higher dimensional analogues of fuchsian subgroups of 2 the problem of classification of 2 2 indefinite hermitian matrices over orders in clifford algebras is considered the unit groups of these matrices are analogous to maximal arithmetic fuchsian subgroups of psl 2 o where o is an order in a quadratic number field

symbol calculus and zeta function regularized determinants in this work we use semigroup integral to evaluate zeta function regularized determinants this is especially powerful for nonpositive operators such as the dirac operator in order to understand fully the quantum effective action one should know not only the potential term but also the leading kinetic term in this purpose we use the weyl type of symbol calculus to evaluate the determinant as a derivative expansion the technique is applied both to a spin 0 bosonic operator and to the dirac operator coupled to a scalar field

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

generalized yang mills actions from dirac operator determinants we consider the quantum effective action of dirac fermions on four dimensional flat euclidean space coupled to external vector and axial yang mills fields i e the logarithm of the regularized determinant of a dirac operator on flat r4 twisted by generalized yang mills fields according to physics folklore the logarithmic divergent part of this effective action in the pure vector case is proportional to the yang mills action we present a simple explicit computation proving this fact and extending it to the chiral case we use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo differential operators and which yields an expansion of the logarithm of dirac operators in local and quasi gauge invariant polynomials of decreasing scaling dimension

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

noncommutative integration calculus a noncommutative integration calculus arising in the mathematical description of schwinger terms of fermion yang mills systems is discussed the differential complexes of forms u0 e u1 e un with e a grading operator on a hilbert space h and ui bounded operators on h which naturally contains the compactly supported de rham forms on rd i e e is the sign of the free dirac operator on rd and h is a l2 space on rd are considered an elementary proof is presented in which the integral of d forms rdtrn x0dx1 dxd for xi c0 rd gln is equal up to a constant to the conditional hilbert space trace of x0 e x1 e xd where 1 for d odd and d 1 5 matrix a spin matrix anticommuting with e for d even this result provides a natural generalization of integration of de rham forms to the setting of connes noncommutative geometry which involves the ordinary hilbert space trace rather than the dixmier trace

high order finite difference schemes for the solution of second order bvps we introduce new methods in the class of boundary value methods bvms to solve boundary value problems bvps for a second order ode these formulae correspond to the high order generalizations of classical finite difference schemes for the first and second derivatives in this research we carry out the analysis of the conditioning and of the time reversal symmetry of the discrete solution for a linear convection diffusion ode problem we present numerical examples emphasizing the good convergence behavior of the new schemes finally we show how these methods can be applied in several space dimensions on a uniform mesh

quaternionic modular groups abstract matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n space or hyperbolic n 1 space h n 1 in particular groups operating in the hyperbolic plane or hyperbolic 3 space may be represented by 2 2 matrices whose entries are rational integers or real or imaginary quadratic integers the theory is extended here to groups operating in h 4 or h 5 and matrices over one of the three basic systems of quaternionic integers quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of h 4 and h 5 for four dimensional groups the division ring of quaternions is treated as a clifford algebra results in hyperbolic 5 space derive from the homeomorphism of inversive 4 space and the quaternionic projective line

spectral collocation variational integrators abstract spectral methods are a popular choice for constructing numerical approximations for smooth problems as they can achieve geometric rates of convergence and have a relatively small memory footprint in this paper we introduce a general framework to convert a spectral collocation method into a shooting based variational integrator for hamiltonian systems we also compare the proposed spectral collocation variational integrators to spectral collocation methods and galerkin spectral variational integrators in terms of their ability to reproduce accurate trajectories in configuration and phase space their ability to conserve momentum and energy as well as the relative computational efficiency of these methods when applied to some classical hamiltonian systems in particular we note that spectrally accurate variational integrators such as the galerkin spectral variational integrators and the spectral collocation variational integrators combine the computational efficiency of spectral methods together with the geometric structure preserving and long time structural stability properties of symplectic integrators

dirac functional determinants in terms of the eta invariant and the noncommutative residue the zeta and eta functions associated with massless and massive dirac operators in a d dimensional d odd or even manifold without boundary are rigorously constructed several mathematical subtleties involved in this process are stressed as the intrinsic ambiguity present in the definition of the associated fermion functional determinant in the massless case and also the unavoidable presence in some situations of a multiplicative anomaly that can be conveniently expressed in terms of the non commutative residue the ambiguity is here proven to disappear in the massive case giving rise to a phase of the dirac determinant that agrees with very recent calculations which appeared in the mathematical literature and to a multiplicative anomaly also in agreement with other calculations in the coinciding cases in fact our results cover much more general situations a number of physically relevant explicit examples are worked out after explicit nontrivial resummation of the mass series expansions involving zeta and eta functions our results are finally expressed in terms of quite simple formulae

determinants of dirac operators with local boundary conditions we study functional determinants for dirac operators on manifolds with boundary we give for local boundary conditions an explicit formula relating these determinants to the corresponding green s functions we finally apply this result to the case of a bidimensional disk under baglike conditions

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

the relativistic lee model on riemannian manifolds we study the relativistic lee model on static riemannian manifolds the model is constructed nonperturbatively through its resolvent which is based on the so called principal operator and the heat kernel techniques it is shown that making the principal operator well defined dictates how to renormalize the parameters of the model the renormalization of the parameters is the same in the light front coordinates as in the instant form moreover the renormalization of the model on riemannian manifolds agrees with the flat case the asymptotic behavior of the renormalized principal operator in the large number of bosons limit implies that the ground state energy is positive in 2 1 dimensions the model requires only a mass renormalization we obtain rigorous bounds on the ground state energy for the n particle sector of the 2 1 dimensional model

scattering matrix in external field problems we discuss several aspects of second quantized scattering operators s for fermions in external time dependent fields we derive our results on a general abstract level having in mind as a main application potentials of the yang mills type and in various dimensions we present a new and powerful method for proving the existence of s which is also applicable to other situations like external gravitational fields we also give two complementary derivations of the change of phase of the scattering matrix under generalized gauge transformations which can be used whenever our method of proving the existence of s applies the first is based on a causality argument i e s including phase is determined from a time evolution and the second exploits the geometry of certain infinite dimensional group extensions associated with the second quantization of one particle operators as a special case we obtain a hamiltonian derivation of the axial fermion yang mills anomaly and the schwinger terms related to it via the descent equations which is on the same footing and traces them back to a common root

spectral variational integrators in this paper we present a new variational integrator for problems in lagrangian mechanics using techniques from galerkin variational integrators we construct a scheme for numerical integration that converges geometrically and is symplectic and momentum preserving furthermore we prove that under appropriate assumptions variational integrators constructed using galerkin techniques will yield numerical methods that are arbitrarily high order in particular if the quadrature formula used is sufficiently accurate then the resulting galerkin variational integrator has a rate of convergence at the discrete time steps that is bounded below by the approximation order of the finite dimensional function space in addition we show that the continuous approximating curve that arises from the galerkin construction converges on the interior of the time step at half the convergence rate of the solution at the discrete time steps we further prove that certain geometric invariants also converge with high order and that the error associated with these geometric invariants is independent of the number of steps taken we close with several numerical examples that demonstrate the predicted rates of convergence

commutator anomalies and the fock bundle we show that the anomalous finite gauge transformations can be realized as linear operators acting on sections of the bundle of fermionic fock spaces parametrized by vector potentials and more generally by splittings of the fermionic one particle space into a pair of complementary subspaces on the lie algebra level we show that the construction leads to the standard formula for the relevant commutator anomalies

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

high order multisymplectic runge kutta methods we study the spatial semidiscretizations obtained by applying runge kutta rk and partitioned runge kutta prk methods to multisymplectic hamiltonian pdes these methods can be regarded as multisymplectic hp finite element methods for wave equations all the methods we consider are multisymplectic we determine their properties with regard to existence of solutions dispersion and order the lobatto iiia iiib prk method can lead to explicit odes and exhibits a surprisingly high order of 2 higher than the number of degrees of freedom per cell

generalized multi symplectic integrators for a class of hamiltonian nonlinear wave pdes nonlinear wave equations such as burgers equation and compound kdv burgers equation are a class of partial differential equations pdes with dissipation in hamiltonian space the numerical method of which plays an important role in complex fluid analysis based on the multi symplectic idea a new theoretical framework named generalized multi symplectic integrator for a class of nonlinear wave pdes with small damping is proposed in this paper the generalized multi symplectic formulation is introduced and a twelve point generalized multi symplectic scheme which satisfies two discrete modified conservation laws approximately as well as the local momentum conservation law accurately is constructed to solve the first order pdes that derived from the compound kdv burgers equation to test the generalized multi symplectic scheme several numerical experiments on the travelling front solution are carried out the results of which imply that the generalized multi symplectic scheme can simulate the travelling front solution accurately and satisfy the modified conservation laws well when step sizes and the damping parameter satisfy the inequality 41 it is more remarkable that the scheme 36 can be used to capture the shock wave structure of the compound kdv burgers equation within one data point which can further illustrate the good structure preserving property of the generalized multi symplectic scheme 36 from the results of this paper we can conclude that similar to the multi symplectic scheme the generalized multi symplectic scheme also has two remarkable advantages the excellent long time numerical behavior and the good conservation property for the existing of the excellent numerical properties the generalized multi symplectic method can be used to exposit some specific phenomena in the complex fluid

smoothing schemes for reaction diffusion systems with nonsmooth data cox and matthews s m cox p c matthews exponential time differencing for stiff systems j comput phys 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic equations kassam and trefethen a k kassam ll n trefethen fourth order time stepping for stiff pdes siam j sci comput 26 2005 1214 1233 have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order etdrk4 scheme they use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials but this approach creates new difficulties in choosing and evaluating the contour for larger problems neither treatment addresses problems with nonsmooth data where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications including computational chemistry and financial engineering we introduce a new version of the fourth order cox matthews kassam trefethen etdrk4 scheme designed to eliminate the remaining computational difficulties this new scheme utilizes an exponential time differencing runge kutta etdrk scheme using a diagonal pade approximation of matrix exponential functions while to deal with the problem of nonsmooth data we use several steps of an etdrk scheme using a sub diagonal pade formula the new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel in this approach it is only required that several backward euler linear problems be solved in serial or parallel numerical experiments are described to support the new scheme

current algebras in d 1 dimensions and determinant bundles over infinite dimensional grassmannians we extend the methods of pressley and segal for constructing cocycle representations of the restricted general linear group in infinite dimensions to the case of a larger linear group modeled by schatten classes of rank 1 p an essential ingredient is the generalization of the determinant line bundle over an infinite dimensional grassmannian to the case of an arbitrary schatten rank p 1 the results are used to obtain highest weight representations of current algebras with the operator schwinger terms ind 1 dimensions when the space dimensiond is any odd number

functional determinants for radial operators we derive simple new expressions in various dimensions for the functional determinant of a radially separable partial differential operator thereby generalizing the one dimensional result of gel fand and yaglom to higher dimensions we use the zeta function formalism and the results agree with what one would obtain using the angular momentum cutoff method based on radial wkb the final expression is numerically equal to an alternative expression derived in a feynman diagrammatic approach but is considerably simpler

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

spectral variational integrators for semi discrete hamiltonian wave equations abstract in this paper we present a highly accurate hamiltonian structure preserving numerical method for simulating hamiltonian wave equations this method is obtained by applying spectral variational integrators svi to the system of hamiltonian odes which are derived from the spatial semi discretization of the hamiltonian pde the spatial variable is discretized by using high order symmetric finite differences an efficient implementation of svi for high dimensional systems of odes is presented

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

fermion current algebras and schwinger terms in 3 1 dimensions we discuss the restricted linear group in infinite dimensions modeled by the schatten class of rank 2p 4 which contains the 3 1 dimensional analogs of the loop groups and is closely related to yang mills theory with fermions in 3 1 dimensions we give an alternative to the construction of the highest weight representation of this group found by mickelsson and rajeev our approach is close to quantum field theory with the elements of this group regarded as bogoliubov transformations for fermions in an external yang mills field though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra we argue that they can be implemented by sesquilinear forms and that there is a regularized product of forms providing an appropriate group structure on the lie algebra level this gives an explicit non perturbative construction of fermion current algebras in 3 1 space time dimensions which explicity shows that the wave function renormalization required for a consistent definition of the currents and their lie bracket naturally leads to the schwinger term identical with the mickelsson rajeev cocycle though the explicit form of the schwinger term is given only for the casep 2 our arguments apply also to the restricted linear groups modeled by schatten classes of rank 2p 6 8 corresponding to current algebras in d 1 dimensions d 5 7

the noncommutative lorentzian cylinder as an isospectral deformation we present a new example of a finite dimensional noncommutative manifold namely the noncommutative cylinder it is obtained by isospectral deformation of the canonical triple associated with the euclidean cylinder we discuss connes character formula for the cylinder in the second part we discuss noncommutative lorentzian manifolds here the definition of spectral triples involves krein spaces and operators on krein spaces a central role is played by the admissible fundamental symmetries on the krein space of square integrable sections of a spin bundle over a lorentzian manifold finally we discuss isospectral deformation of the lorentzian cylinder and determine all admissible fundamental symmetries of the noncommutative cylinder

current algebra representation for the 3 1 dimensional dirac yang mills theory the structure of the current algebra representation in the state space of fermions in an external yang mills field in 3 1 space time dimensions is analyzed the topology of the vector space is determined by a countable family of semi definite inner products we show that there is no hermitian non trivial hilbert space representation such that the energy is bounded from below the structure of the hilbert space for the quantized coupled dirac yang mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite dimensional grassmannians

mickelsson rajeev cocycle corresponding to dimension five we construct a mickelsson rajeev type cocycle corresponding to five space dimensions in the three dimensional case this cocycle is shown to be equivalent to the original cocycle proposed by mickelsson and rajeev furthermore we construct a local representative for this cocycle and evaluate it explicitly on a five dimensional torus

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

higher dimensional analogues of the modular and picard groups clifford algebras are used to describe arithmetic groups which are generalizations of the modular and picard groups

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

functional determinants for general sturm liouville problems simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest we extend the range of situations for which these hold to cover systems of self adjoint operators of the sturm liouville type with arbitrary linear boundary conditions the results hold whether or not the operators have negative eigenvalues the physically important case of functional determinants of operators with a zero mode but where that mode has been extracted is studied in detail for the same range of situations as when no zero mode exists the method of proof uses the properties of generalized zeta functions the general form of the final results is the same for the entire range of problems considered