noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

the green s function for the radiative transport equation in the slab geometry the radiative transport equation is solved in the three dimensional slab geometry by means of the method of rotated reference frames in this spectral method the solution is expressed in terms of analytical functions such as spherical harmonics and wigner d functions in addition the eigenvalues and eigenvectors of a tridiagonal matrix and certain coefficients which are determined from the boundary conditions must also be computed the green s function for the radiative transport equation is computed and the results are compared with diffusion approximation and monte carlo simulations we find that the diffusion approximation is not quite correct inside the slab even when the light emitted from the slab is well described by the diffusion approximation the solutions we obtain are especially convenient for solving inverse problems associated with radiative transport

n ary algebras a review with applications this paper reviews the properties and applications of certain n ary generalizations of lie algebras in a self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras particular attention will be paid to generalized lie algebras which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized jacobi identity and to filippov or n lie algebras which are defined by fully antisymmetric n brackets that satisfy the filippov identity three lie algebras have surfaced recently in multi brane theory in the context of the bagger lambert gustavsson model because of this filippov algebras will be discussed at length including the cohomology complexes that govern their central extensions and their deformations it turns out that whitehead s lemma extends to all semisimple n lie algebras when the skewsymmetry of the lie or n lie algebra bracket is relaxed one is led to a more general type of n algebras the n leibniz algebras these will be discussed as well since they underlie the cohomological properties of n lie algebras the standard poisson structure may also be extended to the n ary case we shall review here the even generalized poisson structures whose generalized jacobi identity reproduces the pattern of the generalized lie algebras and the nambu poisson structures which satisfy the filippov identity and determine filippov algebras finally the recent work of bagger lambert and gustavsson on superconformal chern simons theory will be briefly discussed emphasis will be made on the appearance of the 3 lie algebra structure and on why the a4 model may be formulated in terms of an ordinary lie algebra and on its nambu bracket generalization

ternary hom nambu lie algebras induced by hom lie algebras the need to consider n ary algebraic structures generalizing lie and poisson algebras has become increasingly important in physics and it should therefore be of interest to study the mathematical concepts related to n ary algebras the purpose of this paper is to investigate ternary multiplications as deformations of n lie structures constructed from the binary multiplication of a hom lie algebra a linear twisting map and a trace function satisfying certain compatibility conditions we show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of hom nambu lie algebras obtained using this construction

deformed noncommutative tori we recall a construction of non commutative algebras related to a one parameter family of deformed spheres and tori and show that in the case of tori the algebras can be completed into c algebras isomorphic to the standard non commutative torus as the former was constructed in the context of matrix or fuzzy geometries it provides an important link to the framework of non commutative geometry and opens up for a concrete way to study deformations of non commutative tori furthermore we show how the well known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite dimensional representations of the deformed algebras and their projective modules are described together with connections of constant curvature

detecting unequal marginal scales in a bivariate population abstract a conditional test is proposed that uses the null hypothesis of bivariate symmetry to detect unequal marginal scales in a bivariate population the asymptotic normality of the test statistic and the consistency of the test are established asymptotic relative efficiency comparisons are made with the standard parametric test in both the bivariate normal case and in a trivariate reduction model case and with two of sen s 1967 tests in the bivariate normal case the good performance of the test is also illustrated via a monte carlo study

the analysis of neuronal discharge sequences change point estimation and comparison of variances a key problem in neurophysiology is to determine whether after presentation of a stimulus there has been a modification in the discharge of a recorded neuron and if so an attempt is made to estimate the latency of the response the estimation problem can be considered as that of the estimation of a change point in a sequence of random variables the gamma distribution is adequate to model the distribution of intervals between action potentials for different types of neurons simulations show that the maximum likelihood estimator based on this model is efficient and robust an additional problem in the case of experiments in which a movement follows the stimulus is to determine whether a response is related to the stimulus or to the movement a test based on the comparison of marginal scales of a bivariate distribution is proposed the whole procedure has been tested in simulation and with real examples

noncommutative differential geometry of matrix algebras the noncommutative differential geometry of the algebra mn c of complex n n matrices is investigated the role of the algebra of differential forms is played by the graded differential algebra c sl n c mn c mn c sl n c sl n c acting by inner derivations on mn c a canonical symplectic structure is exhibited for mn c for which the poisson bracket is to within a factor i the commutator also a canonical riemannian structure is described for mn c finally the analog of the maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge

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

multi linear formulation of differential geometry and matrix regularizations we prove that many aspects of the differential geometry of embedded riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions in particular we find algebraic expressions for weingartens formula the ricci curvature and the codazzi mainardi equations for matrix analogues of embedded surfaces we define discrete curvatures and euler characteristics and a non commutative gauss bonnet theorem is shown to follow we derive simple expressions for the discrete gauss curvature in terms of matrices representing the embedding coordinates and explicit examples are provided furthermore we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete gauss curvature implies a bound on the eigenvalues of the discrete laplace operator

noncommutative riemann surfaces by embeddings in mathbb r 3 we introduce c algebras of compact riemann surfaces sigma as non commutative analogues of the poisson algebra of smooth functions on sigma representations of these algebras give rise to sequences of matrix algebras for which matrix commutators converge to poisson brackets as n for a particular class of surfaces interpolating between spheres and tori we completely characterize even for the intermediate singular surface all finite dimensional representations of the corresponding c algebras

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

hypersymmetry a z3 graded generalization of supersymmetry we propose a generalization of non commutative geometry and gauge theories based on ternary z3 graded structures in the new algebraic structures we define all products of two entities are left free the only constraining relations being imposed on ternary products these relations reflect the action of the z3 group which may be either trivial i e abc bca cab generalizing the usual commutativity or non trivial i e abc jbca with j e 2 i 3 the usual z2 graded structures such as grassmann lie and clifford algebras are generalized to the z3 graded case certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

curvature and geometric modules of noncommutative spheres and tori when considered as submanifolds of euclidean space the riemannian geometry of the round sphere and the clifford torus may be formulated in terms of poisson algebraic expressions involving the embedding coordinates and a central object is the projection operator projecting tangent vectors in the ambient space onto the tangent space of the submanifold in this note we point out that there exist noncommutative analogues of these projection operators which implies a very natural definition of noncommutative tangent spaces as particular projective modules these modules carry an induced connection from euclidean space and we compute its scalar curvature

generalization of the hamilton ivey estimate to the higher dimensional ricci flow with a vanishing weyl tensor abstract in this study we consider curvature estimates of the ricci flow and we generalize the hamilton ivey pinching estimate to a higher dimensional ricci flow with a vanishing weyl tensor we also prove a pinching estimate for the ricci curvature under the ricci flow

representation theory of c algebras for a higher order class of spheres and tori we construct c algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree by using the directed graph associated with a matrix the representation theory can be understood in terms of loop and string representations which are closely related to the dynamics of an iterated map in the plane as a particular class of algebras we introduce the henon algebras for which the dynamical map is a generalized henon map and give an example where irreducible representations of all dimensions exist

construction of n lie algebras and n ary hom nambu lie algebras as n ary operations generalizing lie and poisson algebras arise in many different physical contexts it is interesting to study general ways of constructing explicit realizations of such multilinear structures generically they describe the dynamics of a physical system and there is a need of understanding their quantization hom nambu lie algebras provide a framework that might be an appropriate setting in which n lie algebras n ary nambu lie algebras can be deformed and their quantization studied we present a procedure to construct n 1 ary hom nambu lie algebras from n ary hom nambu lie algebras equipped with a generalized trace function it turns out that the implications of the compatibility conditions that are necessary for this construction can be understood in terms of the kernel of the trace function and the range of the twisting maps furthermore we investigate the possibility of defining n k lie algebras from n lie algebras and a k form satisfying certain conditions

lipschitz volume rigidity in alexandrov geometry abstract we prove a lipschitz volume rigidity theorem in alexandrov geometry that is if a 1 lipschitz map f x x l y between alexandrov spaces preserves volume then it is a path isometry and an isometry when restricted to the interior of x we furthermore characterize the metric structure on y with respect to x when f is also onto this implies the converse of petrunin s gluing theorem if a gluing of two alexandrov spaces via a bijection between their boundaries produces an alexandrov space then the bijection must be an isometry

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

generalization of n ary nambu algebras and beyond the aim of this paper is to introduce n ary hom algebra structures generalizing the n ary algebras of lie type including n ary nambu algebras n ary nambu lie algebras and n ary lie algebras and n ary algebras of associative type including n ary totally associative and n ary partially associative algebras we provide examples of the new structures and present some properties and construction theorems we describe the general method allowing one to obtain an n ary hom algebra structure starting from an n ary algebra and an n ary algebra endomorphism several examples are derived using this process also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three dimensional hom nambu lie structures with diagonal homomorphism

categorical perspective on quantization of poisson algebra we propose a generalization of quantization using a categorical approach for a fixed poisson algebra quantization categories are defined as subcategories of the r module category equipped with the structure of classical limits we then construct the generalized quantization categories including matrix regularization strict deformation quantization prequantization and poisson enveloping algebra it is shown that the categories of strict deformation quantization prequantization and matrix regularization with certain conditions are equivalent categories on the other hand the categories of poisson enveloping algebra are not equivalent to the other categories

bayesian analysis of a poisson process with a change point summary a bayesian approach to estimation and hypothesis testing for a poisson process with a change point is developed and an example given

symmetries inversion formulas and image reconstruction for optical tomography we consider the image reconstruction problem for optical tomography with diffuse light the associated inverse scattering problem is analyzed by making use of particular symmetries of the scattering data the effects of sampling and limited data are analyzed for several different experimental modalities and computationally efficient reconstruction algorithms are obtained these algorithms are suitable for the reconstruction of images from very large data sets

on special classes of n algebras we define n algebras as linear spaces on which the internal composition law involves n elements m v n v it is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics using the notion of a commutant of two subalgebras of an n algebra we distinguish certain classes of n algebras with reasonable properties semisimple abelian nilpotent solvable we also consider a few examples of n algebras of different types and show their properties

radiative transport equation in rotated reference frames a novel method for solving the linear radiative transport equation rte in a three dimensional homogeneous medium is proposed and illustrated with numerical examples the method can be used with an arbitrary phase function with the constraint that it depends only on the angle between the angular variables and this assumption corresponds to spherically symmetric on average random medium constituents boundary conditions are considered in the slab and half space geometries the approach developed in this paper is spectral it allows for the expansion of the solution to the rte in terms of analytical functions of angular and spatial variables to relatively high orders the coefficients of this expansion must be computed numerically however the computational complexity of this task is much smaller than in the standard method of spherical harmonics the solutions obtained are especially convenient for solving inverse problems associated with radiative transfer

the bad truth about laplace s transform inverting the laplace transform is a paradigm for exponentially ill posed problems for a class of operators including the laplace transform we give forward and inverse formulae that have fast implementations using the fast fourier transform these formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the laplace transform

on the higher order generalizations of poisson structures the characterization of the nambu poisson n tensors as a subfamily of the generalized poisson ones recently introduced and here extended to the odd order case is discussed the homology and cohomology complexes of both structures are compared and some physical considerations are made