Ambient connections realising conformal Tractor holonomy

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham. Current address for first author: Erwin Schr?dinger International Institute for Mathematical Physics (ESI), Boltzmanngasse 9, 1090 Vienna, Austria Current address for second author: Department of Mathematics, University of Hamburg, Bundesstra?e 55, 20146 Hamburg, Germany     

Ordered *-Algebras with an Order Unit

We develop a theory of ordered *-algebras with an order unit. These are complex algebras equipped with a conjugate-linear vector space involution whose hermitian elements form an ordered real algebra with an order unit but not necessarily with a multiplicative identity. Our main result is a representation theorem for ordered *-algebras with an order unit. This can be viewed as the complex and the non-unital version of the representation theorem of Stone and Kadison for ordered real algebras containing an order unit which is a multiplicative identity. A key role in our approach is played by certain weighted function algebras.     

Auto-balancing of a rotor with an orthotropic elastic shaft

The self-balancing of a statically unbalanced orthotropic elastic rotor equipped with a ball auto-balancing device is investigated. Equations of motion in fixed and rotating systems of coordinates, as well as equations describing steady motions of the regular precession type, are derived using a simple model of a Jeffcott rotor. Formulae for calculating the amplitude-frequency and phase-frequency characteristics of the precessional motion of the rotor are obtained. It is established that the conditions for a steady balanced mode of motion for an orthotropic rotor to exist have the same form as for an isotropic rotor, but the stability region of such a mode for an orthotropic rotor is narrower than the stability region for an isotropic rotor. The unsteady modes of motion of the rotor in the case of rotation with constant angular velocity and in the case of passage through critical velocities with constant angular acceleration is investigated numerically. It is established that the mode of slow passage through the critical region for an orthotropic rotor is far more dangerous than the similar mode for an isotropic rotor.     

Additive functionals of semi-Markov processes with absorption

Given a semi-Markov process with an arbitrary set of states, a criterion is obtained for the attainability of a certain isolated subset of states and for finiteness of the average attainment time. An equation is given for the average of an additive functional of a process with absorption, existence and uniqueness conditions are deduced for the solution of that equation in a given class of functions, and an integral representation is obtained for the solution.     

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of the Newton-type formulation, we derive an a posteriori estimate of the norm for the infinite-dimensional operator. In numerical examples, we show that the proposed method should be more efficient than the existing method. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.     

On a contact problem for a viscoelastic von Karman plate and its semidiscretization

We deal with the system describing moderately large deflections of thin viscoelastic plates with an inner obstacle. In the case of a long memory the system consists of an integro-differential 4th order variational inequality for the deflection and an equation with a biharmonic left-hand side and an integro-differential right-hand side for the Airy stress function. The existence of a solution in a special case of the Dirichlet-Prony series is verified by transforming the problem into a sequence of stationary variational inequalities of von Karman type. We derive conditions for applying the Banach fixed point theorem enabling us to solve the biharmonic variational inequalities for each time step.     

Existence of Solutions for Impulsive Neutral Integro-Differential Inclusions with Nonlocal Initial Conditions via Fractional Operators

This paper is mainly concerned with existence of mild solutions for first-order impulsive neutral integro-differential inclusions with nonlocal initial conditions in α-norm. We assume that the undelayed part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed point theorem for condensing multivalued maps, a main existence theorem is established. As an application of this main theorem, a practical consequence is derived for the sublinear growth case. Finally, we present an application to a neutral partial integro-differential equation with Dirichlet and nonlocal initial conditions.     

Dynamics of solutions of the Cauchy problem for semilinear parabolic stochastic partial differential equations with power-law singularities

We prove a comparison theorem for bounded solutions of the Cauchy problem for stochastic partial differential equations of the parabolic type with linear leading part. The drift and diffusion coefficients have locally bounded derivatives with respect to the state variable. We use this comparison theorem to study the dynamics of solutions of an equation with an absorber and an equation with a source.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

On error estimates in the Galerkin method for hyperbolic equations

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

Primal and dual predicted decrease approximation methods

We introduce the notion of predicted decrease approximation (PDA) for constrained convex optimization, a flexible framework which includes as special cases known algorithms such as generalized conditional gradient, proximal gradient, greedy coordinate descent for separable constraints and working set methods for linear equality constraints with bounds. The new scheme allows the development of a unified convergence analysis for these methods. We further consider a partially strongly convex nonsmooth model and show that dual application of PDA-based methods yields new sublinear convergence rate estimates in terms of both primal and dual objectives. As an example of an application, we provide an explicit working set selection rule for SMO-type methods for training the support vector machine with an improved primal convergence analysis.     

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Let X be an algebraic surface defined over the complex field and endowed with an -fibration. As such a surface we have a Platonic -fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed -action and its fixpoint deleted off. We consider an étale endomorphism and show that is an automorphism in most cases. Of particular interest is the case of a Platonic -fiber space, for which being an automorphism is closely related to the Jacobian Problem for the affine plane . We also investigate the automorphism group of such surfaces. Received January 7, 1999 / Published online October 30, 2000     

The Darboux transformation of some Manakov systems

The Darboux transformation is explicitly constructed for a coupled system of an arbitrary number of focusing or defocusing nonlinear Schrödinger fields with cubic nonlinearity (i.e., a Manakov system with an arbitrary number of fields). An immediate consequence is an explicit formula for an auto-Bäcklund transformation for the Manakov system.     

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.     

Optimal repair-cost limit for a consumer following expiry of a warranty

This paper develops a methodology for obtaining the optimalrepair-cost limit for the repair-replace decision that a consumerfaces on the expiry of a product warranty, where the producthas been minimally repaired on failure during the warranty.Three cases are considered; (i) an exponentially distributedtime to failure with an arbitrary repair-cost distribution,(ii) an exponentially distributed time to failure with a uniformrepair-cost distribution, and (iii) a Weibull-distributed timeto failure with a beta-distributed repair cost. Numerical optimizationsare performed for the third case, and conclusions drawn as tothe sensitivity of the optimal repair-cost limit on the parametervalues of the distributions involved.     

A mathematical model of market competition

We consider a mathematical model of decision making by a company attempting to win a market share. We assume that the company releases its products to the market under the competitive conditions that another company is making similar products. Both companies can vary the kinds of their products on the market as well as the prices in accordance with consumer preferences. Each company aims to maximize its profit. A mathematical statement of the decision-making problem for the market players is a bilevel mathematical programming problem that reduces to a competitive facility location problem. As regards the latter, we propose a method for finding an upper bound for the optimal value of the objective function and an algorithm for constructing an approximate solution. The algorithm amounts to local ascent search in a neighborhood of a particular form, which starts with an initial approximate solution obtained simultaneously with an upper bound. We give a computational example of the problem under study which demonstrates the output of the algorithm.     

Generalized Variational Relation Problems with Applications

In this paper, we first obtain an existence theorem of the solutions for a variational relation problem. An existence theorem for a variational inclusion problem, a KKM theorem and an extension of the well know Ky Fan inequality will be established, as particular cases. Some applications concerning a saddle point problem with constraints, existence of a common fixed point for two mappings and an optimization problem with constraints, will be given in the last section of the paper.     

Pricing in an equilibrium based model for a large investor

We study a financial model with a non-trivial price impact effect. In this model we consider the interaction of a large investor trading in an illiquid security, and a market maker who is quoting prices for this security. We assume that the market maker quotes the prices such that by taking the other side of the investor’s demand, the market maker will arrive at maturity with the maximal expected utility of the terminal wealth. Within this model we provide an explicit recursive pricing formula for an exponential utility function, as well as an asymptotic expansion for the price for a “small” simple demand.     

The Stress Boundary Value Problem for an Ideal Fibre-reinforced Rectangular Plate with a Hole

The title problem is examined for plane deformations of an idealfibre-reinforced rectangular plate with a hole. The solutionis found to depend on simultaneous ordinary differential equationswhich may be integrated without difficulty. This is in markedcontrast with the corresponding problems for an anisotropicelastic body and enables approximate solutions for stronglyanisotropic bodies to be derived without difficulty. The particularcases of a line crack in the plate at an angle to the fibresand an arbitrary symmetrical hole are examined in detail.