a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

a note on lattice simplices and toric varieties in this note we answer the question on terminal singularities of toric 4 folds raised by k matsuki in 1 p 857 we thank david cox for suggesting this problem to us a nonsingular integer d x d matrix a with column vectors ai a2 ad is called primitive if the d simplex conv 0 a a a contains no lattice points except its vertices given any rational matrix or vector x we write den x for the smallest positive integer v such that v x is an integer matrix we abbreviate ed 1 1 1 e zd

configuration spaces transfer and 2 nodal solutions of a semiclassical nonlinear schrodinger equation we establish new lower bounds for the number of nodal bound states for the semiclassical nonlinear schrodinger equation varepsilon 2 delta u a x u u p 2 u with bounded and uniformly continuous potential a the solutions we obtain have precisely two nodal domains and their positive and negative parts concentrate near the set of minimum points of a our approach is independent of penalization techniques and yields in some cases the existence of infinitely many nodal solutions for fixed varepsilon via a dynamical systems approach we exhibit positively invariant sets of sign changing functions for the negative gradient flow of the associated energy functional we analyze these sets on the cohomology level with the help of dold s fixed point transfer in particular we estimate their cuplength in terms of the cuplength of equivariant configuration spaces of subsets of mathbb r n we also provide new estimates of the cuplength of configuration spaces

filtering for partially observed diffusion and its applications abstract this paper provides an analytic method of filtering for partially observed diffusions which can be also used for parameter estimation with the quasi maximum likelihood method the filtering is shown to have consistency in a weak sense in addition using the stochastic volatility models a comparative simulation study is carried out to see how well the proposed method numerically works the performance of the proposed method is basically better than that of the extended kalman filtering

nodal solutions for singularly perturbed equations with critical exponential growth abstract the existence and concentration behavior of nodal solutions are established for the equation 2 u v z u f u in where is a domain in r 2 not necessarily bounded v is a positive holder continuous function and f c 1 is an odd function having critical exponential growth

the fully implicit stochastic α method for stiff stochastic differential equations a fully implicit integration method for stochastic differential equations with significant multiplicative noise and stiffness in both the drift and diffusion coefficients has been constructed analyzed and illustrated with numerical examples in this work the method has strong order 1 0 consistency and has user selectable parameters that allow the user to expand the stability region of the method to cover almost the entire drift diffusion stability plane the large stability region enables the method to take computationally efficient time steps a system of chemical langevin equations simulated with the method illustrates its computational efficiency

pade approximation for the exponential of a block triangular matrix abstract in this work we obtain improved error bounds for pade approximations to e a when a is block triangular as a result improved scaling strategies ensue which avoid some common overscaling difficulties

termination of flops for 4 folds the purpose of this note is to prove the termination of flops for 4 folds i e the second part of the so called flop conjecture in dimension 4 which asserts that there is no infinite sequence of flops with respect to a fixed reference divisor for projective 4 folds having only terminal singularities our idea for the proof stems from crossbreeding the ideas of kollar in ko3 where he proves the termination of flops for 3 folds with those of kawamata in kmm where he shows the termination of flips for 4 folds we should mention that the notion of difficulty due to shokurov si plays the most basic role in the ar

existence of a least energy nodal solution for a schrodinger kirchhoff equation with potential vanishing at infinity in this paper we study a class of nonlocal schrodinger kirchhoff problems involving only continuous functions using a minimization argument and a quantitative deformation lemma we find a least energy nodal or sign changing solution to this problem moreover when the problem presents symmetry we show that it has infinitely many nontrivial solutions

generalised filtering and stochastic dcm for fmri this paper is about the fitting or inversion of dynamic causal models dcms of fmri time series it tries to establish the validity of stochastic dcms that accommodate random fluctuations in hidden neuronal and physiological states we compare and contrast deterministic and stochastic dcms which do and do not ignore random fluctuations or noise on hidden states we then compare stochastic dcms which do and do not ignore conditional dependence between hidden states and model parameters generalised filtering and dynamic expectation maximisation respectively we first characterise state noise by comparing the log evidence of models with different a priori assumptions about its amplitude form and smoothness face validity of the inversion scheme is then established using data simulated with and without state noise to ensure that dcm can identify the parameters and model that generated the data finally we address construct validity using real data from an fmri study of internet addiction our analyses suggest the following i the inversion of stochastic causal models is feasible given typical fmri data ii state noise has nontrivial amplitude and smoothness iii stochastic dcm has face validity in the sense that bayesian model comparison can distinguish between data that have been generated with high and low levels of physiological noise and model inversion provides veridical estimates of effective connectivity iv relaxing conditional independence assumptions can have greater construct validity in terms of revealing group differences not disclosed by variational schemes finally we note that the ability to model endogenous or random fluctuations on hidden neuronal and physiological states provides a new and possibly more plausible perspective on how regionally specific signals in fmri are generated

positive and sign changing clusters around saddle points of the potential for nonlinear elliptic problems we study the existence and asymptotic behavior of positive and sign changing multipeak solutions for the equation varepsilon 2 delta v v x v f v quad rm in mathbb r n where e is a small positive parameter f a superlinear subcritical and odd nonlinearity v a uniformly positive potential no symmetry on v is assumed it is known kang and wei in adv differ equ 5 899 928 2000 that this equation has positive multipeak solutions with all peaks approaching a local maximum of v it is also proved that solutions alternating positive and negative spikes exist in the case of a minimum see d aprile and pistoia in ann inst h poincare anal non lineaire 26 1423 1451 2009 the aim of this paper is to show the existence of both positive and sign changing multipeak solutions around a nondegenerate saddle point of v

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

least energy nodal solution for nonlinear schrodinger equation without ar condition abstract in this paper we are concerned with the existence of least energy nodal solution for the following stationary nonlinear schrodinger equation 0 1 u x v x u x f u x r n n 3 where 0 is a parameter v has double potential well and f has asymptotically linear growth or superlinear growth without ar condition by using variational methods we show that 0 1 has a least energy nodal solution

on the cohomology of flows of stochastic and random differential equations we consider the flow of a stochastic differential equation on d dimensional euclidean space we show that if the lie algebra generated by its diffusion vector fields is finite dimensional and solvable then the flow is conjugate to the flow of a non autonomous random differential equation i e one can be transformed into the other via a random diffeomorphism of d dimensional euclidean space viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory can be an advantage when dealing with asymptotic properties of the system to illustrate this we give sufficient criteria for the existence of global random attractors in terms of the random differential equation which are applied in the case of the duffing van der pol oscillator with two independent sources of noise

localized nodal solutions for quasilinear schrodinger equations abstract in this paper we study the existence of localized nodal solutions for a class of semiclassical quasilinear schrodinger equations including as a special case the modified nonlinear schrodinger equation mnls e 2 v 1 2 v v 2 v x v v q 2 v 0 in r n v x 0 as x we establish for small e the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function by developing new variational perturbation method to treat this class of non smooth variational problems the new method allows the perturbed variational functionals to share critical points with the original functional this method allows us to avoid any limiting process from the perturbed problems to the original problem and it is effective in dealing with multiple existence of solutions

numerical simulation of nonlinear dynamical systems driven by commutative noise the local linearization ll approach has become an effective technique for the numerical integration of ordinary random and stochastic differential equations one of the reasons for this success is that the ll method achieves a convenient trade off between numerical stability and computational cost besides the ll method reproduces well the dynamics of nonlinear equations for which other classical methods fail however in the stochastic case most of the reported works has been focused in stochastic differential equations sde driven by additive noise this limits the applicability of the ll method since there is a number of interesting dynamics observed in equations with multiplicative noise on the other hand recent results show that commutative noise sdes can be transformed into a random differential equation rde by means of a random diffeomorfism conjugacy this paper takes advantages of such conjugacy property and the ll approach for defining a ll scheme for sdes driven by commutative noise the performance of the proposed method is illustrated by means of numerical simulations

on the location and profile of spike layer nodal solutions to nonlinear schrodinger equations abstract the existence and concentration behavior of a nodal solution are established for the equation e 2 u v x u f u in where is a domain in r n not necessarily bounded v is a positive holder continuous function and f c 1 r is an odd function having subcritical and superlinear growth under additional local condition on v it is proved that such a solution has just one positive and one negative peaks which are located around local minima of v