Confluence of Several Regular Singular Points into an Irregular Singular One

Consider the systems and , where s and t are variables, is a parameter, and A and B = diag(₁,..., _n ) are n by n matrices. (1) has only regular singular points, and (2) has an irregular singular point at t = . Several kinds of special solutions having particular behavior near singular points were selected in previous papers. In the present paper, the author shows how (2) results from (1) in a process of confluence as . It is analyzed how the special solutions of (1) converge to those of (2) in that process. As a consequence new proofs of earlier results about connection problems are obtained.     

C ∞ and L ∞ Observability of Single-Input C ∞-Systems

For single-input multi-output C -systems, we state conditions under which observability for every C -input implies observability for every almost everywhere continuous, bounded input (for every L -input in the control-affine case). A normal system is then defined as a system whose only bad inputs are smooth on some nonempty open set. When the state space is compact, normality turns out to be generic and enables us to extend some results of genericity of observability to nonsmooth inputs.     

On the Summability of Formal Solutions in Liouville-Green Theory

We consider the second-order differential equation ² y = (1+²(x, ))y with a small parameter , where is even with respect to . It is well known that it has two formal solutions y ^±(x, ) = e ^±x/ h ^±(x, ), where h ^±(x, ) is a formal series in powers of whose coefficients are functions of x.It has been shown [4] that one resp. both of these solutions are 1-summable in certain directions if satisfies certain conditions, in particular, concerning its x-domain. In the present article we give necessary (and sufficient) conditions for 1-summability of one or both of the above formal solutions in terms of .The method of proof involves a certain inverse problem, i.e., the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to .     

Transformations Having Homogeneous Spectra

We show that for a generic automorphism T, the Cartesian product T × T has homogeneous spectrum of multiplicity two. New examples of automorphisms with the property * are presented.     

Mappings of Bounded Φ-Variation with Arbitrary Function Φ

We develop the general theory of mappings of bounded -variation in the sense of L. C. Young that are defined on a subset of the real line and take values in metric or normed spaces. We single out the characterizing properties for these mappings, prove the structural theorem for them, and study their continuity properties. We obtain the existence of a geodesic path of bounded -variation between two points of a compact set with certain regularity of its modulus of continuity. The classical Helly selection principle from the theory of functions of bounded variation is generalized for mappings of bounded -variation. Under natural restrictions on the function , we show that the space of all normed space-valued mappings under consideration can be endowed with a metric. Finally, we consider the problem of existence of selections of a continuous set-valued mapping Fof bounded -variation with respect to the Hausdorff distance. We show that if (0) is finite> 0, then Fhas a continuous selection of bounded -variation; if (0) = , then Fis a constant mapping; and if (0) = 0, then, under additional assumptions on , we give examples of mappings Fwith no continuous selection and with no selection of bounded -variation.     

Groups of Flagged Homotopies and Higher Gauge Theory

Groups _k (X;) of flagged homotopies are introduced of which the usual (abelian for k > 1) homotopy groups _k (X;p) is the limit case for flags contracted to a point p. The calculus of exterior forms with values in an algebra A is developped of which the limit cases are the differential forms calculus (for A = ) and gauge theory (for 1-forms). Moduli space of integrable forms with respect to higher gauge transforms (cohomology with coefficients in A) is introduced with elements giving representations of _k in G = expA.     

Rational Approximations of Transfer Functions of Some Viscoelastic Rods with Applications to Robust Control

We study rational approximations of the transfer function of a uniform or nonuniform viscoelastic rod undergoing torsional vibrations that are excited and measured at the same end. The approximation is to be carried out in a way that is appropriate, with respect to stability and performance, for the construction of suboptimal rational stabilizing compensators for the rod. The function can be expressed as , where g is an infinite product of fractional linear transformations and is a (generally transcendental) function that characterizes a particular viscoelastic material. First, g(²) is approximated by its partial products g _N(²). For relevant values of ², convergence rates for g _N are analyzed in detail. Convergence suitable for our problem requires the introduction of a new irrational convergence factor, which must be approximated separately. In addition, the fractional linear factors in ²(s) that appear in g _N(²(s)) must be replaced by something rational. When the damping is weak it is possible to do this by separating the oscillatory modes from the creep modes and ignoring the latter; in general, this step remains incomplete. Some numerical data illustrating all the stages of the process as well as the final results for various viscoelastic constitutive relations are presented.     

Number of Singularities of a Generic Web on the Complex Projective Plane

Given a generic d-web W_d of degree n in ², we associate with it a triple (S_Wd, |S_Wd, F_Wd), where S_Wd is a surface in T*², the projective cotangent bundle of ², |S_Wd is the restriction of the natural projection T*^{2 2} to S_Wd and F_Wd is a foliation on S_Wd given by a special meromorphic 1-form. The main objective of this article is to calculate the total number of singularities and the sum of the indices of Baum–Bott for the foliation F_Wd in terms of d and n. These results are compared with the case d = 1 (foliation in ²). We also calculate the total number of nodes and cusps of the projection |S_Wd in terms of d and n.2000 Mathematics Subject Classification. Primary: 37F75, Secondary: 34M45.     

Nilpotency of Zinbiel Algebras

Zinbiel algebras are defined by the identity (a b) c = a(bc+cb). We prove an analog of the Nagata–Higman theorem for Zinbiel algebras. We establish that every finite-dimensional Zinbiel algebra over an algebraically closed field is solvable. Every solvable Zinbiel algebra with solvability length N is a nil-algebra with nil-index 2^N if p = char K = 0 or p > 2^N – 1. Conversely, every Zinbiel nil-algebra with nil-index N is solvable with solvability length N if p = 0 or p > N – 1. Every finite-dimensional Zinbiel algebra over complex numbers is nilpotent, nil, and solvable.2000 Mathematics Subject Classification. 17A32.     

On Some Aspects of the Deligne–Simpson Problem

The Deligne-Simpson problem in the multiplicative version is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes C _j GL(n, ), so that there exist irreducible (p + 1)-tuples of matrices M _j C _j satisfying the equality M ₁ . . . M _p+1 = I.We solve the problem for generic eigenvalues in the case where all the numbers _j,m() of Jordan blocks of a given matrix M _j with a given eigenvalue and of a given size m (taken over all j, , and m) are divisible by d > 1. Generic eigenvalues are defined by explicit algebraic inequalities of the form a 0. For such eigenvalues there exist no reducible (p + 1)-tuples.The matrices M _j are interpreted as monodromy operators of regular linear systems on the Riemann sphere.     

Generalized Rolle theorem in ℝ n and ℂ

The Rolle theorem for functions of one real variable asserts that the number of zeros off on a real connected interval can be at most that off plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if [0,1] ⁿ ,tx(t), is a closed smooth spatial curve and L() is the length of its spherical projection on a unit sphere, then for thederived curve [0,1], ⁿ , the following inequality holds: L() L(). For the analytic functionF(z) defined in a neighborhood of a closed plane curve ² this inequality implies that (F) (F) + (), where (F) is the total variation of the argument ofF along , and () is the integral absolute curvature of .As an application of this inequality, we find an upper bound for the number of complex isolated zeros of quasipolynomials. We also establish a two-sided inequality between the variation index (F) and another quantity, called theBernstein index, which is expressed in terms of the modulus growth of an analytic function.     

Integration of Complex Differential Equations

Let X be a polynomial vector field in ²; then it defines an algebraic foliation on P(2). If admits a Liouvillian first integral on P(2), then it is transversely affine outside some algebraic invariant curve S P(2). If, moreover, for some irreducible component S₀ S, the singularities q Sing S are generic, then either is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation This result has several applications such as the study of foliations with algebraic limit sets on P(2)(2), the classification polynomial complete vector fields over ², and topological rigidity of foliations on P(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.     

Output Feedback Robust <Emphasis Type="Italic">H</Emphasis>∞ Control with <Emphasis Type="Italic">D</Emphasis>-Stability and Variance Constraints: Parametrization Approach

In this paper, we study the problem of robust H controller design for uncertain continuous-time systems with variance and D-stability constraints. The parameter uncertainties are allowed to be unstructured but norm-bounded. The aim of this problem is the design of an output feedback controller such that, for all admissible uncertainties, the closed-loop poles be placed within a specified disk, the H norm bound constraint on the disturbance rejection attenuation be guaranteed, and the steady-state variance for each state of the closed-loop system be no more than the prescribed individual upper bound, simultaneously. A parametric design method is exploited to solve the problem addressed. Sufficient conditions for the existence of the desired controllers are derived by using the generalized inverse theory. The analytical expression of the set of desired controllers is also presented. It is shown that the obtained results can be readily extended to the dynamic output feedback case and the discrete-time case.2000 Mathematics Subject Classification. 93E15, 93B36, 93B55.This work was partially supported by the EPSRC under Grant GR/S27658/01, the Nuffield Foundation under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany.     

Extremal Synthesis for Generic Planar Systems

For a generic single-input planar control system we analyze the structure of the set of extremals for the time-optimal problem. Generically all extremals are finite concatenations of regular arcs that are bang or correspond to a smooth feedback. Moreover, the support of extremals is a Whitney stratified set. We collect these information in the definition of extremal synthesis. In the cotangent bundle, we give a topological classification of the singularities of the extremal synthesis and study the projections of the support of extremals (regarded as a two-dimensional object, after normalization) from ² × S ¹ to the plane. With respect to the Whitney classical singularities here we deal with a stratified set with edges and corners, and along with cusps and folds, we find other stable singularities.     

On a Loewner Umbilic-Index Conjecture for Surfaces Immersed in ℝ4

Loewner's umbilic-index conjecture cannot be extended to surfaces immersed in ⁴. More precisely, given n , there exists an analytic surface immersed in ⁴ having an isolated umbilic point of index n/2.     

Newton's Problem of Minimal Resistance for Bodies Containing a Half-Space

We consider Newton's problem of minimal resistance for unbounded bodies in Euclidean space ^d, d 2. A homogeneous flow of noninteracting particles of velocity v falls onto an immovable body containing a half-space {x : (x, n) < 0} ^d, (v, n) < 0. No restriction is imposed on the number of (elastic) collisions of the particles with the body. For any Borel set A {v} of finite measure, consider the flow of cross-section A: the part of initial flow that consists of particles passing through A.We construct a sequence of bodies that minimize resistance to the flow of cross-section A, for arbitrary A. This sequence approximates the half-space; any particle collides with any body of the sequence at most twice. The infimum of resistance is always one half of corresponding resistance of the half-space.     

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

Bifurcations of Cycles in Systems of Differential Equations with a Finite Symmetry Group – I

One-parameter bifurcations of periodic solutions of differential equations in ⁿ with a finite symmetry group are studied. The following three types of periodic solutions x(t) with the symmetry group H are considered separately. F-cycles: H consists of transformations that do not change the periodic solution, h(x(t)) x(t); S-cycles: H consists of transformations that shift the phase of the solution,

Stabilization of Linear Parabolic Equations Defined in an Exterior of a Bounded Domain by a Boundary Control

In this paper, we study the stabilization problem for linear parabolic equation defined in an exterior of a bounded domain. For any positive k > 0 and for a given initial datum we prove the existence of a boundary condition which is the control such, that the solution of this boundary-value problem tends to 0 as t with the rate 1/t ^k .     

Tangent, Quasitangent, and Peritangent Directions to the Trajectories of Differential Inclusions

We introduce several new types of tangent and generalized tangent directions to the trajectories of autonomous differential inclusions and prove some refinements of the existing results on lower estimates and even exact characterizations of these sets in the case where the differential inclusion is either locally Lipschitz or continuously parametrized.