on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

synchronized chaotic intermittent and spiking behavior in coupled map chains we study phase synchronization effects in a chain of nonidentical chaotic oscillators with a type i intermittent behavior two types of parameter distribution linear and random are considered the typical phenomena are the onset and existence of global all to all and cluster partial synchronization with increase of coupling increase of coupling strength can also lead to desynchronization phenomena i e global or cluster synchronization is changed into a regime where synchronization is intermittent with incoherent states then a regime of a fully incoherent nonsynchronous state spatiotemporal intermittency appears synchronization desynchronization transitions with increase of coupling are also demonstrated for a system resembling an intermittent one a chain of coupled maps replicating the spiking behavior of neurobiological networks

transition to weak turbulence via spatiotemporal intermittency in the taylor dean system the transition to weak turbulence via spatiotemporal intermittency has been studied in the taylor dean system when the inner cylinder is fixed and only the outer cylinder is rotating the spatiotemporal intermittency regime is characterized by the coexistence of laminar domains and turbulent patches for the same value of the control parameter the transition is supercritical but a particular boundary condition induces an imperfection statistical analysis of the spatiotemporal intermittency shows that the distribution of laminar domains by widths follows a power law decay near the onset while it is exponential for higher values of the control parameter the exponential distribution of the laminar domains has a definite threshold near which the correlation lengths and correlation times of laminar and turbulent domains diverge reinforcing the analogy with second order phase transitions the exponents measured near the critical points in this experiment differ from those observed in other systems thus showing that the spatiotemporal intermittency is not universal in character textcopyright 1996 the american physical society

weak solutions of the landau lifshitz bloch equation abstract the landau lifshitz bloch llb equation is a formulation of dynamic micromagnetics valid at all temperatures treating both the transverse and longitudinal relaxation components important for high temperature applications we study llb equation in case the temperature raised higher than the curie temperature the existence of weak solution is showed and its regularity properties are also discussed in this way we lay foundations for the rigorous theory of llb equation that is currently not available

regularity properties of the solutions to the 3d maxwell landau lifshitz system in weighted sobolev spaces this paper is concerned with special regularity properties of the solutions to the maxwell landau lifshitz mll system describing ferromagnetic medium besides the classical results on the boundedness of tm te and th in the spaces l i l 2 w and l 2 i w 1 2 w we derive also estimates in weighted sobolev spaces this kind of estimates can be used to control the taylor remainder when estimating the error of a numerical scheme

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics abstract the tangent plane scheme is a time marching scheme for the numerical solution of the nonlinear parabolic landau lifshitz gilbert equation which describes the time evolution of ferromagnetic configurations exploiting the geometric structure of the equation the tangent plane scheme requires only the solution of one linear variational form per time step which is posed in the discrete tangent space determined by the nodal values of the current magnetization we develop an effective solution strategy for the arising constrained linear systems which is based on appropriate householder reflections we derive possible preconditioners which are essentially independent of the time step and prove linear convergence of the preconditioned gmres algorithm numerical experiments underpin the theoretical findings

current induced magnetization dynamics global existence of weak solutions abstract we consider a model of current induced magnetization dynamics described by the landau lifshitz gilbert llg equations of the magnetization in which an additional current dependent term is added two methods namely faedo galerkin penalty fgp method and hyperbolic regularization method are used to show the existence of finite energy global weak solutions

spatio temporal chaos and intermittency in a 1 dimensional energy conserving coupled map lattice abstract the klein gordon equation with cubic nonlinearity the 4 equation is considered and an energy conserving difference scheme is proposed for its solution the scheme extended to finite time increments and spacing is then used to define a coupled map lattice for which an energy like functional is conserved the case of linear instability of the vacuum state is considered when this energy is not positive definite and found to lead under certain additional conditions to spatio temporal chaos the statistical properties of this type of solution such as probability densities and correlation functions are calculated strong intermittency whereby the process wanders between two sub manifolds is found and studied in detail

analysis of a coupled spin drift diffusion maxwell landau lifshitz system abstract the existence of global weak solutions to a coupled spin drift diffusion and maxwell landau lifshitz system is proved the equations are considered in a two dimensional magnetic layer structure and are supplemented with dirichlet neumann boundary conditions the spin drift diffusion model for the charge density and spin density vector is the diffusion limit of a spinorial boltzmann equation for a vanishing spin polarization constant the maxwell landau lifshitz system consists of the time dependent maxwell equations for the electric and magnetic fields and of the landau lifshitz gilbert equation for the local magnetization involving the interaction between magnetization and spin density vector the existence proof is based on a regularization procedure l 2 type estimates and moser type iterations which yield the boundedness of the charge and spin densities furthermore the free energy is shown to be nonincreasing in time if the magnetization spin interaction constant in the landau lifshitz equation is sufficiently small

regular and chaotic phase synchronization of coupled circle maps we study the effects of regular and chaotic phase synchronization in ensembles of coupled nonidentical circle maps cms and find phase locking regions for both types of synchronization we show that synchronization of chaotic cms is crucially influenced by the three quantities i rotation number difference ii variance of the phase evolution and iii relative duration of intervals of phase increase respect decrease in the case of regular cms only variance and rotation number difference are important it is demonstrated that with increase of noncoherence of phase evolutionsin the regular and chaotic regime the regions of the main 1 1 synchronization are usually decreased we present a chaotic synchronization in the systems of coupled nonidentical circle maps where phase entrainment occurs and it is not accompanied by bifurcations of the chaotic set for ensembles chains of coupled cms with linear and random distributions of the individual frequencies soft and hard transitions to global synchronization are found

convergence of an implicit explicit midpoint scheme for computational micromagnetics abstract based on lowest order finite elements in space we consider the numerical integration of the landau lifschitz gilbert equation llg the dynamics of llg is driven by the so called effective field which usually consists of the exchange field the external field and lower order contributions such as the stray field the latter requires the solution of an additional partial differential equation in full space following bartels and prohl 2006 we employ the implicit midpoint rule to treat the exchange field however in order to treat the lower order terms effectively we combine the midpoint rule with an explicit adams bashforth scheme the resulting integrator is formally of second order in time and we prove unconditional convergence towards a weak solution of llg numerical experiments underpin the theoretical findings

spatio temporal intermittency in coupled map lattices abstract the transition to turbulence in a one dimensional array of maps coupled by diffusion is shown to display critical properties resembling those of directed percolation the analogy is supported by the reconstruction of a probabilistic cellular automation with closely similar statistical properties numerical results suggest however that spatio temporal intermittency does not belong to the same universality class as directed percolation

a reduced theory for thin film micromagnetics micromagnetics is a nonlocal nonconvex variational problem its minimizer represents the ground state magnetization pattern of a ferromagnetic body under a specified external field this paper identifies a physically relevant thin film limit and shows that the limiting behavior is described by a certain reduced variational problem our main result is the 0 convergence of suitably scaled three dimensional micromagnetic problems to a two dimensional reduced problem this implies in particular convergence of minimizers for any value of the external field the reduced problem is degenerate but convex as a result it determines some but not all features of the ground state magnetization pattern in the associated thin film limit c 2002 wiley periodicals inc

sums of heights of algebraic numbers for at x f x tg x we consider the set at 0 h t q the polynomials f x g x are in z x with only mild restrictions and h is the weil height of we show that this set is dense in d for some effectively computable limit point d

a landau lifshitz type equation in studying thin film micromagnetics the paper is concerned with a landau lifshitz type equation which was first raised by desimone and otto etc when studying thin film micromagnetics we get the existence of a continuous solution using penalty approximation compensated compactness and commutator estimate

partial regularity to the landau lifshitz equation with spin accumulation abstract in this paper we consider a model for the spin magnetization system that takes into account the diffusion process of the spin accumulation this model consists of the landau lifshitz equation describing the precession of the magnetization coupled with a quasi linear parabolic equation describing the diffusion of the spin accumulation this paper establishes the global existence and uniqueness of weak solutions for large initial data in r 2 moreover partial regularity is shown in particular the solution is regular on r 2 0 with the exception of at most finite singular points

spatiotemporal intermittency and scaling laws in inhomogeneous coupled map lattices we study the phenomenon of intermittency in an inhomogeneous lattice of coupled maps where the inhomogeneity appears in the form of different values of the map parameter at adjacent sites this system exhibits spatiotemporal intermittency as well as purely spatial intermittency accompanied by temporal periodicity in different regions of the parameter space both types of intermittency appear as a result of bifurcations of codimension two in such systems we identify the types of bifurcations that are seen the intermittency near the bifurcation points and lines is associated with power law distributions for the laminar lengths the scaling laws for the laminar length distributions are obtained two distinct types of scaling behavior characterized by power laws with exponents that fall in two distinct ranges can be seen in the neighborhood of codimension two bifurcation points additionally we find two crossover exponents

higher order regularity results in 3d for the landau lifshitz equation with an exchange field abstract we derive and prove higher order regularity results for the landau lifshitz ll equation describing the evolution of spin fields in continuum ferromagnets these results are based on the weight s min 1 s which helps to get nonlinear parts under control

a finite element approximation for the stochastic landau lifshitz gilbert equation abstract the stochastic landau lifshitz gilbert llg equation describes the behaviour of the magnetisation under the influence of the effective field containing random fluctuations we first transform the stochastic llg equation into a partial differential equation with random coefficients without the ito term the resulting equation has time differentiable solutions we then propose a convergent linear scheme for the numerical solution of the reformulated equation as a consequence we show the existence of weak martingale solutions to the stochastic llg equation a salient feature of this scheme is that it does not involve solving a system of nonlinear algebraic equations and that no condition on time and space steps is required when 1 2 1 numerical results are presented to show the applicability of the method

hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems the paper presents a qualitative analysis of an array of diffusively coupled identical continuous time dynamical systems the effects of full partial anti phase and in phase anti phase chaotic synchronization are investigated via the linear invariant manifolds of the corresponding differential equations existence of various invariant manifolds a self similar behavior a hierarchy and embedding of the manifolds of the coupled system are discovered sufficient conditions for the stability of the invariant manifolds are obtained via the method of lyapunov functions conditions under which full global synchronization can not be achieved even for the largest coupling constant are defined the general rigorous results are illustrated through examples of coupled lorenz like and coupled rossler systems

heat flow for the square root of the negative laplacian for unit length vectors abstract the paper is concerned with heat flow for the square root of the negative laplacian for unit length vectors motivated by the model of desimone kohn otto and muller we show the existence of weak solutions for the periodic case by means of the penalty approximation and commutator estimate

on a hyperbolic parabolic system arising in magnetoelasticity the evolution of a magnetoelastic material is described by a nonlinear hyperbolic parabolic system we introduce a simplified but nontrivial model and prove the existence of a unique solution to the corresponding initial boundary value problem

orbits of algebraic numbers with low heights we prove that the two smallest values of h h 1 1 h 1 1 are 0 and 0 4218 for any algebraic integer

hausdorff measure of the singular set of landau lifshitz equations with a nonlocal term this paper is concerned with the partial regularity of the stationary solutions to the landau lifshitz system of ferromagnetic spin chain with gilbert damping and a nonlocal term in 3 dimensions the hausdorff measure of the singular set is discussed

the gilbert equation with dry friction type damping a modified gilbert equation for micromagnetics is considered obtained by augmenting the standard viscous like dissipation with a rate independent term we prove existence of a weak solution both with and without viscous dissipation by scaling time we show that if the applied field varies very slowly then gyromagnetic effects and viscous dissipation become negligible in the infinitesimally slow loading limit the system thus becomes fully rate independent

two new points in the spectrum of the absolute mahler measure of totally positive algebraic integers for totally positive algebraic integers 0 1 of degree d we consider the set l of values of m 1 d where m is the mahler measure of c j smyth has found the four smallest values of l and conjectured that the fifth point is 2 cos 2 60 2 we prove that this is so and moreover we give the sixth point of l

mutual synchronization and clustering in randomly coupled chaotic dynamical networks we introduce and study systems of randomly coupled maps rcm where the relevant parameter is the degree of connectivity in the system global almost synchronized states are found equivalent to the synchronization observed in globally coupled maps until a certain critical threshold for the connectivity is reached we further show that not only the average connectivity but also the architecture of the couplings is responsible for the cluster structure observed we analyse the different phases of the system and use various correlation measures in order to detect ordered non synchronized states finally it is shown that the system displays a dynamical hierarchical clustering which allows the definition of emerging graphs

loss of coherence in a system of globally coupled maps we study the formation of symmetric i e equally sized or nearly symmetric clusters in an ensemble of globally coupled identical chaotic maps it is shown that the loss of synchronization for the coherent state and the emergence of subgroups of oscillators with synchronized behavior are two distinct processes and that the type of behavior that arises after the loss of total synchronization depends sensitively on the dynamics of the individual map for our system of globally coupled logistic maps symmetric two cluster formation is found to proceed through a periodic state associated with the stabilization either of an asynchronous period 2 cycle or of an asynchronous period 4 cycle with further reduction of the coupling strength each of the principal clustering states undergoes additional bifurcations leading to cycles of higher periodicity or to quasiperiodic and chaotic dynamics if desynchronization of the coherent chaotic state occurs before the formation of stable clusters becomes possible high dimensional chaotic motion is observed in the intermediate parameter interval

wong zakai approximation for the stochastic landau lifshitz gilbert equations abstract in this work we study stochastic landau lifshitz gilbert equations sllges in one dimension with non zero exchange energy only firstly by introducing a suitable transformation we convert the sllges to a highly nonlinear time dependent partial differential equation with random coefficients which is not fully parabolic we then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property following regular approximation of the brownian motion and using reverse transformation we show existence of strong solution of sllges taking values in a two dimensional unit sphere s 2 in r 3 the construction of the solution and its corresponding convergence results are based on wong zakai approximation

homogenization of the demagnetization field operator in periodically perforated domains in this paper we study the homogenization of the demagnetization field operator in periodically perforated domains using the two scale convergence method as an application we homogenize the landau lifshitz equation in such domains we consider regular homothetic holes

numerical methods for the landau lifshitz equation we discuss numerical schemes for various forms of the landau lifshitz equation a new simple projection method is introduced and is shown to be unconditionally stable the advantages over other schemes are also demonstrated numerically

maxwell s equations with vector hysteresis electromagnetic processes in magnetic materials are described by maxwell s equations in ferrimagnetic insulators assuming that d e e we have the equation

numerical scheme for augmented landau lifshitz equation in heat assisted recording the motivation of heat assisted recording is to improve the thermal stability of recorded data bits by increasing the strength of the uniaxial anisotropy during the recording process the medium is heated by a laser reducing the coercivity and allowing the head to write data bits we present a micromagnetic model based on the augmented landau lifshitz equation taking into account a phenomenological power law describing the dependence of the saturation magnetization on the temperature a full discrete numerical scheme is presented and the convergence of approximate solutions to a weak one is shown

symplectic flow for the square root of the negative laplacian abstract the paper is concerned with symplectic flow for the square root of the negative laplacian which is motivated when studying a one dimensional model about ferromagnetic thin film without gilbert damping term only gyromagnetic term remained derived from the stationary model of a desimone r v kohn s muller and f otto existence of weak solution to the model is proved by viscosity approximation and commutator estimate

the weak solutions to the evolution problems of harmonic maps on demontre l existence de solutions faibles globales a des problemes d evolution des applications harmoniques de varietes de riemann compactes sans bord dans des spheres

global solutions of a dynamical equation in ferrimagnet abstract we consider global solutions of a dynamical equation in ferrimagnet we show that it admits a global weak solution by using the penalty method by the energy estimates method we show there exists a unique global smooth solution finally we establish the relationship between this equation and wave maps

numerical analysis of an explicit approximation scheme for the landau lifshitz gilbert equation the landau lifshitz gilbert equation describes magnetic behavior in ferromagnetic materials construction of numerical strategies to approximate weak solutions for this equation is made difficult by its top order nonlinearity and nonconvex constraint in this paper we discuss necessary scaling of numerical parameters and provide a refined convergence result for the scheme first proposed by alouges and jaisson 2006 as an application we numerically study discrete finite time blowup in two dimensions

directed percolation universality in asynchronous evolution of spatiotemporal intermittency we present strong evidence that a coupled map lattice model for spatiotemporal intermittency belongs to the universality class of directed percolation when the updating rules are asynchronous i e when only one randomly chosen site is evolved at each time step in contrast when the system is subjected to parallel updating available numerical evidence suggest that it does not belong to this universality class and that it is not even universal we argue that in the absence of periodic external forcing the asynchronous rule is the more physical

optimal error estimates of a linearized backward euler fem for the landau lifshitz equation we present a fully discrete linearized backward euler finite element method for the landau lifshitz equation in which a new linearization is proposed for the gyromagnetic term optimal almost unconditional error estimates of the scheme in l 2 and h 1 norms are proved i e when the stepsizes h and tau are smaller than given constants the analysis relies on the new linearization and an error splitting argument proposed by li and sun int j numer anal model 10 2013 pp 622 633 and siam j numer anal 51 2013 pp 1959 1977 numerical results in both two and three dimensional spaces are provided to confirm our theoretical analysis and show the unconditional stability convergence of the scheme

initial boundary value problems for anisotropic landau lifshitz equations in three or four space dimensions abstract in this paper we will show the existence of partially regular solutions to the initial boundary value problem for landau lifshitz equations with nonpositive anisotropy constants in three or four space dimensions the partial regularity is proved up to the boundary both for the dirichlet problem and for the neumann problem in addition for the neumann case a generalized stability condition which ensures the partial regularity is given for equations with positive or negative anisotropy coefficients we will give two results of existence and uniqueness for the solutions corresponding to ground states

global weak solutions of the fractional landau lifshitz maxwell equation abstract the fractional landau lifshitz maxwell equation is considered in this paper we show the global existence of suitable weak solutions by galerkin method and the vanishing viscosity method the main difficulties in this study are due to the loss of compactness of this system and the fact that the nonlinear term is nonlocal and of the same order of the equation to overcome these difficulties we introduce the commutator estimates and some cancellation properties to this equation which prove to be proper tools in the study of landau lifshitz type equations

partial synchronization and spontaneous spatial ordering in coupled chaotic systems a model of many symmetrically and locally coupled chaotic oscillators is studied partial chaotic synchronizations associated with spontaneous spatial ordering are demonstrated very rich patterns of the system are revealed based on partial synchronization analysis the stabilities of different partially synchronous spatiotemporal structures and some dynamical behaviors of these states are discussed both numerically and analytically

experimental observation of the characteristic relations of type i intermittency in the presence of noise recently it has been reported that the characteristic relation of type i intermittency in the presence of noise is deformed nontrivially as the channel width epsilon changes from the positive region to the negative in order to verify it experimentally as a real phenomenon we study the characteristic relations both for epsilon 0 in a simple inductor resistor diode circuit that is under noisy circumstances the experimental results agree well with the theoretical expectation that the characteristic relations are proportional to epsilon 1 4 for epsilon 0 and proportional to exp alpha epsilon 3 2 for epsilon 0

on the weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations in this paper we deal with weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations first we prove that these solutions satisfy some weak strong uniqueness property then we investigate the validity of energy identities in particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation in the case of the hall magneto hydrodynamic equations we also give a sufficient condition to guarantee the magneto helicity identity our conditions correspond to the same heuristic scaling as the one introduced by onsager in hydrodynamic theory finally we examine the sign locally of the anomalous dissipations of weak solutions obtained by some natural approximation processes

a convergent implicit finite element discretization of the maxwell landau lifshitz gilbert equation we propose an implicit fully discrete scheme for the numerical solution of the maxwell landau lifshitz gilbert equation which is based on linear finite elements and satisfies a discrete sphere constraint as well as a discrete energy law as numerical parameters tend to zero solutions weakly accumulate at weak solutions of the maxwell landau lifshitz gilbert equation a practical linearization of the nonlinear scheme is proposed and shown to converge for certain scalings of numerical parameters computational studies are presented to indicate finite time blow up behavior and to study combined electromagnetic phenomena in ferromagnets for benchmark problems

existence of partially regular weak solutions to landau lifshitz maxwell equations abstract in this paper the authors establish the existence of partially regular weak solutions to the landau lifshitz equations coupling with static maxwell systems in 3 dimensions by ginzburg landau approximation it is proved that the hausdorff measure of the singular set is locally finite this extends the similar results of ding and guo s ding b guo hausdorff measure of the singular set of landau lifshitz equations with a nonlocal term comm math phys 250 1 2004 95 117 from the stationary solutions to weak solutions and the results of wang c wang on landau lifshitz equations in dimensions at most four indiana univ math j 55 5 2006 1615 1644 from landau lifshitz equations to landau lifshitz maxwell equations

existence regularity and local uniqueness of the solutions to the maxwell landau lifshitz system in three dimensions abstract we study a model of ferromagnetism governed by maxwell s equations coupled with the non linear landau lifshitz equation of micromagnetism we are interested in the cases of space periodic solutions for 3d domains obtaining the regularity of the solution m in w per 2 2 space and of the solutions e h in w per 1 2 space we state the existence theorem finally we prove the local uniqueness of the solutions