Automorphism groups of power functions

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous space. The naturally reductive case is also treated. This completes the geometric calculation of the isometry group of naturally reductive spaces. In addition, we prove that for normal homogeneous spaces the set of fixed points of the full isotropy is a torus. As an application of our results it follows that the holonomy group of a homogeneous fibration is contained in the group of (canonically) affine transformations of the fibers, in particular this holonomy group is a Lie group (this is a result of Guijarro and Walschap).

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transformations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.

Recently Cunningham and Valentine gave in (3) an axiomatic description of the one-dimensional real affine space in terms of its order structure and the (abstract) group of affine transformations It is the purpose of the present note to show that the system of axioms in (3) (cf. (L. 1)–(L. 5) of this note) leads in a natural way to a model of the real number field. Our method is suggested by a result of Hall ((4), p. 382), namely, that an infinite doubly transitive Frobenius group is isomorphic to the group of affine transformations in a near-field, provided that there is at most one transformation displacing all points and taking a given point a into a given point b. The salient point of our investigation is the redundancy of the latter condition in the case where the underlying space is endowed with a certain linear order structure which is invariant under the transformations of the given group.

The paper puts forward two theorems underlying a method of coordination of the descriptions of contours for the equivalence class with the group of affine transformations. High speed performance of the method is achieved by excluding the calculations of similarity estimates. Fundamentals of the method are explained by examples of coordination of the descriptions of contours of trapezoids.

Let M be a compact complex manifold and ∇ an arbitrary complex (not necessarily Riemannian) connection. In this paper we study the relation between the geometry of (M, ∇) and the topology of M, that is, we are interested in the following problem: To what extent does the topology of M determine the relations between the group of holomorphically projective transformations, the group of projective transformations and the group of affine transformations on M? Under assumptions on the Ricci-type tensors of ∇ and Chern numbers of M we show that a holomorphically projective transformation and a projective transformation are in fact affine transformations on M. A family of interesting examples of connections of this kind are constructed. Also, the case when M is a Kähler manifold is studied.

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$be a countable group of $\nu$-preserving invertible maps of $X$ intoitself. To a probability measure $\mu$ on $\Gamma$ corresponds arandom walk on $X$ with Markov operator $P$ given by $P\psi(x) =\sum_{a} \psi(ax) \, \mu(a)$. We consider various examples ofergodic $\Gamma$-actions and random walks and their extensions by avector space: groups of automorphisms or affine transformations oncompact nilmanifolds, random walks in random scenery on non amenablegroups, translations on homogeneous spaces of simple Lie groups,random walks on motion groups. A powerful tool in this study is thespectral gap property for the operator $P$ when it holds. We use itto obtain limit theorems, recurrence/transience property andergodicity for random walks on non compact extensions of thecorresponding dynamical systems.

A task in statistics is to find meaningful associations or dependencies between multivariate random variables or in multivariate, time-dependent stochastic processes. Hawkes (1971) introduced the powerful multivariate point process model of mutually exciting processes (Hawkes model) to explain causal structure in data. Therefore, we discuss several causality concepts and show that causal structure is fully encoded in the corresponding Hawkes kernels. Hence, for causal inference and for establishing graphical models induced by causality it is necessary to estimate the Hawkes kernels. We provide a nonparametric, consistent and asymptotically normal estimator of the Hawkes kernels depending on the increments on a time scale with mesh $\Delta$ using methods from infinite order regression and time series analysis. To illustrate our results we apply our method to EEG data from the spinal dorsal horn of a rat. To tackle the problem for random samples of random vectors we examine a new dependence measure, namely distance correlation (Sz\'ekely, Rizzo and Bakirov; 2007). Distance correlation provides a strikingly simple sample version in order to test for independence between two random vectors of arbitrary dimensions and finite first moments. However, distance correlation is not well understood on the population side and it fails to be invariant under the group of all invertible affine transformations. Hence, we introduce the affinely invariant distance correlation and compute the analytic usual distance correlation and affinely invariant distance correlation in various settings: for multivariate normal distributions and for Lancaster probabilities (e.g. the bivariate gamma distribution) explicitly. Furthermore, we generalize an integral which is at the core of distance correlation.

We prove the covering radius of the third-order Reed-Muller code RM(3, 7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of RM(6, 6)/RM(3, 6), we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove that any function in the remaining 4 types can be transformed into a type (6,10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve the third-order nonlinearity 21 (with the assistance of computers). By the way, we prove that the affine transformation group over any finite field can be generated by two elements.

We present a new pattern similarity measure that behaves well under affine transformations. Our similarity measure is useful for pattern matching since it is defined on patterns with multiple components, satisfies the metric properties, is invariant under affine transformations, and is robust with respect to perturbation and occlusion. We give an algorithm, based on hierarchical subdivision of transformation space, which minimises our measure under the group of affine transformations, given two patterns. In addition, we present results obtained using an implementation of this algorithm.

The standard closed convex hull of a set is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. In this paper we propose a generalisation of this classical notion, that we call a (K, ℍ)-hull, and which is obtained from the above construction by replacing a half-space with some other closed convex subset K of the Euclidean space, and a group of rigid motions by a subset ℍ of the group of invertible affine transformations. The main focus is on the analysis of (K, ℍ)-convex hulls of random samples from K.

Maximally positive polynomial systems supported on circuits

Monomial Modular Representations and Construction of the Held Group

We apply a construction of G. A. Margulis to show that there exists a free non-abelian properly discontinuous group of affine transformations of $$\mathbb {R}^3$$ with both linear and translational parts having integer entries and acting on $$\mathbb {R}^3$$ without fixed points.

This paper is devoted to the substantiation of a criterion for the quasisymmetric conjugacy of an arbitrary group of homeomorphisms of the real line to a group of affine transformations (the Ahlfors problem). In a criterion suggested by Hinkkanen the constants in the definition of a quasisymmetric homeomorphism were assumed to be uniformly bounded for all elements of the group. Subsequently, for orientation-preserving groups this author put forward a more relaxed criterion, in which one assumes only the uniform boundedness of constants for each cyclic subgroup. In the present paper this relaxed criterion is proved for an arbitrary group of line homeomorphisms, which do not necessarily preserve the orientation. Bibliography: 4 titles. Introduction A homeomorphism g : R → R is said to be quasisymmetric [1] if it satisfies the condition M−1 g g(x+ t)− g(x) g(x) − g(x− t) Mg . (1) If g is a quasisymmetric homeomorphism, then the homeomorphism g−1 is also quasisymmetric. For arbitrary quasisymmetric homomorphisms g1 and g2 their composite is also a quasisymmetric homeomorphism and Mg1g2 Mg1Mg2 [1]. Since the constant Mg in condition (1) for a homeomorphism g is not unique, this means that there exists a constant Mg1g2 for the homeomorphism g1g2 such that the inequality holds. We say that a group G consisting of quasisymmetric homeomorphisms is quasisymmetric. The following basic result for a quasisymmetric group of line homeomorphisms was obtained in [2]. Theorem 1. Let G be a group of line homeomorphisms. Then a quasisymmetric homeomorphism η such that η ◦G ◦ η−1 is a group of affine transformations exists if and only if G is a quasisymmetric group such that Mg = M for all g ∈ G, where M is a fixed constant. This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 53-01-00174) and the Programme of Support of Leading Scientific Schools of RF (grant no. NSh-457.2003.1). AMS 2000 Mathematics Subject Classification. Primary 54H15; Secondary 20F38, 28D99.