Volume 24, Issue 2

Abstract. In this paper, the authors establish uniform weighted estimates for certain oscillatory singular integrals T$T_{\lambda }$ on Lp(1<p<)$L^p\,(1&lt;p&lt;\infty )$ and H1$H^1$, respectively. The kernel function K$K$ satisfies only an Hq$H_q$-condition for 1<q$1&lt;q\le \infty $ and weight function wAp$w\in A_p$.

Abstract. It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta-function) can be presented as a class of solutions of the Fourier transformed spatially homogeneous linear Boltzmann equation with a special Maxwell-type collision kernel. The result is based on an explicit integral representation of solutions to the Cauchy problem for the Boltzmann equation. Possible applications to the theory of Dirichlet series are briefly discussed.

Abstract. A unital $\ell $-group (G,u)$(G,u)$ is an abelian group G$G$ equipped with a translation-invariant lattice-order and a distinguished element u$u$, called order-unit, whose positive integer multiples eventually dominate each element of G$G$. It is shown that, for direct systems $\mathcal {S}$ and $\mathcal {T}$ of finitely presented unital $\ell $-groups, confluence is a necessary condition for limlim$\lim \mathcal {S} \cong \lim \mathcal {T}$. (Sufficiency is an easy byproduct of a general result). When (G,u)$(G,u)$ is finitely generated we equip it with a sequence (G,u)=(W0,W1,...)$\mathcal {W}_{(G,u)} = (W_{0},W_{1},\ldots )$ of weighted abstract simplicial complexes, where Wt+1$W_{t+1}$ is obtained from Wt$W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of Wt$W_{t}$. We show that the map (G,u)(G,u)$(G,u)\mapsto \mathcal {W}_{(G,u)}$ has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital $\ell $-groups.

Abstract. Using the notion of flat covers and proper flat resolutions, we study modules with periodic flat resolutions. It follows that, equivalently, we may study modules with periodic homology. We specialize our results to the category of modules over integral group ring $\mathbb {Z} \Gamma $, where $\Gamma $ is an arbitrary group. Among other results, we show that if a group $\Gamma $ is in a certain class of groups, then $\Gamma $ has periodic homology of period q$q$ after some steps with the periodicity isomorphisms of homology groups induced by the cap product with an element in Hq(,C)$H^q(\Gamma ,C)$, where C$C$ is the cotorsion envelope of the trivial $\Gamma $-module $\mathbb {Z} $, if and only if it has periodic cohomology of period q$q$ after some steps with the periodicity isomorphisms of cohomology groups induced by the cup product with an element in Hq(,)$H^q(\Gamma ,\mathbb {Z} )$.

Abstract. We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

Abstract. In this paper, we introduce the notion of a free profinite p$p$-ring over a boolean space (over a set). We prove that free profinite p$p$-rings over infinite boolean spaces (over infinite sets) are prime. As a consequence, we obtain that every profinite pro-p$p$-ring with 1 is a continuous homomorphic image of a profinite prime ring with 1. In addition, we construct examples of profinite prime rings with non-open radical. For a profinite prime ring that satisfies a polynomial identity over its centroid, the radical is open. Furthermore, we prove that every profinite ring of prime characteristic is a subring of a profinite prime ring.

Abstract. Recall the well-known notion of a cellular cover e:HG$e: H \longrightarrow G$ from algebraic topology (here for groups and R$R$-modules). The map e$e$ is a homomorphism such that any homomorphism :HG$\phi : H \longrightarrow G$ factors uniquely through e$e$. The existence of cellular covers of groups and R$R$-modules was shown in [14, 31] and a more explicit construction was given in [8, Section 6.3]. We want to base the existence and uniqueness of cellular covers on classical results in group and module theory; see Baer's [2, 3, 4] and Dickson's [13]. With the help of an argument from [8] (partly from [14]) we shift the result from R$R$-modules to the existence of cellular covers over any variety ${\mathcal {V}}$ of groups and get the following result. If G$G$ belongs to some variety ${\mathcal {V}}$, then any cellular cover 1KCG$1\longrightarrow K\longrightarrow C\longrightarrow G$ is a short exact sequence with Cc$C\in {\mathcal {V}}^c$ (and hence K,C,Gc$K,C,G\in {\mathcal {V}}^c$). Here c${\mathcal {V}}^c$ is the variety generated by all central extensions of groups V/V$V/{\mathfrak {z}}V$ with V$V\in {\mathcal {V}}$. This unifies older arguments (from [14, 8]) and adds 20$2^{{\aleph _0}}$ new group varieties. Moreover, the following interesting problem arrives naturally. Can we find group varieties which are not closed under taking cellular covers? The positive answer to this question needs work and application of deep results from combinatorial group theory and on the Schur multiplier; see [29, 34]. We will show that for each prime p>1075$p &gt; 10^{75}$ the Burnside-variety p${\mathcal {B}}_p$ satisfying xp=1$x^p =1$ has cellular covers with free abelian subgroups of infinite rank, thus surely these cellular covers of groups from p${\mathcal {B}}_p$ are outside p${\mathcal {B}}_p$.

Abstract. Given a (symmetric) recurrent local regular Dirichlet form with state space E$E$ and an associated symmetric diffusion Xtt[0,)$\lbrace X_{t}\rbrace _{t\in [0,\infty )}$ on E$E$, we consider a function h$h$ which belongs to the extended Dirichlet space, is harmonic outside F1F2$F_{1}\cup F_{2}$ and equal to a$a$ on F1$F_{1}$ and to b$b$ on F2$F_{2}$, where F1,F2E$F_{1},F_{2}\subset E$ are ($\mathcal {E}$-quasi-)closed sets and a,b$a,b\in \mathbb {R}$, a<b$a&lt;b$. We prove that the time change of the real-valued process h(Xt)t[0,)$\lbrace h(X_{t})\rbrace _{t\in [0,\infty )}$ by the energy measure h$\mu _{\langle h\rangle }$ of h$h$ is a reflecting Brownian motion on [a,b]$[a,b]$. As an application, we also discuss asymptotic analysis of the heat kernel on the harmonic Sierpinski gasket.

Abstract. We give a new construction of a universal G$G$-connection, where G$G$ is a finite dimensional Lie group.

Abstract. Infinite fields are not finitely generated rings. A similar question is considered for further algebraic structures, mainly commutative semirings. In this case, purely algebraic methods fail and topological properties of integral lattice points turn out to be useful. We prove that a commutative semiring that is a group with respect to multiplication can be two-generated only if it belongs to the subclass of additively idempotent semirings; this class is equivalent to $\ell $-groups.

Abstract. Let X$X$ be a complex quasiprojective variety. A result of Noguchi–Winkelmann–Yamanoi shows that if X$X$ admits a Zariski dense entire curve, then its quasi-Albanese map is a fiber space. We show that the orbifold structure induced by a proper birationally equivalent map on the base is special in this case. As a consequence, if X$X$ is of log-general type with q(X)dimX$\bar{q}(X)\ge \dim X$, then any entire curve is contained in a proper subvariety in X$X$.

Abstract. Let Ed$E\subset \mathbb {R}^d$, d2$d\ge 2$, be an unbounded domain that is either open or closed. If it is closed, we assume that the boundary is locally the boundary of an extension domain. We present conservativeness criteria for (possibly reflected) diffusions with state space E$E$ and generator L$L$ which in the interior of E$E$ is given in the following suggestive form: Lf=12i,j=1dj(aijif)+i=1dBiif.$ Lf=\frac{1}{2} \sum _{i,j=1}^{d}\partial _j(a_{ij}\partial _i f)+\sum _{i=1}^{d}B_i\,\partial _i f. $ Here the diffusion matrix (aij)$(a_{ij})$ is allowed to be non-symmetric, is merely assumed to consist of measurable functions, and satisfies locally a strict ellipticity condition. Moreover, B=(B1,...,Bd)$B=(B_1,\ldots ,B_d)$ is a divergence free vector field that satisfies some sector condition. Our main tool is a recently extended forward and backward martingale decomposition, which reduces to the well-known Lyons–Zheng decomposition in the symmetric case.