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.     

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:

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.     

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.     

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 .     

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.     

The Connectedness of Some Varieties and the Deligne—Simpson Problem

The Deligne—Simpson problem (DSP) (respectively, the weak DSP) is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes C_j GL(n, ) or c_j gl(n, ) so that there exist irreducible (respectively, with trivial centralizer) (p + 1)-tuples of matrices M_j C_j or A_j c_j satisfying the equality M₁ ... M_p+1 = I or A₁ + ... + A_p+1 = 0. The matrices M_j and A_j are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on the Riemann sphere. For ((p + 1))-tuples of conjugacy classes one of which is with distinct eigenvalues we prove that the variety {(M₁, ..., M_p+1) | M_j C_j, M₁ ... M_p+1 = I} or {(A₁, ..., A_p+1) | A_j c_j, A₁ + ... + A_p+1 = 0| is connected if the DSP is positively solved for the given conjugacy classes and give necessary and sufficient conditions for the positive solvability of the weak DSP.2000 Mathematics Subject Classification. 15A30, 15A24, 20G05.     

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.     

The Asymptotic Behavior for the Streaming Operator in Slab Geometry

In this paper, we complete Generation theorem for the streaming operator in slab geometry, where we have proved that the one-dimensional streaming operator in slab domain, with general boundary conditions described by a linear operator K, generates a strongly continuous semigroup. Here, under the positivity and irreducibility of the boundary operator K, we prove the positivity and the irreducibility of the generated semigroup. We also study the spectral properties of the streaming operator and characterize its spectral bound. Under the compactness of the boundary operator K, we determine the essential type of the generated semigroup and find its behavior asymptotic in the operator norm topology.     

Existence and Uniqueness of Optimal Control for a Distributed-Parameter Bilinear System

In this paper, we study an optimal control problem of bilinear type. The system is governed by a fourth-order parabolic operator. The performance index is of the form J(u) = ₀ ^T L(z(t), u(t)) dt. Under suitable hypotheses, it is shown that there exists an optimal control and it satisfies an appropriate optimality system. Further, for a small initial state is unique.     

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.     

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.     

On the C^r-Closing Lemma and the Koebe–Morse Coding of Geodesics on Surfaces

We consider the -closing lemma for vector fields with finitely many singularities on an orientable closed surface M of genus g2. Given a nontrivially recurrent trajectory, there is the corresponding geodesic having the same asymptotic directions (both negative and positive). Using the Koebe–Morse coding for the corresponding geodesic, we introduce the notion of p-expansions in the form of two sequences of nonnegative integers. The main result is the following. Suppose a vector field , r, has a nontrivially recurrent trajectory l through a point m; then there exists arbitrarily close to X (in the -topology) having a periodic trajectory through m provided that the Koebe–Morse coding of the corresponding geodesic g(l) has p-expansions of unrestricted type.     

H ∞ Reliable Control of Uncertain Linear State Delayed Systems

This paper deals with the problem of robust and reliable H control design for linear uncertain time-delay systems with time-varying norm-bounded parameter uncertainty, and also with actuator failures among a specified subset of actuators. A state feedback control design is presented that stabilizes the plant and guarantees an H -norm bound constraint on attenuation of the augmented disturbances, including failure signals, for all admissible uncertainties as well as actuator failures. It is shown that the existence of the desired controllers is related to the positive definite solution of a parameter-dependent Riccati-like matrix equation, whose solving algorithm is also discussed in detail. Two illustrative examples are provided to demonstrate the applicability of the proposed method.     

On the Codimension One Motion Planning Problem

In this paper, we improve the results of [5] related to motion planning problems for corank one sub-Riemannian (SR) metrics. First, we give the exact estimate of the metric complexity, in the generic 3-dimensional case. (Only bounds from above and from below were given in [5].) Second, we show that the general expression for the metric complexity (that was proven to hold generically in the C case, or under certain nonvanishing condition (C) in the analytic case) is, in fact, always true under condition (C), on the complement of a subset of codimension infinity, in the set of C motion planning problems. Both results are constructive, i.e., an asymptotic optimal synthesis is exhibited in both cases.2000 Mathematics Subject Classification. 53C17, 49J15, 34H05.     

Exact Controllability of a Hybrid System, Membrane with Strings, on General Polygon Domains

We use here HUM (cf. Lions [9]–[l0]) to study the Neumann controllability of a two-dimensional hybrid system membrane with strings on general convex polygon domains (cf. Lee and You [1], Littman [11] for a related version of this model). This system is governed by u _tt – u = 0 in on on ₂ × (0,T), u = 0 on ₃ × (0,T); u(A _j ) = 0 if if e _{j 2} and e _{j+1 1}, 0<t<T, and if e _{j 1} and e _{j+1 2}, 0<t<T (see Sec. 1 for notations). An inverse inequality of the energy has been derived when satisfies certain geometric conditions and T is sufficiently large. As a consequence, an exact control in or is respectively obtained. Some other interesting properties (such as the uniqueness of the solution and a Carleman type inequality) of the above problems are also presented.     

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,

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.     

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.     

Sub-Riemannian Geometry: One-Parameter Deformation of the Martinet Flat Case

Consider a sub-Riemannian geometry (U,D,g), where U is a neighborhood of 0 in ³, D is a Martinet type distribution identified to ker , being the one-form and g is a metric on D which can be written as a(q)dx ² + 2b(q)dxdy + c(q)dy ², whereq = (x,y,z). In a previous article [1] we proved that g can be written in a normal form where b 0, a = 1 + yF(q ), c = 1 + G(q ), where . Moreover we analyzed the flat case a = c = 1. In this article we study the following one-parameter deformation of the flat case: a = l, c = (1 + y)² where µ . We parametrize the set of geodesics using elliptic functions. This allows us to compute the trace of the sphere and the wave front of small radius on the plane y = 0. We show that the sphere of small radius is not sub-analytic. This analysis clarifies the role of one of the functional invariants in the normal form.