LANGLANDS DUALITY AND G2 SPECTRAL CURVES

We first demonstrate how duality for the fibres of the so-calledHitchin fibration works for the Langlands dual groups Sp(2m)and SO(2m + 1). We then show that duality for G₂ is implementedby an involution on the base space which takes one fibre toits dual. A formula for the natural cubic form is given andshown to be invariant under the involution.     

The Gallery Length Filling Function and a Geometric Inequality for Filling Length

We exploit duality considerations in the study of singular combinatorial2-discs (diagrams) and are led to the following innovationsconcerning the geometry of the word problem for finite presentationsof groups. We define a filling function called gallery lengththat measures the diameter of the 1-skeleton of the dual ofdiagrams; we show it to be a group invariant and we give upperbounds on the gallery length of combable groups. We use gallerylength to give a new proof of the Double Exponential Theorem.Also we give geometric inequalities relating gallery lengthto the space-complexity filling function known as filling length.2000 Mathematics Subject Classification 20F05 (primary), 20F06,57M05, 57M20 (secondary).     

Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory anda nonlinear dual transformation method for solving the fullynonlinear, non-convex parametric variational problem inf{W(u- µ) - F(u)}, and associated nonlinear boundary valueproblems, where is a nonlinear operator, W is either convexor concave functional of p = u, and µ is a given parameter.Detailed mathematical proofs are provided for the complementaryextremum principles proposed recently in finite deformationtheory. A method for obtaining truly dual variational principles(without a dual gap and involving the dual variable p* of uonly) in n-dimensional problems is proposed. It is proved thatfor convex W(p), the critical point of the associated LagrangianL_µ(u, p*) is a saddle point if and only if the so-calledcomplementary gap function is positive. In this case, the systemhas only one dual problem. However, if this gap function isnegative, the critical point of the Lagrangian is a so-calledsuper-critical point, which is equivalent to the Auchmuty'sanomalous critical point in geometrically linear systems. Wediscover that, in this case, the system may have more than oneprimal-dual set of problems. The critical point of the Lagrangianeither minimizes or maximizes both primal and dual problems.An interesting triality theorem in non-convex systems is proved,which contains a minimax complementary principle and a pairof minimum and maximum complementary principles. Applicationsin finite deformation theory are illustrated. An open problemleft by Hellinger and Reissner is solved completely and a purecomplementary energy principle is constructed. It is provedthat the dual Euler-Lagrange equation is an algebraic equation,and hence, a general analytic solution for non-convex variational-boundaryvalue problems is obtained. The connection between nonlineardifferential equations and algebraic geometry is revealed.     

Dual Variational Principles and Pad?-type Approximants

Families of approximants for <?, f> are derived from dualvariational functionals associated with the linear equation(1+yL)?=f. Expansions in both ascending and descending powersof y are considered, and the approximants are identified eitheras one- or two-point Pad? approximants, or as approximants ofa closely related type. Compact formulae are obtained for theapproximants, and their duality and bounding properties areexhibited. Attention is paid to the special situations occurringwhen the non-negative self-adjoint operator L (i) has a zeroeigenvalue, (ii) is bounded.     

Localization Algebras and Duality

The paper studies the dual algebras of localization Roe algebrasover proper metric spaces and develops a localization versionof Paschke duality for K-homology. It is shown that the localizationK-homology groups are isomorphic to Kasparov's K-homology groupsfor the Rips complex of proper metric spaces with bounded geometry.It follows that the obstruction groups to the coarse Baum–Connesconjecture can also be derived from the dual localization algebras.     

DUALITY IN LINEAR CONTROL SYSTEMS

The present paper establishes various duality relations-equalities which exist in linear control systems. By means of these relations, we demonstrate various duality properties in the theory of linear control systems. In particular we discuss the duality of controllability and observability of linear systems, the concepts of controllability and observability in stochastic systems the duality of regulator and observer of time-invariant systems and the duality property of reduced dimension observers etc. The duality relation established here is used to solve the problem of min-max state estimation in systems with uncertainty.     

Two Examples Concerning Martingales in Banach Spaces

The analytic concepts of martingale type p and cotype q of aBanach space have an intimate relation with the geometric conceptsof p-concavity and q-convexity of the space under consideration,as shown by pisier. In particular, for a banach space X, havingmartingale type p for some p > 1 implies that X has martingalecotype q for some q < . The generalisation of these concepts to linear operators wasstudied by the author, and it turns out that the duality aboveonly holds in a weaker form. An example is constructed showingthat this duality result is best possible. So-called random martingale unconditionality estimates, introducedby Garling as a decoupling of the unconditional martingale differences(UMD) inequality, are also examined. It is shown that the random martingale unconditionality constantof for martingales of length n asymptotically behaves like n. This improves previous estimatesby Geiss, who needed martingales of length 2ⁿ to show this asymptotic.At the same time the order in the paper is the best that canbe expected.     

A Numerical Algorithm for the Most Economical Structural Synthesis of Linear Control Systems

Given a time-invariant state matrix A, the Most Economical StructuralSynthesis (MESS) calls for the construction of the controllermatrix B and observer matrix C are as sparse as possible and(II) the resulting linear time-invariant control systems arecompletely controllable and completely observable. A numericallystable algorithm for the MESS problem is suggested. Three numericalexamples are given. Generalizations of MESS are also discussed.     

Stability of Runge-Kutta Methods for Trajectory Problems

A solution of a system of m autonomous differential equationsdefines a trajectory in m-dimensional space and, in particular,may give a closed orbital path. Typical trajectories are describedby a model nonlinear problem introduced in this article. Forthis problem, a trajectory lies on a surface characterized bya real symmetric matrix. It is shown that some Runge-Kutta methodspossess a property which ensures that, for this model problem,the numerical solution lies on the same surface as the trajectory.When m = 2, the numerical solution lies on the trajectory. Thisproperty is related to algebraic stability. A weaker propertysuffices for normalized differential systems.     

Duality involving the mock theta function f(q)

We show that the coefficients of Ramanujan's mock theta functionf(q) are the first non-trivial coefficients of a canonical sequenceof modular forms. This fact follows from a duality which equatescoefficients of the holomorphic projections of certain weight1/2 Maass forms with coefficients of certain weight 3/2 modularforms. This work depends on the theory of Poincaré series,and a modification of an argument of Goldfeld and Sarnak onKloosterman–Selberg zeta functions.     

Conjugate Duality and the Control of Linear Discrete Systems

In this paper we deal with the minimization of a convex function over the solution set of a range inclusion problem determined by a multivalued operator with convex graph. We attach a dual problem to it, provide regularity conditions guaranteeing strong duality and derive for the resulting primal–dual pair necessary and sufficient optimality conditions. We also discuss the existence of optimal solutions for the primal and dual problems by using duality arguments. The theoretical results are applied in the context of the control of linear discrete systems.     

Observability of Nonlinear Systems

We consider the observability of systems of the form = Ax +Nx, y = Fx, where A is a linear operator and N and F are nonlinear.We show that if the system is linearized about an equilibriumpoint x_e and the linearized system is continuously initiallyobservable, then the nonlinear system is continuously initiallyobservable in some neighbourhood of x_e. We then look at conditionsunder which solutions of the nonlinear system can be extendedfor all time and consider the problem of stabilizing the systemby feedback controls such that the solutions are eventuallyin the observability neighbourhood of x_e. Finally, we applythese ideas to two systems: a wave equation and a diffusionequation with nonlinear perturbations and nonlinear observations.     

Time-Varying Linear Control Systems: A Geometric Approach

The concept of [A, B]-invariance for time-varying (not necessarilycontinuous) linear systems is introduced. A natural approachvia time-varying subspaces is given. A disturbance decouplingand a noninteracting problem is studied.     

Jaffard-Ohm correspondence and Hochster duality

We study categorical aspects of the Jaffard–Ohm correspondencebetween abelian l-groups and Bézout domains and showthat this correspondence is close to a localization. For thispurpose, we establish a general extension theorem for valuationswith value group that is an abelian l-group. As an application,we prove Anderson's conjecture which refines the Jaffard–Ohmcorrespondence. We then extend the correspondence to sheaveson spectral spaces and show that the spectrum of a Bézoutdomain and the spectrum of its corresponding abelian l-groupprovide a concrete example for Hochster's duality of spectralspaces.     

Noncanonical time transformations relating finite-dimensional integrable systems

We consider dual Stäckel schemes related to each other by a noncanonical transformation of the time variable. We prove that this duality of different integrable systems arises from the multivaluedness of the Abel mapping. We construct the Lax matrices and the r-matrix algebras for some integrable systems on a plane. The integrable deformations of the Kepler problem and the Holt-type systems are considered in detail.     

The Penrose Transform for Compactly Supported Cohomology

Let the manifold X parametrise a family of compact complex submanifoldsof the complex (or CR) manifold Z. Under mild conditions thePenrose transform typically provides isomorphisms between acohomology group of a holomorphic vector bundle V Z and thekernel of a differential operator between sections of vectorbundles over X. When the spaces in question are homogeneousfor a group G the Penrose transform provides an intertwiningoperator between representations. The paper develops a Penrose transform for compactly supportedcohomology on Z. It provides a number of examples where a compactlysupported cohomology group is shown to be isomorphic to thecokernel of a differential operator between compactly supportedsections of vector bundles over X. It considers also how theSerre duality pairing carries through the transform.     

Linear optimal control problem for discrete 2-D systems with constraints

Optimal control problem for linear two-dimensional (2-D) discrete systems with mixed constraints is investigated. The problem under consideration is reduced to a linear programming problem in appropriate Hubert space. The main duality relations for this problem is derived such that the optimality conditions for the control problem are specified by using methods of the linear operator theory. Optimality conditions are expressed in terms of solutions for conjugate system.     

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.     

Hierarchical generating method for large-scale multiobjective systems

This paper investigates large-scale multiobjective systems in the context of a general hierarchical generating method which considers the problem of how to find the set of all noninferior solutions by decomposition and coordination. A new, unified framework of the hierarchical generating method is developed by integrating the envelope analysis approach and the duality theory that is used in multiobjective programming. In this scheme, the vector-valued Lagrangian and the duality theorem provide the basis of a decomposition of the overall multiobjective system into several multiobjective subsystems, and the envelope analysis gives an efficient approach to deal with the coordination at a high level. The following decomposition-coordination schemes for different problems are developed: (i) a spatial decomposition and envelope coordination algorithm for large-scale multiobjective static systems; (ii) a temporal decomposition and envelope coordination algorithm for multiobjective dynamic systems; and (iii) a three-level structure algorithm for large-scale multiobjective dynamic systems.This work was supported by NSF Grant No. CEE-82-11606.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.