Covalence sequences of planar vertex-homogeneous maps

Given a cyclic d-tuple of integers at least 3, we consider the class of all 1-ended 3-connected d-valent planar maps such that every vertex manifests this d-tuple as the (clockwise or counterclockwise) cyclic order of covalences of its incident faces. We obtain necessary and/or sufficient conditions for the class to contain a Cayley map, a non-Cayley map whose underlying graph is a Cayley graph, a vertex-transitive graph whose subgroup of orientation-preserving automorphisms acts (or fails to act) vertex-transitively, a non-vertex-transitive map, or no planar map at all.     

Regular Cayley maps for finite abelian groups

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex. Supported in part by the N.Z. Marsden Fund (grant no. UOA0124).     

Balancing Transformations for Infinite-Dimensional Control Systems

In order to facilitate model reduction by balanced truncation, we introduce state space transformations that can be used to construct an ℓ₂-balanced realization of a regular, linear input-ouput map with nuclear Hankel-operator directly from the system generators of an arbitrary, given realization. These balancing transformations are based on factors of the Gramians and, for infinite-dimensional systems, they are usually unbounded operators. Subsequently the ℓ₂-balanced realization can be truncated in a non-trivial way to obtain an approximating, finite-dimensional model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A small minimal blocking set in PG(n, p t ), spanning a (t − 1)-space, is linear

In this paper, we show that a small minimal blocking set with exponent e in PG(n, p ^t ), p prime, spanning a (t/e ? 1)-dimensional space, is an ${\mathbb{F}_{p^e}}$ -linear set, provided that p > 5(t/e)?11. As a corollary, we get that all small minimal blocking sets in PG(n, p ^t ), p prime, p > 5t ? 11, spanning a (t ? 1)-dimensional space, are ${\mathbb{F}_p}$ -linear, hence confirming the linearity conjecture for blocking sets in this particular case.     

Faddeev invariants for central simple algebras over rational function fields

Let p be a prime, k a field, containing a primitive pth root of unity, char k ≠ p. We give an upper bound for the Faddeev index of a central simple algebra of exponent p over the rational function field k(t) in the case where the ramification set of the algebra consists of rational points. This bound depends only on the number of ramification points and in certain cases turns out to be strict. In the case where p =  2 and the ramification set in \({\mathbb{A}_k^1}\) consists of three rational points we compute the Faddeev index, using the language of quadratic forms. Let X be a smooth geometrically irreducible complete curve over k. We show that there exist algebras of exponent p over k(X) with the prescribed Faddeev index, provided there are algebras of exponent p and arbitrarily large index over k. In the last section of the paper we consider another invariant of a central simple algebra of prime exponent p over k(t), the so called Faddeev cyclic length. In certain cases we compute this invariant, using triviality of the divided power operations on central simple cyclic algebras of exponent p.     

Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

A Cayley map is a Cayley graph embedded in an orientable surface such that. the local rotations at every vertex are identical. In this paper, balanced regular Cayley maps for cyclic groups, dihedral groups, and generalized quaternion groups are classified.     

Semilinear evolution equations in Banach spaces

We study the evolution equation u′(t) = Au(t) + J(u(t)), t ? 0, where e^tA is a C₀ semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In Sections 1 and 2 of the paper we suppress the map J and instead consider maps K_t: E → E, t > 0, which heuristically are just e^tAJ. Under certain integrability conditions on the K_t we prove existence and uniqueness of local solutions to the integral equation u(t) = e^tAφ + ∝₀^tK_{t ? s}(u(s)) ds for all φ in E, and investigate the regularity of the solutions. Conditions which insure existence of global solutions are given. In Section 3 we recover the map J from the maps K_t, and show that the generator of the semi-flow on E induced by the integral equation has dense domain. Finally, we apply these results to a large class of examples which includes polynomial perturbations to elliptic operators on a domain in Rⁿ.     

Triangulations of Cayley and Tutte polytopes

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun’s conjecture  , which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t$t$-Cayley   and t$t$-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph.     

Cyclic Vertex-Connectivity of Cayley Graphs Generated by Transposition Trees

Let G be a graph. Then ${T\subseteq V(G)}$ is called a cyclic vertex-cut if G ? T is disconnected and at least two components in G ?T contain a cycle. The cyclic vertex-connectivity is the size of a smallest cyclic vertex-cut. In this paper, we determine this number for Cayley graphs generated by transposition trees as well as classify all the minimum cyclic vertex-cuts.     

Normed Topological Pseudovector Groups

A normed topological pseudovector group (NTPVG for short) is a valued topological group (V,?+?,||·||) (not necessarily Abelian) endowed with a continuous scalar multiplication \({\mathbb R}_+ \times V \ni (t,x) \mapsto t \cdot x \in V\) such that 0 ·x?=?e (e denotes the neutral element of V), 1 ·x?=?x, (st) ·x?=?s ·(t ·x), t ·(x?+?y)?=?(t ·x)?+?(t ·y) and ||t ·x||?=?t ||x|| for each t, \(s \in {\mathbb R}_+\) and x, y?∈?V. It is shown that every valued topological group can be isometrically and group-homomorphically embedded in a NTPVG as a closed subset by means of a functor. Locally compact NTPV groups are fully classified. It is shown that the (unbounded) Urysohn universal metric space can be endowed with a structure of a NTPV group of exponent 2.     

Hyperzyklische gruppen

Definition: (a)G is called hypercyclic «iff each epimorphic imageH≠1 ofG possesses a cyclic normal subgroupA≠1». (b)G is called hypercentral «iff each epimorphic imageH≠1 ofG hasZ(H)≠1». (c) the set of prime numbers which divide the orders of the torsion elements (≠1) ofG is called «the characteristic ofG». Baer has shown that each hypercyclic groupG is a subdirect product of hypercyclic groups of finite characteristic. In this note we will characterize hypercentral groups by abelian torsion groups of finite exponent.     

Self-intersections in combinatorial topology: statistical structure

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m letters, then for large m the distribution of the self-intersection number approaches the Gaussian distribution. The theorem was strongly suggested by a computer experiment with four million curves producing a very nearly Gaussian distribution.     

Simulation de lois conjuguées et application à l'estimation de la transformée de Cramer dans ℝd

Let X, X₁, X₂,… be a sequence of i.i.d. ℝ^d-valued random variables with distribution F. An algorithm for the simulation of random vectors with distribution dF_t (x):= e〈t,x〉dF(x)/(t), where (t):= Ee^ť,X〉 (cf. [2]) is used for the estimation of the Cramer transform H (x):= sup_t (〈t, X〉 − log(t)). This method, which belongs to the class of “acceptance-rejection” techniques, is fast and uses a random sieve on the sequence (X_i)_{i ≥ 1}; it does not assume any prior knowledge on F or . We state the asymptotic properties of this estimator calculated on a n-sample of simulated r.v. 's with distribution F_t. We also present some numerical simulations.     

Regular orientable imbeddings of complete graphs

This paper classifies the regular imbeddings of the complete graphs K_n in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p^e, his examples being Cayley maps based on the finite field F = GF(n). We show that these are the only examples, and that there are $\text{φ(n ? 1)}\text{e}$ isomorphism classes of such maps (where φ is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Φ_{n ? 1}(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations H_i, and we determined for which n they are reflexible or self-dual.     

Computing Teichmüller Maps Between Polygons

By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle-preserving manner). However, when this map is extended to the boundary, it need not necessarily map the vertices of P to those of Q. For many applications, it is important to find the “best” vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichmüller maps. There are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a \((1+\varepsilon )\)-approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichmüller maps. This paper solves the problem of finding a finite-time procedure for approximating Teichmüller maps in the continuous setting. Our construction is via an iterative procedure that is proven to converge in \(O(\text {poly}(1/\varepsilon ))\) iterations to a map whose dilatation is at most \(\varepsilon \) more than that of the Teichmüller map, for any \(\varepsilon >0\). We reduce the problem of finding an approximation algorithm for computing Teichmüller maps to two basic subroutines, namely, computing discrete (1) compositions and (2) inverses of discretely represented quasiconformal maps. Assuming finite-time solvers for these subroutines, we provide an approximation algorithm with an additive error of at most \(\varepsilon \).     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

Groups of finite exponent acting regulary on an abelian group

Let G be a group of squarefree exponent e which acts regularly on some abelian p-group V. If V is the union of a finite number of G-orbits and e divides p ? 1, we show that G is the cyclic group of order e.     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.