On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.     

Estimate of an Entire Function in the Union of Two Angles

We obtain an estimate of an entire function expressed as the limit of a sequence of polynomials in exponentials in the union of two angles via an estimate of the function and its derivatives on a set lying on a horizontal strip.     

On Unitary Extensions of Multiplicative Families of Partial Isometries with a Generating Subspace

We consider a two parameter family of operators on a Hilbert space given by a multiplicative family of partial isometries with a generating subspace that commutes with an unitary group. The first parameter runs on a generalized interval of an abelian ordered group and the second on an abelian group. We show that it can be extended to a two parameter group of unitary operators on a larger Hilbert space. Applications to the problem of extending generalized semigroups of isometries and locally defined positive definite functions are given.     

Application of the Armijo stepsize rule to the solution of a nonlinear system of equalities and inequalities

In this paper, we present an application of the Armijo procedure to an algorithm for solving a nonlinear system of equalities and inequalities. The stepsize procedure contained in a quadratically convergent algorithm is replaced by an Armijo procedure on a nondifferentiable function, without any substantial extra computational work and without losing the quadratic rate of convergence. The use of the Armijo procedure on a nondifferentiable function opens the possibility of using it on nondifferentiable and minimax functions.     

On the interaction of deep water waves and exponential shear currents

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.     

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.     

Genus embeddings of a type of graph

In 2003, Liu set up the joint tree model such that an embedding of a graph on a surface can be represented by a joint tree, further by an associated surface of it. In this paper, by dividing an associated surface into segments layer by layer and doing a sequence of operations on the layer division, we obtain the genus of complete bipartite graphs with a different method as before. Based on it, firstly, the genus of a new type of graph is derived. Then the number of non-homeomorphic genus embeddings of some complete bipartite graph is easily estimated.     

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.     

On the interaction of deep water waves and exponential shear currents

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.     

可定向曲面上两个地图的交叉数

In this paper, we discuss the crossing numbers of two one-vertex maps on orientable surfaces. By using a reductive method, we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface. This provides a lower bound for the crossing number of two general maps on an orientable surface.     

Upper semicontinuous representations of interval orders

Given an interval order on a topological space, we characterize its representability by means of a pair of upper semicontinuous real-valued functions. This characterization is only based on separability and continuity conditions related to both the interval order and one of its two traces. As a corollary, we obtain the classical Rader’s theorem concerning the existence of an upper semicontinuous representation for an upper semicontinuous total preorder on a second countable topological space.     

An iterative penalty method for a nonlinear problem of equilibrium of a Timoshenko-type plate with a crack

A variational problem of equilibrium of an elastic Timoshenko-type plate in a domain with a slit is considered. A nonpenetration boundary condition in the form of an inequality is specified on the edges of the slit. A penalized equation and an iterative linear equation in integral and differential forms are constructed. Some results on solution convergence and an error estimate are obtained.     

Problems of the concentration of elastic stresses near defects in spherically multilayered media

A spherically multilayered medium, whose elastic parameters change abruptly on the spherical surfaces, with defects in the form of cracks or thin rigid inclusions, is considered. The method of solving problems of the stress concentration near such defects is based on the introduction of linear combinations of the displacements and stresses as the fundamental unknowns. This enables the difficulties related to the presence of an arbitrary number of layers to be effectively overcome. The method is described initially for an unbounded elastic medium and defects of spherical form, situated on the surfaces where the elastic parameters change (interphase defects) and a way of extending this to the case of an elastic medium of finite dimensions, defects of other forms and not situated on these surfaces, is indicated. The method is described in detail as it applies to the case of a two-layer medium with an interphase crack when a torsion centre at the origin of coordinates acts on the medium. The problem is reduced to an integral equation, an effective method of solving it is given, and a formula is obtained for the stress intensity factor.     

On a method of obtaining spectral relationships for integral operators of mixed problems of mechanics of continuous media

The method of orthogonal polynomials, and its generalization, the method of orthogonal functions /1,2/ applied for the investigation of complex mixed problems of the mechanics of continuous media, are based on the utilization of spectral relationships that invert the main (singular) part of the kernel of the integral equation of the problem under consideration. A sufficiently general approach to the derivation of spectral relationships that is based on potential theory is proposed. Eigenfunctions are obtained in the problem of impressing a strip stamp in an elastic halfspace as are also the odd eigenfunctions of a logarithmic series in the case of two symmetric intervals. An an application of the results obtained, the solution is constructed for any value of a certain dimensionless parameter, for the plane contact problem of the impression of a rigid stamp into the surface of an elastic strip which is under an interlayer of the type of a covering resting on an undeformable foundation.     

Global and local convergence of a class of penalty-free-type methods for nonlinear programming

We present a class of trust region algorithms without using a penalty function or a filter for nonlinear inequality constrained optimization and analyze their global and local convergence. In each iteration, the algorithms reduce the value of objective function or the measure of constraints violation according to the relationship between optimality and feasibility. A sequence of steps focused on improving optimality is referred to as an f-loop, while some restoration phase focuses on improving feasibility and is called an h-loop. In an f-loop, the algorithms compute trial step by solving a classic QP subproblem rather than using composite-step strategy. Global convergence is ensured by requiring the constraints violation of each iteration not to exceed an progressively tighter bound on constraints violation. By using a second order correction strategy based on active set identification technique, Marato’s effect is avoided and fast local convergence is shown. The preliminary numerical results are encouraging.     

Remarks on the solvability of a convolution integral equation on a finite interval

We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.     

Maximum face size in an arrangement of curves

The purpose of this note is to establish a bound on the number of edges on a face of an arrangement of curves in the plane, and to correct thereby an error in an earlier formulation by Grünbaum.     

Locomotion based on the control of the shape of magnetic fluid surfaces and of magnetizable media

The realization of locomotion based on the deformation of a free surface of a magnetic fluid layer in a traveling magnetic field is studied. A plane flow of an incompressible viscous magnetic fluid layer on a horizontal surface in a nonuniform magnetic field and a plane two-layers flow of incompressible viscous magnetic fluids between two parallel solid planes in a magnetic field is considered. Also the flow of an incompressible viscous magnetic fluid layer on a cylinder in a nonuniform magnetic field is an object of investigation. The deformation and the motion of a body made by a magnetizable polymer in an alternating magnetic field are experimentally studied. The cylindrical body (worm) which is located in a cylindrical tube is analyzed. These effects can be used in designing autonomous mobile robots without a hard cover. Such robots can be employed in clinical practice and biological investigations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach to the covering radius of codes

We introduce a new approach which facilitates the calculation of the covering radius of a binary linear code. It is based on determining the normalized covering radius ϱ. For codes of fixed dimension we give upper and lower bounds on ϱ that are reasonably close. As an application, an explicit formula is given for the covering radius of an arbitrary code of dimension ⩽4. This approach also sheds light on whether or not a code is normal. All codes of dimension ⩽4 are shown to be normal, and an upper bound is given for the norm of an arbitrary code. This approach also leads to an amusing generalization of the Berlekamp-Gale switching game.     

Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave

We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L ₁[?r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.