dynamical noether invariants for time dependent nonlinear systems dynamical invariants are derived for time dependent systems with nonlinear equations of motion including nonharmonic damped systems the concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated finally we show the existence of dynamical invariants for some nonlinear quantum systems

algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

on barnes multiple zeta and gamma functions abstract we show how various known results concerning the barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions the pertinent functions involve laplace transforms and their asymptotics is obtained by exploiting this we also demonstrate how barnes multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations both of these new approaches to the barnes functions give rise to novel integral representations

the barrier transmission problem treated by the amplitude phase method and expressed in terms of an invariant of the ermakov lewis type transmission and reflection of a quantal particle by a single hump potential barrier are analysed by means of an amplitude phase decomposition of the wavefunction on both sides of the barrier the amplitude phase analysis of the wavefunction provides a particular invariant of the ermakov lewis type which originates in the matching process the transmission and reflection coefficients turn out to be simple functions of this invariant numerical calculations of the invariant for an eckart epstein potential barrier provide very accurate results

a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

correction to the application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrarily oriented wire antennas the method of conjugate gradients is applied to the analysis of radiation from thin wire antennas with this iterative technique it is possible to solve electrically large arbitrarily oriented wire structures without storing any matrices as is conventionally done in the method of moments the basic difference between the proposed method and galerkin s method for the same expansion functions is that for the iterative technique we are solving a least squares problem hence as the order of the approximation is increased the proposed technique guarantees a monotonic decrease of the least squared error parallel ai y parallel 2 whereas galerkin s method does not even though the method converges for any initial guess a good one may significantly reduce the time of computation also explicit error formulas are given for the rate of convergence of this method hence any problem can be solved to a prespecified degree of accuracy it is shown that the method has the advantage of a direct solution as the final solution is obtained in a finite number of steps the method is also suitable for solving singular operator equations in which case the method monotonically converges to the least squares solution with minimum norm numerical results are presented for the thin wire antennas and are compared with the solution obtained by the method of moments

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

the farrell hsiang method revisited we present a sufficient condition for groups to satisfy the farrell jones conjecture in algebraic k theory and l theory the condition is formulated in terms of finite quotients of the group in question and is motivated by work of farrell hsiang

shintani barnes zeta and gamma functions abstract we show that shintani s work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations we re prove many of shintani s formulas and prove several new ones among the latter is a generalization to the shintani barnes gamma functions of raabe s 1843 formula 0 1 log x dx log 2 and a further generalization to the shintani zeta functions these explicit formulas can be interpreted as vanishing period integral side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions we also relate barnes triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

iterative computational techniques in scattering based upon the integrated square error criterion an iterative technique is developed to rigorously compute the electromagnetic time and frequency domain scattering problems the method is based upon a wave function expansion technique this also includes the integral representation techniques in which the electromagnetic field equations and causality conditions are satisfied analytically while the boundary conditions or the constitutive relations have to be satisfied in a computational manner the latter is accomplished by an iterative minimization of the integrated square error for the solution of an integral equation it is shown how to obtain optimum convergence some numerical results pertaining to a number of representative problems illustrate the numerical advantages and disadvantages of the iterative method

warped products admitting a curvature bound abstract this paper completes a fundamental construction in alexandrov geometry previously we gave a new construction of metric spaces with curvature bounds either above or below namely warped products with intrinsic metric space base and fiber and with possibly vanishing warping functions thereby extending the classical cone and suspension constructions from interval base to arbitrary base and furthermore encompassing gluing constructions this paper proves the converse namely all conditions of the theorems are necessary note that in the cone construction both the construction and its converse are widely used we also show that our theorems for curvature bounded above and below respectively are dual we give the first systematic development of basic properties of warped products of metric spaces with possibly vanishing warping functions including new properties

a generalization of van den berg s integral square error iterative computational technique for scattering van den berg shows a way of using spectral iteration si within a scheme that shows good convergence his method can be reinterpreted as a global expansion of the field quantities in terms of a set of basis functions it is shown here that the orthogonalization of these basis functions can considerably improve convergence rates this involves extra storage requirements but only a negligible decrease in speed of computation

the k theoretic farrell jones conjecture for hyperbolic groups we prove the k theoretic farrell jones conjecture for hyperbolic groups with twisted coefficients in any associative ring with unit

the li yau hamilton estimate and the yang mills heat equation on manifolds with boundary the paper pursues two connected goals firstly we establish the li yau hamilton estimate for the heat equation on a manifold m with nonempty boundary results of this kind are typically used to prove monotonicity formulas related to geometric flows secondly we establish bounds for a solution t of the yang mills heat equation in a vector bundle over m the li yau hamilton estimate is utilized in the proofs our results imply that the curvature of t does not blow up if the dimension of m is less than 4 or if the initial energy of t is sufficiently small

array decomposition method for the accurate analysis of finite arrays presented in this paper is a fast method to accurately model finite arrays of arbitrary three dimensional elements the proposed technique referred to as the array decomposition method adm exploits the repeating features of finite arrays and the free space green s function to assemble a nonsymmetric block toeplitz matrix system the toeplitz property is used to significantly reduce storage requirements and allows the fast fourier transform fft to be applied in accelerating the matrix vector product operations of the iterative solution process each element of the array is modeled using the finite element boundary integral fe bi technique for rigorous analysis consequently we demonstrate that the complete lu decomposition of the matrix system from a single array element can be used as a highly effective block diagonal preconditioner on the larger array matrix system this rigorous method is compared to the standard fe bi technique for several tapered slot antenna tsa arrays and is demonstrated to generate the same accuracy with a fraction of the storage and solution time

characterizing complete operatorname cat kappa spaces kappa 0 with geodesic boundary we investigate the bourdon and hamenstadt boundaries of complete cat spaces 0 and characterize those with geodesic hamenstadt boundary up to isometry

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

k theory of solvable groups we first prove that the whitehead group of a torsion free virtually solvable linear group vanishes next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable mathbb q linear groups finally we prove an l theory analogue for elementary amenable groups

higher order narrow tube quantizations of quasienergies the narrow tube quasienergy quantization allows an identification of part of the spectrum due to stable elliptic islands of the period map with its transparent connection to the classical dynamics it also sets an absolute scale to the understanding of the brillouin zone spectrum of quasienergies in combination with a recent time dependent normal form expansion we show that the applicability and accuracy of the narrow tube quantization is improved its analytic form helps us understand the quasiregular motion centred near periodic responses we numerically consider harmonic and subharmonic elliptic islands of the weakly nonlinear duffing oscillator

the farrell jones conjecture for the solvable baumslag solitar groups in this paper we prove the farrell jones conjecture for the solvable baumslag solitar groups with coefficients in an additive category we also extend our results to groups of the form z 1 p semidirect product with any virtually cyclic group where p is a prime number

narrow tube approximation of semiclassical quasi energies application to the weakly nonlinear duffing oscillator an exact transformation of semiclassical quantization conditions determining quantal quasi energies of time periodic hamiltonian systems is suggested for motion on vortex tubes centred at stable periodic orbits the classical quasi energy of the underlying periodic orbit separates out as a single term which is a lower bound for the quantized quasi energies a particular approximation involving the classical lewis invariant for calculating semiclassical quasi energies near the centre of a perturbed linear response is discussed in some detail

the ermakov lewis invariants for coupled linear oscillators we consider n coupled linear oscillators with time dependent coefficients an exact complex amplitude real phase decomposition of the oscillatory motion is constructed this decomposition is further used to derive n exact constants of motion which generalize the so called ermakov lewis invariant of a single oscillator in the floquet problem of periodic oscillator coefficients we discuss the existence of periodic complex amplitude functions in terms of existing floquet solutions

geometric curvature bounds in riemannian manifolds with boundary an alexandrov upper bound on curvature for a riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior as corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii the hadamard cartan theorem and yau s isoperimetric inequality for spaces with negative curvature are generalized

solution stability of iterative schemes utilizing the discrete fourier transform em scattering the fredholm integral equation of the first kind can be solved numerically using iterative schemes which minimize the integral square error when the kernal is of the convolution type the discrete fourier transform dft can be used when evaluating the operator equation in the iterative schemes however the dft imposes an artificial periodicity thus changing the nature of the problem the effect of this on the solution has been studied and convergence investigated by comparison with a method of moments solution a proposed method of avoiding the periodicity problem has been studied the effect of introducing losses in the medium surrounding the scatterer has been investigated provided losses are low there is little effect on the solution

time dependent normal form hamiltonian for dynamical equilibria a new time dependent normal form procedure for dynamical equilibria undergoing parametric excitation of one dimensional 1d hamiltonian systems is developed with the method of lie transforms the expansion is based on the lewis invariant for the linearized motion the time dependent hamiltonian normal form reduces smoothly to the usual representation in the autonomous limit illustrative examples of the formalism are focused on time periodic systems and the dynamics of hamiltonian switching

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

on the limitations of the birkhoff gustavson normal form approach recently the birkhoff gustavson normal form bgnf approach has been used in both classical and semiclassical calculations the authors consider the limitations associated with this approach using as an illustration the anharmonic oscillator at the classical level they emphasise the need to properly take into account the divergence properties of the bgnf series by reviewing an earlier work they demonstrate that this divergence problem may in some cases be resolved however after considering several standard quantisation rules they find that it is not possible to obtain the correct quantisation of a system via the bgnf as a result they maintain that care should be used when working with the bgnf

the conjugate gradient spectral iterative technique for planar structures it is shown that using the spectral iterative technique sit to solve the first kind integral equation is equivalent to the neumann iterative solution of a related second kind integral equation it is thus shown that sit only converges when the norm of the operator in the second kind equation is small enough applying a conjugate gradient technique to the second kind equation results in a convergent iterative scheme some representative numerical results show a superiority in the rate of convergence of the conjugate gradient scheme for the second kind equation cgsit scheme when compared with the convergence of the conjugate scheme for the original first kind equation cg scheme the cgsit scheme combines the advantages of the conjugate gradient method with those of the spectral iterative technique

direct extrapolation of a causal signal using low frequency and early time data in this paper we provide three direct procedures to extrapolate the early time and the low frequency response of a causal signal simultaneously in the time and frequency domain compared with the extrapolation by orthonormal basis functions direct extrapolation is straightforward and we do not need to evaluate the basis functions and search for the optimal scaling factor and the optimal number of basis functions we show that the extrapolation introduced by adve and sarkar is equivalent to a neumann series solution of an integral equation of the second kind it is further shown that this iterative neumann expansion is an error reducing method we propose to solve this integral equation efficiently by employing a conjugate gradient iterative scheme the convergence of this scheme is also demonstrated we provide the matrix equations and show the equivalence to the integral equations and demonstrate that the method of singular value decomposition svd of solving the matrix equation provides accurate and stable results finally a number of illustrative numerical examples are presented and the performances of the three direct methods are compared

semiclassical narrow tube quantizations of time periodic hamiltonian systems an approximate semiclassical quantization procedure for time periodic hamiltonians is formulated which applies to motion near stable periodic orbits of the system corresponding to elliptic islands of the period map the theory involves canonical transformations and the use of a particular dynamical lewis invariant the basic assumption in the narrow tube approximation is the localization of the quantal wavefunction inside the elliptic islands

nonlinear phase integral approximations of stationary waves in nonhomogeneous systems a recent normal form approximation for dynamical equilibria of one dimensional hamiltonian systems is shown to provide a phase integral wkb approximation to solutions of nonlinear differential equations in the present paper a restricted class of ordinary differential equations is considered the integrability of the truncated normal form allows for expressing the solutions as trigonometric expansions in terms of an amplitude and phase the method is applied to a dirichlet boundary value problem on the interval for n 3 and n 4 where the coefficient functions depend on an additional parameter as in the constant coefficient case we obtain approximate expressions for eigenvalues and eigensolutions near the linear limit the results show that the interpretative and the predictive power of the linear wkb solutions carry through to the nonlinear regime of small amplitude wavelike solutions we further analyse the mechanism by which the odd n nonlinearity in general causes a splitting of the linear eigenvalues in particular we discuss the singular threshold behaviour of the doubling mechanism for nonlinearities with n 3 if the coefficient functions become constants the doubling of eigenvalues corresponding to standing waves of odd numbers of nodes gradually disappears the method of approximation can be worked out similarly for any perturbing polynomial in

functional determinants generalized btz geometries and selberg zeta function we continue the study of a special entry in the ads cft dictionary namely a holographic formula relating the functional determinant of the scattering operator in an asymptotically locally anti de sitter space to a relative functional determinant of the scalar laplacian in the bulk a heuristic derivation of the formula involves a one loop quantum effect in the bulk and the corresponding sub leading correction at large n on the boundary we presently explore the formula in the background of a higher dimensional version of the euclidean btz black hole obtained as a quotient of hyperbolic space by a discrete subgroup of isometries generated by a loxodromic or hyperbolic element consisting of dilation temperature and torsion angles twist the bulk computation is done using heat kernel techniques and fractional calculus at the boundary we acquire a recursive scheme that allows us to successively include rotation blocks in spacelike planes in the embedding space the determinants are compactly expressed in terms of an associated patterson selberg zeta function and a connection to quasi normal frequencies is discussed

thin riemannian manifolds with boundary mathematics subject classi cation 1991 53c20 53c291 introductionlet m be a complete connected riemannian manifold with boundary b k bethe sectional curvature of the interior and be the normal curvature of b theinradius of m isinr supfd p b p2mg thus for every r inr there is some metric ball of radius r in m that does notintersect b byathin manifold with boundary m we mean one for which thescale free invariantinr

determinants and conformal anomalies of gjms operators on spheres the conformal anomalies and functional determinants of the branson gjms operators p2k on the d dimensional sphere are evaluated in explicit terms for any d and k such that k d 2 if d is even the determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly which vanishes for odd d taking the mode system on the sphere as the union of neumann and dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non existence of super critical operators 2k d for even d significant use is made of the barnes zeta function the results are given in terms of ratios of determinants of operators on a d 1 dimensional bulk dual sphere for odd dimensions the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations yielding an integral over a plancherel measure the n d determinant ratio is also found explicitly for even dimensions ehrhart polynomials are encountered