Electron optics in multi-stage lens system I

Zusammenfassung Die Fokussierungseigenschaften eines aus schwachen Linsen bestehenden Systems unterhalb eines Beschleunigungsrohres sind studiert worden. Das System wird als dicke Linse in der Näherung der paraxialen Bahn behandelt. Die Kardinalelemente des Linsensystems können als Ganzes für den Fall, dass das Feld periodisch ist, in einfacher Form dargestellt werden.Im besondern sind die Ausdrücke für die Kardinalelemente eines Systems, das aus den Zweirohr-Immersionslinsen mit bestimmter Länge der Lücke zwischen den Rohren besteht, wiedergegeben. Die beigefügten Figuren erlauben, diese Elemente bequem zu berechnen.     

A nonconvex dissipative system and its applications (II)

This paper presents exact solutions in terms of implicit functions and hyperbolic functions to a nonconvex dissipative system, controlled by a Duffing–van der Pol nonlinear equation with a fifth-order nonlinearity. Applications to the complex Ginzburg–Landau equation are illustrated and several classes of uniformly translating solutions are obtained accordingly. Part of the work was announced at the International Conference on Complementarity, Duality, and Global Optimization in Science and Engineering, Virginia Tech. University, Blacksburg, Virginia, August 15–17, 2005. This work is supported by NSF Grant CCF–0514768.     

Optimal batch sizing in a multi-stage production system with rework consideration

In a production system, rework process plays an important role in eliminating waste and effectively controlling the cost of manufacturing. Determining the optimal batch size in a system that allows for rework is, therefore, a worthwhile objective to minimize the inventory cost of work-in-processes and the finished goods. In this paper, models for the optimum batch quantity in a multi-stage system with rework process have been developed for two different operational policies. Policy 1 deals with the rework within the same cycle with no shortage and policy 2 deals with the rework done after N cycles, incurring shortages in each cycle. The major components that play a role in minimizing this cost of the system are manufacturing setups, work-in-processes, storage of finished goods, rework processing, waiting-time, and penalty costs to discourage the generation of defectives. The mathematical structure of this rework processing model falls under a nonlinear convex programming problems for which a closed-form solution has been proposed and results are demonstrated through numerical examples, followed by sensitivity analyses of different important parameters. It is concluded that the total cost in policy 2 tends to be smaller than that in policy 1 at lower proportion of defectives if the in-process carrying cost is low. Policy 2 may be preferred when the work-in-process carrying cost is low and the penalty cost is negligible.     

Riemann problem for a geometrical optics system

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics,in which the flux contains the nonconvex function possessing an infinite number of inflection points.Firstly,the generalized Rankine–Hugoniot relations and entropy condition of delta shock waves and left(right)-contact delta shock waves are proposed and clarified.Secondly,with the help of the convex hull,seven kinds of structures of Riemann solutions are obtained.The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities,vacuums and delta shock waves.Finally,numerical experiments confirm the theoretical analysis.     

On optimal batch sizing in a multi-stage production system with rework consideration

Recently, Sarker et al. [Sarker, B.R., Jamal, A.M.M., Mondal, S., 2008. Optimal batch sizing in a multi-stage production system with rework consideration. European Journal of Operational Research 184(3) 915–929] presented an EPQ inventory model for a multi-stage manufacturing system with rework process; basically they proposed two operational inventory policies. In the paper, there are some mathematical expressions which are to be corrected. At first, this paper presents the mathematical expressions corrected and the appropriate solution to the numerical example. We also established the closed forms for the optimal total inventory cost, the conditions for which there is an optimal solution, and the mathematical expressions for determining the total additional cost for working with a non optimal solution for both policies that were not given by Sarker et al. (2008).     

Bilevel parameters identification for the multi-stage nonlinear impulsive system in microorganisms fed-batch cultures

Considering the abrupt jump of the substrate and different characters of the bacterial in the lag phase, the exponential phase and the stationary phase this paper proposes the multi-stage nonlinear impulsive system for the fed-batch fermentation from glycerol to 1,3-propanediol (1,3-PD) and establishes the bilevel identification system for its sensitive parameters. The properties of the solutions for the nonlinear multi-stage dynamical system are investigated and identifiability of the parameters is proved. Finally an optimal algorithm is constructed to obtain the optimal solution of the identification model and the numerical example is then discussed to illustrate the algorithm.     

Polynomial deformations of the Lie algebrasl(2) in problems of quantum optics

It is shown that specific (polynomial) deformations of Lie algebras arise naturally as dynamical symmetry algebrasg ^ds of second-quantized models with nonquadratic HamiltoniansH invariant with respect to certain groupsG ^inv(H). Such deformationssl _d(2) of the Lie algebrasl(2) are found in a number of models of quantum optics (multiphoton processes, generalized Dicke model, and frequency conversion), and ways to apply thesl(2) formalism to the solution of physics problems are indicated.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 3–19, April, 1993.     

Trigonometry (II)

This paper considers geodesic triangies in a Riemannian manifoldM. First we imbed the set of geodesic triangles inM into a big spaceE, then find some equations inE satisfied by tangent vectors of . Finally we give an application of the result.     

Coordination properties of vic-isonitrosoimines in their copper (II) and palladium (II) complexes

Preparation and structural characterization of palladium (II) complexes of ligands III-V and copper (II) complexes of III are reported. The elemental analyses of the complexes show that the metal: ligand ratio is 1:2. The electrical conductance in acetone shows the non-electrolytic nature of the complexes. The diamagnetic character suggests a gross square-planar geometry for the palladium (II) complexes. Copper (II) complexes are paramagnetic with¼eff.～1·90 B.M. Spectral data suggest that in all the complexes the ligand coordinates to the metal (II) symmetrically through isonitroso-nitrogen and imine-nitrogen, forming a five membered chelate ring. Amine-exchange reactions of the complexes are discussed and compared on the basis of their structures.     

Optimizing a multi-stage production/inventory system by DC programming based approaches

This paper deals with optimizing the cost of set up, transportation and inventory of a multi-stage production system in presence of bottleneck. The considered optimization model is a mixed integer nonlinear program. We propose two methods based on DC (Difference of Convex) programming and DCA (DC Algorithm)—an innovative approach in nonconvex programming framework. The mixed integer nonlinear problem is first reformulated as a DC program and then DCA is developed to solve the resulting problem. In order to globally solve the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). A convex minorant of the objective function is introduced. DCA is used to compute upper bounds while lower bounds are calculated from a convex relaxation problem. The numerical results compared with those of COUENNE (http://www.coin-or.org/download/binary/Couenne/), a solver for mixed integer nonconvex programming, show the rapidity and the ?-globality of DCA in almost cases, as well as the efficiency of the combined DCA-Branch and Bound algorithm. We also propose a simple heuristic algorithm which is proved by experimental results to be better than an existing heuristic in the literature for this problem.     

Coordination compounds of Cu(II), Ni(II) and Co(II) with sulphamethazine-5-bromo-salicylaldimine

Coordination compounds of Cu (II), Ni (II) and Co (II) with sulphamethazine salicylaldimine (an antitubercular) have been prepared with a view to study their antibacterial activity. These complexes are granular, stable and are quantitatively formed and characterised by elemental analysis. Structures have been assigned based on their infrared, electronic absorption spectral and magnetic susceptibility studies. The antibacterial activity was tested against eleven available pathogens and in some cases complexes are found to be more potent.     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.