate gradient method is considered stabilized and block versions of the ainv factorized approximate inverse preconditioner are presented and tested on a variety of difficult problems comparisons with other preconditioning methods are also included

darboux transformations for a 6 point scheme we introduce the jonas transformation a transformation of the darboux type for the general second order differential equation in two independent variables we present a discrete version of the transformation for a 6 point difference scheme the scheme is appropriate for the solution of a hyperbolic type initial boundary value problem we discuss several reductions and specifications of the transformations

geometric discretization of the koenigs nets we introduce the koenigs lattice which is a new integrable reduction of the quadrilateral lattice discrete conjugate net and provides natural integrable discrete analog of the koenigs net we construct the darboux type transformation of the koenigs lattice and we show permutability of superpositions of such transformations thus proving integrability of the koenigs lattice we also investigate the geometry of the discrete koenigs transformation in particular we characterize the koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice

vbarms a variable block algebraic recursive multilevel solver for sparse linear systems sparse matrices arising from the solution of systems of partial differential equations often exhibit a perfect block structure meaning that the nonzero blocks in the sparsity pattern are fully dense and typically small e g when several unknown quantities are associated with the same grid point similar block orderings can be sometimes unravelled also on general unstructured matrices by ordering consecutively rows and columns with a similar sparsity pattern and treating some zero entries of the reordered matrix as nonzero elements with a little sacrifice of memory we show how we can take advantage of these frequently occurring structures in the design of the multilevel incomplete lu factorization preconditioner arms saad and suchomel 2002 14 and maximize computational efficiency

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

integrable discretization of hodograph type systems hyperelliptic integrals and whitham equations based on the well established theory of discrete conjugate nets in discrete differential geometry we propose and examine discrete analogues of important objects and notions in the theory of semi h

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

precise asymptotics of certain wiener functionals abstract in this article we present a method for developing certain wiener integrals in an asymptotic expansion as the variance of the wiener goes to 0 the basic ideas are a combination of methods borrowed from the theory of large deviations together with techniques from the calculus of wiener space

discrete line complexes and integrable evolution of minors based on the classical plcker correspondence we present algebraic and geometric properties of discrete integrable line complexes in cp3 algebraically these are encoded in a discrete integrable s

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

the use of supernodes in factored sparse approximate inverse preconditioning in recent years the growing popularity of supercomputers has fostered the development of algorithms able to take advantage of the massive parallelism offered by multiple processors direct methods though robust and computationally efficient hardly exploit high degrees of parallelism by contrast krylov methods preconditioned by factored sparse approximate inverses fsai provide at least in principle a perfectly parallel approach but are often thwarted by an excessive set up cost in this paper we extend the concept of supernode from sparse lu factorizations to approximate inverses and use it to accelerate the computation of an fsai type preconditioner the numerical experiments on real world problems show that the overall fsai efficiency can be significantly increased while preserving its intrinsic parallelism

lattice and q difference darboux zakharov manakov systems via delta dressing method a general scheme is proposed for the introduction of lattice and q difference variables to integrable hierarchies in the frame of the delta dressing method using this scheme lattice and q difference darboux zakharov manakov systems of equations are derived darboux backlund and combescure transformations and exact solutions for these systems are studied

the curvature line parametrization from circular nets on a surface we study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets circular nets and planar quadrilateral nets respectively with edges of order both smooth and discrete geometries are described by integrable systems it is shown that one can obtain an order 2 approximation globally with points of the discrete nets on the smooth surface a new simple geometric construction of principal directions of smooth surfaces is given we study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets circular nets and planar quadrilateral nets respectively with edges of order both smooth and discrete geometries are described by integrable systems it is shown that one can obtain an order 2 approximation globally with points of the discrete nets on the smooth surface a new simple geometric construction of principal directions of smooth surfaces is given

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

on the unification of classical and novel integrable surfaces ii difference geometry a novel class of discrete integrable surfaces is recorded this class of discrete o surfaces is shown to include discrete analogues of classical surfaces such as isothermic linear weingarten guichard and petot surfaces moreover natural discrete analogues of the gaussian and mean curvatures for surfaces parametrized in terms of curvature coordinates are used to define surfaces of constant discrete gaussian and mean curvatures and discrete minimal surfaces remarkably these turn out to be prototypical examples of discrete o surfaces it is demonstrated that the construction of a backlund transformation for discrete o surfaces leads in a natural manner to an associated parameter dependent linear representation canonical discretizations of the classical pseudosphere and breather pseudospherical surfaces are generated connections with pioneering work by bobenko and pinkall are established