Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii’s work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno’s condition (A). In dimension 2, no condition is required since, according to Stró?yna–?o?ladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations.     

On an invariant Möbius measure and the Gauss-Kuzmin face distribution

We study Möbius measures of the manifold of n-dimensional continued fractions in the sense of Klein. By definition any Möbius measure is invariant under the natural action of the group of projective transformations PGL(n + 1) and is an integral of some form of the maximal dimension. It turns out that all Möbius measures are proportional, and the corresponding forms are written explicitly in some special coordinates. The formulae obtained allow one to compare approximately the relative frequencies of the n-dimensional faces of given integer-affine types for n-dimensional continued fractions. In this paper we make numerical calculations of some relative frequencies in the case of n = 2.     

Embedding and splitting ordinary differential equations in normal form

We introduce an embedding of real or complex n-dimensional space Kⁿ as an algebraic variety V which is determined by the action of a linear one-parameter group. Every analytic vector field on Kⁿ corresponds to some embedded vector field on V. For a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non-autonomous linear systems. Examples and applications are mostly concerned with Poincaré-Dulac normal forms. Embeddings provide a natural setting for perturbations of symmetric systems, in particular of systems in normal form up to some degree.     

Polynomial normal forms with exponentially small remainder for analytic vector fields

A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in R^m. More precisely we prove that for any vector field in R^m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small. We also give several examples of non-semi-simple linearization for which this result is still true.     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.     

Uniqueness theorem for integral equations and its application

This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn.     

Complète intégrabilité singulière

We show that an holomorphic vector field in a neighbourhood of its singular point 0 0 ∈ ℂⁿ is analytically normalizable as soon as it has a sufficiently large number of commuting holomorphic vector fields, a sufficiently large number of formal first integrals, and that a Diophantian small divisors condition related to its linear part is satisfied.     

Classification of pairs consisting of a linear and a semilinear map

L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices [5, Chapter XII]. See also Dickson's paper [3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

Convergence of Normal Form Transformations: The Role of Symmetries

We discuss the convergence problem for coordinate transformations which take a given vector field into Poincaré–Dulac normal form. We show that the presence of linear or nonlinear Lie point symmetries can guarantee convergence of these normalizing transformations in a number of scenarios. As an application, we consider a class of bifurcation problems.     

Unique normal forms for area preserving maps near a fixed point with neutral multipliers

We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers ±1 at ? = 0. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain non-degeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.     

Comparing Zagreb indices for connected graphs

It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M₂(G)/m≥M₁(G)/n, where M₁ and M₂ are the first and second Zagreb indices. Hansen and Vuki?evi? proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m−n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M₂/(m−1)>M₁/n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.     

Generation of finite tight frames by Householder transformations

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ?ⁿ or ?ⁿ only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ?ⁿ or ?ⁿ, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?     

Infinite-dimensional triangularization

The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector space, without placing any restrictions on the dimension of the space or on the base field. We define a transformation T of a vector space V to be triangularizable if V has a well-ordered basis such that T sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that the following conditions (among others) are equivalent: (1) T is triangularizable, (2) every finite-dimensional subspace of V is annihilated by $f(T)$ for some polynomial f that factors into linear terms, (3) there is a maximal well-ordered set of subspaces of V that are invariant under T, (4) T can be put into a crude version of the Jordan canonical form. We also show that any finite collection of commuting triangularizable transformations is simultaneously triangularizable, we describe the closure of the set of triangularizable transformations in the standard topology on the algebra of all transformations of V, and we extend to transformations that satisfy a polynomial the classical fact that the double-centralizer of a matrix is the algebra generated by that matrix.     

Average condition number for solving linear equations

We study an average condition number and an average loss of precision for the solution of linear equations and prove that the average case is strongly related to the worst case. This holds if the perturbations of the matrix are measured in Frobenius or spectral norm or componentwise. In particular, for the Frobenius norm we show that one gains about log₂n+0.9 bits on the average as compared to the worst case, n being the dimension of the system of linear equations.     

Generic polynomial vector fields are not integrable

We study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. In particular, using a well-suited presentation of Darboux polynomials at some Darboux point as power series in local Darboux coordinates, it is possible to show, by algebraic means only, that the Jouanolou derivation in four variables has no polynomial first integral for any integer value s ≥ 2 of the parameter.Using direct sums of derivations together with our previous results we show that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables.     

Topologically equivalent N-dimensional isometries

Let G be the group of isometries of the n-sphere, Euclidean n-space, or hyperbolic n-space, the group of similarities of Euclidean n-space, or the group of Möbius transformations of the n-sphere. In each case we attempt to determine the conjugacy classes in G which are amalgamated when we allow conjugation of the elements of G by homeomorphisms of the space on which G acts. We are successful modulo undetermined amalgamation among certain periodic orthogonal transformations.     

Geometry of D’Atri spaces of type k

A Riemannian n-dimensional manifold M is a D’Atri space of type k (or k-D’Atri space), 1 ≤ k ≤ n ? 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D’Atri spaces for all possible k ≥ 1 and the property 1-D’Atri is the D’Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D’Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D’Atri spaces for all k = 1, . . ., l satisfy that ${{\rm{tr}}(R_{v}^{k})}$ , v a unit vector in TM, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D’Atri for all k = 1, . . ., n ? 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D’Atri spaces for all k = 1, . . ., n ? 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D’Atri.