Explicit analytical results for one-mode oblique penetration into a periodic array of screens

An analytical approach to scattering of a plane scalar wavethrough a periodic array of rigid screens is studied. The one-modeproblem with oblique incidence is considered and reduced toan integral equation (of the first kind) holding over each opening.An approximated form is given to the kernel so as to deriveexplicit results for the reflection and transmission coefficients.Numerical investigation and comparison with previous resultsare finally made and reflected on some figures.     

An investigation of the impact and penetration of solids of revolution into soft earth

The numerical method proposed earlier in [1] is developed to solve problems of the impact and penetration of rigid and deformable bodies of revolution into soft soil, which are described by Grigoryan's model [2]. The effect of the surface Coulomb friction, the bulk compressibility and the shear strength of soft soil on the forces of resistance and contact pressures in the contact zone is analysed. The results of numerical solutions of problems in a coherent formulation are compared with analytical relations and experimental data on the determination of the forces and coefficients of resistance to the penetration of impactors of different shapes into soft soil.     

Application of fuzzy soft set in decision making problems based on grey theory

There are many uncertain problems in practical production and life which need decisions made with soft sets and fuzzy soft sets. However, the basis of evaluation of the decision method is single and simple, the same decision problem can obtain different results from using a different evaluation basis. In this paper, in order to obtain the right result, we discuss fuzzy soft set decision problems. A new algorithm based on grey relational analysis is presented. The evaluation bases of the new algorithm are multiple. There is more information in a decision result based on multiple evaluation bases, which is more easily accepted and logical to one’s thinking. For the two cases examined, the results show that the new algorithm is efficient for solving decision problems.     

Application of fuzzy soft set in decision making problems based on grey theory

There are many uncertain problems in practical production and life which need decisions made with soft sets and fuzzy soft sets. However, the basis of evaluation of the decision method is single and simple, the same decision problem can obtain different results from using a different evaluation basis. In this paper, in order to obtain the right result, we discuss fuzzy soft set decision problems. A new algorithm based on grey relational analysis is presented. The evaluation bases of the new algorithm are multiple. There is more information in a decision result based on multiple evaluation bases, which is more easily accepted and logical to one’s thinking. For the two cases examined, the results show that the new algorithm is efficient for solving decision problems.     

Mixture surrogate models based on Dempster-Shafer theory for global optimization problems

Recent research in algorithms for solving global optimization problems using response surface methodology has shown that it is in general not possible to use one surrogate model for solving different kinds of problems. In this paper the approach of applying Dempster-Shafer theory to surrogate model selection and their combination is described. Various conflict redistribution rules have been examined with respect to their influence on the results. Furthermore, the implications of the surrogate model type, i.e. using combined, single or a hybrid of both, have been studied. The suggested algorithms were applied to several well-known global optimization test problems. The results indicate that the used approach leads for all problems to a thorough exploration of the variable domain, i.e. the vicinities of global optima could be detected, and that the global minima could in most cases be approximated with high accuracy.     

Solution of two problems on wavelets

We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space ?²(?). The second is the existence, given any regular wavelet basis for $\mathbb{H}^2 (\mathbb{R})$ , of aMulti-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for $\mathbb{L}^2 (\mathbb{R}^n )$ arises from a Multi-Resolution Analysis that is regular modulo the action of a unitary operator, which is whenn = 2 a Calderón-Zygmund operator of convolution type.     

Non‐local approach in mathematical problems of fluid–structure interaction

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯₁⊂ℝ³ while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯₂=ℝ³\Ω₁ which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω₁. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Solution of some combinatorial problems which arise in calculating correlators in exactly solvable models

In this paper the recursion relations which arise in calculating correlators in exactly solvable models by the method of V. E. Korepin are explicitly solved. The solution is given in the form of determinants of certain matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 67–79, 1987.     

Numerical simulation of nonlinear consolidation problems by models based on the network method

The nonlinear consolidation problem in saturated soils, for any type of constitutive dependences of the hydraulic permeability and the void ratio on the effective pressure, has been numerically simulated by the network method. Three different network models, based on logarithmic and/or potential constitutive dependences, called the Davis and Raymond, Juárez-Badillo and Cornetti and Battaglio models, as well as a fourth one with dependences in tabulated form, are solved. In addition, new network models that delete the two restrictive hypotheses assumed by these authors are presented. These hypotheses are the influence of the void ratio changes in the term of contraction of the governing equation and the influence of the thickness change of the volume element as consolidation progresses. Only a few rules based on elementary theory of circuits are required for the design of the models, whose solution is reliable with relatively small grids and computational times. After verifying the results of the network method with the solutions of the classic authors, the extended models have been used to address a real case of consolidation.     

Computation of forces acting on bodies in plane and axisymmetric cavitation flow problems

Plane and axisymmetric cavitation flow problems are considered using Riabouchinsky’s scheme. The incoming flow is assumed to be irrotational and steady, and the fluid is assumed to be inviscid and incompressible. The flow problems are solved by applying the boundary element method with quadrature formulas without saturation. The free boundary is determined using a gradient descent technique based on Riabouchinsky’s principle. The drag force acting on the cavitator is expressed in terms of the Riabouchinsky functional. As a result, for small cavitation numbers, the force is calculated with a fairly high accuracy. Dependences of the drag coefficient are investigated for variously shaped cavitators: a wedge, a cone, a circular arc, and a spherical segment.     

A solution approach of production planning problems based on compact formulations for single-item lot-sizing models

We survey the main results presented in the authors PhD Thesis presented in June 2003 at the Université catholique de Louvain and supervised by Y. Pochet and L. A. Wolsey. The dissertation is written in English and is available from the author. In the first part of the thesis, we investigate the complexity and the polyhedral structure of various extensions of the uncapacitated single-item lot-sizing problem (Barany et al. 1984). In particular, we study models involving fixed charges on stocks, constant capacity and backlogging, and lower bounds on production. We describe algorithms, extended formulations, (facet-defining) valid inequalities and separation algorithms. Emphasis is placed on compact (i.e. of polynomial size) exact extended formulations. In a second part, we show how such extended reformulations for single-item problems can help to improve the solution of much more general production planning problems.Received: August 2003,     

Error estimators for advection–reaction–diffusion equations based on the solution of local problems

This paper deals with a posteriori error estimates for advection–reaction–diffusion equations. In particular, error estimators based on the solution of local problems are derived for a stabilized finite element method. These estimators are proved to be equivalent to the error, with equivalence constants eventually depending on the physical parameters. Numerical experiments illustrating the performance of this approach are reported.     

Solution of boundary value problems in cylinders separated by a three-layer film into two semicylinders

Boundary value problems are considered for the class of equations ? _x ² u + L[u] = 0 in cylinders D = (x ?? R, y ?? Q ? R ^m ) with an infinitely thin film at x = 0 consisting of three sublayers with alternating high and low permeability (L-linear differential operator with respect to y _i ). The solutions of the problems are expressed in terms of those of the corresponding classical boundary value problems in homogeneous cylinders D with no film. The resulting formulas have the form of simple quadrature rules, which are amenable to numerical computations.     

Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates

The local averaging technique has become a popular tool in adaptive finite element methods for solving partial differential boundary value problems since it provides efficient a posteriori error estimates by a simple postprocessing. In this paper, the technique is introduced to solve a class of symmetric eigenvalue problems. Its efficiency and reliability are proved by both the theory and numerical experiments structured meshes as well as irregular meshes. Dedicated to Charles A. Micchelli on his 60th birthday Mathematics subject classifications (2000) 65N15, 65N25, 65N30, 65N50. Subsidized by the Special Funds for Major State Basic Research Projects, and also supported in part by the Chinese National Natural Science Foundation and the Knowledge Innovation Program of the Chinese Academy of Sciences.     

Solution of three-dimensional problems of gas dynamics on multiprocessor computers

We give the results of solving a number of three-dimensional nonstationary problems of gas dynamics, whose programmatic implementation on multiprocessor computers is carried out using the NORMA language. We compare the effectiveness of different computing systems. Nine figures. Bibliography: 9 titles. Translated fromMetody Matematicheskoi Fiziki, 1998, pp. 172–183.     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Quasiclassical approximation for models of spin-spin interaction on a one-dimensional lattice

The one-dimensional anisotropic Heisenberg model in an external field is considered. For the corresponding energy operator with parameter, the quasiclassical (h0) spectral series are constructed using the method of the canonical operator. In the integrable case (in the absence of the external field) the results obtained coincide with the exact ones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 63–76, 1984.