Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

Regular and well-posed formulation of the boundary integral method for a singular biharmonic problem

By adequate choice of a fundamental solution, the singular point of the solution is excluded from the integral equations. The use of a special differential operator yields a well-posed formulation of the system of two integral equations. Moreover, the application of the symmetry principle for biharmonic functions improves the efficiency of the method. Finally, the results are used to compute the coefficients of the William's series (stress intensity factors) which is the eigenfunction expansion of the solution around the singular point.The research was supported in part by the Technion VPR Fund-M. R. Saulson Research Fund.     

Conic coconuts: the pricing of contingent capital notes using conic finance

In this paper we introduce a fundamental model under which we will price contingent capital notes using conic finance techniques. The model is based on more realistic balance-sheet models recognizing the fact that asset and liabilities are both risky and have to be treated differently taking into account bid and ask prices in a prudent fashion. The underlying theory makes use of conic finance which is based on the concept of acceptability and distorted expectations. We recall the theory and give a brief overview of the related literature. Next, we discuss and propose some potential funded and unfunded contingent capital notes. Traditionally, the conversion trigger of contingent capital notes is in terms of the Core-Tier 1 ratio. We argue that this ratio is maybe not optimal, certainly when taking into account the presence of risky liabilities. As an alternative we introduce triggers based on capital shortfall. The pricing of seven variations of funded as well as unfunded notes is overviewed. We further investigate the effect of the dilution factor and the grace factor. In an appendix we show conic balance sheets including contingent capital instruments.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

Generic structure of the Lagrangian manifold in chattering problems

This paper studies the structure of the singularity of Lagrangian manifolds in a neighborhood of the surface of singular extremals of second order in optimal control problems. For the Fuller classical problem, the structure of the Lagrangian manifold is explicitly constructed: it is shown that it has a singularity of conic type at the origin of coordinates. In the general case, it is proved that the Lagrangian manifold is a locally trivial fiber bundle over the surface of singular extremals with each fiber having a singularity of a similar conic type at the point of exit of the singular extremals.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.     

Conic systems and sublinear mappings: equivalent approaches

It is known that linear conic systems are a special case of set-valued sublinear mappings. Hence the latter subsumes the former. In this note we observe that linear conic systems also contain set-valued sublinear mappings as a special case. Consequently, the former also subsumes the latter.     

A semi-analytic solution for multiple curved cracks emanating from circular hole using singular integral equation

This paper provides an elastic solution for an infinite plate containing multiple curved edge cracks emanating from a circular hole. A fundamental solution is suggested, which represents a particular solution for a concentrated dislocation in an infinite plate with the traction free hole. The generalized image method and the concept of the modified complex potentials are used in the derivation of the fundamental solution. After using the fundamental solution and placing the distributed dislocations at the prospective sites of cracks, a singular integral equation is formulated. The singular integral equation is solved by using the curve length method in conjunction with the semi-opening quadrature rule. By taking an additional point dislocation at the hole center, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. Finally, several numerical examples are given to illustrate the efficiency of the method presented. Numerical examinations are carried out and sufficient accurate results have been found.     

Robust synchronization of singular complex switched networks with parametric uncertainties and unknown coupling topologies via impulsive control

This paper presents a general model of singular complex switched networks, in which the nodes can be singular dynamic systems and switching behaviors act on both nodes and edges. The parametric uncertainties and unknown coupling topologies are also considered in this model. Two robust synchronization schemes are discussed respectively. In one scheme, the network is synchronized to a homogeneous orbit and in the other one the network is synchronized to a weighted average of all the nodes. Based on the Lyapunov stability theory, different robust synchronization conditions for the two schemes are obtained for this singular complex switched network model via impulsive control. The similarities and differences between these synchronization conditions for the two schemes are discussed. In addition, three useful robust results for the special cases of the singular complex switched networks are presented. Two systematic-design procedures are presented for the two schemes, and three numerical examples are provided for illustrations.     

An approach to singular perturbation problems insoluble by asymptotic methods

Singular perturbation problems not amenable to solution by asymptotic methods require special treatment, such as the method of Carrier and Pearson. Rather than devising special methods for these problems, this paper suggests that there may be a uniform way to solve singular perturbation problems, which may or may not succumb to asymptotic methods. A potential mechanism for doing this is the author's boundary-value technique, a nonasymptotic method, which previously has only been applied to singular perturbation problems that lend themselves to asymptotic techniques. Two problems, claimed by Carrier and Pearson to be insoluble by asymptotic methods, are solved by the boundary-value method.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.     

Two different approaches for the treatment of boundary singularities

The finite element treatment of boundary singularities in elliptic problems has demanded that special techniques be developed. Many of these use some form of singular element in the neighborhood of a singular point. For a homogenous second-order problem, defined in a domain with a polygonal boundary, we studied three cases with different singularity orders. Some results about the accuracy of the solution are presented. Numerical results have been obtained using Akin singular triangular elements with three and six nodes and quadrilateral elements having four and eight nodes. The behavior of the gradient in each of these elements is also analyzed. However, these elements are not completely satisfactory, so that an alternative technique using curved isoparametric elements is given here. The results obtained with the two methods are compared. Conclusions about numerical accuracy of each method, the order of integration and the simplicity of application are made.     

构造非线性微分差分方程精确解的一种方法

把最近提出的G′/G展开法推广到了非线性微分差分方程,利用该方法成功构造了一种修正的Volterra链和Toda链的双曲函数、三角函数以及有理函数三类涉及任意参数的行波解,当这些参数取特殊值时,可得这两个方程的扭状孤立波解、奇异行波解以及三角函数状的周期波解等.研究结果表明,该算法探讨非线性微分差分方程精确解十分有效、简洁.     

An asymptotic method of solving transient dynamic contact problems

Using a special approximation in the complex plane of the symbol of the kernel of the contact-problem integral equation, an asymptotic form of its solution is constructed which is the fundamental solution of the transient dynamic plane contact problem of the impact of a rigid punch with an elastic half-plane for short interaction times. The proposed approximation of the kernel symbol enables it to be approximated in the complex plane with any previously specified accuracy. Unlike existing approaches [1, 2, etc.], the approximation of the kernel symbol of the integral equation employed here enables the solution of this problem to be obtained in the form of simple formulae not containing singular quadratures.     

The fundamental theorem for singular surfaces with limiting tangent planes

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundamental forms with local properties in the differential of its parametrization and decompositions in the matrices associated to the fundamental forms. In particular, we introduce new types of curvatures that can be used to characterize wave fronts. The only restriction on the parametrizations that is assumed in several occasions is that the singular set has empty interior.     

High-order nonlinear boundary value problems admitting multiple exact solutions with application to the fluid flow over a sheet

We frame a hierarchy of nonlinear boundary value problems which are shown to admit exponentially decaying exact solutions. We are able to convert the question of the existence and uniqueness of a particular solution to this nonlinear boundary value problem into a question of whether a certain polynomial has positive real roots. Furthermore, if such a polynomial has at least two distinct positive roots, then the nonlinear boundary value problem will have multiple solutions. In certain special cases, these boundary value problems arise in the self-similar solutions for the flow of certain fluids over stretching or shrinking sheets; examples given include the flow of first and second grade fluids over such surfaces.     

On the explicit parametric description of waves in periodic media

A method for the parameterization of the one-dimensional wave equation is proposed that makes it possible to find its solution by quadratures under an arbitrary dependence of the refraction index on the current wave phase. The form of the solution found is used to investigate the structure of the wave function for a periodic refraction index. Explicit expressions for the fundamental system of solutions and for the Floquet index are obtained. Examples of applying the proposed method to the optimal synthesis of multilayer interference mirrors and Bragg waveguides are discussed.     

Automorphisms of the solution spaces of special double-confluent Heun equations

Two new linear operators determining automorphisms of the solution space of a special double-confluent Heun equation in the general case are obtained. This equation has two singular points, both of which are irregular. The obtained result is applied to solve the nonlinear equation of the resistively shunted junction model for an overdamped Josephson junction in superconductors. The new operators are explicitly expressed in terms of structural polynomials, for which recursive computational algorithms are constructed. Two functional equations for the solutions of the special double-confluent Heun equation are found.     

Biorthogonal sequences of solutions of the generalized feller equation

The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.     

Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n.