bifurcation phenomena in coupled chemical oscillators normal form analysis and numerical simulations abstract a class of diffusively coupled chemical oscillators is mapped into a problem of two interacting hopf bifurcations the normal form analysis predicts a cascade of steady state limit cycle 2 torus 3 torus bifurcations as well as the coexistence of two stable limit cycles numerical simulations of the original system confirm these predictions and in particular show that this system provides an example of bifurcation leading to a stable quasiperiodic regime with three incommensurate frequencies

a lanczos approach to the inverse square root of a large and sparse matrix i construct a lanczos process on a large and sparse matrix and use the results of this iteration to compute the inverse square root of the same matrix the algorithm is a stable version of an earlier proposal by the author it can be used for problems related to the matrix sign and polar decomposition the application here comes from the theory of chiral fermions on the lattice

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

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

verified error bounds for solutions of sylvester matrix equations abstract we develop methods for computing verified solutions of sylvester matrix equations ax xb c to this purpose we propose a variant of the krawczyk interval operator with a factorized preconditioner so that the complexity is reduced to cubic when a and b are dense and diagonalizable block diagonalizations can be used in cases where a or b are not diagonalizable the lyapunov equation as a special case is also considered

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

efficient matrix valued algorithms for solving stiff riccati differential equations in the time varying case a classical approach which has been widely used to compute the solution of the riccati matrix equation of for example size n n is to unroll the matrices into vectors and integrate the resulting system of n sup 2 vector differential equations directly if the system of vectorized differential equations is stiff the cost computation time and storage requirements of applying the popular backward differentiation formulas bdfs to the stiff equations will be very high for large n because a linear system of algebraic equations of size n sup 2 n sup 2 must be solved at each time step new matrix valued algorithms based on a matrix generalization of the bdfs are proposed for solving stiff riccatti differential equations the amount of work required to compute the solution per time step is only o n sup 3 flops by using the matrix valued algorithms whereas the classical approach requires o n sup 6 flops per time step

map dependence of the fractal dimension deduced from iterations of circle maps every orientation preserving circle mapg with inflection points including the maps proposed to describe the transition to chaos in phase locking systems gives occasion for a canonical fractal dimensiond namely that of the associated set of for whichf g has irrational rotation number we discuss how this dimension depends on the orderr of the inflection points in particular in the smooth case we find numerically thatd r d r 1 r 1 8

methods for verified stabilizing solutions to continuous time algebraic riccati equations abstract we describe a procedure based on the krawczyk method to compute a verified enclosure for the stabilizing solution of a continuous time algebraic riccati equation a x x a q x g x building on the work of hashemi 2012 and adding several modifications to the krawczyk procedure we show that after these improvements the krawczyk method reaches results comparable with the current state of the art algorithm miyajima 2015 and surpasses it in some examples moreover we introduce a new direct method for verification which has a cubic complexity in term of the dimension of x employing a fixed point formulation of the equation inspired by the adi procedure the resulting methods are tested on a number of standard benchmark examples

separating eigenvalues using the matrix sign function abstract howland has used the matrix sign function to separate the eigenvalues of a given matrix the derived method required complex arithmetic even in the case of real matrices a modification of the method is presented that does not suffer from this drawback moreover it is shown that the problem may be decomposed into smaller subproblems which leads to considerable savings in computational work

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

hom lie 2 algebras abstract in this paper we introduce the notions of hom lie 2 algebras which is the categorification of hom lie algebras hl algebras which is the hom analogue of l algebras and crossed modules of hom lie algebras we prove that the category of hom lie 2 algebras and the category of 2 term hl algebras are equivalent we give a detailed study on skeletal hom lie 2 algebras in particular we construct the hom analogues of the string lie 2 algebras associated to any semisimple involutive hom lie algebras we also proved that there is a one to one correspondence between strict hom lie 2 algebras and crossed modules of hom lie algebras we give the construction of strict hom lie 2 algebras from hom left symmetric algebras and symplectic hom lie algebras

accurate sum and dot product algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time we show that the computed results are as accurate as if computed in twice or k fold working precision k ge 3 for twice the working precision our algorithms for summation and dot product are some 40 faster than the corresponding xblas routines while sharing similar error estimates our algorithms are widely applicable because they require only addition subtraction and multiplication of floating point numbers in the same working precision as the given data higher precision is unnecessary algorithms are straight loops without branch and no access to mantissa or exponent is necessary

a numerical study of the likelihood of phase locking abstract we report the results of numerical computation of the size of the set of s for which the hemeomorphism of the circle f k x x k 2 sin 2 x mod 1 is phased locked for values of k ranging from 0 1 to 1 the results suggest that for moderate values of k phase locking is unlikely except for small periods but that for k 1 almost all s give phase locking with large periods accounting for a surprisingly large portion of the total

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

fast verification of solutions of matrix equations in this paper we are concerned with a matrix equation ax b where a is an n times n real matrix and x and b are n vectors assume that an approximate solution tilde x is given together with an approximate lu decomposition we will present fast algorithms for proving nonsingularity of a and for calculating rigorous error bounds for a 1 b tilde x infty the emphasis is on rigour of the bounds the purpose of this paper is to propose different algorithms the fastest with frac 2 3 n 3 flops computational cost for the verification step the same as for the lu decomposition the presented algorithms exclusively use library routines for lu decomposition and for all other matrix and vector operations

fast enclosure for a matrix inverse square root abstract a fast algorithm for computing an interval matrix containing the inverse square root of an n n matrix a is proposed this algorithm utilizes numerical spectral decomposition of a and requires only o n 3 operations the principal property and uniqueness of the contained inverse square root can moreover be verified by the proposed algorithm numerical results show the effectiveness and efficiency of this algorithm

quasi hom lie algebras central extensions and 2 cocycle like identities this paper introduces the notion of a quasi hom lie algebra or simply a qhl algebra which is a natural generalization of hom lie algebras introduced in a previous paper j t hartwig d larsson s d silvestrov deformations of lie algebras using sigma derivations math qa 0408064 quasi hom lie algebras include also as special cases color lie algebras and superalgebras and can be seen as deformations of these by maps twisting the jacobi identity and skew symmetry the natural realm for these quasi hom lie algebras is generalizations deformations of the witt algebra delta of derivations on the laurent polynomials c t t 1 we also develop a theory of central extensions for qhl algebras which can be used to deform and generalize the virasoro algebra by centrally extending the deformed witt type algebras constructed here in addition we give a number of other interesting examples of quasi hom lie algebras among them a deformation of the loop algebra less

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

complete devil s staircase fractal dimension and universality of mode locking structure in the circle map it is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil s staircase at the onset of chaos the complementary set to the stability intervals is a cantor set of fractal dimension d 0 87 this exponent is found to be universal for a large class of functions

vibration of fast trains palindromic eigenvalue problems and structure preserving doubling algorithms the vibration of fast trains is governed by a quadratic palindromic eigenvalue problem l 2a 1 email protected 0 a 1 x 0 where a 0 a email protected c n x n and a 0 t a 0 accurate and efficient solution can only be obtained using algorithms which preserve the structure of the eigenvalue problem this paper reports on the successful application of the structure preserving doubling algorithms

scaling exponents relations and order dependence for circle maps abstract scaling relations describing devil s staircase structures obtained by iterations of orientation preserving circle maps with minimal slope converging to zero have been investigated in the limit the circle map has a zero slope inflection point and the influence of the order at this point on the scaling exponents is analyzed we study the behavior of the step sizes and the minimal distance to an integer from a non zero cycle point for the super stable orbit and in particular we treat the average behavior this leads to a definition of an average crossover scale exponent and an average measure exponent besides the earlier defined average dimension d we find and d d moreover 1 d finally as the order increases we observe that hourglass structures evolve in the arnold tongues

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

torus maps and the problem of a one dimensional optical resonator with a quasiperiodically moving wall abstract we study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically with d 2 incommensurate frequencies we show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d dimensional torus that preserves a foliation by irrational straight lines in particular the jacobian of this map has d 1 eigenvalues equal to 1 we present rigorous and numerical results about several dynamical features of such maps we also show how these dynamical features translate into properties for the field in the cavity in particular we show that when the torus map satisfies a kam theorem which happens for a cantor set of positive measure of parameters the energy of the electromagnetic field remains bounded when the torus map is in a resonant region which happens in open sets of parameters inside the gaps of the previous cantor set the energy grows exponentially

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

o operators on hom lie algebras o operators also known as relative rota baxter operators on lie algebras have several applications in integrable systems and the classical yang baxter equations in this article we study o operators on hom lie algebras we define a cochain complex for o operators on hom lie algebras with respect to a representation any o operator induces a hom pre lie algebra structure we express the cochain complex of an o operator in terms of the specific hom lie algebra cochain complex if the structure maps in a hom lie algebra and its representation are invertible then we can extend the above cochain complex to a deformation complex for o operators by adding the space of zero cochains subsequently we study formal deformations of o operators on regular hom lie algebras in terms of the deformation cohomology in the end we deduce deformations of s rota baxter operators of weight 0 and skew symmetric r matrices on hom lie algebras as particular cases of o operators on hom lie algebras

the pade iterations for the matrix sign function and their reciprocals are optimal abstract it is proved that among the rational iterations locally converging with order s 1 to the sign function the pade iterations and their reciprocals are the unique with the lowest sum of the degrees of numerator and denominator

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

computing enclosures for the inverse square root and the sign function of a matrix abstract we study computational methods for obtaining rigorous a posteriori error bounds for the inverse square root and the sign function of an n n matrix a given a computed approximation for the inverse square root of a our methods work by using interval arithmetic to obtain a narrow interval matrix which with mathematical certainty is known to contain the exact inverse square root particular emphasis is put on the computational efficiency of the method which has complexity o n 3 and which uses almost exclusively matrix matrix operation a key to the efficient use of available software for interval computations the standard formulation of the method assumes that a can be diagonalized and that the eigenvector matrix of a is well conditioned a modification relying on a stable similarlity transformation to block diagonal form is also developed

theory of circle maps and the problem of one dimensional optical resonator with a periodically moving wall we consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions notably well known results in the theory of circle maps which we review briefly imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially even if these intervals are dense for typical motions of the reflecting boundary in the complement there is a positive measure set of parameters where the energy remains bounded

palindromic matrix polynomials matrix functions and integral representations abstract we study the properties of palindromic quadratic matrix polynomials z p qz pz 2 i e quadratic polynomials where the coefficients p and q are square matrices and where the constant and the leading coefficients are equal we show that for suitable choices of the matrix coefficients p and q it is possible to characterize by means of z well known matrix functions namely the matrix square root the matrix polar factor the matrix sign and the geometric mean of two matrices finally we provide some integral representations of these matrix functions

computing primary solutions of equations involving primary matrix functions abstract the matrix equation f x a where f is an analytic function and a is a square matrix is considered some results on the classification of solutions are provided when f is rational a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of a for real data the algorithm yields the real solutions using only real arithmetic numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic

improvement of error free splitting for accurate matrix multiplication recently new algorithms for accurate matrix multiplication have been developed by the authors a characteristic of the algorithms is a high dependency on level 3 blas routines which are highly optimized for several architectures an error free splitting for floating point matrices is a key technique in the algorithms in this paper an improvement of the error free splitting is focused on it is shown by numerical examples that the accuracy of computed results of matrix products can be improved by the modified error free splitting compared to that by the previous algorithms

accurate floating point summation part ii sign k fold faithful and rounding to nearest in part ii of this paper we first refine the analysis of error free vector transformations presented in part i based on that we present an algorithm for calculating the rounded to nearest result of s sum p i for a given vector of floating point numbers p i as well as algorithms for directed rounding a special algorithm for computing the sign of s is given also working for huge dimensions assume a floating point working precision with relative rounding error unit eps we define and investigate a k fold faithful rounding of a real number r basically the result is stored in a vector mathtt res nu of k nonoverlapping floating point numbers such that sum mathtt res nu approximates r with relative accuracy mathtt eps k and replacing mathtt res k by its floating point neighbors in sum mathtt res nu forms a lower and upper bound for r for a given vector of floating point numbers with exact sum s we present an algorithm for calculating a k fold faithful rounding of s using solely the working precision furthermore an algorithm for calculating a faithfully rounded result of the sum of a vector of huge dimension is presented our algorithms are fast in terms of measured computing time because they allow good instruction level parallelism they neither require special operations such as access to mantissa or exponent they contain no branch in the inner loop nor do they require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithms are proved to be optimal

a p q deformed virasoro algebra a conformal dimension delta dependent p q deformed virasoro p q virasoro algebra with two independent deformation parameters p q is constructed the comultiplication rule for the generating functional for the delta 0 1 case is established and found to be depending on p and q individually the central charge term for the p q virasoro algebra is described a p q deformed nonlinear equation p q kdv corresponding to p q virasoro algebra for delta 0 1 case is obtained

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

fast enclosure for solutions of sylvester equations abstract fast algorithms for enclosing solutions of sylvester equations ax xb c a c m m b c n n x c c m n are proposed the results obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account for developing these algorithms theories which directly supply error bounds for numerical solutions are established the proposed algorithms require only o m 3 n 3 operations if a and b are diagonalizable techniques for accelerating the enclosure and obtaining smaller error bounds are introduced numerical results show the properties of the proposed algorithms

fast enclosure for all eigenvalues in generalized eigenvalue problems a fast method for enclosing all eigenvalues in generalized eigenvalue problems email protected is proposed firstly a theorem for enclosing all eigenvalues which is applicable even if a is not hermitian and or b is not hermitian positive definite is presented next a theorem for accelerating the enclosure is presented the proposed method is established based on these theorems numerical examples show the performance and property of the proposed method as an application of the proposed method an efficient method for enclosing all eigenvalues in polynomial eigenvalue problems is also sketched