Alternative formulations of a combined trip generation,trip distribution,modal split,and trip assignment model

The traditional four-step model has been widely used in travel demand forecasting by considering trip generation, trip distribution, modal split and traffic assignment sequentially in a fixed order. However, this sequential approach suffers from the inconsistency among the level-of-service and flow values in each step of the procedure. In the last two decades, this problem has been addressed by many researchers who have sought to develop combined (or integrated) models that can consider travelers’ choice on different stages simultaneously and give consistent results. In this paper, alternative formulations, including mathematical programming (MP) formulation and variational inequality (VI) formulations, are provided for a combined travel demand model that integrates trip generation, trip distribution, modal split, and traffic assignment using the random utility theory framework. Thus, the proposed alternative formulations not only allow a systematic and consistent treatment of travel choice over different dimensions but also have behavioral richness. Qualitative properties of the formulations are also given to ensure the existence and uniqueness of the solution. Particularly, the model is analyzed for a special but useful case where the probabilistic travel choices are assumed to be a hierarchical logit model. Furthermore, a self-adaptive Goldstein–Levitin–Polyak (GLP) projection algorithm is adopted for solving this special case.     

On the Caginalp system with dynamic boundary conditions and singular potentials

This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in H ², the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Lojasiewicz inequality in order to prove the convergence of solutions to steady states.      

Superconvergence estimates of finite element methods for American options

In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp L ²-, L ^∞-norm error estimates and an H ¹-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator. This work was supported in part by the National Natural Science Foundation of China (10471103 and 10771158), the National Basic Research Program (2007CB814906), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), Tianjin Natural Science Foundation (07JCY-BJC14300), and Tianjin University of Finance and Economics.     

The guaranteed cost control for uncertain nonlinear large-scale stochastic systems via state and static output feedback

The guaranteed cost control (GCC) problem involved in decentralized robust control of a class of uncertain nonlinear large-scale stochastic systems with high-order interconnections is considered. After determining the appropriate conditions for the stochastic GCC controller, a class of decentralized local state feedback controllers is derived using the linear matrix inequality (LMI). The extension of the result of the study to the static output feedback control problem is discussed by considering the Karush-Kuhn-Tucker (KKT) conditions. The efficiency of the proposed design method is demonstrated on the basis of simulation results.     

On the stochastic p-Laplace equation

The p-Laplace equation with random perturbation is studied for the singular case 1<p2 in this paper. Some properties of the invariant measure and transition semigroups are obtained. The main tool is the dimension-free Harnack inequality, which is established by using the coupling argument. As consequences, some ergodicity, compactness and contractive properties are derived for the associated transition semigroups. The main results are applied to stochastic reaction–diffusion equations and the stochastic p-Laplace equation in Hilbert space.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of Hölder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

On Jensen-McShane’s inequality

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.     

Three solutions for an obstacle problem for a class of variational–hemivariational inequalities

In this paper, we consider a class of obstacle problems for variational–hemivariational inequalities, by using nonsmooth version of three points critical theory in [S.A. Marano, D. Motreanu, On a three critical points theorem for non-differentiable functions and application to nonlinear boundary value problems, Nonlinear Anal. 48 (2002) 37–52], the existence of three solutions for the problem is obtained.     

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].