a sparse collocation method for solving time dependent hjb equations using multivariate b splines this paper presents a sparse collocation method for solving the time dependent hamilton jacobi bellman hjb equation associated with the continuous time optimal control problem on a fixed finite time horizon with integral cost functional through casting the problem in a recursive framework using the value iteration procedure the value functions of every iteration step is approximated with a time varying multivariate simplex b spline on a certain state domain of interest in the collocation scheme the time dependent coefficients of the spline function are further approximated with ordinary univariate b splines to yield a discretization for the value function fully in terms of piece wise polynomials the b spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems the proposed algorithm is demonstrated on a pair of benchmark example problems simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation

a radial basis collocation method for hamilton jacobi bellman equations in this paper we propose a semi meshless discretization method for the approximation of viscosity solutions to a first order hamilton jacobi bellman hjb equation governing a class of nonlinear optimal feedback control problems in this method the spatial discretization is based on a collocation scheme using the global radial basis functions rbfs and the time variable is discretized by a standard two level time stepping scheme with a splitting parameter q a stability analysis is performed showing that even for the explicit scheme that q 0 the method is stable in time since the time discretization is consistent the method is also convergent in time numerical results performed to verify the usefulness of the method demonstrate that the method gives accurate approximations to both of the control and state variables

multiple correction and continued fraction approximation abstract the main aim of this paper is to further develop a multiple correction method formulated in a previous work 6 as applications we find a kind of hybrid type finite continued fraction approximations in two cases of landau constants and lebesgue constants in addition we refine the previous results of lu 29 and xu and you 48 for the euler mascheroni constant

improved convergence towards generalized euler mascheroni constant we propose new simple sequences approximating the euler mascheroni constant and its generalization which converge faster towards their limits than those considered by detemple d w detemple a quicker convergence to euler s constant am math monthly 100 5 1998 468 470 sintamarian a sintamarian a generalization of euler s constant numer algorithms 46 2 2007 141 151 and vernescu a vernescu a new accelerate convergence to the constant of euler gazeta matem ser a bucharest xvii xcvi 4 1999 273 278

derivatives pricing with market impact and limit order book abstract this paper investigates the derivatives pricing under the existence of liquidity costs and market impact for the underlying asset in continuous time first we formulate the charge for the liquidity costs and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark to market value of the portfolio less the quadratic variation multiplied by a risk aversion parameter during the hedging period and the liquidation cost at maturity then we obtain the derivatives price by reduction of this charge from the premium in the bachelier model second we consider a second order semilinear partial differential equation pde of parabolic type associated with the control problem which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear pde finally we present the numerical examples of the pricing for a variance option and a european call option and show comparative static analyses

on inequalities and asymptotic expansions for the landau constants we demonstrate how arbitrarily sharp asymptotic expansions for the landau constants can be derived and we prove some related inequalities the main tool used is brouncker s continued fraction formula we also show that the implied series expansion of the landau constants is in fact divergent

likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates summary we consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical with no missing responses or covariates one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters unfortunately incomplete data in the responses and covariates are a common occurrence in longitudinal studies here we assume the missing data are missing at random rubin 1976 biometrika63 581 592 since all of the missing data responses and covariates are categorical a useful technique for obtaining maximum likelihood parameter estimates is the em algorithm by the method of weights proposed in ibrahim 1990 journal of the american statistical association85 765 769 in using the em algorithm with missing responses and covariates one specifies the joint distribution of the responses and covariates here we consider the parameters of the covariate distribution as a nuisance in data sets where the percentage of missing data is high the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest we propose a conditional model for the covariate distribution that has several modeling advantages for the em algorithm and provides a reduction in the number of nuisance parameters thus providing more stable estimates in finite samples

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

sharp bounds of the landau constants the aim of this paper is to establish new bounds of the landau constants 1 introduction and motivation the landau constants de ned for all positive integers n by gn 1 1 2 2 1 3 2 4 2 1 3 5 2 4 6 2 1 3 2n 1 2 4 2n 2 play an important role in some extremal problems in complex analysis and in the theory of fourier series more precisely in 1913 landau 6 proved that gn is the maximum of the expression j pn k 1 akj with respect to all functions of the form f z p1 k 0 akz k which is analytic in the unit disk and satis es jf z j 1 for every jzj 1 in consequence the problem of approximation of the landau constants have attracted the attention of many authors in particular landau himself studied the asymptotic behaviour of gn and showed that gn 1 lnn then watson 7 established the following asymptotic formula gn c0 1 ln n 1 1 4 n 1 o 1 n2 n 1 here and in what follows c0 1 4 ln 2 1 06627 and 0 577215 is the euler mascheroni constant this problem of approximation of the landau constants was continued in the works of brutman 3 and falaleev 5 who proved that for every non negative integer n 1 1 ln n 1 gn 1 0663 1 ln n 1 respective 1 0662 1 ln n 3 4 gn 1 0916 1 ln n 3 4 we improve the upper bound in the following way that also shows that the constant 3 4 is the best possible date november 18 2009 1991 mathematics subject classi cation primary 41a60 secondary 26d15

a multivariate adaptive regression b spline algorithm bmars for solving a class of nonlinear optimal feedback control problems in this paper we present a novel method for solving a class of nonlinear optimal feedback control problems with moderately high dimensional state spaces based on an adapted version of the bmars algorithm numerical experiments were performed using problems with up to six state variables the numerical results clearly demonstrate the efficiency and potential of the method for solving high dimensional problems

convergence of meshfree collocation methods for fully nonlinear parabolic equations we prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step a numerical experiment with a radial basis function demonstrates the convergence property

compactly supported radial basis functions based collocation method for level set evolution in image segmentation the partial differential equation driving level set evolution in segmentation is usually solved using finite differences schemes in this paper we propose an alternative scheme based on radial basis functions rbfs collocation this approach provides a continuous representation of both the implicit function and its zero level set we show that compactly supported rbfs csrbfs are particularly well suited to collocation in the framework of segmentation in addition csrbfs allow us to reduce the computation cost using a kd tree based strategy for neighborhood representation moreover we show that the usual reinitialization step of the level set may be avoided by simply constraining the l1 norm of the csrbf parameters as a consequence the final solution is topologically more flexible and may develop new contours i e new zero level components which are difficult to obtain using reinitialization the behavior of this approach is evaluated from numerical simulations and from medical data of various kinds such as 3 d ct bone images and echocardiographic ultrasound images

inequalities for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 by gn k 0n1 16k 2 and ln 12 sin n t sin t dt respectively we establish sharp inequalities for gn and ln 2 in terms of the logarithmic derivative of the gamma function further we prove that the sequence gn is completely monotonic we provide best possible upper and lower bounds for the ratios gn 1 gn 1 gn and l n 1 2 l n 1 2 ln 2 and we present sharp bounds for ln 2 gn and ln 2 gn

asymptotic expansions for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 in order by gn k 0n116k 2kk 2andln 12 sin n 12 t sin 12t dt which play important roles in the theories of complex analysis and fourier series respectively certain inequalities and asymptotic expansions for the constants gn and ln have been investigated by many authors here we aim at establishing new asymptotic expansions for the constants gn and ln of landau and lebesgue respectively by mainly using bell polynomials and the partition function

new sequence converging towards the euler mascheroni constant we propose new sequences containing a modified logarithmic term which converge to the euler mascheroni constant faster than sequences known from the literature

optimal rate of convergence for sequences of a prescribed form abstract we provide a highly convergent version of the generalized euler sequence and we determine the element with the optimal rate of convergence in several classes of sequences of a prescribed form the new approach extends and unifies many previous efforts in this direction in addition we answer an open problem on the asymptotic expansion of the harmonic number sequence

inequalities and asymptotic expansions for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 in order by g n k 0 n 1 16 k 2 k k 2 and l n 1 2 sin n 1 2 t sin 1 2 t d t which play important roles in the theories of complex analysis and fourier series respectively diverse inequalities and approximations for these constants have been investigated and developed by many authors here in this paper we establish new asymptotic expansions for the constants g n and l n 2 of landau and lebesgue respectively in terms of the digamma and polygamma functions based on our expansion for the landau constants g n we present new bounds for the landau constants g n in terms of the digamma and polygamma functions we also establish inequalities for the lebesgue constants l n 2 which are applied to derive an asymptotic expansion for l n 2 in terms of 1 n 1 furthermore by giving numerical calculations to be compared among several developed asymptotic expansions for the constants g n and l n 2 it is shown that our expansions presented here would be best ones

proofs of two conjectures on the landau constants abstract in this paper we derive a family of asymptotic expansions for the nth landau constant g n the proof is based on a new general integral representation of the constants in terms of a hypergeometric function as a consequence we prove two conjectures related to earlier asymptotic developments of the g n s

conditional empirical processes defined by φ mixing sequences for a random vector x y in r 2 let m y x p y y x x be the conditional distribution function it is assumed that x 1 y 1 x 2 y 2 form a strictly stationary mixing sequence of random vectors with the same distribution as x y considering an estimate mn x of m x it is shown that nan 1 2 mn x 0 m x 0 converges to a certain gaussian process depending on x 0 where an is a sequence of bandwiths

on application of an alternating direction method to hamilton jacobin bellman equations this paper presents a numerical method for the approximation of viscosity solutions to a hamilton jacobi bellman hjb equation governing a class of optimal feedback control problems the first order hjb equation is first perturbed by adding a diffusion term with a singular perturbation parameter the time and spatial variables in the resulting equation are then discretized respectively by an implicit modified method of characteristics and the alternating direction ad scheme we show that the ad procedure s perturbation error is virtually negligible due to the small perturbation parameter and the efficient ad scheme can be applied to our hjb equation without generating splitting error numerical results performed to verify the usefulness of the method show that the method gives accurate approximate solutions to both of the control and the state variables

on an open problem of chen and mortici concerning the euler mascheroni constant abstract chen and mortici ch p chen c mortici new sequence converging towards the euler mascheroni constant comput math appl 2011 http dx doi org 10 1016 j camwa 2011 03 099 proposed an open problem for a given positive integer s find the constants a i i 0 1 2 s such that k 1 n 1 k log n i 0 s a i n i is the fastest sequence which would converge to the euler mascheroni constant using logarithmic type bell polynomials we solve this problem the main result shows that the a i s can be recursively determined

some sharp estimates of the constants of landau and lebesgue we establish several sharp two sided inequalities involving the constants of landau and lebesgue which occur and play important roles in two related extremal problems in complex analysis and in fourier analysis respectively the main results improves essentially the watson asymptotic expansion formulas for these constants

a note on the landau constants abstract in this paper we establish new asymptotic expansions for the n th landau constant g n an open problem is indicated as well

some quicker classes of sequences convergent to euler s constant in this paper some quicker classes of sequences convergent to euler s constant are provided finally for demonstrating the superiority of our new convergent sequences over detemple s sequence vernescu s sequence and mortici s sequences some numerical computations are also given