Estimates of Jacobians by subdeterminants

Let ƒ: Ω → ℝⁿ be a mapping in the Sobolev space W^1,n−1(Ω,ℝⁿ), n ≥ 2. We assume that the determinant of the differential matrix Dƒ (x) is nonnegative, while the cofactor matrix D^#ƒ satisfies , where L^p(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L _loc ¹ (Ω). Estimates above and below L _loc ¹ (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.     

New mixed integer-programming model for the pickup-and-delivery problem with transshipment

In recent years, many important real-world applications are studied as “rich” vehicle routing problems that are variants and generalizations of the well-known vehicle routing problem. In this paper we address the pickup-and-delivery version of this problem and consider further generalization by allowing transshipment in the network. Moreover, we allow heterogenous vehicles and flexible fleet size. We describe mixed integer-programming formulations for the problem with and without time windows for services. The number of constraints and variables in the models are bounded by polynomial size of the problem. We discuss several problem variants that are either captured by our models or can be easily captured through simple modifications. Computational work gave promising results and confirms that transshipment in network can indeed enhance optimization.     

二维复杂目标电磁散射的计算机仿真

采用点线模型法建立二维复杂目标的几何模型 ,运用 Mo M- CGM- FFT相结合的混合技术 ,结合可视化编程技术 ,形成“二维复杂目标的电磁极化散射的建模与仿真”软件。这使得几何模型的建立和仿真的速度有了很大地提高 ,大大降低了内存需求和仿真时间。最后运用该软件对一个复合柱体进行了仿真 ,给出了仿真结果     

The Ribaucour transformation for flat Lagrangian submanifolds

We study the Ribaucour transformation for flat Lagrangian submanifolds in complex flat space and complex projective space. As a consequence, we obtain a process to generate a new family of such submanifolds from a given one. Analytically, this provides a method to construct new solutions of the corresponding systems of PDEs from a given one. We also show that such transformation always comes with a permutability formula.     

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric g_Q and second fundamental form L_Q on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data g_Q and L_Q; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.     

Weakly elliptic systems with obstacle constraints Part II — An N × N model problem

Existence, uniqueness (even stability), and regularity are established for a special system of variational inequalities of obstacle type. The system is only considered to be elliptic in the weakest possible sense. The system includes a version of the biharmonic and polyharmonic obstacle problems. The main aspect of this problem and the approach here is its reduction via algebraic invarients to special canonical forms. One might view this work as the beginnings of a “group analysis” of variational inequalities. This paper is dedicated to the memory of Guido Stampacchia (1922–1978).     

Modeling employees behavior in workplace dynamics

During the past years, Cellular Automata (CAs) have been extensively used for modeling of many complex systems and processes with great success. In this paper, we study a Cellular Automaton (CA) model for the influence of employees’ behavior in a parameterized workplace environment taking into account different behavioral characteristics. In specific, we model employees’ interactions based on their influence radius, the degree of their willingness on adaption of organizational norms and the employee's attitude in general in the under study workplace. The proposed CA model is taking into account employee loyalty, a combined statistic of the employee behavior and her/his insistence and company policies applied to the employees so as to restrain unwanted or impose desirable behavioral patterns in correspondence to the organization norms. Conclusively, the CA model facilitates the presentation and simulation of a workplace with a variety of employee behavioral characteristics and under adaptable company policies. Different workplaces were used to illustrate the simulation of employee behavior with CA model. As a result, the proposed model was practically used on two levels, firstly to estimate the workplace robustness and secondly to illustrate workspace dynamics. Finally, the CA model has been utilized to simulate behavioral patterns at a small enterprise in Greece. In specific, based on the employees answers to detailed surveys the CA model was initialized and then applied to describe the behavioral traits of the under study company employees. Finally, the proposed model, in all the examined cases can be utilized in conjunction with applied employee management techniques to facilitate managerial decisions and forecast the impact of employee behavioral changes and company decisions.     

On the structure of pseudo-holomorphic discs with totally real boundary conditions

In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points, which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic trajectories for Lagrangian intersections in symplectic geometry. This paper is supported in part by NSF Grant DMS 9215011.     

Regularity of ghosts in tensor tomography

We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L²-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.