Affine Embeddings Research Articles

The Bonnet theorem for statistical manifolds

We prove the Bonnet theorem for statistical manifolds, which states that if a statistical manifold admits tensors satisfying the Gauss–Codazzi–Ricci equations, then it is locally embeddable to a flat statistical manifold (or a Hessian manifold). The proof is based on the notion of statistical embedding to the product of a vector space and its dual space introduced by Lauritzen. As another application of Lauritzen’s embedding, we show that a statistical manifold admitting an affine embedding of codimension 1 or 2 is locally embeddable to a flat statistical manifold of the same codimension.

New dimension bounds for αβ sets

In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.

Affine embeddings of Cantor sets in the plane

Let F, E ⊆ ℝ2 be two self-similar sets. First, assuming F is generated by an IFS Ф with strong separation, we characterize the affrne maps g: ℝ2 → ℝ2 such that g(F) ⊆ F. Our analysis depends on the cardinality of the group GФ generated by the orthogonal parts of the similarities in Ф. When |GФ| = ∞ we show that any such self-embedding must be a similarity, and so (by the results of Elekes, Keleti and Maathe [9]) some power of its orthogonal part lies in GФ. When |GФ| = ∞ and Ф has a uniform contraction λ, we show that the linear part of any such embedding is diagonalizable, and the norm of each of its eigenvalues is a rational power of λ. We also study the existence and properties of affrne maps g such that g(F) ⊆ E, where E is generated by an IFS Ψ In this direction, we provide more evidence for a Conjecture of Feng, Huang and Rao [17], that such an embedding exists only if the contraction ratios of the maps in O are algebraically dependent on the contraction ratios of the maps in Ψ Furthermore, we show that, under some conditions, if |GФ| = ∞ then |GΨ| = ∞ and if |GФ| = ∞ then |GΨ lt; ∞.

Affine embeddings of Cantor sets on the line

Let s \in (0,1) , and let F \subset \mathbb R be a self similar set such that 0 < \mathrm {dim}_H F \leq s . We prove that there exists \delta = \delta (s) > 0 such that if F admits an affine embedding into a homogeneous self similar set E and 0 \leq \mathrm {dim}_H E – \leq \mathrm {dim}_H F < \delta then (under some mild conditions on E and F ) the contraction ratios of E and F are logarithmically commensurable. This provides more evidence for Conjecture 1.2 of Feng, Huang, and Rao [7], that states that these contraction ratios are logarithmically commensurable whenever F admits an affine embedding into E (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao in [7] with a new result by Hochman [10], which is related to the increase of entropy of measures under convolutions.

Affine embeddings of Cantor sets and dimension of αβ-sets

Let E,F ⊂ Rd be two self-similar sets, and suppose that F can be affinely embedded into E. Under the assumption that E is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of E and F. This gives a partial affirmative answer to Conjecture 1.2 in [9]. The proof is based on our study of the boxcounting dimension of a class of multi-rotation invariant sets on the unit circle, including the αβ-sets initially studied by Engelking and Katznelson.

SELF-SIMILAR SUBSETS OF SYMMETRIC CANTOR SETS

This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set [Formula: see text] can be affinely embedded into a symmetric Cantor set [Formula: see text], then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062–1079].

Affine analogues of the Sasaki-Shchepetilov connection

AbstractFor two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.

Affine embeddings and intersections of Cantor sets

Let E,F⊂Rd be two self-similar sets. Under mild conditions, we show that F can be C1-embedded into E if and only if it can be affinely embedded into E; furthermore if F cannot be affinely embedded into E, then the Hausdorff dimension of the intersection E∩f(F) is strictly less than that of F for any C1-diffeomorphism f on Rd. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of E and F if F can be affinely embedded into E. As an application, we show that dimH⁡E∩f(F)<min⁡{dimH⁡E,dimH⁡F} when E is any Cantor-p set and F any Cantor-q set, where p,q⩾2 are two integers with log⁡p/log⁡q∉Q. This is related to a conjecture of Furstenberg about the intersections of Cantor sets.

From the attempt of certain classical reformulations of quantum mechanics to quasi-probability representations

The concept of an injective affine embedding of the quantum states into a set of classical states, i.e., into the set of the probability measures on some measurable space, as well as its relation to statistically complete observables is revisited, and its limitation in view of a classical reformulation of the statistical scheme of quantum mechanics is discussed. In particular, on the basis of a theorem concerning a non-denseness property of a set of coexistent effects, it is shown that an injective classical embedding of the quantum states cannot be supplemented by an at least approximate classical description of the quantum mechanical effects. As an alternative approach, the concept of quasi-probability representations of quantum mechanics is considered.

A semi-linear group which is not affine

In this short note we provide an example of a semi-linear group G which does not admit a semi-linear affine embedding; in other words, there is no semi-linear isomorphism between topological groups f : G → G ′ ⊆ M m , such that the group topology on G ′ coincides with the subspace topology induced by M m .

Minimal affine coordinates for [formula omitted] character varieties of free groups

Let X r be the moduli of SL ( 3 , C ) representations of a rank r free group. In this paper we determine minimal generators of the coordinate ring of X r . This at once gives explicit global coordinates for the moduli and determines the dimension of the moduli's minimal affine embedding. Along the way, we utilize results concerning the moduli of r-tuples of matrices in gl ( 3 , C ) . Consequently, we also state general invariant theoretic correspondences between the coordinate rings of the moduli of r-tuples of elements in gl ( n , C ) , sl ( n , C ) , and SL ( n , C ) .

Locating Servers for Reliability and Affine Embeddings

Consider the problem of locating servers in a network for the purpose of storing data, performing an application, etc., so that at least one server will be available to clients even if up to $k$ component failures occur throughout the network. Letting $G = (V,E)$ be the graph with vertex set $V$ and edge set $E$ representing the topology of the network, and letting $L \subseteq V$ be a set of potential locations for the servers, a fundamental problem is to determine a minimum-size set $S \subseteq L$ such that the network remains connected to $S$ even if up to $k$ component failures occur throughout the network. We say that such a set $S$ is $k$-fault-tolerant. In this paper we present an algebraic characterization of $k$-fault-tolerant sets in terms of affine embeddings of $G$ in $k$-dimensional Euclidean space. Employing this characterization, we present a polynomial-time Monte Carlo algorithm for computing a minimum-size $k$-fault-tolerant subset $S$ of $L$. In fact, we solve the following more general problem for directed networks: given a digraph $G = (V,E)$ (an undirected graph is equivalent to a symmetric digraph) and a subset $L \subseteq V$, we find a $k$-fault-tolerant subset $S$ of $L$ having minimum cost, where a unary integer cost $c(v)$ is associated with locating a server at vertex $v \in V$.

On Affinely Closed, Homogeneous Spaces

Affinely closed, homogeneous spaces G/H, i.e., affine homogeneous spaces that admit only the trivial affine embedding, are characterized for an arbitrary affine algebraic group G. A description of affine G-algebras with finitely generated invariant subalgebras is obtained.

Quadratic functions on torsion groups

We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characteristic form, a quadratic function on a torsion group. When the associated symmetric bilinear pairing is fixed, we construct an affine embedding of a quotient of the set of characteristic forms into the set of all quadratic functions and determine explicitly its cokernel. We determine a suitable class of torsion groups so that quadratic functions defined on them are classified by the stable class of their lift. This refines results due to A.H. Durfee, V. Nikulin, and E. Looijenga and J. Wahl. Finally, we show that on this class of torsion groups, two quadratic functions q , q ′ are isomorphic if and only if they have equal associated Gauss sums and there is an isomorphism between the associated symmetric bilinear pairings b q and b q ′ which sends d q to d q ′ , where d q is the homomorphism defined by d q ( x ) = q ( x ) - q ( - x ) . This generalizes a classical result due to V. Nikulin. Our results are elementary in nature and motivated by low-dimensional topology.

Algebras with finitely generated invariant subalgebras

We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

Reductive embeddings are Cohen-Macaulay

In this paper we prove that in positive characteristics normal embeddings of connected reductive groups are Frobenius split. As a consequence, they have rational singularities and are thus Cohen-Macaulay varieties. As an application, we study the particular case of reductive monoids, which are affine embeddings of their unit group. In particular, we show that the algebra of regular functions of a normal irreducible reductive monoid M has a good filtration for the action of the unit group of M.

On modality and complexity of affine embeddings

Let G be a reductive algebraic group and let H be a reductive subgroup of G. The modality of a G-variety X is the largest number of the parameters in a continuous family of G-orbits in X. A precise formula for the maximum value of the modality over all affine embeddings of the homogeneous space G/H is obtained.

Affine embeddings with a finite number of orbits

LetG be a reductive algebraic group and letH be a reductive subgroup ofG. We describe all pairs (G, H) such that, for any affineG-varietyX with a denseG-orbit isomorphic toG/H, the number ofG-orbits inX is finite.

Nonlinear affine embedding of the Dirac field from the multiplicity-free SL(4, ) unirreps

The correspondence between the linear multiplicity-free unirreps of SL(4, studied by Ne'eman and Sijacki and the nonlinear realizations of the affine group is derived. The results obtained clarify the inclusion of spinorial fields in a nonlinear affine gauge theory of gravitation.

The Symmetry of the Weierstrass Generalized Semigroups and Affine Embeddings

The Symmetry of the Weierstrass Generalized Semigroups and Affine Embeddings