Volume 29, Issue 4

We prove that the existence of a Hurewicz fibration between certain spaces with the homotopy type of a CW-complex implies some topological restrictions on their universal coverings. This result is used to deduce differentiable and metric properties of maps between compact Riemannian manifolds under curvature restrictions.

Within crystallization theory, two interesting PL invariants for d -manifolds have been introduced and studied, namely, gem-complexity and regular genus . In the present paper we prove that for any closed connected PL 4-manifold M , its gem-complexity k⁢(M){k(M)} and its regular genus 𝒢⁢(M){\mathcal{G}(M)} satisfy k⁢(M)≥3⁢χ⁢(M)+10⁢m-6 and 𝒢⁢(M)≥2⁢χ⁢(M)+5⁢m-4,k(M)\geq 3\chi(M)+10m-6\quad\text{and}\quad\mathcal{G}(M)\geq 2\chi(M)+5m-4, where r⁢k⁢(π1⁢(M))=m{rk(\pi_{1}(M))=m}. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of “standard type”, involved in existing crystallization catalogs, and their connected sums.

Let Lt:=Δt+Zt{L_{t}:=\Delta_{t}+Z_{t}}, t∈[0,Tc){t\in[0,T_{c})} on a differential manifold equipped with a complete geometric flow (gt)t∈[0,Tc){(g_{t})_{t\in[0,T_{c})}}, where Δt{\Delta_{t}} is the Laplacian operator induced by the metric gt{g_{t}} and (Zt)t∈[0,Tc){(Z_{t})_{t\in[0,T_{c})}} is a family of C1,∞{C^{1,\infty}}-vector fields. In this article, we present a number of equivalent inequalities for the lower bound curvature condition, which include gradient inequalities, transportation-cost inequalities, Harnack inequalities and other functional inequalities for the semigroup associated with diffusion processes generated by Lt{L_{t}}. To this end, we establish derivative formulae for the associated semigroup and construct coupling processes for these diffusion processes by parallel displacement and reflection.

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.

In this paper we consider the Milnor fiber F associated to a reduced projective plane curve C . A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of F , also known as the Alexander polynomial of the curve C , is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.

We prove that for any constant K>0{K>0}, there exists a separable group equipped with a complete bi-invariant metric bounded by K , which is isometric to the Urysohn sphere of diameter K and of ‘almost-universal disposition’. It is thus an object in the category of separable groups with bi-invariant metric, analogous in its properties to the Gurarij space from the category of separable Banach spaces. We show that this group contains an isometric copy of any separable group equipped with a bi-invariant metric bounded by K . As a consequence, we get that it is a universal Polish group admitting a compatible bi-invariant metric or a universal second countable SIN group. Moreover, the almost-universal disposition shows that the automorphism group of this group is rich and it characterizes the group uniquely up to isometric isomorphism. We also show that this group is in a certain sense generic in the class of separable groups with bi-invariant metric (bounded by K ). On the other hand, we prove that there is no metrically universal separable group with bi-invariant metric when there is no restriction on the diameter. The same is true for separable locally compact groups with bi-invariant metric. Assuming the generalized continuum hypothesis (GCH), we prove that there exists a metrically universal (unbounded) group of density κ with bi-invariant metric for any uncountable cardinal κ. Moreover, under GCH, we deduce that there exists a universal SIN group of weight κ for any infinite cardinal κ.

We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet L -functions at the critical point, in the function field setting. We fix the ground field 𝔽q{\mathbb{F}_{q}}, and assume for simplicity that q is a prime with q≡1⁢(mod⁢ 4){q\equiv 1~{}(\mathrm{mod}\,4)}. We compute the second and third moment of L⁢(1/2,χD){L(1/2,\chi_{D})}, when D is a monic square-free polynomial of degree 2⁢g+1{2g+1}, as g→∞{g\to\infty}. The answer we get for the second moment agrees with Andrade and Keating’s conjectured formula in [4]. For the third moment, we check that the leading term agrees with the conjecture.

We prove that a one-dimensional Noetherian domain is slender if and only if it is not a local complete ring. The latter condition for a general Noetherian domain characterizes the domains that are not algebraically compact. For a general Noetherian domain R we prove that R is algebraically compact if and only if R satisfies a condition slightly stronger than not being slender. In addition we enlarge considerably the number of classes of rings for which the question of slenderness can be answered. For instance we prove that any domain, not a field, essentially of finite type over a field is slender.

A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random graph, infinite disjoint unions of complete graphs Kn{K_{n}} with n∈ℕ{n\in\mathbb{N}} vertices, the Kn{K_{n}}-free graphs, finite unions of the infinite complete graph Kω{K_{\omega}}, and duals of such graphs. The groups Aut⁡(Γ){\operatorname{Aut}(\Gamma)} of automorphisms of such graphs Γ have a natural topology, which is compatible with multiplication and inversion, i.e. the groups Aut⁡(Γ){\operatorname{Aut}(\Gamma)} are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups Aut⁡(Γ){\operatorname{Aut}(\Gamma)} where Γ is ultrahomogeneous. We show that if Γ is ultrahomogeneous, then Aut⁡(Γ){\operatorname{Aut}(\Gamma)} has 2-generated dense subgroups, and that under certain conditions given f∈Aut⁡(Γ){f\in\operatorname{Aut}(\Gamma)} there exists g∈Aut⁡(Γ){g\in\operatorname{Aut}(\Gamma)} such that the subgroup generated by f and g is dense. We also show that, roughly speaking, g can be chosen with a high degree of freedom. For example, if Γ is either an infinite disjoint union of Kn{K_{n}} or a finite union of Kω{K_{\omega}}, then g can be chosen to have any given finite set of orbit representatives.

We consider a non-negative self-adjoint operator L satisfying generalized Gaussian estimates on a doubling metric measure space, and show that if L has a spectral gap, then the local and global Hardy spaces, defined by means of appropriate square functions, coincide.

We investigate IPA{\mathrm{IPA}}-real closed fields, that is, real closed fields which admit an integer part whose non-negative cone is a model of Peano arithmetic. We show that the value group of an IPA{\mathrm{IPA}}-real closed field is an exponential group in the residue field, and that the converse fails in general. As an application, we classify (up to isomorphism) value groups of countable recursively saturated exponential real closed fields. We exploit this characterization to construct countable exponential real closed fields which are not IPA{\mathrm{IPA}}-real closed fields.

We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space ℍn{\mathbb{H}^{n}}. We are primarily interested in the existence of Weinstein functional maximizers or, in other words, existence of extremal functions for the best constant of the Gagliardo–Nirenberg inequality. The main result is that the supremum of the Weinstein functional on ℍn{\mathbb{H}^{n}} is the same as that on ℝn{\mathbb{R}^{n}} and the related fact that the said supremum is not attained on ℍn{\mathbb{H}^{n}}, when functions are chosen from the Sobolev space H1⁢(ℍn){H^{1}(\mathbb{H}^{n})}. This proves a conjecture made in [8] (see also [3]). We also prove an analogous version of the conjecture for the Weinstein functional defined with the fractional Laplacian.

In this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd square order. We investigate various geometric and combinatorial properties of these planes, such as the self-duality, the existence of O’Nan configurations, Wilbrink’s conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).