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.     

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,

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.     

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.     

Closed Principal Lines of Surfaces Immersed in the Euclidean 4-Space

In this paper, -principal cycles of surfaces immersed in ⁴ are studied. In terms of geometric invariants, an integral expression for the first derivative of the Poincaré mapping associated with a -principal cycle is obtained.     

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:

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.     

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.     

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.     

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.     

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.     

Normalizable, Integrable, and Linearizable Saddle Points for Complex Quadratic Systems in \mathbb{C}^2

In this paper, we consider complex differential systems in the neighborhood of a singular point with eigenvalues in the ratio 1 : – with . We address the questions of orbital normalizability, normalizability (i.e., convergence of the normalizing transformation), integrability (i.e., orbital linearizability), and linearizability of the system. As for the experimental part of our study, we specialize to quadratic systems and study the values of for which these notions are distinct. For this purpose we give several tools for demonstrating normalizability, integrability, and linearizability.Our main interest is the global organization of the strata of those systems for which the normalizing transformations converge, or for which we have integrable or linearizable saddles as and the other parameters of the system vary. Many of the results are valid in the larger context of polynomial or analytic vector fields. We explain several features of the bifurcation diagram, namely, the existence of a continuous skeleton of integrable (linearizable) systems with sequences of holes filled with orbitally normalizable (normalizable) systems and strata finishing at a particular value of . In particular, we introduce the Ecalle-Voronin invariants of analytic classifcation of a saddle point or the Martinet-Ramis invariants for a saddle-node and illustrate their role as organizing centers of the bifurcation diagram.     

Rigidity Theorem for Degenerated Singular Points of Germs of Holomorphic Vector Fields in the Complex Plane

We consider the class _n of germs of holomorphic vector fields in ( , 0) with vanishing (n – 1)-jet at zero. We prove that the formal equivalence of two generic germs in _n implies their analytic equivalence. This result is analogous to the one obtained in [16] for the case of orbital analytic equivalence.     

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.     

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.     

Limit Cycles for Multidimensional Vector Fields. The Elliptic Case

We consider polynomial vector fields of the form and their polynomial perturbations of degree n. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y ² – x ³ + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R ₁(h)I ₁ + + R _e(h)I _e, where R _j (h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x,y) we estimate the number of zeros of the integral by const n.