Volume 27, Issue 5

We classify by numerical invariants the finite subgroups H of a primary abelian group G for which every homomorphism or monomorphism of H into G , or every endomorphism of H , extends to an endomorphism of G . We apply these results to show that for finitely generated subgroups of general abelian groups, the extendibility of monomorphisms implies the extendibility of all homomorphisms.

This paper introduces the frame of partially defined real numbers and the lattice-ordered ring of partial real functions on a frame. This is then used to construct the order completion of rings of pointfree continuous real functions. The bounded and integer-valued cases are also analysed. The application of this pointfree approach to the classical case C( X ) of the ring of continuous real-valued functions on a topological space X yields a new construction for the Dedekind completion of C( X ), considerably more direct and natural than the known procedure using Hausdorff continuous functions.

We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we introduce a new notion of left coreflexivity for counital coalgebras, namely, a coalgebra is left coreflexive if C is isomorphic canonically to the finite dual of its left rational dual Rat (C*C*)${{\rm Rat}(_{C^*}C^*)}$. We show that right semiperfectness for coalgebras is in fact essentially equivalent to this left reflexivity condition, and we give the connection to usual coreflexivity. As application, we give a generalization of some recent results connecting dual objects such as quiver or incidence algebras and coalgebras, and show that Hopf algebras with non-zero integrals (compact quantum groups) are coreflexive.

In this paper we establish a limit theorem for contractive means on a Hadamard space. For a contractive mean satisfying an extended metric inequality, the corresponding geometric power means varying over t∈(0,1]${t\in (0,1]}$ converge to the Karcher mean (center of gravity or Cartan mean) as t→0+.${t\rightarrow 0^{+}.}$ This provides not only a deterministic approach to the Karcher mean, but also a simple proof of some important properties of the Karcher mean, which have been established by Sturm using probabilistic methods on the metric structure of Hadamard spaces. We obtain further new results for the Karcher mean via this deterministic method, especially we derive a characteristic property of the Karcher mean and effective and explicit upper bounds for distances between Karcher means.

The maximal regularity properties of parameter dependent abstract convolution-elliptic equations are investigated. Here find sufficient conditions that guarantee the uniform separability of these problems in L p spaces. It is established that the corresponding convolution-elliptic operator is sectorial and is also a generator of an analytic semigroup. Finally, these results applied to obtain the uniform maximal regularity for the Cauchy problem for abstract parabolic equation in mixed L𝐩${L_{\mathbf {p}}}$ norms, boundary value problems for anisotropic integro-differential equations and infinite systems of elliptic integro-differential equations with parameters.

Let C be a model category with an initial object ⌀${\varnothing }$ and functorial factorizations. Let S:C→C${S : C \rightarrow C}$ be the suspension functor. An object X of C is said to be charged if it comes equipped with a map S⌀→X${S \varnothing \rightarrow X}$. If Y is any object of C , then S Y has a preferred charge, given by applying suspension to the map ⌀→Y${\varnothing \rightarrow Y}$. This motivates the question of whether a given charged object is a suspension up to a weak equivalence in a way that preserves charge structures. We study this question in the context of spaces over a given space, where we give a complete obstruction in a certain metastable range. As an application we show how this can be used to study when an embedding into a smooth manifold of the form N × I compresses to an embedding into N .

In this article, we explore new aspects in the classification of non-associative division algebras. By a detailed study of the representations of the Lie group of autotopies of real division algebras we show that, if the group of autotopies has a sufficiently rich structure, then the algebra is isotopic to one of the classical real division algebras. This turns out to be the case for large classes of real division algebras, including many that are defined by identities. In several cases, a classification up to isomorphism can be worked out from this information.

Let C be a semidualizing complex over a noetherian local ring A . If there exists a local homomorphism A→B${A \rightarrow B}$ vanishing at the level of the first Koszul homology modules (e.g. the Frobenius endomorphism in positive characteristic, or any homomorphism factorizing through a regular local ring) and of finite Gorenstein dimension relative to C , then C is dualizing.

In this paper, we study oscillation and asymptotic behaviour of higher-order neutral delay dynamic equations, and establish comparison with first-order delay dynamic equations. We also present some examples to show applicability and significance of the new results.

Let L1,L2${L_1,L_2}$ be a pair of one-to-one commuting sectorial operators such that each L i for i∈{1,2}${i\in \lbrace 1,2\rbrace }$ satisfies the m i order L 2 off-diagonal estimates and m1≥m2>0${m_1\ge m_2>0}$. Let HLip(ℝn)${H_{L_i}^p(\mathbb {R}^n)}$, i∈{1,2}${i\in \lbrace 1,2\rbrace }$, and HL1+L˜2p(ℝn)${H_{L_1+\widetilde{L}_2}^p(\mathbb {R}^n)}$ be the Hardy spaces associated, respectively, to the operators L i and L1+L˜2${L_1+\widetilde{L}_2}$, where L˜2:=L2m1/m2${\widetilde{L}_2:=L_2^{m_1/m_2}}$ is a fractional power of L 2 . In this paper, the authors give out some real-variable properties of these Hardy spaces. More precisely, the authors first establish the bounded joint H ∞ functional calculus in these Hardy spaces and prove that the abstract Riesz transform Dmi(L1+L2)-1/2${D^{m_i}(L_1+L_2)^{-1/2}}$ is bounded from HLip(ℝn)${H_{L_i}^p(\mathbb {R}^n)}$ to the classical Hardy space Hp(ℝn)${H^p(\mathbb {R}^n)}$ for all p∈(nn+mi,1]${p\in (\frac{n}{n+m_i},1]}$, where i∈{1,2}${i\in \lbrace 1,2\rbrace }$. Moreover, for all p∈(0,1]${p\in (0, 1]}$, the authors show that HL1+L˜2p(ℝn)=HL1p(ℝn)+HL2p(ℝn)$ {H_{L_1+\widetilde{L}_2}^p(\mathbb {R}^n)= H_{L_1}^p(\mathbb {R}^n)+H_{L_2}^p(\mathbb {R}^n)}\vspace*{-1.42262pt} $ and give a sufficient condition to guarantee HL1p(ℝn)⊂HL2p(ℝn)${H_{L_1}^p(\mathbb {R}^n)\subset H_{L_2}^p(\mathbb {R}^n)}$.

In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann–Liouville and Weyl.

We show that the sporadic simple group M(22), the exceptional group of Lie type 2 E 6 (2) and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group.

We define an Artin group of circuit rank 1 as an Artin group based on a connected graph with fundamental group a free group of rank 1. For an Artin group G of circuit rank 1 the Bieri–Neumann–Renz–Strebel invariant Σ1(G,ℤ)${\Sigma ^1(G, \mathbb {Z})}$ is calculated.

We compute the average Tristram–Levine signature of any graph link with positive weights in a three sphere. The main tools are Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.

The classical theorem of Peetre characterizes linear partial differential operators. We generalize this result for operators between spaces of C r -functions defined on a finite dimensional domain with values in infinite dimensional locally convex vector spaces. We also give a variant of the theorem for continuous functions on infinite dimensional metrizable domains.

It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

We introduce and study a new class of homotopy spheres called Farrell–Jones spheres. Using Farrell–Jones sphere we construct examples of closed negatively curved manifolds M2n${M^{2n}}$, where n = 7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C. S. Aravinda and F. T. Farrell. We show that every exotic sphere not bounding a spin manifold (Hitchin sphere) is a Farrell–Jones sphere. We also discuss the relationship between inertia groups of ℂℙn${\mathbb {C}\mathbb {P}^n}$ and Farrell–Jones spheres.

Let q be a cube-free positive integer and χ(modq)${\chi ~(\text{mod}~q)}$ be a non-principal Dirichlet character. Our main result is a Burgess-type estimate for ∑n∈Aχ(n)${\sum _{n\in A}\chi (n)}$, where A⊂[1,q]${A\subset [1,q]}$ is the union of s disjoint intervals I1,...,Is${I_1,\ldots ,I_s}$. We obtain a nontrivial estimate for the character sum over A whenever |A|s-1/2>q1/4+ϵ${|A|s^{-1/2}>q^{1/4+\epsilon }}$ and each interval I j (1≤j≤s${1\le j\le s}$) has length |Ij|>qϵ${|I_j|>q^{\epsilon }}$ for any ϵ>0${\epsilon >0}$. This follows from an improvement of a mean value Burgess-type estimate studied by Heath-Brown [Number Theory and Related Fields, Springer Proc. Math. Statist. 43, New York (2013), 199–213].

We observe interesting interactions among isolated orderings of groups, exceptional prime segments of cones, and exceptional prime ideals of chain domains. We study the structure of certain isolated left orderings and their cones, and use this to construct a new family of chain domains with prime but not completely prime ideals.

We reprove Wolff's L 5/2 -bound for the ℝ 3 -Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right.

In this paper, we discussed about the boundedness of the fractional type Marcinkiewicz integral operators, and improved a result given by Chen, Fan and Ying in 2002. They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel–Lizorkin spaces F˙pqα(ℝn)${\dot{F}_{pq}^{\alpha }({\mathbb {R}^n})}$ to Lp(ℝn)${L^p({\mathbb {R}^n})}$. We greatly weakened their assumptions and extended their results to a pretty much more general way.

Let U be an open set of ℝ n , m be a positive Radon measure on U such that supp [m]=U${{\rm supp}[m]=U}$, and (Pt)t>0${(P_t)_{t>0}}$ be a strongly continuous contraction sub-Markovian semigroup on L2(U;m)${L^2(U;m)}$. We investigate the structure of (Pt)t>0${(P_t)_{t>0}}$. (i) Denote respectively by (A,D(A))${(A,D(A))}$ and (A^,D(A^))${(\hat{A},D(\hat{A}))}$ the generator and the co-generator of (Pt)t>0${(P_t)_{t>0}}$. Under the assumption that C0∞(U)⊂D(A)∩D(A^)${C^{\infty }_0(U)\subset D(A)\cap D(\hat{A})}$, we give an explicit Lévy–Khintchine type representation of A on C0∞(U)${C^{\infty }_0(U)}$. (ii) If (Pt)t>0${(P_t)_{t>0}}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form (ℰ,D(ℰ))${({\cal E}, D({\cal E}))}$, we give an explicit characterization of ℰ on C0∞(U)${C^{\infty }_0(U)}$ under the assumption that C0∞(U)⊂D(ℰ)${C^{\infty }_0(U)\subset D({\cal E})}$. We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.

In this paper, we consider two related problems concerning homogeneous (α,β)-metrics. In the first part we consider homogeneous (α,β)-spaces of Douglas type. We prove that a homogeneous (α,β)-metric is a Douglas metric if and only if either F is a Berwald metric or F is a Douglas metric of Randers type. In the second part, we prove that if F is a homogeneous (α,β)-metric which is neither a Riemannian metric nor a Minkowski metric, then F is locally projectively flat if and only if F is a locally projectively flat left invariant Randers metric on the hyperbolic space 𝐇n${\mathbf {H}^n}$ as a solvable Lie group. We also give all the explicit forms of the metric F in the second case. This result also provides new examples of locally projectively flat metrics which have not been described before in the literature, presenting some new insight in the study of the Hilbert's fourth problem.