Remarks on lines and minimal rational curves

We determine all of lines in the moduli space M of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section. This work was supported by the Competitive Earmarked Research Grant (Grant No. HKU7025/03P) of the Research Grant Council, Hong Kong     

Blow-up Results for Some Nonlinear Delay Differential Equations

In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations. The first author is supported by a Grant from TWAS under contract No: 03-030 RG/MATHS/AF/AC. The second author is supported by a grant from the Lebanese National Council for Scientific Research.     

Properties of Distortion Risk Measures

The current literature does not reach a consensus on which risk measures should be used in practice. Our objective is to give at least a partial solution to this problem. We study properties that a risk measure must satisfy to avoid inadequate portfolio selections. The properties that we propose for risk measures can help avoid the problems observed with popular measures, like Value at Risk (VaR _α ) or Conditional VaR _α (CVaR _α ). This leads to the definition of two new families: complete and adapted risk measures. Our focus is on risk measures generated by distortion functions. Two new properties are put forward for these: completeness, ensuring that the distortion risk measure uses all the information of the loss distribution, and adaptability, forcing the measure to use this information adequately. This research was partially funded by ^1,3 Welzia Management, SGIIC SA, RD Sistemas SA, Comunidad Autónoma de Madrid Grant s-0505/tic/000230, and MEyC Grant BEC2000-1388-C04-03 and by ² the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 36860-06.     

Proximal Point Algorithms for General Variational Inequalities

This paper presents a unified framework of proximal point algorithms (PPAs) for solving general variational inequalities (GVIs). Some existing PPAs for classical variational inequalities, including both the exact and inexact versions, are extended to solving GVIs. Consequently, several new PPA-based algorithms are proposed. M. Li was supported by NSFC Grant 10571083 and SRFDP Grant 200802861031. L.Z. Liao was supported in part by grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong. X.M. Yuan was supported in part by FRG/08-09/II-40 from Hong Kong Baptist University and NSFC Grant 10701055.     

Subgame Consistent Solutions of a Cooperative Stochastic Differential Game with Nontransferable Payoffs

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by the prior optimal behavior of the players would remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In this paper, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs. The previously intractable subgame consistent solution for games with nontransferable payoffs is rendered tractable.This research was supported by the Research Grant Council of Hong Kong, Grant HKBU2056/99H and by Hong Kong Baptist University, Grant FRG/02-03/II16.Communicated by G. Leitmann     

An elementary and direct proof of the Painlevé property for the Painlevé equations I,II and IV

We present a direct and elementary proof that all the solutions of the Painlevé Equations I, II and IV are meromorphic functions on the whole complex plane. The proof uses some ideas from the existing proofs but applies the ideas in a different setting. Research was supported by Hong Kong Research Grant Council Earmark Grant HKUST6123/00P. Research was supported by Hong Kong Research Grant Council Earmark Grant HKUST6107/02P.     

Portfolio theory for the recourse certainty equivalent maximizing investor

The portfolio selection problem with one safe andn risky assets is analyzed via a new decision theoretic criterion based on the Recourse Certainty Equivalent (RCE). Fundamental results in portfolio theory, previously studied under the Expected Utility criterion (EU), such as separation theorems, comparative static analysis, and threshold values for inclusion or exclusion of risky assets in the optimal portfolio, are obtained here. In contrast to the EU model, our results for the RCE maximizing investor do not impose restrictions on either the utility function or the underlying probability laws. We also derive a dual portfolio selection problem and provide it with a concrete economic interpretation.Research partly supported by ONR Contracts N0014-81-C-0236 and N00014-82-K-0295, and NSF Grant SES-8408134 with the Center for Cybernetic Studies, The University of Texas at Austin.Partly supported by NSF Grant DDM-8896112.Partly supported by AFOSR Grant 0218-88 and NSF Grant ECS-8802239 at the University of Maryland, Baltimore Campus.     

Optimization Techniques for State-Constrained Control and Obstacle Problems

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples.The first author was supported by the Russian Foundation for Basic Research, Grant 03-01-00663, by the program Universities of Russia, Grant 03.03.007, and by the program of the Russian Federation President for the support of scientific research in leading scientific schools, Grant NSh-1889.2003.1.The second author was supported by the National Science and Engineering Research Council of Canada and by ONR MURI Contract 79846-23800-44-NDSAS.The third and first authors were supported by NSF Grants ECS-0099824 and ECS-0424445.Communicated by G. Leitmann     

Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points

Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-region technique to select the time steps of a second-order Rosenbrock method for a special initial-value problem, namely, a gradient system obtained from an unconstrained optimization problem. The technique is different from the local error approach. Both local and global convergence properties of the new method for solving an equilibrium point of the gradient system are addressed. Finally, some promising numerical results are also presented. This research was supported in part by Grant 2007CB310604 from National Basic Research Program of China, and #DMS-0404537 from the United States National Science Foundation, and Grant #W911NF-05-1-0171 from the United States Army Research Office, and the Research Grant Council of Hong Kong.     

Explicit classes of permutation polynomials of $$ \mathbb{F}_{3^{3m} } $$

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over are presented. These polynomials have simple shapes, and they are related to planar functions. This work was supported by Australian Research Council (Grant No. DP0558773), National Natural Science Foundation of China (Grant No. 10571180) and the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No. 612405)     

Area requirement and symmetry display of planar upward drawings

In this paper we investigate the problem of constructing planar straight-line drawings of acyclic digraphs such that all the edges flow in the same direction, e.g., from bottom to top. Our contribution is twofold. First we show the existence of a family of planar acyclic digraphs that require exponential area for any such drawing. Second, motivated by the preceding lower bound, we relax the straight-line constraint and allow bends along the edges. We present a linear-time algorithm that produces drawings of planarst-graphs with a small number of bends, asymptotically optimal area, and such that symmetries and isomorphisms of the digraph are displayed. If the digraph has no transitive edges, then the drawing obtained has no bends. Also, a variation of the algorithm produces drawings with exact minimum area.Research supported in part by Cadre Technologies Inc., by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM), by the National Science Foundation under Grant CCR-9007851, by the Office of Naval Research and the Defense Advanced Research Projects Agency under Contract N00014-83-K-0146 and ARPA order 6320, amendment 1, by the Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo of the Italian National Research Council, by the Texas Advanced Research Program under Grant No. 3972, and by the U.S. Army Research Office under Grant DAAL03-91-G-0035.     

Explicit Evaluation of Euler and Related Sums

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a (presumably) new proof of the classical Euler sum. We show that several interesting analogues of the Euler sums can be evaluated by systematically analyzing some known summation formulas involving hypergeometric series. Many other identities related to the Euler sums are also presented. Research of the first author was supported by Korea Science and Engineering Foundation Grant R05-2003-10441-0. Research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada Grant OGP0007353. 2000 Mathematics Subject Classification: Primary–11M06, 33B15, 33E20; Secondary–11M35, 11M41, 33C20     

Sensitivity of Pareto Solutions in Multiobjective Optimization

The paper presents a sensitivity analysis of Pareto solutions on the basis of the Karush-Kuhn-Tucker (KKT) necessary conditions applied to nonlinear multiobjective programs (MOP) continuously depending on a parameter. Since the KKT conditions are of the first order, the sensitivity properties are considered in the first approximation. An analogue of the shadow prices, well known for scalar linear programs, is obtained for nonlinear MOPs. Two types of sensitivity are investigated: sensitivity in the state space (on the Pareto set) and sensitivity in the cost function space (on the balance set) for a vector cost function. The results obtained can be used in applications for sensitivity computation under small variations of parameters. Illustrative examples are presented.Research of this author was partially supported by Grant BEC2003-09067-C04-03.Research of this author was partially supported by NSERC Grant RGPIN-3492-00.Research of this author was partially supported by Grant BEC2003-09067-C04-02.     

Dynamically Consistent Cooperative Solution in a Differential Game of Transboundary Industrial Pollution

This paper presents a cooperative differential game of transboundary industrial pollution. A noted feature of the game model is that the industrial sectors remain competitive among themselves while the governments cooperate in pollution abatement. It is the first time that time consistent solutions are derived in a cooperative differential game on pollution control with industries and governments being separate entities. A stochastic version of the model is presented and a subgame-consistent cooperative solution is provided. This is the first study of pollution management in a stochastic differential game framework. This research was supported by the Research Grant Council of Hong Kong Grant HKBU2103/04H and Hong Kong Baptist University Grant FRG/05-06/II22.     

Boolean products of lattices

A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with properties of its individual stalks.Research supported by the Natural Sciences and Engineering Research Council of Canada.Research supported by ONR Grant N00014-90-J-1008.     

Merit functions in vector optimization

We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence of error bounds for merit functions are also given. This research was supported by a direct grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong.     

On the zeros and poles of Padé approximants toe z

Summary In this paper, we study the location of the zeros and poles of general Padé approximats toe ^z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Fla. Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT(11-1)-2075.     

The multi-soliton solutions of the CH-γ equation

This paper solves the integrable CH-γ equation for analytical multiple soliton solutions with the Darboux transformation method. Some properties of the soliton solutions are different from the CH equation. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10401022) and the Research Grants Council of Hong Kong     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

Martingale representation and the Malliavin calculus

Using stochastic flows and the Itô differentiation rule, the integrand in the representation of a martingale as a stochastic integral is identified. By iterating this representation result a homogeneous chaos type expansion is obtained. Using the stochastic integral representation, an integration by parts formula is obtained without using any calculus of variations in function space. If the inverse of the Malliavin matrix belongs to all spacesL ^p() it follows that a random variable has a smooth density.The research of R. J. Elliott was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-7964 and the Air Force Office of Scientific Research, United States Air Force, under Grant AFOSR-86-0332. The research of M. Kohlmann was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-7964.