State estimation of linear dynamical systems under bounded control

The problem of estimation of all possible states that a linear system under bounded control may take (namely, the reachable or attainable set) is addressed. A number of previously developed Lyapunov techniques for estimating the reachable set of ann-dimensional linear system are extended and compared. The techniques produce over-estimates in the form ofn-dimensional ellipsoids. Illustrative examples are solved.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Operating Grant Nos. A-0621 and A-4080, and by Research Personnel Support from the Dean of Science, Memorial University of Newfoundland, Canada.     

Estimates of reachable sets of multidimensional control systems with nonlinear interconnections

The paper is devoted to the problem of constructing external estimates for the reachable set of a multidimensional control system by means of vector estimators. A system is considered that permits a decomposition into several independent subsystems with simple structure (for example, linear subsystems), which are connected to each other by means of nonlinear interconnections. For each of the subsystems, an external estimate of the reachable set is assumed to be known; this estimate is representable in the form of a level set of some function satisfying a differential inequality. An estimate for the reachable set of the combined system is constructed with the use of estimates for subsystems. The method of deriving the estimates is based on constructing comparison systems for analogs of vector Lyapunov functions (value functions).     

Projections of Bodies and Hereditary Properties of Hypergraphs

We prove that for every n-dimensional body K, there is a rectangularparallelepiped B of the same volume as K, such that the projectionof B onto any coordinate subspace is at most as large as thatof the corresponding projection of K. We apply this theorem to projections of finite set systems andto hereditary properties. In particular, we show that everyhereditary property of uniform hypergraphs has a limiting density.     

Properties of Higher-Dimensional Shintani Generating Functions and Cocycles on Pgl3(Q)

In an earlier paper the second author used the formal, algebraicproperties of 2-dimensional Shintani generating functions toconstruct a 1-cocycle on PGL₂{Q}. We aim to generalise theseresults by using such functions in dimension n to obtain an(n–1)-cocycle on PGL_n{Q}, presumably related to the Bernoulliand Eisenstein cocycles of R. Sczech. By improving our methodswe achieve this goal for n=3. For n>3 we encounter obstaclesrelated to degenerate configurations of hyperplanes in n-space.Nevertheless, we obtain partial results closely connected toreciprocity laws for certain n-dimensional Dedekind sums. 1991Mathematics Subject Classification: 11F20, 11F75.     

Minimal Submanifolds in a Sphere II

We use the method of H. Gauchman [2] to show that if the sectionalcurvature of a compact minimal n-dimensional submanifold ina unit sphere is everywhere bigger than n/2(n+1) then it mustbe totally geodesic. We also discuss some related results andconjectures.     

Reachability properties of constrained linear systems

This paper presents reachability results for a linear control system and arbitrary terminal pointsp inR ⁿ with controls constrained within a compact setU containing the origin. The well-known results forp=0 are then a special case of our results.Geometric properties of the reachable set are presented and include: general containment properties which describe conditions that guarantee the inclusion of a reachable set in another reachable set; classification of the set of all pointsp that ensure the equivalence of two (different) reachable sets and the properties of this set.The topological properties of the reachable set to a pointp depends on the control setU, the final pointp, and the location of the spectrum of the system in the complex plane. We characterize the geometric properties of the reachable set when the spectrum lies in the closed right-half plane, the open left-half plane, or a combination of both.This work was supported in part by NSF Grant ECS-86-09586.     

The Heat Kernel on Hyperbolic Space

The purpose of this note is to provide a new proof for the explicitformulas of the heat kernel on hyperbolic space. By definition,the hyperbolic space Hⁿ is a (unique) simply connected completen-dimensional Riemannian manifold with a constant negative sectionalcurvature –1. 1991 Mathematics Subject Classification58G32, 58G11.     

Non-Regular Tangential Behaviour of a Monotone Measure

A Radon measure µ on Rⁿ is said to be k-monotone if is a non-decreasing function on (0,) for every x Rⁿ. (If µ is the k-dimensional Hausdorffmeasure restricted to a k-dimensional minimal surface then thisimportant property is expressed by the monotonicity formula.)We give an example of a 1-monotone measure µ in R² withnon-unique and non-conical tangent measures at a point. Furthermore,we show that µ can be the one-dimensional Hausdorff measurerestricted to a closed set A R². 2000 Mathematics Subject Classification49Q05, 49Q20 (primary), 28A75, 53A10 (secondary).     

Invariants of Finite Reflection Groups and the Mean Value Problem for Polytopes

Let P be an n-dimensional polytope admitting a finite reflectiongroup G as its symmetry group. Consider the set H_P(k) of allcontinuous functions on Rⁿ satisfying the mean value propertywith respect to the k-skeleton P(k) of P, as well as the setH_G of all G-harmonic functions. Then a necessary and sufficientcondition for the equality H_P(k) = H_G is given in terms of adistinguished invariant basis, called the canonical invariantbasis, of G. 1991 Mathematics Subject Classification 20F55,52B15.     

Quasi-Permutation Representations of p-Groups of Class 2

If G is a finite linear group of degree n, that is, a finitegroup of automorphisms of an n-dimensional complex vector space(or, equivalently, a finite group of non-singular matrices oforder n with complex coefficients), we shall say that G is aquasi-permutation group if the trace of every element of G isa non-negative rational integer. The reason for this terminologyis that, if G is a permutation group of degree n, its elements,considered as acting on the elements of a basis of an n-dimensionalcomplex vector space V, induce automorphisms of V forming agroup isomorphic to G. The trace of the automorphism correspondingto an element x of G is equal to the number of letters leftfixed by x, and so is a non-negative integer. Thus, a permutationgroup of degree n has a representation as a quasi-permutationgroup of degree n. See [8].     

Complete Manifolds with Sectional Curvature Bounded Below and Large Volume Growth

This paper provides a proof that an n-dimensional complete openRiemannian manifold M with sectional curvature K_M –1is diffeomorphic to a Euclidean n-space Rⁿ if the volume growthof geodesic balls in M is close to that of the balls in an n-dimensionalhyperbolic space Hⁿ(–1) of sectional curvature –1.     

Hyperkahler Structures and Group Actions

A 4n-dimensional Riemannian manifold (M, g) is hyperkählerif it possesses three anti-commuting complex structures I, J,K such that the metric g is Kähler with respect to eachof them. The reduced holonomy group of such a manifold is necessarilya subgroup of Sp(n) so the Ricci tensor of g vanishes and (M,g) can be regarded as a positive definite solution to Einstein'sequations in vacuum.     

Minimal Lagrangian 2-Tori in CP2 Come in Real Families of Every Dimension

It is shown that for every non-negative integer n, there isa real n-dimensional family of minimal Lagrangian tori in CP²,and hence of special Lagrangian cones in C³ whose link is atorus. The proof utilises the fact that such tori arise fromintegrable systems, and can be described using algebro-geometric(spectral curve) data.     

On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space

Many new universal relations are obtained between the Eulernumbers of manifolds of singular supporting hyperplanes of anarbitrary generic smooth closed k-dimensional submanifold inRⁿ where n 7 or k = 1. These relations are applied to Barner-convexcurves in an odd-dimensional space Rⁿ. A universal (nontrivial)linear relation is established between the numbers of singularsupporting hyperplanes of various types but of the same totalmultiplicity n of tangency with a given generic smooth closedconnected Barner-convex curve in Rⁿ. The coefficients of thisrelation are defined by Catalan numbers.     

Motion planning in tori

Let X be a subcomplex of the standard CW-decomposition of then-dimensional torus. We exhibit an explicit optimal motion planningalgorithm for X. This construction is used to calculate thetopological complexity of complements of general position arrangementsand Eilenberg–Mac Lane spaces associated to right-angledArtin groups.     

Coincidence and The Colouring of Maps

In [8, 6] it was shown that for each k and n such that 2k >n, there exists a contractible k-dimensional complex Y and acontinuous map : Sⁿ Y without the antipodal coincidence property,that is, (x)(–x) for all x Sⁿ. In this paper it is shownthat for each k and n such that 2k > n, and for each fixed-pointfree homeomorphism f of an n-dimensional paracompact Hausdorffspace X onto itself, there is a contractible k-dimensional complexY and a continuous map :X Y such that (x)(f(x)) for all xX.Various results along these lines are obtained. 1991 MathematicsSubject Classication 55M10, 54C05.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

An optimal control problem for the multidimensional diffusion equation with a generalized control variable

The present paper is concerned with an optimal control problem for then-dimensional diffusion equation with a sequence of Radon measures as generalized control variables. Suppose that a desired final state is not reachable. We enlarge the set of admissible controls and provide a solution to the corresponding moment problem for the diffusion equation, so that the previously chosen desired final state is actually reachable by the action of a generalized control. Then, we minimize an objective function in this extended space, which can be characterized as consisting of infinite sequences of Radon measures which satisfy some constraints. Then, we approximate the action of the optimal sequence by that of a control, and finally develop numerical methods to estimate these nearly optimal controls. Several numerical examples are presented to illustrate these ideas.     

ON THE SUM OF THE FIRST n PRIMES

In this note, we show that the set of n such that the arithmeticmean of the first n primes is an integer is of asymptotic densityzero. We use the same method to show that the set of n suchthat the sum of the first n primes is a square is also of asymptoticdensity zero. We also prove that both the arithmetic mean ofthe first n primes as well as the square root of the sum ofthe first n primes are well distributed modulo 1.     

The space clcv(? n ) with the Hausdorff-Bebutov metric and statistically invariant sets of control systems

We obtain conditions that allow one to evaluate the relative frequency of occurrence of the reachable set of a control system in a given set. If the relative frequency of occurrence in this set is 1, then the set is said to be statistically invariant. It is assumed that the images of the right-hand side of the differential inclusion corresponding to the given control system are convex, closed, but not necessarily compact. We also study the basic properties of the space clcv(? ⁿ ) of nonempty closed convex subsets of ? ⁿ with the Hausdorff-Bebutov metric.