ground state solutions for choquard equations with hardy littlewood sobolev upper critical growth and potential vanishing at infinity abstract in this paper we investigate the following non autonomous choquard equation u v x u i u 2 k x f u 2 u 2 2 u k x f u in r n where n 3 i is the riesz potential of order 0 n 2 n n 2 is the upper critical exponent due to the hardy littlewood sobolev inequality the functions v k c r n r may vanish at infinity and the function f c r r is noncritical and f t 0 t f s d s based on variational methods and the hardy type inequalities using the mountain pass theorem and the nehari manifold approach we prove the existence of ground state solution

a semi analytical approach to molecular dynamics despite numerous computational advances over the last few decades molecular dynamics still favors explicit and thus easily parallelizable time integrators for large scale numerical simulation as a consequence computational efficiency in solving its typically stiff oscillatory equations of motion is hampered by stringent stability requirements on the time step size in this paper we present a semi analytical integration scheme that offers a total speedup of a factor 30 compared to the verlet method on typical md simulation by allowing over three orders of magnitude larger step sizes by efficiently approximating the exact integration of the strong harmonic forces of covalent bonds through matrix functions far improved stability with respect to time step size is achieved without sacrificing the explicit symplectic time reversible or fine grained parallelizable nature of the integration scheme we demonstrate the efficiency and scalability of our integrator on simulations ranging from dna strand unbinding and protein folding to nanotube resonators

shadow hybrid monte carlo an efficient propagator in phase space of macromolecules shadow hybrid monte carlo shmc is a new method for sampling the phase space of large molecules particularly biological molecules it improves sampling of hybrid monte carlo hmc by allowing larger time steps and system sizes in the molecular dynamics md step the acceptance rate of hmc decreases exponentially with increasing system size n or time step t this is due to discretization errors introduced by the numerical integrator shmc achieves an asymptotic o n1 4 speedup over hmc by sampling from all of phase space using high order approximations to a shadow or modified hamiltonian exactly integrated by a symplectic md integrator shmc satisfies microscopic reversibility and is a rigorous sampling method shmc requires extra storage modest computational overhead and a reweighting step to obtain averages from the canonical ensemble this is validated by numerical experiments that compute observables for different molecules ranging from a small n alkane butane with four united atoms to a larger solvated protein with 14 281 atoms in these experiments shmc achieves an order magnitude speedup in sampling efficiency for medium sized proteins sampling efficiency is measured by monitoring the rate at which different conformations of the molecules dihedral angles are visited and by computing ergodic measures of some observables

an integral formula adapted to different boundary conditions for arbitrarily high dimensional nonlinear klein gordon equations with its applications in this paper we are concerned with the initial boundary value problem of arbitrarily high dimensional klein gordon equations posed on a bounded domain d for d 1 and equipped with the requirement of boundary conditions we derive and analyze an integral formula which is proved to be adapted to different boundary conditions for general klein gordon equations in arbitrarily high dimensional spaces the formula gives a closed form solution to arbitrarily high dimensional homogeneous linear klein gordon equations which is totally different from the well known d alembert poisson and kirchhoff formulas some applications are included as well

ground state solutions for a choquard equation with lower critical exponent and local nonlinear perturbation abstract this paper deals with the following choquard equation with a local nonlinear perturbation u u i u n 1 u n 1 u f u x r n u h 1 r n where i r n r is the riesz potential n 3 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f is only required to satisfy some weak assumptions near 0 and our results improve the previous related ones in the literature

groundstates for a local nonlinear perturbation of the choquard equations with lower critical exponent abstract we prove the existence of ground state solutions by variational methods to the nonlinear choquard equations with a nonlinear perturbation u u i u n 1 u n 1 u f x u in r n where n 1 i is the riesz potential of order 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions

trigonometric collocation methods based on lagrange basis polynomials for multi frequency oscillatory second order differential equations abstract in the present work a kind of trigonometric collocation methods based on lagrange basis polynomials is developed for effectively solving multi frequency oscillatory second order differential equations q t m q t f q t the properties of the obtained methods are investigated it is shown that the convergent condition of these methods is independent of m which is very crucial for solving oscillatory systems a fourth order scheme of the methods is presented numerical experiments are implemented to show the remarkable efficiency of the methods proposed in this paper

ground states and non existence results for choquard type equations with lower critical exponent and indefinite potentials abstract in this paper we study following choquard equation with lower critical exponent u v x u i u n 1 u n 1 u f x u x r n u h 1 r n where n 1 i is the riesz potential and v r n r allows to be sign changing under some mild assumptions imposed on the nonlinearity f we prove that above equation has a ground state solution for the periodic case and asymptotically periodic case respectively the characterization of the ground states is also investigated by a direct approach that are constrained to the nehari manifold moreover we show a non existence result for the equation via a generalized poho aev identity established for the non autonomous nonlinearity f the results extend and improve some recent ones in the literature

what makes molecular dynamics work the equations of motion for deterministic molecular dynamics md are chaotic creating problems for their numerical treatment due to the exponential growth of error with time indeed modeling and computational errors overwhelm numerical trajectories in typical simulations consequently accuracy is expected only in a statistical sense based on random initial conditions of great interest then is the relationship between errors in the dynamics and their effects on the accuracy of statistical quantities specifically expectations this article provides a formula for the effect of a perturbation on an ensemble average which explains the accuracy of such calculations it also provides a formula for the effect of a perturbation on a time correlation function which however fails to explain accuracy for these calculations additionally this article clarifies the relationships among various dynamical properties of md and provides an extension to a theory of non hamiltonian md

variational integrators for electric circuits in this contribution we develop a variational integrator for the simulation of stochastic and multiscale electric circuits when considering the dynamics of an electric circuit one is faced with three special situations 1 the system involves external control forcing through external controlled voltage sources and resistors 2 the system is constrained via the kirchhoff current kcl and voltage laws kvl 3 the lagrangian is degenerate based on a geometric setting an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived a time discrete variational formulation provides an iteration scheme for the simulation of the electric circuit dependent on the discretization the intrinsic degeneracy of the system can be canceled for the discrete variational scheme in this way a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits in particular a comparison to bdf methods which are usually the method of choice for the simulation of electric circuits shows that even for simple lcr circuits a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator

numerical algorithms for highly oscillatory dynamic system based on commutator free method in the present paper an efficiently improved modified magnus integrator algorithm based on commutator free method is proposed for the second order dynamic systems with time dependent high frequencies firstly the second order dynamic systems are transferred to the frame of reference by introducing new variable so that highly oscillatory behaviour inherited from the entries then the modified magnus integrator method based on local linearization is appropriately designed for solving the above new form and some optimized strategies for reducing the number of function evaluations and matrix operations are also suggested finally several numerical examples for highly oscillatory dynamic systems such as airy equation bessel equation mathieu equation are presented to demonstrate the validity and effectiveness of the proposed method

a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the nonrelativistic limit regime a multiscale time integrator fourier pseudospectral mti fp method is proposed and analyzed for solving the klein gordon schrodinger kgs equations in the nonrelativistic limit regime with a dimensionless parameter 0 varepsilon le 1 0 1 which is inversely proportional to the speed of light in fact the solution of the kgs equations propagates waves with wavelength at o varepsilon 2 o 2 and o 1 in time and space respectively when 0 varepsilon ll 1 0 1 which brings significantly numerical burden in practical computation the mti fp method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives respectively we rigorously establish two independent error bounds for the mti fp at o tau 2 varepsilon 2 h m 0 o 2 2 hm0 and o varepsilon 2 h m 0 o 2 hm0 for varepsilon in 0 1 0 1 with tau time step size h mesh size and m 0 ge 4 m0 4 an integer depending on the regularity of the solution which imply that the mti fp converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau o for varepsilon in 0 1 0 1 in addition the mti fp method converges optimally with quadratic convergence rate at o tau 2 o 2 in the regime when 0 tau lesssim varepsilon 2 0 2 and the error is at o varepsilon 2 o 2 independent of tau in the regime when 0 varepsilon lesssim tau 1 2 0 1 2 thus the meshing strategy requirement or varepsilon scalability of the mti fp is tau o 1 o 1 and h o 1 h o 1 for 0 varepsilon ll 1 0 1 which is significantly better than that of classical methods numerical results demonstrate that our error bounds are optimal and sharp finally the mti fp method is applied to study numerically convergence rates of the kgs equations to its limiting models in the nonrelativistic limit regime

verlet i r respa impulse is limited by nonlinear instabilities this paper shows that in molecular dynamics md when constant energy nve simulations of newton s equations of motion are attempted using the multiple time stepping mts integrator verlet i r respa impulse there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period s of the fastest motion s in the system this is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem the numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein but also the constrained dynamics simulation of a solvated protein representing a range of simulation protocols commonly in use by biomolecular modelers the observed and predicted instabilities match exactly previous work has identified and explained a linear instability for verlet i r respa impulse at around half the period of the fastest motion mandziuk and schlick discovered nonlinear resonances in single time stepping md integrators but unstable nonlinear resonances for mts integrators are reported here for the first time this paper also offers an explanation on the instability of mts constrained molecular dynamics simulations of explicitly solvated proteins more aggressive multiple step sizes are possible with mild langevin coupling or targeted langevin coupling and its combination with the mollified impulse method permits step sizes 3 to 4 times larger thanverlet i r respa impulse while still retaining some accuracy

filters of residuated lattices and triangle algebras an important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic is that of a filter filters can be used amongst others to define congruence relations specific kinds of filters include boolean filters and prime filters in this paper we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections triangle algebras characterize interval valued residuated lattices

monitoring energy drift with shadow hamiltonians the application of a symplectic integrator to a hamiltonian system formally conserves the value of a modified or shadow hamiltonian defined by some asymptotic expansion in powers of the step size an earlier article describes how it is possible to construct highly accurate shadow hamiltonian approximations using information readily available from the numerical integration this article improves on this construction by giving formulas of order up to 24 not just up to 8 and by greatly reducing both storage requirements and roundoff error more significantly these high order formulas yield remarkable results not evident for 8th order formulas even for systems as complex as the molecular dynamics of water these numerical experiments not only illuminate theoretical properties of shadow hamiltonians but also give practical information about the accuracy of a simulation by removing systematic energy fluctuations they reveal the rate of energy drift for a given step size and uncover the ill effects of using switching functions that do not have enough smoothness

on nonlocal choquard equations with hardy littlewood sobolev critical exponents abstract we consider the following nonlinear choquard equation with dirichlet boundary condition u u 2 x y d y u 2 2 u f u in where is a smooth bounded domain of r n 0 n 3 0 n and 2 is the critical exponent in the sense of the hardy littlewood sobolev inequality under suitable assumptions on different types of nonlinearities f u we are able to prove some existence and multiplicity results for the equation by variational methods

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

a filon type asymptotic approach to solving highly oscillatory second order initial value problems abstract in this paper we analyze and design filon type asymptotic methods for solving highly oscillatory second order initial value problems q t mq t f t q t q t where m is a non singular and diagonalizable matrix having large eigenvalues and m 1 we first derive filon type asymptotic methods for highly oscillatory linear systems and then extend the methods to highly oscillatory nonlinear systems the integrators are a blend of the existing filon type methods and asymptotic methods based on the variation of constants formula it is shown that the new integrators are convergent and the accuracy of the new integrators is improved as m grows large this point is significant and powerful for effectively dealing with highly oscillatory systems numerical experiments are carried out and the numerical results show the advantage and efficiency of our new integrators as compared with existing numerical methods for highly oscillatory problems from the scientific literature

on filter theory of residuated lattices the aim of this paper is to develop the filter theory of general residuated lattices first we extend some particular types of filters and fuzzy filters in bl algebras and mtl algebras naturally to general residuated lattices and further enumerate some relative results obtained in bl algebras or mtl algebras which still hold in general residuated lattices next we introduce the concepts of regular filters and fuzzy regular filters to general residuated lattices which are two new types of filters and fuzzy filters and derive some of their characterizations finally we discuss the relations between fuzzy regular filters and several other special fuzzy filters and also characterize some special classes of residuated lattices by filters or fuzzy filters

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

time integrators based on approximate discontinuous hamiltonians summary we introduce a class of time integration algorithms for finite dimensional mechanical systems whose hamiltonians are separable by partitioning the system s configuration space to construct an approximate potential energy we define an approximate discontinuous hamiltonian adh whose resulting equations of motion can be solved exactly the resulting integrators are symplectic and precisely conserve the approximate energy which by design is always close to the exact one we then propose two adh algorithms for finite element discretizations of nonlinear elastic bodies these result in two classes of explicit asynchronous time integrators that are scalable and because they conserve the approximate hamiltonian could be considered to be unconditionally stable in some circumstances in addition these integrators can naturally incorporate frictionless contact conditions we discuss the momentum conservation properties of the resulting methods and demonstrate their performance with several problems such as rotating bodies and multiple collisions of bodies with rigid boundaries copyright 2011 john wiley sons ltd

practical construction of modified hamiltonians one of the most fruitful ways to analyze the effects of discretization error in the numerical solution of a system of differential equations is to examine the modified equations which are equations that are exactly satisfied by the approximate discrete solution these do not actually exist in general but rather are defined by an asymptotic expansion in powers of the discretization parameter nonetheless if the expansion is suitably truncated the resulting modified equations have a solution which is remarkably close to the discrete solution in the case of a hamiltonian system of ordinary differential equations the modified equations are also hamiltonian if and only if the integrator is symplectic evidence for the existence of a hamiltonian for a particular calculation is obtained by calculating modified hamiltonians and monitoring how well they are conserved also energy drifts caused by numerical instability are better revealed by evaluating modified hamiltonians doing this calculation would normally be complicated and highly dependent on the details of the method even if differences are used to approximate derivatives a relatively simple procedure is presented here nearly independent of the internal structure of the integrator for obtaining highly accurate estimates for modified hamiltonians as a bonus of the method of construction the modified hamiltonians are exactly conserved by a numerical solution in the case of a quadratic hamiltonian

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations in this chapter we derive and analyse an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations which can exactly preserve the energy of the underlying hamiltonian wave equations to this end we first define and discuss the bounded operator argument functions on the underlying domain we then introduce an operator variation of constants formula based on which we present an energy preserving scheme for nonlinear hamiltonian wave equations the scheme preserves the energy of the original continuous hamiltonian system exactly in comparison with the existing work on this topic such as the well known average vector field avf formula for hamiltonian ordinary differential equations the energy preserving scheme avoids the semi discretisation of spatial derivatives and exactly preserves the hamiltonian of the original continuous hamiltonian wave equation this point is very significant in comparison with the avf formula since the avf formula can preserve only the energy of hamiltonian ordinary differential equations hence the main theme of this chapter is to establish a scheme which can exactly preserve the energy of the nonlinear hamiltonian wave equation the chapter is also accompanied by some examples

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

on error bounds for the gautschi type exponential integrator applied to oscillatory second order differential equations this paper studies a numerical method for second order oscillatory differential equations in which high frequency oscillations are generated by a linear time and or solution dependent part for constant linear part it is known that the method allows second order error bounds independent of the product of the step size with the frequencies and is therefore a long time step method most real world problems are not of that kind and it is important to study more general equations the analysis in this paper shows that one obtains second order error bounds even in the case of a time and or solution dependent linear part if the matrix is evaluated at averaged positions

geometric integrators for multiple time scale simulation in this paper we review and extend recent research on averaging integrators for multiple time scale simulation such as are needed for physical n body problems including molecular dynamics materials modelling and celestial mechanics a number of methods have been proposed for direct numerical integration of multiscale problems with special structure such as the mollified impulse method garcia archilla sanz serna and skeel 1999 siam j sci comput 20 930 63 and the reversible averaging method leimkuhler and reich 2001 j comput phys 171 95 114 features of problems of interest such as thermostatted coarse grained molecular dynamics require extension of the standard framework at the same time in some applications the computation of averages plays a crucial role but the available methods have deficiencies in this regard we demonstrate that a new approach based on the introduction of shadow variables which mirror physical variables has promised for broadening the usefulness of multiscale methods and enhancing accuracy of or simplifying computation of averages the shadow variables must be computed from an auxiliary equation while a geometric integrator in the extended space is possible in practice we observe enhanced long term energy behaviour only through use of a variant of the method which controls drift of the shadow variables using dissipation and sacrifices the formal geometric properties such as time reversibility and volume preservation in the enlarged phase space stabilizing the corresponding properties in the physical variables the method is applied to a gravitational three body problem as well as a partially thermostatted model problem for a dilute gas of diatomic molecules

some types of filters in mtl algebras the aim of this work is to introduce the concepts of imtl filters and strong mtl filters in mtl algebras and show that these filters are related to imtl algebras and strong mtl algebras respectively we also introduce the concepts of eimtl and associative filters in mtl algebras the relationships between these filters and quotient algebras that are constructed via these filters are described we clarify that eimtl imtl and fantastic filters coincide in bl algebras whereas they have different properties in mtl algebras moreover the relations between them are delineated we prove that strong filters include some current filters such as implicative positive implicative and fantastic filters

a liouville operator derived measure preserving integrator for molecular dynamics simulations in the isothermal isobaric ensemble the constant pressure constant temperature npt molecular dynamics approach is re examined from the viewpoint of deriving a new measure preserving reversible geometric integrator for the equations of motion the underlying concepts of non hamiltonian phase space analysis measure preserving integrators and the symplectic property for hamiltonian systems are briefly reviewed in addition current measure preserving schemes for the constant volume constant temperature ensemble are also reviewed a new geometric integrator for the npt method is presented is shown to preserve the correct phase space volume element and is demonstrated to perform well in realistic examples finally a multiple time step version of the integrator is presented for treating systems with motion on several time scales

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

error analysis of exponential integrators for oscillatory second order differential equations in this paper we analyse a family of exponential integrators for second order differential equations in which high frequency oscillations in the solution are generated by a linear part conditions are given which guarantee that the integrators allow second order error bounds independent of the product of the step size with the frequencies our convergence analysis generalizes

infinitely many bound states for choquard equations with local nonlinearities abstract in this paper we consider the following choquard equation ch u u i u p u p 2 u v x u q 2 u in r n u h 1 r n where n 3 n 4 n p 2 n n 2 q 2 2 n n 2 1 p 1 n 2 n 2 1 2 n n n 2 p 1 n 2 and i is the riesz potential under some suitable decay assumptions but without any symmetry property on v x we prove that the problem has infinitely many solutions whose energy can be arbitrarily large

a new framework for solving partial differential equations using semi analytical explicit rk n type integrators this paper develops a new explicit semi analytical approach to solving pdes based on the variation of constants formula or the equivalent integration equation these new schemes avoid the discretization of spatial derivatives therefore the accuracy of the semi analytical explicit scheme depends entirely on time integration we first establish a semi analytical formula for wave partial differential equations and show the consistency of the semi analytical formula with different boundary conditions we then present semi analytical explicit rkn type integrators for solving wave pdes this methodology is also extended to dealing with first order hyperbolic pdes and parabolic pdes and for both of them the corresponding semi analytical explicit rk type integrators are derived as well numerical simulations are implemented and the results show the effectiveness of the new semi analytical explicit schemes meanwhile an operator variation of constants formula with applications is introduced for high dimensional nonlinear wave equations before presenting the idea of semi analytical explicit rkn integrators

a multiscale method for highly oscillatory ordinary differential equations with resonance a multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations odes is presented the oscillations may be in resonance with on

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

time parallel multiscale multiphysics framework we introduce the time parallel compound wavelet matrix method tpcwm for modeling the temporal evolution of multiscale and multiphysics systems the method couples time parallel tp and cwm methods operating at different spatial and temporal scales we demonstrate the efficiency of our approach on two examples a chemical reaction kinetic system and a non linear predator prey system our results indicate that the tpcwm technique is capable of accelerating time to solution by 2 3 orders of magnitude and is amenable to efficient parallel implementation

multiple time stepping generalized hybrid monte carlo methods performance of the generalized shadow hybrid monte carlo gshmc method 1 which proved to be superior in sampling efficiency over its predecessors 2 4 molecular dynamics and hybrid monte carlo can be further improved by combining it with multi time stepping mts and mollification of slow forces we demonstrate that the comparatively simple modifications of the method not only lead to better performance of gshmc itself but also allow for beating the best performed methods which use the similar force splitting schemes in addition we show that the same ideas can be successfully applied to the conventional generalized hybrid monte carlo method ghmc the resulting methods mts ghmc and mts gshmc provide accurate reproduction of thermodynamic and dynamical properties exact temperature control during simulation and computational robustness and efficiency mts ghmc uses a generalized momentum update to achieve weak stochastic stabilization to the molecular dynamics md integrator mts gshmc adds the use of a shadow modified hamiltonian to filter the md trajectories in the hmc scheme we introduce a new shadow hamiltonian formulation adapted to force splitting methods the use of such hamiltonians improves the acceptance rate of trajectories and has a strong impact on the sampling efficiency of the method both methods were implemented in the open source md package protomol and were tested on a water and a protein systems results were compared to those obtained using a langevin molly lm method 5 on the same systems the test results demonstrate the superiority of the new methods over lm in terms of stability accuracy and sampling efficiency this suggests that putting the mts approach in the framework of hybrid monte carlo and using the natural stochasticity offered by the generalized hybrid monte carlo lead to improving stability of mts and allow for achieving larger step sizes in the simulation of complex systems

characterizations and new subclasses of i filters in residuated lattices abstract filters play an important role in studying logical systems and the related algebraic structures various filters have been proposed in the literature in this paper we aim to develop a unifying definition for some specific filters called i filters which provide us with a meaningful method to study these filters and corresponding logical algebras in particular trivial characterizations of i filters non trivial characterizations of classes of i filters such as implicative fantastic and boolean filters and characterizations of homologous logical algebras are obtained next three new types of i filters named divisible filters strong and n contractive filters in residuated lattices are introduced particularly it is verified that n fold implicative bl algebras and n contractive bl algebras coincide finally we investigate the relationships between these specific i filters it is shown that a filter is a fantastic filter if and only if it is both a divisible filter and a regular filter

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

radicals in mtl algebras we study some special sets in mtl algebras the notion of the radical of a filter in mtl algebras is defined and several characterizations of the radical of a filter are given the sets of dense elements and double complemented elements in mtl algebras are defined and we state and prove some theorems on the properties of these sets and filters of mtl algebras we also prove that the radical of a positive implicative filter of an mtl algebra is a subset of double complemented elements

instabilities and inaccuracies in the integration of highly oscillatory problems by means of a two frequency test problem we analyze the instabilities and inaccuracies that may impair the performance of multiple time steps split operator integrators in highly oscillatory situations such as those encountered in molecular dynamics astrophysics or partial differential equations describing waves considered are the impulse verlet i r respa method the mollified impulse method and the reversible averaging integrator the analysis covers errors in positions momenta and energy

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

gshmc an efficient method for molecular simulation the hybrid monte carlo hmc method is a popular and rigorous method for sampling from a canonical ensemble the hmc method is based on classical molecular dynamics simulations combined with a metropolis acceptance criterion and a momentum resampling step while the hmc method completely resamples the momentum after each monte carlo step the generalized hybrid monte carlo ghmc method can be implemented with a partial momentum refreshment step this property seems desirable for keeping some of the dynamic information throughout the sampling process similar to stochastic langevin and brownian dynamics simulations it is however ultimate to the success of the ghmc method that the rejection rate in the molecular dynamics part is kept at a minimum otherwise an undesirable zitterbewegung in the monte carlo samples is observed in this paper we describe a method to achieve very low rejection rates by using a modified energy which is preserved to high order along molecular dynamics trajectories the modified energy is based on backward error results for symplectic time stepping methods the proposed generalized shadow hybrid monte carlo gshmc method is applicable to nvt as well as npt ensemble simulations

heterogeneous multiscale methods for stiff ordinary differential equations the heterogeneous multiscale methods hmm is a general framework for the numerical approximation of multiscale problems it is here developed for ordinary differential equations containing differe

existence and concentration of ground state solutions for choquard equations involving critical growth and steep potential well abstract in the present paper we are interested in the following choquard type equation u v x u i u 2 u 2 2 u u p 2 u in r 3 where p 4 6 r r is a constant such that the operator l v x is non degenerate i is a riesz potential of order 0 3 2 3 is the upper critical exponent due to the hardy littlewood sobolev inequality the function v c r 3 r is nonnegative and has a potential well int v 1 0 additionally the operator has a sequence of dirichlet eigenvalues in h 0 1 expressed as 0 1 2 n n if 1 via the nehari manifold techniques and the ekeland variational principle we prove the existence and concentration of positive ground state solutions for sufficiently large if 1 and j for all j n employing the nehari pankov manifold methods and the constrained minimization arguments we obtain the existence of ground state solution for large enough and verify the asymptotic behaviour of ground state solutions as

improved filon type asymptotic methods for highly oscillatory differential equations with multiple time scales abstract in this paper we consider multi frequency highly oscillatory second order differential equations x t m x t f t x t x t where high frequency oscillations are generated by the linear part m x t and m is positive semi definite not necessarily nonsingular it is known that filon type methods are effective approach to numerically solving highly oscillatory problems unfortunately however existing filon type asymptotic methods fail to apply to the highly oscillatory second order differential equations when m is singular we study and propose an efficient improvement on the existing filon type asymptotic methods so that the improved filon type asymptotic methods can be able to numerically solving this class of multi frequency highly oscillatory systems with a singular matrix m the improved filon type asymptotic methods are designed by combining filon type methods with the asymptotic methods based on the variation of constants formula we also present one efficient and practical improved filon type asymptotic method which can be performed at lower cost accompanying numerical results show the remarkable efficiency

explicit symplectic integrators using hessian vector products in 1991 rowlands proposed an effectively fourth order effectively two stage explicit symplectic integrator based on using a hessian vector product to modify the force evaluation in the leapfrog method and evidence indicates that for modest accuracy this method is highly competitive here we explore the possible existence of even more efficient fourth order explicit symplectic integrators also based on the use of hessian vector products and the concept of effective order first it is shown that the cost of a force evaluation plus a hessian vector product is less than twice the cost of the force alone for a sum of two body interactions then a new method is found that is generally better than both the method of rowlands and that of calvo according to both a theoretical measure of the error and limited numerical experiments the basic motivation behind the new method is quite simple do a hessian vector computation only every other step significantly cutting costs while only marginally increasing the error the idea of effective order means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibility of postprocessing the values obtained by the basic method but at output points only for some applications processing is unnecessary but in any case processing has been shown to be possible at low additional cost the derivation of the new method illustrates how to simplify by the use of h series the determination of parameters for methods of increased effective accuracy