functional determinants for the dirac equation with mixed pseudodifferential boundary conditions over finite cylinders in this note we explicitly compute the functional determinant of a dirac laplacian with nonlocal pseudodifferential boundary conditions over a finite cylinder in terms of the function of the dirac operator on the cross section and the pseudodifferential operators defining the boundary conditions in particular this result reduces to our previous formula j phys ajphac5 37 7381 2004 for the special case of generalized atiyah patodi singer conditions to prove our main result we use the gluing and comparison formulas established by the present authors in refs 14 and 15

eigenvalues of an elliptic system we describe the spectrum of a non self adjoint elliptic system on a finite interval under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines the eigenfunctions need not generate a basis of the relevant hilbert space and the larger eigenvalues are extremely sensitive to small perturbations of the operator we show that the leading term in the spectral asymptotics is closely related to a certain convex polygon and that the spectrum does not determine the operator up to similarity two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics while our study makes no claim to generality the results obtained will have to be incorporated into any future general theory

zeta determinants on manifolds with boundary abstract through a general theory for relative spectral invariants we study the determinant of global boundary problems of aps type in particular we compute the determinant ratio for dirac laplacian boundary problems in terms of a scattering fredholm determinant over the boundary

functorial qft gauge anomalies and the dirac determinant bundle using properties of the determinant line bundle for a family of elliptic boundary value problems we explain how the fock space functor defines an axiomatic quantum field theory which formally models the fermionic path integral the sewing axiom of the theory arises as an algebraic pasting law for the determinant of the dirac operator we show how representations of the boundary gauge group fit into this description and that this leads to a fock functor description of certain gauge anomalies

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

the ζ determinant and the additivity of the η invariant on the smooth self adjoint grassmannian in this paper we discuss the existence of the determinant of a dirac operator on a compact manifold with boundary we show that the determinant is well defined in the case of the operator with a domain determined by a boundary condition from the smooth self adjoint grassmannian discussed in the papers 5 13 29 we prove a generalization of a pasting formula for the invariant see 34 the results of the paper are used in the recent proof of the projective equality of the determinant and quillen determinant on see 30 31

numerical solutions of coupled kleingordonzakharov equations by quintic b spline differential quadrature method numerical solutions of the coupled kleingordonzakharov equations are obtained by using quintic b spline based differential quadrature method a rungekutta fourth method is used for time integration stability of the scheme is studied using matrix stability analysis the accuracy and efficiency of the presented method is shown by conducting some numerical experiments on test problems which includes the motion of single soliton and interactions of two solitons the numerical results are found in good agreement with the exact solutions

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

adiabatic limits of the η invariants the odd dimensional atiyah patodi singer problem we study the invariant of boundary value problems of atiyah patodi singer type we prove the formula for the spectral flow of the families overs1 assuming a product structure in a collar neighbourhood of the boundary we show that the invariant behaves the same way as on a closed manifold we also study the adiabatic limit of the spectral invariants in nice cases we are able to split them into a contribution from the interior one from the cylinder and an error term then we show that the error term disappears with the increasing length of the cylinder

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

on the determinant of one dimensional elliptic boundary value problems we discuss the regularized determinant of elliptic boundary value problems on a line segment our framework is applicable for separated and non separated boundary conditions

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

a quadratic b spline finite element method for solving nonlinear schrodinger equation a numerical solution of the nonlinear schrodinger nls equation is obtained by using the quadratic b spline finite element method test problems concerning motion birth and bound states of solitons of the nlg equation are studied to evaluate the method results are compared with various numerical methods

on the gluing problem for the spectral invariants of dirac operators abstract in this paper we solve the gluing problem for the determinant of a dirac laplacian to do so we develop a new approach to solve such problems which relies heavily on the theory of elliptic boundary problems the analysis of the resolvent of the dirac operator and the introduction of an auxiliary model problem moreover as a byproduct of our approach we obtain a new gluing formula for the eta invariant au gratis

the origin of universality making and invalidating a free category we propose a model of evolutionary systems characterized by internal measurement or endo physics that is based on a particular graph morphism if we take an internal stance we have to focus on the influence of the outside of the domain of choice the outside of the domain is not explicitly expressed as a particular ensemble in advance although a formal expression generally requires an explicit form of the outside as the possible forms in our model a system is expressed by a directed graph and the time transition is expressed as a particular graph morphism referring to the outside of the domain of the morphism referring to the outside of the domain is expressed by dynamical decomposition and synthesis of the relationship between a graph and its possible forms expressed as an underlying free category because the transformations from a graph to a category and vice versa are expressed as two adjunctive functors a graph morphism is defined by a sequence of the operation of a free functor and a special operator standing for invalidation of the operation of the free functor time development of this system can generate a particular universal structure like a limit and co limit as a part of a directed graph which stands for generation of a higher order we also argue that the process toward the generation of a universal structure can reveal intermittency and that an emergent boundary in pattern formation can be demonstrated by the generation of this universal structure in addition we discuss emergent computation such as the origin of primitive recursive functions through the origination of a universal structure

stability satisfied numerical approximates to the non analytical solutions of the cubic schrodinger equation abstract the time dependent complex schrodinger equation with cubic non linearity is solved by constructing differential quadrature algorithm based on sinc functions reduction the equation to a coupled system of real equations enables to approximate the space derivative terms by the proposed method the resultant ordinary differential equation system is integrated with respect to the time variable by using various explicit methods of lower and higher orders some initial boundary value problems containing some analytical and non analytical initial data are solved for experimental illustrations the computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solutions exist the two conserved quantities are calculated by using the numerical results in all cases the matrix stability analysis is implemented to control the time step size

η invariants and determinant lines the invariant of an odd dimensional manifold with boundary is investigated the natural boundary condition for this problem requires a trivialization of the kernel of the dirac operator on the boundary the dependence of the invariant on this trivialization is best encoded by the statement that the exponential of the invariant lives in the determinant line of the boundary our main results are a variational formula and a gluing law for this invariant these results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of dirac operators also known as the global anomaly formula the ideas developed here fit naturally with recent work in topological quantum field theory in which gluing which is a characteristic formal property of the path integral and the classical action is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary

dynamically changing interface as a model of measurement in complex systems abstract we propose and sketch a novel approach toward the study of complex systems by considering a basic type of measurement problem hidden in any system we call this approach hyper dilation and cover it under the label of measurement oriented physics mop as compared with state oriented physics sop the essential difference between the two concerns the concepts of state to which they refer mop deals with two different concepts of state non measured states with infinite precision and measured states with finite precision the measurement process is expressed as a dynamically changing interface between them sop deals with one single concept of state and does not comprise a corresponding distinction we show fundamental differences between mop and sop with respect to the noise characteristics of complex systems around critical states mop can give rise to exact and universal 1 noise while sop shows 1 noise with 1 in general in mop a self similar return map can be degenerate with a cantor set rendering a solution for a basic measurement paradox the significance of this degeneracy as a model for emergent properties is discussed

b spline finite element studies of the non linear schrodinger equation abstract the non linear schrodinger equation is solved numerically by a b spline finite element method the approach used is based on collocation of cubic b splines over spatial finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range time integration of the resulting system of ordinary differential equations is effected using a crank nicolson approximation standard problems are used to validate the algorithm comparisons are made with published numerical and analytical solutions the proposed method exhibits good conservation properties and performs well the development of temporally recurrent states from spatially periodic initial conditions is examined through computer experiments the birth of mobile as well as static solitons out of maxwellian initial data is observed

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

formal model of internal measurement alternate changing between recursive definition and domain equation abstract we sketch a paradox generally resulting from recursivity and propose a novel model to express evolutionary processes that requires identification of an interaction with internal measurement in this model a paradox is not resolved and the notion of relativity of any resolution is implicit in a dynamical system a certain transition rule is used recursively along time if one takes the foundation or context of recursivity into consideration one obtains a fixed point or one confronts a paradox in order to resolve this paradox we adopt scott s technical way to identify the form of a fixed point with a domain equation and to obtain a reflective domain however we simultaneously show that any resolution is destined to be relative in utilizing this notion we construct a model of dynamical process by embedding a measurement process in one time step any time transition involves the process of doubting the foundation of a transition rule leading to a fixed point solving it and obtaining a reflexive domain is used as a new transition rule also this process perpetually proceeds along time and then the system perpetually proceeds while any solution is destined to be relative we illustrate this type of model by using a dynamically changing contraction mapping as the interface of state and transition rule finally we show that one can formalize emergent properties by using this model and discuss the relationship between endo physics and internal measurement

observational heterarchy enhancing active coupling heterarchy and the agent of self organization are ones of most challengeable topics in complex systems when the notion of heterarchy is generalized and dilated to general phenomena and is replaced by observational heterarchy the agent of self organization i e internal observer carrying measurement is found even in non hierarchical system a usual dynamics in the sense of complex system is articulated into two kinds of dynamics intent and extent dynamics and the interaction between them where intent corresponds to an attribute to a given phenomenon and extent corresponds to a collection of objects satisfying the phenomenon we formalize observational heterarchy by introducing pre equivalence pre adjunction between intent and extent and apply it to a coupled map system since the model inherits evoluvability of dynamics we call it active coupling to distinguish usual passive coupling from it we show that perturbation can modify the dynamics itself and enhance robust behavior opened to emergence in the system of active coupling and discuss the significance of observational heterarchy in complex systems

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

orthomodular lattice obtained from addressing a fixed point abstract one of us previously proposed a model of internal measurement that is not only an interaction between internal elements but also involves the notion of an indefinite observer measuring the interaction in this model all interactions encompass the notion of detection the latter can be expressed as a process generating a contradiction hereby a fixed point becomes invalid after inducing the contradiction instead of embedding this process as an instance along the time axis we here represent the logical jump via a contradiction fixed point specifically a new operation is defined which replaces the operation by which a fixed point is induced and we show that both a set as a fixed point and this particular operation on a set can give rise to an orthomodular lattice that is a model of quantum logic the new procedure is a mixture of inducing a fixed point and addressing a fixed point it can be compared to mixing up indicating an element with indicating a set consisting of elements that is to russel s paradox finally we point out that orthomodularity can ubiquitously be found in the evolutionary complex systems because it is generally relevant to resolving a paradox or a logical jump

ζ determinants of laplacians with neumann and dirichlet boundary conditions in this paper we derive a formula for the ratio of the determinants of the laplacian with neumann and dirichlet boundary conditions over a noncompact manifold with an infinite cylindrical end and a compact boundary in terms of the determinant of the dirichlet to neumann map

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

quartic and quintic b spline methods for advection diffusion equation differential quadrature methods based on b spline functions of degree four and five have been introduced to solve advection diffusion equation numerically two initial boundary value problems modeling the transportation of a concentration and distribution of an initial pulse are simulated using both methods the errors of the numerical results obtained by both methods have been computed stability analysis for both methods is also studied by the use of matrix stability

η forms and determinant lines abstract we construct a canonical zeta function quillen connection on the determinant line bundle for a family of aps elliptic boundary value problems and show that its curvature is the 2 form component of a relative eta form

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

splitting of the family index we establish a general splitting formula for index bundles of families of dirac type operators among the applications our result provides a positive answer to a question of bismut and cheeger bc2

poles of zeta and eta functions for perturbations of the atiyah patodi singer problem the zeta and eta functions of a differential operator of dirac type on a compact n dimensional manifold provided with a well posed pseudodifferential boundary condition have been shown in g99 to be meromorphic on with simple or double poles on the real axis extending results from g99 we show how perturbations of the boundary condition of order j affect the poles in particular they preserve a possible regularity of zeta at 0 and a possible simple pole of eta at 0 when j n this applies to perturbations of spectral boundary conditions also when the structure is non product and the problem is non selfadjoint

a differential quadrature algorithm for simulations of nonlinear schrodinger equation numerical simulations of nonlinear schrodinger equation are studied using differential quadrature method based on cosine expansion propogation of a soliton interaction of two solitons birth of standing and mobile solitons and bound state solutions are simulated the accuracy of the method dq is measured using maximum error norm the results are compared with some earlier works the lowest two conserved quantities are computed numerically for all cases

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

an engineer s guide to soliton phenomena application of the finite element method abstract the paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton type solutions namely the korteweg de vries equation the nonlinear schrodinger equation and the sine gordon equation throughout this study a semidiscrete galerkin method is applied using a finite element discretization in space and a step by step time integration of the resulting system of nonlinear ordinary differential equations depending upon the special type of the evolutionary equation the application of a petrov galerkin procedure may increase significantly the numerical stability accuracy and effectivity of the different approaches are demonstrated on a series of computer plots

determinants of dirac boundary value problems over odd dimensional manifolds we present a canonical construction of the determinant of an elliptic selfadjoint boundary value problem for the dirac operatord over an odd dimensional manifold for 1 dimensional manifolds we prove that this coincides with the function determinant this is based on a result that elliptic self adjoint boundary conditions ford are parameterized by a preferred class of unitary isomorphisms between the spaces of boundary chiral spinor fields with respect to a decompositions 1 x 0 x 1 we explain how the determinant of a dirac type operator overs 1 is related to the determinants of the corresponding boundary value problems overx 0 andx 1

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained