Regular quantizations and covering maps

Let $\tilde{M} \rightarrow MLet be a holomorphic (unbranched) covering map between two compact complex manifolds, with . We prove that if and M both admit regular K?hler forms and ω respectively then, up to homotheties, and (M, ω) are biholomorphically isometric. This work was supported by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

Regular versus continuous rational maps

We prove that a continuous map from a compact nonsingular real algebraic variety X into the unit 2-sphere can be approximated by regular maps if and only if it is homotopic to a continuous map which is regular in the complement of a Zariski closed subvariety A of X of codimension at least 3. The assumption on the codimension of A is essential.     

Regular maps on non-orientable surfaces

It is well known that regular maps exist on the projective plane but not on the Klein bottle, nor the non-orientable surface of genus 3. In this paper several infinite families of regular maps are constructed to show that such maps exist on non-orientable surfaces of over 77 per cent of all possible genera.     

Regular maps of 2-power order

Journal of Algebraic Combinatorics - In this paper, we consider the possible types of regular maps of order $$2^n$$ , where the order of a regular map is the order of automorphism group of the map....     

Regular incidence quasi-polytopes and regular maps

We define incidence quasi-polytopes and give a procedure for constructing regular incidence quasi-polytopes. We use this procedure to construct a finite map of type {a,b} for all even a and b, and infinitely many such maps when a or b is divisible by 4 and both are greater than or equal to 4.     

Regular cyclic coverings of regular affine maps

The regular coverings of regular affine algebraic maps are considered, and a large family of totally ramified coverings—the so-called Steinberg and Accola coverings—are fully classified.     

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).     

Regular components of moduli spaces of stable maps

We construct regular components of the moduli space of stable maps from curves of genus to a product of two projective spaces. These components are generically smooth and have the expected dimension predicted by deformation theory. This result can be seen as a general position theorem for loci in consisting of curves carrying exceptional linear series.

     

A combinatorial way of counting unicellular maps and constellations

Our work is devoted to the bijective enumeration of the set of factorizations of a permutation into m factors with a given number of cycles. Previously, this major problem in combinatorics and its various specializations were considered mainly from the character theoretic or algebraic geometry point of view. Let us specially mention here the works of Harer and Zagier or Kontsevich. In 1988, Jackson reported a very general formula solving this problem. However, to the author’s own admission this result left little room for combinatorial interpretation and no bijective proof of it was known yet. In 2001, Lass found a combinatorial proof of the celebrated special case of Jackson’s formula known as the Harer–Zagier formula. This work was followed by Goulden and Nica, who presented in 2004 another combinatorial proof involving a direct bijection. In the past two years, we have introduced new sets of objects called partitioned maps and partitioned cacti, the enumeration of which allowed us to construct bijective proofs for more general cases of Jackson’s formula.     

Regular abelian Banach algebras of linear maps of operator algebras

With H a complex Hilbert space we study regular abelian Banach subalgebras of the Banach algebra of bounded linear maps of B(H) into itself. If a ? b denotes the map x → axb, a, b, x ? B(H), it is shown that normalized positive maps in algebras of the form A ? A with A an abelian C¹-algebra, can be described by a generalized Bochner theorem.     

Regular or stochastic dynamics in real analytic families of unimodal maps

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, hybrid classes. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family. To Jacob Palis on his 60th birthdayMathematics Subject Classification (2000) 37E05, 37F45, 30D05     

Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions

A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in , the Jack parameter shifted by . More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families.

The coefficients of a second series, essentially the logarithm of the first, specialize at values and of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in , and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.

     

Some combinatorial properties of words in discrete dynamical systems from antisymmetric cubic maps

In this paper, we study the combinatorial properties ol words m chscrete dynamical systems from antisymmetric cubic maps. We also discuss the relationship of primitive kneading sequences of length n and period-doubling kneading sequences of length 2n, and then determine the number of all kneading sequences of length n.     

Regular maps whose groups do not act faithfully on vertices,edges, or faces

Regular maps whose automorphism groups do not have faithful action on vertices, edges, or faces are analysed in detail.     

A combinatorial proof of a combinatorial theorem

Acta Mathematica Hungarica -