double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem this paper presents a preconditioner for non overlapping schwarz methods applied to the helmholtz problem starting from a simple analytic example we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain the preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains and can be interpreted as a coarse grid for the domain decomposition method the resulting algorithm is shown to converge very fast independently of the number of subdomains and frequency the preconditioner has the advantage that like the original schwarz algorithm it can be implemented as a matrix free routine with no additional preprocessing

stability of the one step replica symmetry broken phase in neural networks the stability of the phase with one step replica symmetry breaking is studied in fully connected neural networks with modified pseudo inverse interactions the interaction matrix has an intermediate form between the hebb learning rule and the pseudo inverse one at low temperature there is a region of parameters where the one step replica symmetry broken solution exists fluctuations around this solution are analysed by a replica group representations approach and it is found that the solution is stable for all ranges of the parameters where it exists

three letters to walter feit on group representations and quaternions abstract representations of the quaternion group by 2 2 matrices with coefficients in the ring of integers of an imaginary quadratic field

finite size scaling in the p state mean field potts glass a monte carlo investigation the p state mean field potts glass with bimodal bond distribution j is studied by monte carlo simulations both for p 3 and p 6 states for system sizes from n 5 to n 120 spins considering particularly the finite size scaling behavior at the exactly known glass transition temperature t c it is shown that for p 3 the moments q k of the spin glass order parameter satisfy a simple scaling behavior q k alpha n k 3 tilde f k n 1 3 1 t t c text k 1 2 3 tilde f k being the appropriate scaling function and t the temperature also the specific heat maxima have a similar behavior c v max alpha text const n 1 3 while moments of the magnetization scale as m k alpha n k 2 the approach of the positions t max of these specific heat maxima to t c as n is nonmonotonic for p 6 the results are compatible with a first order transition q k q jump k as n but since the order parameter q jump at t c is rather small a behavior q k n k 3 as n also is compatible with the data thus no firm conclusions on the finite size behavior of the order parameter can be drawn the specific heat maxima c v max behave qualitatively in the same way as for p 3 consistent with the prediction that there is no latent heat a speculative phenomenological discussion of finite size scaling for such transitions is given for small n n 15 for p 3 n 12 for p 6 the monte carlo data are compared to exact partition function calculations and excellent agreement is found we also discuss ratios r x equiv langle x rangle t langle x rangle t text av 2 text av langle x rangle t text av 2 for various quantities x to test the possible lack of self averaging at t c

integrability of rotationally symmetric n harmonic maps abstract a rotationally symmetric n harmonic map is a rotationally symmetric p harmonic map between two n dimensional model spaces such that p n we show that rotationally symmetric n harmonic maps can be integrated and are n harmonic diffeomorphism and apply such results to investigate the asymptotic behaviors of these maps we also derive this integrability using lie theory

rotationally symmetric biharmonic maps between models abstract the main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m dimensional models in the sense of greene and wu we obtain a complete classification of rotationally symmetric proper biharmonic conformal diffeomorphisms in the special case that m 4 and the models have constant sectional curvature then by introducing the hamiltonian associated to this problem we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is r 4 in the second part of the paper we carry out a stability study with respect to equivariant variations equivariant stability in particular we prove that i the inverse of the stereographic projection from the open 4 dimensional euclidean ball to the hyperbolic space is equivariant stable ii the inverse of the stereographic projection from the closed 4 dimensional euclidean ball to the sphere is equivariant stable with respect to variations which preserve the boundary data

locally self similar phase diagram of the disordered potts model on the hierarchical lattice we study the critical behavior of the random q state potts model in the large q limit on the diamond hierarchical lattice with an effective dimensionality d eff 2 by varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points according to our renormalization group study the phase boundary in the vicinity of the multicritical point is self similar it is well represented by a logarithmic spiral we expect an infinite number of reentrances in the thermodynamic limit consequently one cannot define standard thermodynamic phases in this region

on the asymptotic behavior of rotationally symmetrical harmonic maps abstract we study a second order ordinary differential equation which is the euler lagrange equation of the energy functional for maps with prescribed rotational symmetry we obtain liouville s type theorems for symmetric harmonic maps into ellipsoids euclidean and hyperbolic spaces and existence results as well

mean field theory and fluctuations in potts spin glasses i a short range potts spin glass is studied a stable non marginal mean field theory is found with one level of replica symmetry breaking and a discontinuous transition for p 4 a complete stability analysis is provided and two different correlation lengths are found above eight dimensions the fluctuations in the ordered phase around this solution are incorporated in a renormalisation group approach and it is found that for a small range of the parameters they restore scaling close to the upper critical dimension for the three state case it appears that fluctuations destroy the stability of the solution for d or 8 and cause the system to undergo a first order phase transition non universal corrections to the equation of state above the upper critical dimension are discussed

statistical and systematic errors in monte carlo sampling we have studied the statistical and systematic errors which arise in monte carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used we find that depending on the degree of self averaging exhibited by the quantities measured the statistical errors can increase decrease or stay the same as the system size is increased the systematic underestimation of response functions due to the finite number of measurements made is also studied we develop a scaling formalism to describe the size dependence of these errors as well as their dependence on the bin length size of the statistical sample both at and away from a phase transition the formalism is tested using simulations of thed 3 ising model at the infinite lattice transition temperature we show that for a 96 96 96 system noticeable systematic errors systematic underestimation of response functions are still present for total run lengths of 106 monte carlo steps site mcs with measurements taken at regular intervals of 10 mcs

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

an initial value approach to rotationally symmetric harmonic maps abstract we study the effect of the varying y 0 on the existence and asymptotic behavior of solutions for the initial value problem y r n 1 f r y r f r n 1 g y r g y r f r 2 0 y 0 0 where f and g are some prescribed functions global solutions of this ode on 0 represent rotationally symmetric harmonic maps with possibly infinite energies between certain class of riemannian manifolds by studying this ode we show among other things that i all rotationally symmetric harmonic maps from r n to the hyperbolic space h n blow up in a finite interval ii all such harmonic maps from h n to r n are bounded and iii a trichotomy phenomenon occurs for such harmonic maps from h n into itself viz they blow up in a finite interval are the identity map or are bounded according as the initial value y 0 1 finally when n 2 the above equation can be solved exactly by quadrature method our results supplement those of ratto and rigoli j differential equations 101 1993 15 27 and tachikawa tokyo j math 11 1988 311 316

uniqueness of positive solutions to the rotationally symmetric p harmonic map equations abstract let and be positive solutions on 0 to the rotationally symmetric p harmonic map equation on model manifolds m f and m g where f is assumed to be sufficiently large near infinity and g y 0 for y 0 we show that if and have the same limit at infinity then on 0

ultrametric matrices and representation theory the consequences of replica symmetry breaking on the structure of ultrametric matrices appearing in the theory of disordered systems is investigated with the help of representation theory and the results are compared with those obtained by temesvari de dominicis and kondor

algebraic invariants of knots and disordered potts model the authors propose the reformulation of the kauffman bracket invariant of the knot in terms of statistical mechanics of the 2d disordered potts model this allows one to put the question of the determination of knot entropy or of the probability of an arbitrary knot formation in terms of usual statistical mechanics to demonstrate the possibilities of their approach they give a constructive estimation for the trivial knot formation probability for a long strongly contracted closed random path confined in a thin slit they use the mean field approximation for the free energy of the potts system at the point of the transition from the paramagnetic phase to the spin glass one

phase transition in an optimal clusterization model abstract an optimal clusterization model resembling the infinite range potts glass type model with j bonds and unrestricted number of states p n is introduced and studied as a function of the q probability of j bonds it is found that the r relative size of the largest cluster or coalition shows a percolation like transition at q 1 2 by a simple renormalization approach and several optimization methods we investigate the r q curves for finite system sizes non trivial consequences for social percolation problems are discussed

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

unbounded positive entire solutions of rotationally symmetric harmonic map equations abstract we study the existence of unbounded positive entire c 2 solutions of the rotationally symmetric harmonic map equations using the existence result we solve the dirichlet problem at infinity for any nonnegative boundary value at infinity

on the static and dynamical transition in the mean field potts glass we study the static as well as the glassy or dynamical transition in the mean field p state potts glass by numerical solution of the saddle point equations we investigate the static and dynamical transitions for all values of p in the non perturbative regime p 4 the static and dynamical edwards anderson parameter increase with p logarithmically this makes the glassy transition temperature lie very close to the static one we compare the main predictions of the theory with numerical simulations

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

finite size scaling in the p state mean field potts glass exact statistical mechanics for small samples the mean field potts glass with bimodal bond distribution is studied for p 3 and p 6 potts states via exact summation of the partition function of systems of up to n 15 sites averages over the disorder realizations are also done exactly up to n 8 while random bond configurations are generated for the larger systems our enumeration method is described in some detail and the applicability of finite size scaling fss for our very small systems is discussed for p 3 second order transition the data agree reasonably with the fss relations given by parisi et al for the p 2 ising case for p 6 no clear signs of the predicted first order behaviour could be observed this case is even more affected by finite size effects since the necessary antiferromagnetic bias of the bond distribution introduces an additional correction in the free energy for both p the data are compatible with a vanishing ground state entropy

high temperature series analysis of the free energy and susceptibility of the 2d random bond ising model abstract we derive high temperature series expansions for the free energy and susceptibility of the two dimensional random bond ising model with a symmetric bimodal distribution of two positive coupling strengths j1 and j2 and study the influence of the quenched random bond disorder on the critical behavior of the model by analysing the series expansions over a wide range of coupling ratios j2 j1 covering the crossover from weak to strong disorder we obtain for the susceptibility with two different methods compelling evidence for a singularity of the form t 7 4 ln t 7 8 as predicted theoretically by shalaev shankar and ludwig for the specific heat our results are less convincing but still compatible with the theoretically predicted log log singularity

an algebraic subgrid scale finite element method for the convected helmholtz equation in two dimensions with applications in aeroacoustics abstract an algebraic subgrid scale finite element method formally equivalent to the galerkin least squares method is presented to improve the accuracy of the galerkin finite element solution to the two dimensional convected helmholtz equation a stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method an appropriate value for this parameter has been obtained by means of a dispersion analysis as an application we have considered the case of aerodynamic sound radiated by incompressible flow past a two dimensional cylinder following lighthill s acoustic analogy we have used the time fourier transform of the double divergence of the reynolds stress tensor as a source term for the helmholtz and convected helmholtz equations and showed the benefits of using the subgrid scale stabilization

the q state potts glass on d dimensional hypercubic lattices we use a high temperature star graph expansion to compute the free energy and the susceptibility of a q state potts glass with bimodal distribution of the random bonds the series are up to order o 20 in k beta j for arbitrary number of states q on d dimensional simple cubic lattices the result for the three state potts glass in three dimensions shows excellent consistency with existing monte carlo data

monte carlo study of the potts glass with nearest neighbour random gaussian interaction the three state potts model with nearest neighbour random gaussian interaction on the simple cubic lattice is investigated by monte carlo simulation both static quantities e g glass ordering susceptibility and the correlation function and dynamic quantities the analogue of the time dependent edwards anderson order parameter q t for potts spins are obtained as for related models for orientational glasses it is found that q t is consistent with the kohlrausch law q t varies as exp t tau y for a wide range of temperatures with a strongly temperature dependent exponent y with y becoming very small as the temperature t to 0 the relaxation time tau increases dramatically as t is lowered thus the system could only be equilibrated for temperatures where the correlation length is rather small since the critical region has not been reached it cannot be distinguished whether the critical temperature tc is non zero or at t 0 if tc 0 the divergences of tau and the ordering susceptibility are probably exponential i e the system is then at its lower critical dimensionality

upper and lower critical dimension of the three state potts spin glass we used a high temperature star graph expansion to compute the susceptibility of a q state potts glass with a bimodal distribution of the random bonds the series are up to order o 20 in k beta j in three dimensions our result shows excellent consistency with existing monte carlo data the results for the three state potts glass in three to ten dimensions are analysed using dlog pade approximants and the adler moshe privman amp method in combination with variable transformations the analysis supports d 3 as lower critical dimension with an exponential singularity at tc 0 as concluded earlier using monte carlo data according to our analysis the upper critical dimension is d 8 which is in contrast to earlier papers

harmonic diffeomorphisms between the annuli with rotational symmetry abstract we give some necessary and sufficient conditions for the existence of rotationally symmetric harmonic diffeomorphism between the annuli with poincare metric or euclidean metric on the target respectively

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

statics and dynamics of the ten state mean field potts glass model a monte carlo study we investigate by means of monte carlo simulations the fully connected p state potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the exactly known behaviour of the system in the thermodynamic limit using p 10 we are able to study the equilibrium dynamics for system sizes as large as n 2560 we find that the static quantities such as the energy the entropy the spin glass susceptibility as well as the distribution of the order parameter p q show very strong finite size effects from p q we calculate the fourth order cumulant g4 n t and the guerra parameter g n t and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated also the spin autocorrelation function c t shows strong finite size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature td we show that the dependence on n and t of the relaxation time can be understood by means of a dynamical finite size scaling ansatz c t does not obey the time temperature superposition principle for temperatures around td but does so for significantly lower t finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin i e that the system is dynamically very heterogeneous which explains the non exponentiality of c t

mean field theory and fluctuations in potts spin glasses ii for pt i see ibid vol 22 p 4971 1989 the fluctuations in the ordered phase around the mean field solution are incorporated in a renormalization group approach and it is found that for a small range of the parameters they restore scaling close to the upper critical dimension for the 3 state case it appears that fluctuations destroy the stability of the solution and cause the system to undergo a first order phase transition non universal fluctuation corrections to the equation of state above six dimensions are obtained

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative