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This is a corrigendum to our paper, “On a class of prime entire functions”, published in Acta Math. Sin., Engl. Ser., 25, 1647–1652 (2009).

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.

In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.

This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)(X):= {T ∈ B(X): T = Σ =1 ⊕T i , T i ∈ (FDI), k ∈ ℕ}.

We study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators, and deal with a question posed by Liu and Yang. The results in this paper extend the corresponding results obtained by Liu-Yang and by Liu-Laine respectively. Examples are provided to show that the results in this paper, in a sense, are the best possible.

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let \(\left\{ {X_a } \right\}_{a \in Z_ + }\) be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T ⊂ S be an arbitrary finite subset of S. Denote by L n the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X 1, ..., X n . In this paper, we obtain a limit law and a week version of an Erdos-Renyi type law for L n . A large deviation result of L n is also discussed.

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be \(O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)\). This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on Ma(S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

The purpose of this paper is to study the weak convergence problems of the implicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [Nonlinear Anal., 68, 2977–2983 (2008)], Chen, et al. [J. Math. Anal. Appl., 314, 701–709 (2006)], Xu and Ori [Numer. Funct. Anal. Optim, 22, 767–773 (2001)] and Osilike [J. Math. Anal. Appl., 294, 73–81 (2004)].

In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemOzay2aaeWaaeaadaWcaaqaaiabdIha4fXafv3y % SLgzGmvETj2BSbacgaGae83kaSIaemyEaKhabaGae8Nmaidaaiab-T % caRiabdQha6bGaayjkaiaawMcaaiab-jHiTiabdEgaNnaabmaabaWa % aSaaaeaacqWG4baEcqWFsislcqWG5bqEaeaacqWFYaGmaaGae83kaS % IaemOEaOhacaGLOaGaayzkaaGae8xpa0JaemiAaGMae8hkaGIaemyE % aKNae8xkaKIae8Nla4caaa!5ABC! $$ f\left( {\frac{{x + y}} {2} + z} \right) - g\left( {\frac{{x - y}} {2} + z} \right) = h(y). $$ This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297–300 (1978).

In order to study the Yamabe changing-sign problem, Bahri and Xu proposed a conjecture which is a universal inequality for p points in ℝm. They have verified the conjecture for p ≤ 3. In this paper, we first simplify this conjecture by giving two sufficient and necessary conditions inductively. Then we prove the conjecture for the basic case m = 1 with arbitrary p. In addition, for the cases when p = 4, 5 and m ≥ 2, we manage to reduce them to the basic case m = 1 and thus prove them as well.

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

Plesnik in 1972 proved that an (m − 1)-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m − 1 edges. Alder et al. in 1999 showed that if G is a regular (2n + 1)-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and (or) excluding a set of edges, where 1 ≤ k ≤ n/2. In particular, we generalize Plesnik’s result and the results obtained by Liu et al. in 1998, and improve Katerinis’ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.

Let G be a Lie group whose Lie algebra $$ \mathfrak{g} $$ is quadratic. In the paper “the non-commutative Weil algebra”, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism , called the quantization map, between the Weil algebra $$ W_\mathfrak{g} = S(\mathfrak{g}*) \otimes \wedge (\mathfrak{g}*) $$ and (which they call the noncommutative Weil algebra) for $$ \mathfrak{g} $$ . They showed that induces an algebra isomorphism between the basic cohomology rings H bas * ( $$ W_\mathfrak{g} $$ ) and H bas * ( ). In this paper, we will interpret the quantization map as the super Duflo map between the symmetric algebra $$ S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right) $$ and the universal enveloping algebra $$ U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right) $$ of a super Lie algebra $$ \widetilde{T\mathfrak{g}\left[ 1 \right]} $$ which is canonically associated with the quadratic Lie algebra $$ \mathfrak{g} $$ . The basic cohomology rings H bas * ( $$ W_\mathfrak{g} $$ ) and H bas * ( ) correspond exactly to $$ S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv} $$ and $$ U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv} $$ , respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.

A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejer (or (C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σnf → f (n → ∞) for functions f ∈ Lp, where p > 1 (Journal of Approximation Theory, 101(1), 1–36, (1999)) and also the a.e. convergence σMnf → f (n → ∞) for functions f ∈ L1 (Journal of Approximation Theory, 124(1), 25–43, (2003)). The aim of this paper is to prove the a.e. relation limn→ σ nf = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its "big elements" are not "too dense".

To guarantee every real-valued convex function bounded above on a set is continuous, how ”thick” should the set be? For a symmetric set A in a Banach space E, the answer of this paper is: Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold: i) spanA has finite co-dimentions and ii) coA has nonempty relative interior. This paper also shows that a subset A ⊂ E satisfying every real-valued convex function bounded above on A is continuous on E if (and only if) every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E.

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.

An excellent introduction to the topic of poset matroids is due to M. Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rank axioms for poset matroids. In this paper, we study the special integral function f and obtain a new class of poset matroids from the old ones, and then we generalize this result according to the properties of f. Almost all of these results can be regarded as the application of global rank axioms for poset matroids. The main results in our paper have, indeed, investigated the restriction of the basis of the poset matroid, and we give them the corresponding geometric interpretation.

In this paper, two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods, their properties and relationships are system- atically discussed. We also analysed the implication of the conditions in previous papers. Then we apply these consequences to B-valued random variables, and greatly improve the original results of the strong convergence of the general Jamison weighted sum. Furthermore, our discussions are useful to the corresponding questions of real-valued random variables.

In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional (k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional (k, m)-deleted graphs improving the existing one. Finally, we state an open problem.

In this paper, we discuss our most recent results on the optimal L2 extension problem and Siu’s lemma as applications of the strong openness property of multiplier ideal sheaves obtained by Guan and Zhou.

In this paper, we provide a common generalization to the well-known Erdos–Ko–Rado Theorem, Frankl–Wilson Theorem, Alon–Babai–Suzuki Theorem, and Snevily Theorem on set systems with L-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with k-wise L-intersections by Furedi and Sudakov [J. Combin. Theory, Ser. A, 105, 143–159 (2004)]. We will also derive similar results on L-intersecting families of subspaces of an n-dimensional vector space over a finite field F q , where q is a prime power.

In this paper, we generalize Rees–Shishikura’s theorem to the class of geometrically finite rational maps.

Let G = (V,E) be a simple graph. A function f: E → {+1,−1} is called a signed cycle domination function (SCDF) of G if Ʃ e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{Ʃ e∈E f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ′sc(G) = n.

In this paper, we consider two kinds of 2-Calabi–Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t = D b (KA(2t+1)(n+1)−3)/τt(n+1)−1[1], where n, t ≥ 1, and Dn,t = D b (KD2t(n+1))/τ(n+1)φ n , where n, t ≥ 1, and φ is induced by an automorphism of D2t(n+1) of order 2. Except the categories An,1, they all contain non-zero maximal rigid objects which are not cluster tilting. An,1 contain cluster tilting objects. We define the cluster complex of An,t (resp. Dn,t) by using the geometric description of cluster categories of type A (resp. type D). We show that there is an isomorphism from the cluster complex of An,t (resp. Dn,t) to the cluster complex of root system of type B n . In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan–Palu–Reiten: Let $${R_{{A_{n,t}}}}$$ , resp. $${R_{{D_{n,t}}}}$$ , be the full subcategory of An,t, resp. Dn,t, generated by the rigid objects. Then $${R_{{A_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ and $${R_{{D_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ as additive categories, for all t ≥ 1.

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on ℂ2×2 and Julia set of one dimensional complex dynamic system.

For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame (WNABF). Under the setting of reducing subspaces of L 2(R d ), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′∑(G). Flandrin et al. proposed the following conjecture that χ′∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < $$\tfrac{{37}} {{12}}$$ and Δ(G) ≥ 7.

In this paper, we prove that if a c.e. Turing degree d is non-low2, then there are two left-c.e. reals β 0, β 1 in d, such that, if β 0 is wtt-reducible to a left-c.e. real α, then β 1 is not computable Lipschitz (cl-) reducible to α. As a corollary, d contains a left-c.e. real which is not cl-reducible to any complex (wtt-complete) left-c.e. real.

In this paper, we present some basic notions of simple ordered semihypergroups and regular ordered Krasner hyperrings and prove some results in this respect. In addition, we describe pure hyperideals of ordered Krasner hyperrings and investigate some properties of them. Finally, some results concerning purely prime hyperideals are proved.

Let S = (a 1, …, a m ; b 1, …, b n ), where a 1, …, a m and b 1, …, b n are two nonincreasing sequences of nonnegative integers. The pair S = (a 1, …, a m ; b 1, …, b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y, E) such that a 1, …, a m and b 1, …, b n are the degrees of the vertices in X and Y, respectively. Let Z 3 be the cyclic group of order 3. Define σ(Z 3, m, n) to be the minimum integer k such that every bigraphic pair S = (a 1, …, a m ; b 1, …, b n ) with a m , b n ≥ 2 and σ(S) = a 1 + ⋯ + a m ≥ k has a Z 3-connected realization. For n = m, Yin [Discrete Math., 339, 2018—2026 (2016)] recently determined the values of σ(Z 3, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z 3, m, n) for m ≥ n ≥ 4.

Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x, f) and P(f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω(x, f) = {y| there exist a sequence of points x k ∈ T and a sequence of positive integers n 1 < n 2 < ··· such that lim k→∞ x k = x and lim k→∞ $$f^{n_{k}}$$ (x k ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω(x, f) = Ω(x, f) for any x ∈ T. (3) ∩ =1 ∞ f n (T) = P(f), and ω(x, f) is a periodic orbit for every x ∈ T and map h: x → ω(x, f) (x ∈ T) is continuous. (4) Ω(x, f) is a periodic orbit for any x ∈ T.

In this paper, we first study the mean ergodicity of random linear operators using some techniques of measure theory and L0-convex analysis. Then, based on this, we give a characterization for a complete random normed module to be mean ergodic.

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different fromthe sumof colors taken on the edges incident to v. Let χ′Σ(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) < 5/2, then χ′Σ(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.

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.

In the present paper, we introduce the concepts of Prufer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prufer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prufer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prufer sheaves.

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.

In this paper, we investigate the intersection numbers of nearly Kirkman triple systems. JN[v] is the set of all integers k such that there is a pair of NKTS(v)s with a common uncovered collection of 2-subset intersecting in k triples. It has been established that \({J_N}\left[ v \right] = \left\{ {0,1,...,\frac{{v\left( {v - 2} \right)}}{6} - 6,\frac{{v\left( {v - 2} \right)}}{6} - 4,\frac{{v\left( {v - 2} \right)}}{6}} \right\}\) for any integers v = 0 (mod 6) and v ≥ 66. For v ≤ 60, there are 8 cases left undecided.

Let X denote a compact metric space with distance d and F : X × ℝ → X or Ft : X → X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C0-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U = (1, 1), down steps D = (1,−1), horizontal steps H = (1, 0), and left steps L = (−1,−1), and such that up steps never overlap with left steps. Let S n be the set of all skew Motzkin paths of length n and let s n = |S n |. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {s n } n ≥0. Then we present several involutions on S n and consider the number of their fixed points. Finally we consider the enumeration of some statistics on S n .

In this paper, the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied. There are three approaches to solve this problem. The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths; the second approach is to find a current assignment of the current graph by the theory of current graph; the third approach is to find exponentially many embedding (or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph. According to this three approaches, we can construct exponentially many minimum genus embeddings of complete graph K 12s+8 in orientable surfaces, which show that there are at least $$\frac{10}{3} \times (\frac{200}{9})^s$$ distinct minimum genus embeddings for K 12s+8 in orientable surfaces. We have also proved that K 12s+8 has at least $$\frac{10}{3} \times (\frac{200}{9})^s$$ distinct minimum genus embeddings in non-orientable surfaces.

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A gc-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a gc-coloring with k colors is called the gc-chromatic index of G and denoted by \(\chi\prime_{g_{c}}\) (G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\) (G) = δg(G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).

Let (X, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood–Paley g-function ġ r (r ∈ [2,∞)) is bounded from Hardy space H 1(μ) into L 1(μ). Moreover, the authors show that, if f ∈ RBMO(μ), then [ġ r (f)] r is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ r (f)] r belongs to RBLO(μ) with the norm no more than ‖f‖RBMO(μ) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of ġ r from RBMO(μ) into RBLO(μ). The vector valued Calderon–Zygmund theory over (X, d, μ) is also established with details in this paper.

Let G be a finite group, Irr1(G) be the set of nonlinear irreducible characters of G and cd1(G) the set of degrees of the characters in Irr1(G). A group G is said to be a D2-group if |cd1(G)| = |Irr1(G)| − 2. In this paper, we give a complete classification of solvable D2-groups.

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.

Let A be a small abelian category. For a closed subbifunctor F of Ext 1 (−,−), Buan has generalized the construction of Verdier’s quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.

We first determine in this paper the structure of the generalized Fitting subgroup F* (G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/O p (L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O 2(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.

A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color. In this paper, it is proved that if G is a planar graph with Δ(G) ≥ 7 and without chordal 7-cycles, then G has a (Δ(G)+1)-total-coloring.