Authors:Ke Feng Liu; Yong Wang Pages: 455 - 469 Abstract: By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on (4r − 1)-dimensional manifolds with no assumption that the 3rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on (4r − 1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on (4r − 1)-dimensional manifolds and the Han–Yu rigidity theorem to the (a, b) case. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-5714-x Issue No:Vol. 33, No. 4 (2017)

Authors:Yi Jie Lin; Jian Zhou Pages: 470 - 494 Abstract: We propose an algorithm to derive tautological relations from Pixton relations. We carry out this algorithm explicitly to derive some results in genus 0, 1, 2, 3 and analyze the possibility to generalize to higher genera. As an application, some results about reconstruction of Gromov–Witten invariants can be derived. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-6155-2 Issue No:Vol. 33, No. 4 (2017)

Authors:Lin Chen; Fang Yan Lu Pages: 495 - 500 Abstract: Let Alg ℒ be a J -subspace lattice algebra on a Banach space X and M be an operator in Alg ℒ. We prove that if δ: Alg ℒ → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B) for all A,B ∈ Alg ℒ with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-6235-3 Issue No:Vol. 33, No. 4 (2017)

Authors:Wei Hua He; Hao Li; Yan Dong Bai; Qiang Sun Pages: 501 - 508 Abstract: A linear directed forest is a directed graph in which every component is a directed path. The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and Péroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs, regular digraphs with high directed girth and random regular digraphs and we improve some well-known results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-5071-9 Issue No:Vol. 33, No. 4 (2017)

Authors:Ye Chen; Xiang Qun Yang; Ying Qiu Li; Xiao Wen Zhou Pages: 509 - 525 Abstract: For a < r < b, the approach of Li and Zhou (2014) is adopted to find joint Laplace transforms of occupation times over intervals (a, r) and (r, b) for a time homogeneous diffusion process before it first exits from either a or b. The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions on the joint Laplace transforms of occupation times for Brownian motion with drift, Brownian motion with alternating drift and skew Brownian motion, respectively. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-5184-1 Issue No:Vol. 33, No. 4 (2017)

Authors:Cheng Jun Hou Pages: 526 - 544 Abstract: We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S 2 l (G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C*-algebra C r * (G) of G when G has property (RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G 0 of G, gives rise to a canonical map τ c on the algebra C c (G) of complex continuous functions with compact support on G. We show that the map τ c can be extended continuously to S 2 l (G) and plays the same role as an n-trace on C r * (G) when G has property (RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C r * (G). PubDate: 2017-04-01 DOI: 10.1007/s10114-016-6264-y Issue No:Vol. 33, No. 4 (2017)

Authors:Zhi Tao Yang; Yu Feng Lu; Qing Jin Cheng Pages: 545 - 553 Abstract: We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull co̅C of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball B X* of X* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space. PubDate: 2017-04-01 DOI: 10.1007/s10114-017-6354-5 Issue No:Vol. 33, No. 4 (2017)

Authors:Qing Hua Xu; Fang Fang; Tai Shun Liu Pages: 554 - 564 Abstract: Let S α * be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szegö inequality for the class S α * , and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in C n . PubDate: 2017-04-01 DOI: 10.1007/s10114-016-5762-2 Issue No:Vol. 33, No. 4 (2017)

Authors:Tao Xu; He Guo Liu Pages: 565 - 570 Abstract: Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G → G defined by g φ = [g,α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C G (α 2) and G/[G, α 2] are both abelian-by-finite. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-4660-y Issue No:Vol. 33, No. 4 (2017)

Authors:Zhao Lin Jiang; Yun Cheng Qiao; Shu Dong Wang Pages: 571 - 590 Abstract: In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered. PubDate: 2017-04-01 DOI: 10.1007/s10114-016-5607-z Issue No:Vol. 33, No. 4 (2017)

Authors:Dong Yang Chen; Lei Li Pages: 311 - 326 Abstract: We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-6009-y Issue No:Vol. 33, No. 3 (2017)

Authors:Xing Fu Zhong; Jie Lü Pages: 327 - 340 Abstract: Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of (X, T) is the system (S(X), F T ), where F T is defined by F T (φ) = T ◦ φ for any φ ∈ S(X). We show that (1) If (Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (2) If (S(Σ), F σ ) is transitive then it is Devaney chaos, where (Σ, σ) is a subshift of finite type; (3) If (Σ, T) has shadowing property, then (S U (Σ), F T ) has shadowing property, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (4) If (X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T: X → X is continuous, then (S U (X), F T ) is sensitive; (5) If Σ is a closed subset of a Cantor set with infinite points and T: Σ → Σ is positively expansive then the entropy ent U (F T ) of the functional envelope of (Σ, T) is infinity. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-6130-y Issue No:Vol. 33, No. 3 (2017)

Authors:Fang Li; Chang Ye Pages: 341 - 361 Abstract: The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-6029-7 Issue No:Vol. 33, No. 3 (2017)

Authors:Zhi Hua Wang; Li Bin Li Pages: 362 - 376 Abstract: The Casimir element of a fusion ring (R,B) gives rise to the so called Casimir matrix C of (R,B). This enables us to construct a generalized Cartan matrix D − C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D−C of finite (resp. affine) type. It turns out that there exists a fusion ring with D−C being of finite (resp. affine) type if and only if D−C has only the form A 2 (resp. A 1 (1) ). We also realize all fusion rings with D−C being a particular generalized Cartan matrix of indefinite type. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-6088-9 Issue No:Vol. 33, No. 3 (2017)

Authors:Feng Juan Chen; Yong Gao Chen Pages: 377 - 382 Abstract: For any integer n ≥ 2, let P(n) be the largest prime factor of n. In this paper, we prove that the number of primes p ≤ x with P(p−1) ≥ p c is more than (1−c+o(1))π(x) for 0 < c < 1/2. This extends a recent result of Luca, Menares and Madariaga for 1/4 ≤ c ≤ 1/2. We also pose two conjectures for further research. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-5121-3 Issue No:Vol. 33, No. 3 (2017)

Authors:Yong Chen; Tao Yu; Yi Le Zhao Pages: 383 - 402 Abstract: In this paper, we study some algebraic and spectral properties of dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space of the unit disk. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-5779-6 Issue No:Vol. 33, No. 3 (2017)

Authors:Jun Xia; Xian Jin Wang Pages: 403 - 418 Abstract: The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-5761-3 Issue No:Vol. 33, No. 3 (2017)

Authors:Shan Zhang; Ling Zhou; Zu Han Liu Pages: 419 - 438 Abstract: For the strongly coupled system of M ≥ 3 competing species: $$ - \Delta \left[ {\left( {{d_i} + \sum\limits_{j = 1}^M {{\beta _{ij}}{u_j}} } \right){u_j}} \right] = \left( {{a_i} - {b_i}} \right){u_i} - k{u_i}\sum\limits_{j \ne i} {{u_j}} ,\;i = 1, \ldots ,M,$$ we prove the uniqueness of the limiting configuration as k →∞ under suitable conditions. Moreover, we prove that the limiting configuration minimizes a variational problem associated to the strongly coupled system among the segregated states with the same boundary conditions. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-5686-x Issue No:Vol. 33, No. 3 (2017)

Authors:Guang Hua Shi Pages: 439 - 448 Abstract: In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models $$\frac{d}{{dt}}\left( {\frac{{\dot x}}{{\sqrt {1 - {{\left {\dot x} \right }^2}} }}} \right) + {\left x \right ^{\alpha - 1}}x = \;p\left( t \right),$$ where p(t) ∈ C 0(R1) is 1-periodic and α > 0. PubDate: 2017-03-01 DOI: 10.1007/s10114-016-4735-9 Issue No:Vol. 33, No. 3 (2017)

Authors:Tai Xiang Sun; Hong Jian Xi; Qiu Li He Pages: 449 - 454 Abstract: Let (X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f) and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold: (1) If x ∈ Ω(f) − Ω(f n ) for some n ≥ 2, then x ∈ EP(f). (2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = N0 (the cardinal number of the set of positive integers). PubDate: 2017-03-01 DOI: 10.1007/s10114-016-5578-0 Issue No:Vol. 33, No. 3 (2017)