the kahler potential of abelian higgs vortices we calculate the kahler potential for the samols metric on the moduli space of abelian higgs vortices on r2 in two different ways the first uses a scaling argument the second depends on a variant of the relationship between accessory parameters and the regularized action in liouville field theory the kahler potential on the moduli space of vortices on h2 is also derived and we are led to a geometrical reinterpretation of these vortices

some inequalities for the poincare metric of plane domains in this paper the poincare or hyperbolic metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain open image in new window in particular another proof of a recent result of gardiner and lakic 7 is given with explicit constant this and some other constants in this paper involve particular values of complete elliptic integrals and related special functions a concrete estimate for the hyperbolic distance near a boundary point is also given from which refinements of littlewood s theorem are derived

expressing the entropy of lattice systems as sums of conditional entropies whether a system is to be considered complex or not depends on how one searches for correlations we propose a general scheme for calculation of entropies in lattice systems that has high flexibility in how correlations are successively taken into account compared to the traditional approach for estimating the entropy density in which successive approximations build on step wise extensions of blocks of symbols we show that one can take larger steps when collecting the statistics necessary to calculate the entropy density of the system in one dimension this means that instead of a single sweep over the system in which states are read sequentially one take several sweeps with larger steps so that eventually the whole lattice is covered this means that the information in correlations is captured in a different way and in some situations this will lead to a considerably much faster convergence of the entropy density estimate as a function of the size of the configurations used in the estimate the formalism is exemplified with both an example of a free energy minimisation scheme for the two dimensional ising model and an example of increasingly complex spatial correlations generated by the time evolution of elementary cellular automaton rule 60

convergence result for the constraint preserving mid point scheme for micromagnetism an important progress was recently done in numerical approximation of weak solutions to a micromagnetic model equation the problem with the nonconvex side constraint of preserving the length of the magnetization was tackled by using reduced integration several schemes were proposed and their convergence to weak solutions was proved all schemes were derived from the landau lifshitz gilbert form of the micromagnetic equation however when the precessional term in the original landau lifshitz ll form of the micromagnetic equation tends to zero the above schemes become unusable we propose a scheme derived from the mid point rule for the ll form of the micromagnetic equation combined with the reduced integration we show convergence to a weak solution of the ll equation and demonstrate the usefulness of the proposed scheme to study the limit process when the precessional parameter of the micromagnetic equation goes to zero

hyperbolic 2 spheres with conical singularities accessory parameters and kahler metrics on ℳ _ 0 we show that the real valued function s on the moduli space m 0 n of pointed rational curves defined as the critical value of the liouville action functional on a hyperbolic 2 sphere with n 3 conical singularities of arbitrary orders 1 n generates accessory parameters of the associated fuchsian differential equation as their common antiderivative we introduce a family of kahler metrics on m 0 n parameterized by the set of orders a explicitly relate accessory parameters to these metrics and prove that the functions s are their kahler potentials

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

twice punctured hyperbolic sphere with a conical singularity and generalized elliptic integral we describe in terms of generalized elliptic integrals the hyperbolic metric of the twice punctured sphere with one conical singularity of prescribed order we also give several monotonicity properties of the metric and a couple of applications

metrics with conical singularities on the sphere and sharp extensions of the theorems of landau and schottky an explicit formula for the generalized hyperbolic metric on the thrice punctured sphere mathbb p backslash z 1 z 2 z 3 with singularities of order j 1 at zj is obtained in all possible cases 1 2 3 2 the existence and uniqueness of such a metric was proved long time ago by picard j reine angew math 130 243 258 1905 and heins nagoya math j 21 1 60 1962 while explicit formulas for the cases 1 2 1 were given earlier by agard ann acad sci fenn ser a i 413 1968 and recently by anderson et al math z to appear doi 10 1007 s00209 009 0560 5 we also establish precise and explicit lower bounds for the generalized hyperbolic metric this extends work of hempel j lond math soc ii ser 20 435 445 1979 and minda complex var 8 129 144 1987 as applications sharp versions of landau and schottky type theorems for meromorphic functions are obtained

local conformal structure of liouville quantum gravity liouville conformal field theory lcft is an essential building block of polyakov s formulation of non critical string theory moreover scaling limits of statistical mechanics models on planar maps are believed by physicists to be described by lcft a rigorous probabilistic formulation of lcft based on a path integral formulation was recently given by the present authors and f david in cite dkrv in the present work we prove the validity of the conformal ward identities and the belavin polyakov zamolodchikov bpz differential equations of order 2 for the correlation functions of lcft this initiates the program started in the seminal work of belavin polyakov zamolodchikov cite bpz in a probabilistic setup for a non trivial conformal field theory we also prove several celebrated results on lcft in particular an explicit formula for the 4 point correlation functions with insertion of a second order degenerate field leading to a rigorous proof of a non trivial functional relation on the 3 point structure constants derived earlier in the physics literature by teschner cite tesc the proofs are based on exact identities which rely on the underlying gaussian structure of lcft combined with estimates from the theory of critical gaussian multiplicative chaos and a careful analysis of singular integrals beurling transforms and generalizations as a by product we give bounds on the correlation functions when two points collide making rigorous certain predictions from physics on the so called operator product expansion of lcft

entropy of phase separated structures abstract the morphology of phase separated structures can often be accurately modelled by level cut gaussian random fields we consider the statistical ensemble of level cut states and its relation to the ising model with a long range interaction the entropy of the ensemble can be numerically calculated using the local cluster method the method is tested on the exact solution of one dimensional ising model with the long range kac potential and illustrated using examples of level cut gaussian fields at least in one dimensional systems the cluster method is easy to apply and produces accurate results

a gauss seidel projection method for micromagnetics simulations abstract one of the main difficulties in micromagnetics simulation is the severe time step constraint introduced by the exchange field using standard explicit integrators leads to a physical time step of sub pico seconds which is often two orders of magnitude smaller than the fastest physical time scales direct implicit integrators require solving complicated coupled systems in this paper we introduce an implicit method whose complexity is comparable to solving the scalar heat equation implicitly this method is based on a combination of a gauss seidel implementation of a fractional step implicit solver for the gyromagnetic term and the projection method for the heat flow of harmonic maps this method allows us to carry out fully resolved calculations for the switching of the magnetization in micron sized elements

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

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

optimal error estimates of linearized crank nicolson galerkin fems for the time dependent ginzburg landau equations in superconductivity in this paper we study linearized crank nicolson galerkin finite element methods for time dependent ginzburg landau equations under the lorentz gauge we present an optimal error estimate for the linearized schemes almost unconditionally i e when the spatial mesh size h and the temporal step tau are smaller than a given constant while previous analyses were given only for some schemes with strong restrictions on the time step size the key to our analysis is the boundedness of the numerical solution in some strong norm we prove the boundedness for the cases tau ge h and tau le h respectively the former is obtained by a simple inequality with which the error functions at a given time level are bounded in terms of their average at two consecutive time levels and the latter follows a traditional way with the induction inverse inequality two numerical examples are investigated to confirm our theoretical analysis and to show clearly that no time step condition is needed

structural information in two dimensional patterns entropy convergence and excess entropy we develop information theoretic measures of spatial structure and pattern in more than one dimension as is well known the entropy density of a two dimensional configuration can be efficiently and accurately estimated via a converging sequence of conditional entropies we show that the manner in which these conditional entropies converge to their asymptotic value serves as a measure of global correlation and structure for spatial systems in any dimension we compare and contrast entropy convergence with mutual information and structure factor techniques for quantifying and detecting spatial structure

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

variation of cone metrics on riemann surfaces we discuss the variational properties of the unique conical metric of constant curvature 1 associated to a compact riemann surface together with a weighted divisor 2005 elsevier inc all rights reserved

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

a monte carlo study of the entropy the pressure and the critical behavior of the hard square lattice gas an approximate technique for estimating the entropys with computer simulation methods suggested recently by meirovitch is applied here to the metropolis monte carlo mc simulation of the hard square lattice gas in both the grand canonical and the canonical ensembles the chemical potential calculated by widom s method ands enable one to obtain also the pressurep the mc results are compared with results obtained with pade approximants pa and are found to be very accurate for example at the critical activityz c the mc and the pa estimates fors deviate by 0 5 beyondz c this deviation decreases to 0 01 and comparable accuracy is detected forp we argue that close toz c our results fors andp are more accurate than the pa estimates independent of the entropy study we also calculate the critical exponents by applying fisher s finite size scaling theory to the results for the long range order the compressibility and the staggered compressibility obtained for several lattices of different size at zc the data are consistent with the critical exponents of the plane ising lattice 1 8 1 7 4 and 0 our values for and agree with series expansion and renormalization group results respectively 0 has been obtained also by matrix method studies it differs however from the estimate of baxteret al 0 09 0 05 as far as we know has not been calculated yet

methods for estimating entropy with computer simulation the simple cubic ising lattice an approximate method for estimating entropy with computer simulation suggested recently is applied here together with a new method to the metropolis monte carlo simulation of the simple cubic ising lattice beyond the critical temperature tc the accuracy of the best results for the entropy and the free energy is estimated to be better than 0 07 and 0 05 and at tc itself it is approximately 1 and 0 2 respectively these results are in very good agreement with results obtained with the stochastic models method which is a computer simulation technique and with series expansion estimates the results for the entropy are significantly more accurate than results obtained with the commonly used reversible thermodynamic integration

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

an iterative approximation scheme for the landau lifshitz gilbert equation a new iterative numerical scheme for a solution of the single landau lifshitz gilbert equation i e that the magnetic field is given is proposed the approach uses the fixed point technique for handling the nonlinearities at each time step the second order of convergence in the time is proved and also confirmed by an example with an exact solution

convergence of an implicit finite element method for the landau lifshitz gilbert equation the landau lifshitz gilbert equation describes the dynamics of ferromagnetism where strong nonlinearity and nonconvexity are hard to tackle so far existing explicit schemes to approximate weak solutions suffer from severe time step restrictions in this paper we propose an implicit fully discrete scheme and verify unconditional convergence

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

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