analytic formulation and numerical implementation of an acoustic pressure gradient prediction two new analytical formulations of the acoustic pressure gradient have been developed and implemented in the psu wopwop rotor noise prediction code the pressure gradient can be used to solve the boundary condition for scattering problems and it is a key aspect to solve acoustic scattering problems the first formulation is derived from the gradient of the ffowcs williams hawkings fw h equation this formulation has a form involving the observer time differentiation outside the integrals in the second formulation the time differentiation is taken inside the integrals analytically this formulation avoids the numerical time differentiation with respect to the observer time which is computationally more efficient the acoustic pressure gradient predicted by these new formulations is validated through comparison with available exact solutions for a stationary and moving monopole sources the agreement between the predictions and exact solutions is excellent the formulations are applied to the rotor noise problems for two model rotors a purely numerical approach is compared with the analytical formulations the agreement between the analytical formulations and the numerical method is excellent for both stationary and moving observer cases

on an integrable discretization of the rayleigh quotient gradient system and the power method with a shift an integrable discretization of the rayleigh quotient gradient system is established the solution of the discrete gradient system is described explicitly and converges exponentially to the same equilibrium point as that of the continuous gradient system for arbitrary large difference step size it is shown that the discrete gradient system is essentially equivalent to the power method with a shift of origin for calculating the largest eigenvalue the power method is then proved to be a discrete gradient method

a geometric convergence theory for the preconditioned steepest descent iteration preconditioned gradient iterations for very large eigenvalue problems are efficient solvers with growing popularity however only for the simplest preconditioned eigensolver namely the preconditioned gradient iteration or preconditioned inverse iteration with fixed step size are sharp nonasymptotic convergence estimates known these estimates require a properly scaled preconditioner in this paper a new sharp convergence estimate is derived for the preconditioned steepest descent iteration which combines the preconditioned gradient iteration with the rayleigh ritz procedure for optimal line search convergence acceleration the new estimate always improves that of the fixed step size iteration the practical importance of this new estimate is that arbitrarily scaled preconditioners can be used the rayleigh ritz procedure implicitly computes the optimal scaling constant

derivation and implementation of the boundary integral formula for the convective acoustic wave equation in time domain kirchhoff s formula for the convective wave equation is derived using the generalized function theory the generalized convective wave equation for a stationary surface is obtained and the integral formulation the convective kirchhoff s formula is derived the formula has a similar form to the classical kirchhoff s formula but an additional term appears due to a moving medium effect for convenience the additional term is manipulated to a final form as the classical kirchhoff s formula the frequency domain boundary integral can be obtained from the current time domain boundary integral form the derived formula is verified by comparison with the analytic solution of source in the uniform flow the formula is also utilized as a boundary integral equation time domain boundary element method bem analysis using the boundary integral equation is conducted and the results show good agreement with the analytical solution the formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain

anomaly of fractal dimensions observed in stochastically switched systems we studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven by stochastically switched inputs we calculated the dimensions for different switching time lengths in two dimensional linear dynamical systems and found that changes in the dimensions due to the switching time length had a singular point when the system matrix had two different real eigenvalues using partial dimensions along each eigenvector we explicitly derived a generalized dimension d q and a multifractal spectrum f alpha to explain this anomalous property the results from numerical calculations agreed well with those from analytical equations we found that this anomaly is caused by linear independence inhomogeneity of eigenvalues and overlapping conditions the mechanism for the anomaly could be identified for various inhomogeneous systems including nonlinear ones and this reminded us of anomalies in some physical values observed in critical phenomena

diffusion limited aggregation and iterated conformal maps the creation of fractal clusters by diffusion limited aggregation dla is studied by using iterated stochastic conformal maps following the method proposed recently by hastings and levitov the object of interest is the function phi sup n which conformally maps the exterior of the unit circle to the exterior of an ital n particle dla the map phi sup n is obtained from ital n stochastic iterations of a function phi that maps the unit circle to the unit circle with a bump the scaling properties usually studied in the literature on dla appear in a new light using this language the dimension of the cluster is determined by the linear coefficient in the laurent expansion of phi sup n which asymptotically becomes a deterministic function of ital n we find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the laurent coefficients copyright ital 1999 ital the american physical society

renormalization theory of stochastic growth an analytical renormalization group treatment is presented of a model which for one value of parameters is equivalent to diffusion limited aggregation the fractal dimension of dla is computed to be 2 1 2 1 5 1 7 higher multifractal exponents are also calculated and found in agreement with numerical results it may be possible to use this technique to describe the dielectric breakdown model as well which is given by different parameter values

continuous methods for extreme and interior eigenvalue problems in this paper continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices the main idea is to convert the extreme and interior eigenvalue problems into some optimization problems then a continuous method which includes both a merit function and an ordinary differential equation ode is introduced for each resulting optimization problem the convergence of each ode solution is proved for any starting point the limit of each ode solution for any starting point is fully studied both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition promising numerical results are also presented

computation of acoustic scattering from a moving rigid surface application of the kirchhoff formula for subsonically moving surfaces to scattering of sound from a moving rigid body is described the formulation leads to a singular integral equation that yields surface values of the field the integral equation is solved using a boundary element technique in conjunction with the galerkin method several checks on the accuracy and validity of the scheme are discussed numerical results are obtained for scattering of sound from a harmonic point source co moving with a thin winglike body

laplacian growth as one dimensional turbulence abstract a new model of laplacian stochastic growth is formulated using conformal mappings the model describes two growth regimes stable and turbulent separated by a sharp phase transition the first few fourier components of the mapping define the web an envelope of the cluster the web is used to study the transition and the dynamics of large scale features of the cluster characterized by evolution from macro to micro scales also we derive scaling laws for the cluster size

convergent calculation of the asymptotic dimension of diffusion limited aggregates scaling and renormalization of small clusters diffusion limited aggregation dla is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes we present a convergent calculation of the fractal dimension d of dla based on a renormalization scheme for the first laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster the theory is applicable from very small 2 3 particles to asymptotically large n infinity clusters the computed dimension is d 1 713 0 003

differential equations and the qr algorithm in this paper we consider a variety of isospectral flows on the set of n times n matrices these flows arise from lax pairs and can all be interpreted in terms of the qr decomposition for nonsingular matrices the asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem

laplacian growth with separately controlled noise and anisotropy conformal mapping models are used to study the competition of noise and anisotropy in laplacian growth for this purpose a family of models is introduced with the noise level and directional anisotropy controlled independently fractalization is observed in both anisotropic growth and growth with varying noise the fractal dimension is determined from the cluster size scaling with cluster area for isotropic growth d 1 7 at both high and low noise for anisotropic growth with reduced noise the dimension can be as low as d 1 5 and apparently is not universal also we study the fluctuations of particle areas and observe in agreement with previous studies that exceptionally large particles may appear during growth leading to pathologically irregular clusters this difficulty is circumvented by using an acceptance window for particle areas

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

dynamical systems and non hermitian iterative eigensolvers simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms we observe that these simple methods can be viewed as forward euler discretizations of well known autonomous differential equations that enjoy appealing geometric properties this connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue provides insight into the role of orthogonality and biorthogonality and suggests the development of new methods and analyses based on more sophisticated discretizations these results also highlight the effect of preconditioning on the convergence and stability of the continuous time system and its discretization

statistical physics of fracture surfaces morphology experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics one finds scaling exponents associated with correlation and structure functions indicating a rich phenomenology of anomalous scaling we argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for we present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces successfully reproducing the multiscaling phenomenology at least in 1 1 dimensions for surfaces in 2 1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the so 2 symmetry group it appears that this approach organizes effectively the rich scaling properties we end up proposing new experiments in which the rotational symmetry is not broken such that the scaling properties should be particularly simple

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

preconditioning constrained eigenvalue problems abstract the purpose of our paper is to introduce a robust preconditioning scheme for the numerical solution of the leftmost eigenvalues and corresponding eigenvectors of a constrained eigenvalue problem this constrained eigenvalue problem is congruent to a nonsymmetric eigenvalue problem with nontrivial jordan blocks associated with infinite eigenvalues the proposed preconditioning scheme is relevant to the application of krylov subspace methods and preconditioned eigensolvers the two key results are a semi orthogonal decomposition and a transformation process that implicitly combines a preconditioning step followed by abstract projection onto the subspace associated with the finite eigenvalues numerical results demonstrate the effectiveness of the preconditioning scheme

a model for branch competition branching or tip splitting is a ubiquitous feature of growth processes in nature we introduce a simple model linear in growth probabilities of branch competition that combines tip splitting processes local screening and extinction of branches whose growth has come to an end this model admits an exact solution which is corroborated by numerical results an extension of the model that depends quadratically upon growth probabilities exhibits a phase transition with non trivial scaling in the neighborhood of that transition

pressure field permeable surface integral formulations for sound scattered by moving bodies abstract this paper presents two novel integral formulations for the prediction of sound scattered by moving bodies derived from the lighthill and the ffowcs williams and hawkings equations for the pressure field these are expressed in the frequency domain over a permeable fictitious surface surrounding the scatterer s and are numerically evaluated through application of a boundary element technique the aims of the paper are the assessment of the influence of the nonlinear terms of lighthill and ffowcs williams and hawkings equations on sound scattering prediction in the presence of nonuniform mean flow due to scatterer motion the assessment of the corresponding limits of applicability of the widely used linear formulations for solid wall boundaries and the development of integral formulations capable to predict accurately and efficiently the noise scattered in the far field by moving bodies the numerical investigation concerns a non lifting wing in uniform translation impinged by an acoustic disturbance generated by a co moving source and includes the comparison of the results obtained through the proposed scattering formulations with those provided by a boundary field velocity potential approach recently validated for moving body problems its main outcomes reveal that when the proposed pressure field formulations include the nonlinear terms through application of a suitable permeable surface their predictions match those provided by the boundary field velocity potential solver whereas the fully linear versions of both pressure field approaches yield underestimated scattered noise predictions significantly less accurate than those given by the linear version of the velocity potential formulation

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

correction to scaling analysis of diffusion limited aggregation diffusion limited aggregation is consistent with simple scaling however strong subdominant terms are present and these can account for various earlier claims of anomalous scaling we show this in detail for the case of multiscaling