Solvable cases of the Bogoliubov equations of superconductivity

It is shown that many solvable special cases of the zero-field, one-dimensional Bogoliubov equations in the WKBJ approximation discovered so far are members of a four-parameter family of solvable pair potentials defined by the differential equationd(z)/dz=a+b(z)+c ² (z), witha, b, andc arbitrary real constants. Furthermore, it is pointed out that any one of these solvable real pair potentials remains solvable after a purely imaginary constant is added to it.Work supported by the National Science Foundation under grant number GH-34509.     

On compatibility conditions and stress boundary value problems in linear nonlocal elasticity

Compatibility conditions for stress boundary value problems in linear nonlocal elasticity theory are obtained. Implications of these conditions as to solvability of problems are considered. The results indicate that most problems are significantly simpler if phrased as displacement boundary value problems, even in those cases where the corresponding local problem is easily solvable as a stress boundary value problem.     

Using elastic—plastic plane stress FEM programs to compute anti-plane strain

Elastic—plastic anti-plane strain problems can be solved by using FEM programs developed for plane stress. The technique does not involve any manipulations with the FEM code, but is merely a matter of specialization and of translation of notation. Two analytically solvable cases are chosen for demonstration.     

A fixed job scheduling problem with machine-dependent job weights

This study considers the identical parallel machines operational fixed job scheduling problem with machine-dependent job weights. A job is either processed in a fixed interval or is not processed at all. Our aim is to maximise the total weight of the processed jobs. We show that the problem with machine eligibility constraints resides as a special case of this problem. We identify some special polynomially solvable cases and propose a branch-and-bound (BB) algorithm that employs efficient bounding schemes and dominance conditions. Computational experience on large-sized problem examples reveals the satisfactory performance of the BB algorithm.     

A chromatic scheduling model with costs

An extension of a preemptive open-shop scheduling problem is introduced. All processing times are integral and in each period i there is a cost c_i, for each task which is processed in that period. Finding a schedule with minimum total cost is shown to be NP-hard; some solvable cases are discussed; bounds on the cost of an optimum schedule are computed. Finally, a special case is studied, namely where each job has at most three tasks and each processor has to work on at most three tasks. It is shown to have theoretical complexity equivalent to the general case.     

Spread time considerations in operational fixed job scheduling

In this study, we consider the operational fixed job scheduling problem on identical parallel machines. We assume that the jobs have fixed ready times and deadlines, and spread time constraints are imposed on machines. Our objective is to select a set of jobs for processing so as to maximise the total weight. We show that the problem is strongly NP-hard, and we investigate several special polynomially solvable cases. We propose a branch and bound algorithm that employs size reduction mechanisms, dominance conditions, and powerful lower and upper bounds. The computational results reveal that the branch and bound algorithm returns optimal solutions for problem instances with up to 100 jobs in reasonable solution times.     

有分离调整和移走时间的两机器流水作业总完工时间问题

对有分离调整和移走时间的两机器流水作业问题,同一工件的调整和移走在不同机器上是可以重叠的,但加工时间不能重叠。本文以总完工时间为准则研究调整时间和移走时间均独立于加工时间的两机器流水作业问题,给出了问题最优解中工件排列应满足的条件;同时也讨论了该类问题多项式可解的情形。     

Designing an Assembly Process with Stochastic Material Arrivals

Due to product proliferation and quick changes in manufacturing materials sourcing partnerships, assembly process in the electronic assembly industry is frequently altered or new processes are introduced. The most appropriate assembly process is critical for companies to maintain an effective supply chain. We study the problem of how to design an assembly process that relies on the stochastic arrival times of parts and components such that the probability of on-time delivery of finished products is maximized. We first provide some analytical results for cases that are polynomially solvable and then provide a branch and bound algorithm for deriving an optimal solution to the problem. We also provide heuristic algorithms to solve the problem. Finally, computational experiments show that our heuristic algorithms perform very well.     

Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem

An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.     

Numerical solution of stefan problems by the method of green functions

A method for reducing a multidimensional Stefan problem to a system of Hammerstein integral equations is proposed. Application of the proposed method to numerical solution of one-dimensional nonstationary Stefan problems formulated for the cases of an internal phase front, coincidence of the phase front with the external boundary, and a movable external boundary is considered. The efficiency of the method is tested on an exactly solvable Stefan problem. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 3, pp. 564–570, May–June, 1998.     

The no-wait two-machine flow shop scheduling problem with convex resource-dependent processing times

We extend the classical no-wait two-machine flow shop scheduling problem to the case where job-processing times are controllable through the allocation of a common, limited and nonrenewable resource. Our objective is to simultaneously determine the sequence of the jobs and the resource allocation for each job on both machines in order to minimize the makespan. By using the equivalent load method to obtain the optimal resource allocation on a series-parallel graph, we reduce the problem to a sequencing one and show that it is equivalent to a new special case of the Traveling Salesman Problem (TSP). We prove that although the reduced problem forms a subclass of the TSP on permuted Monge matrices, it is still strongly NP-hard. We provide an approximation result and present three special cases which are polynomially solvable. We have also tested two different subtour-patching heuristics in large-scale computational experiments on randomly generated instances of the problem. Both heuristics produced close-to-optimal solutions in most cases.     

Order batching in walk-and-pick order picking systems

This paper investigates the order-batching problem both as a theoretical problem, including its special variants, and as a practical problem, with possible heuristic solution procedures. It shows that certain special cases or variants of the order batching problem are solvable, while other cases of the problem are NP-hard. It also develops a new Mixed-Integer Programming (MIP) model to obtain near-exact solutions to the problem. The MIP provides good lower and upper bounds to the problem, which prove useful in the investigation of the heuristic solution procedures. The paper identifies and modifies slightly the order batching heuristic, which shows the strongest results in the numerical experiments. Although good solutions are obtained with this particular heuristic, the results indicate that it is computationally intensive. Furthermore, solutions obtained through the MIP model suggest there is still room to obtain even better solutions to the problem via heuristic procedures, although the challenge is to do so without further increasing the computational burden.     

Scheme for generating entangled states of two field modes in a cavity

This paper considers a two-level atom interacting with two cavity modes with equal frequencies. Applying a unitary transformation, the system reduces to the analytically solvable Jaynes–Cummings model. For some particular field states, coherent and squeezed states, the transformation between the two bare bases, related by the unitary transformation, becomes particularly simple. It is shown how to generate (the highly non-classical) entangled coherent states of the two modes, both in the zero and large detuning cases. An advantage of the zero detuning case is that the preparation is deterministic and no atomic measurement is needed. For the large detuning situation, a measurement is required, leaving the field in either of two orthogonal entangled coherent states.     

Scheduling the fabrication and assembly of components in a two-machine flowshop

In this paper we study the problem of scheduling the fabrication and assembly of components in a two-machine flowshop so as to minimize the makespan. Each job consists of a component unique to that job and a component common to all jobs. Both the unique and the common components are processed on the first machine. While the unique components are processed individually, the common components are processed in batches and a setup is needed to form each batch. The assembly operations of a job is performed on the second machine, and can only begin when both components for the job are available. We first show that the problem is NP-complete with either batch availability or item availability for the common components. We identify several properties of an optimal solution to the problem, and some polynomially solvable special cases.     

关于工期分配与加权误工数的双指标排序问题

排序问题中工期分配的目的是处理分配费用与性能指标的利益平衡,由此提出工期分配的双目标排序问题。关于工期分配与加权误工数的单机双指标排序问题,文献中只研究了其线性组合形式。针对该问题,本文针对约束形式及Pareto优化形式进一步研究了更多的模型。主要结果包括NP-困难性、多项式可解情形以及多项式时间近似方案等结果。通过这些结果,一个多目标优化问题的特征得以完整地刻画。     

Parallel-machine,multiple-product-type,continuous-time scheduling: decomposable cases

This paper considers a manufacturing system where multiple-product-types are produced on a set of parallel machines. The production quantity for each product-type in a planning horizon is predetermined. However, the planning horizon is not fixed, and a cost must be paid for each unit of time in the horizon. Inventory holding costs are incurred due to storing products in the buffer placed after each machine. In addition, a production cost is incurred if a machine is not idle. Our objective is to schedule the production so that inventory, production, and planning horizon costs are minimized. With the aid of the maximum principle, this continuous-time scheduling problem is studied, and the conditions such that the problem can be decomposed into a set of well-structured, discrete-time sub-problems are derived. Consequently, several solvable cases are identified, and their corresponding polynomial-time algorithms are suggested.     

Exactly and quasi-exactly solvable 'discrete' quantum mechanics

A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional 'discrete' quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.     

Topological dereliction in polymers

An exactly solvable toy model is introduced to illustrate the entropie frustrations and their thermodynamic consequences in self-ordering systems.     

Inverse method for viscous flow design using stream-function coordinates

In the paper, a new inverse method for viscous 2D laminar flows is developed. The method is based on incompressible Navier–Stokes equations transformed to the stream-function coordinate system (von Mises coordinates). The flow design problem with appropriate boundary conditions is formulated and solved numerically. The geometrical shape of the boundary is obtained through the integration along streamlines. The method may be coupled with a flow analysis solver to model the influence of known parts of geometry. Results for two analytically solvable cases (the Poiseuille and the Jeffery–Hamel flows) are presented. Then, the foil design problem is considered as an example. Potential applications and developments towards axisymmetric and 3D flows are discussed.