Semisymmetric graphs

Let be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection ℘_N is called p-elementary abelian. The projection ℘_N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of the automorphism group of X lifts along ℘_N, and semisymmetric if it is edge- but not vertex-transitive. The projection ℘_N is minimal semisymmetric if it cannot be written as a composition ℘_N=℘℘_M of two (nontrivial) regular covering projections, where ℘_M is semisymmetric.Malni? et al. [Semisymmetric elementary abelian covers of the Möbius-Kantor graph, Discrete Math. 307 (2007) 2156-2175] determined all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, the Generalized Petersen graph GP(8,3), by explicitly giving the corresponding voltage rules generating the covering projections. It was remarked at the end of the above paper that the covering graphs arising from these covering projections need not themselves be semisymmetric (a graph with regular valency is said to be semisymmetric if its automorphism group is edge- but not vertex-transitive). In this paper it is shown that all these covering graphs are indeed semisymmetric.     

The Poincaré series of a simple complete ideal of a two-dimensional regular local ring

The aim of this work is to introduce both a classical and a motivic Poincaré series associated with a residually rational simple complete m-primary ideal ℘ of a two-dimensional regular local ring (R,m). We describe them in terms of the generators of the value semigroup of ℘, and compare them with the Poincaré series arising from a general element f for ℘.     

Generalizations of a lemma of Freudenburg

Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X₁,…,X_n]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂^αg belong to ℘, then there exists c∈k such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1.     

Stability of homogeneous principal bundles with a classical groupas the structure group

Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k and P ⊂ G a parabolic subgroup without any simple factor. Let H be a connected reductive linear algebraic group defined over the field k such that all the simple quotients of H are of classical type. Take any homomorphism π : P → H such that the image of p is not contained in any proper parabolic subgroup of H. Consider the corresponding principal H-bundle E_P(H) = (G × H)/P over G/P. We prove that E_P (H) is strongly stable with respect to any polarization on G/P.     

Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N_R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M_P, as a vector space over the field R_P, is less than or equal to N_R. If a nonzero indecomposable R-module M is such that all the localizations M_P as vector spaces over the fields R_P have the same dimension r, for every minimal prime P of R, then r=1,2,3,4 or 6. Let n be an integer ≥8. We show that if M is an R-module such that the vector space dimensions of the M_P are between n and 2n−8, then M decomposes non-trivially. For each n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

P-orderings: a metric viewpoint and the non-existence of simultaneous orderings

For a prime ideal ℘ and a subset S of a Dedekind ring R, a ℘-ordering of S is a sequence of elements of S with a certain minimizing property. These ℘-orderings were introduced in Bhargava (J. Reine Angew. Math., 490 (1997) 101) to generalize the usual factorial function and many classical results were thereby extended, including results about integer-valued polynomials. We consider ℘-orderings from the viewpoint of the ℘-adic metric on R. We find that the ℘-sequences of S depend only on the closure of S in . When R is a Dedekind domain and R′ is the integral closure of R in a finite extension of the fraction field of R, we relate the ℘-sequences of R and R′. Lastly, we investigate orderings that are simultaneously ℘-orderings for all prime ideals ℘⊂R, and show that such simultaneous orderings do not exist for imaginary quadratic number rings.     

Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/fin has under CH and in the ℵ₂-Cohen model. We prove a similar result in the category of Banach spaces.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

An explicit formula for the action of a finite group on a commutative ring

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G, the trace (or norm) map is defined. is onto if and only if there exists an element x_H such that . We will show that the existence of x_P for every subgroup P of prime order determines the existence of x_G by exhibiting an explicit formula for x_G in terms of the x_P, where P varies over prime order subgroups. Since is onto if and only if is, where g∈G is an arbitrary element, we need to take only one P from each conjugacy class. We will also show why a formula with less factors does not exist, and show that the existence or non-existence of some of the x_P’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

On the reduction of a non-torsion point of a Drinfeld module

Let ρ be a Drinfeld Fq[T]-module defined over a global function field K. Let z∈K be a non-torsion point. We prove that for almost all monic elements n∈Fq[T] there exists a place ℘ of K such that n is the “order” of the reduction of z modulo ℘.     

A direct bijection for the Harer-Zagier formula

We give a combinatorial proof of Harer and Zagier's formula for the disjoint cycle distribution of a long cycle multiplied by an involution with no fixed points, in the symmetric group on a set of even cardinality. The main result of this paper is a direct bijection of a set B_p,k, the enumeration of which is equivalent to the Harer-Zagier formula. The elements of B_p,k are of the form (μ,π), where μ is a pairing on {1,…,2p}, π is a partition into k blocks of the same set, and a certain relation holds between μ and π. (The set partitions π that can appear in B_p,k are called “shift-symmetric”, for reasons that are explained in the paper.) The direct bijection for B_p,k identifies it with a set of objects of the form (ρ,t), where ρ is a pairing on a 2(p-k+1)-subset of {1,…,2p} (a “partial pairing”), and t is an ordered tree with k vertices. If we specialize to the extreme case when p=k-1, then ρ is empty, and our bijection reduces to a well-known tree bijection.     

The six classes of trees with the largest algebraic connectivity

In this paper, we study the algebraic connectivity α(T) of a tree T. We introduce six Classes (C1)-(C6) of trees of order n, and prove that if T is a tree of order n?15, then if and only if , where the equality holds if and only if T is a tree in the Class (C6). At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that α(T_i)>α(T_j) if 1?i<j?6 and T_i is any tree in the Class (Ci) and T_j is any tree in the Class (Cj). We also give the values of the algebraic connectivity of the trees in these six classes. As a technique used in the proofs of the above mentioned results, we also give a complete characterization of the equality case of a well-known relation between the algebraic connectivity of a tree T and the Perron value of the bottleneck matrix of a Perron branch of T.     

The Hilbert stack

Let π:X→S$\pi :X\to S$ be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack—the Hilbert stack—parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.