Authors:Miloš Arsenović; Miroslav Pavlović Pages: 289 - 296 Abstract: Abstract We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0562-15 Issue No:Vol. 67, No. 2 (2017)

Authors:Abdoreza R. Armakan; Mohammed Reza Farhangdoost Pages: 317 - 328 Abstract: Abstract We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra g by another hom-Lie algebra h and discuss the case where h has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0576-15 Issue No:Vol. 67, No. 2 (2017)

Authors:Zhi-Wei Li Pages: 329 - 337 Abstract: Abstract We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category F such that the homotopy category of this model structure is equivalent to the stable category F as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When F is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact (closed) model structure in the sense of Gillespie (2011). PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0582-15 Issue No:Vol. 67, No. 2 (2017)

Authors:Wenchang Li; Jingshi Xu Pages: 497 - 513 Abstract: Abstract Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre’s maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0037-16 Issue No:Vol. 67, No. 2 (2017)

Authors:Ji-Cai Liu Pages: 525 - 531 Abstract: Abstract Euler’s pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler’s pentagonal number theorem. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0063-16 Issue No:Vol. 67, No. 2 (2017)

Authors:Shaohui Wang; Bing Wei Pages: 533 - 536 Abstract: Abstract Let γ(G) and i(G) be the domination number and the independent domination number of G, respectively. Rad and Volkmann posted a conjecture that i(G)/γ(G) ≤ Δ(G)/2 for any graph G, where Δ(G) is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than Δ(G)/2 are provided as well. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0068-16 Issue No:Vol. 67, No. 2 (2017)

Authors:Tomasz Beberok Pages: 537 - 549 Abstract: Abstract We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w 1,w 2) ∈ C n+2: z 1 2+...+ z n 2+ w 1 q < 1, z 1 2+...+ z n 2+ w 2 r < 1}. We also compute the kernel function for {(z 1,w 1,w 2) ∈ C3: z 1 2/n + w 1 q < 1, z 1 2/n + w 2 r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0073-16 Issue No:Vol. 67, No. 2 (2017)

Authors:Qinghou Zeng; Jianfeng Hou Pages: 1 - 12 Abstract: Abstract Let G be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that G admits a bipartition such that each vertex class meets edges of total weight at least (w 1−Δ1)/2+2w 2/3, where wi is the total weight of edges of size i and Δ1 is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph G (i.e., multi-hypergraph), we show that there exists a bipartition of G such that each vertex class meets edges of total weight at least (w 0−1)/6+(w 1−Δ1)/3+2w 2/3, where w 0 is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with m edges, except for K 2 and K 1,3, admits a tripartition such that each vertex class meets at least [2m/5] edges, which establishes a special case of a more general conjecture of Bollobás and Scott. PubDate: 2017-07-14 DOI: 10.21136/cmj.2017.0147-16

Authors:Keli Zheng; Yongzheng Zhang Pages: 1 - 15 Abstract: Abstract We study some properties of generalized reduced Verma modules over ℤ-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for ℤ-graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules. PubDate: 2017-07-13 DOI: 10.21136/cmj.2017.0050-16

Authors:Yi Liu; Wen Yuan Pages: 1 - 18 Abstract: Abstract Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p 0, p 1 ∈ (1,∞] be such that (1 − θ)/p 0 + θ/p 1 = 1/p, and let φ, φ0, φ1 be some admissible functions such that φ, φ0 p/p0 and φ1 p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L φ0 p0),λ (X), L φ1 p1), λ (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L φ p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces. PubDate: 2017-07-13 DOI: 10.21136/cmj.2017.0081-16

Authors:Xiang Han; Jizhu Nan; Chander K. Gupta Pages: 1 - 44 Abstract: Abstract We investigate the invariant rings of two classes of finite groups G ≤ GL(n, F q) which are generated by a number of generalized transvections with an invariant subspace H over a finite field F q in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings. PubDate: 2017-07-12 DOI: 10.21136/cmj.2017.0044-16

Authors:Malkhaz Ashordia Pages: 1 - 30 Abstract: Abstract A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R n → R n is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R n×n with bounded total variation components, and h: BVs([a, b],R n ) → R n is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t1(x)) = B(x) · x(t 2(x))+c 0, where t i: BVs([a, b],R n ) → [a, b] (i = 1, 2) and B: BVs([a, b], R n ) → R n are continuous operators, and c 0 ∈ R n . PubDate: 2017-07-11 DOI: 10.21136/cmj.2017.0144-11

Authors:Xiaoying Du Pages: 1 - 9 Abstract: Abstract For any positive integer D which is not a square, let (u 1, v 1) be the least positive integer solution of the Pell equation u 2 − Dv 2 = 1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2 ł D and v 1h(4D) ≡ 0 (mod D), then D is called a singular number. In this paper, we prove that if (x, y, z) is a positive integer solution of the equation x y + y x = z z with 2 z, then maximum max{x, y, z} <480000 and both x, y are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions (x, y, z). PubDate: 2017-07-11 DOI: 10.21136/cmj.2017.0645-15

Authors:Songxiao Li; Ruishen Qian; Jizhen Zhou Pages: 1 - 15 Abstract: Abstract In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space. PubDate: 2017-07-11 DOI: 10.21136/cmj.2017.0481-15

Authors:Dumitru Popa Pages: 1 - 11 Abstract: Abstract We study the presence of copies of l p n’s uniformly in the spaces Π2(C[0, 1],X) and Π1(C[0, 1],X). By using Dvoretzky’s theorem we deduce that if X is an infinite-dimensional Banach space, then Π2(C[0, 1],X) contains \(\lambda \sqrt 2 \) -uniformly copies of l ∞ n ’s and Π1(C[0, 1],X) contains λ-uniformly copies of l 2 n ’s for all λ > 1. As an application, we show that if X is an infinite-dimensional Banach space then the spaces Π2(C[0, 1],X) and Π1(C[0, 1],X) are distinct, extending the well-known result that the spaces Π2(C[0, 1],X) and N(C[0, 1],X) are distinct. PubDate: 2017-05-04 DOI: 10.21136/cmj.2017.0009-16

Authors:Peng-Jie Wong Pages: 1 - 5 Abstract: Abstract We give a simple proof that critical values of any Artin L-function attached to a representation ℓ with character χℓ are stable under twisting by a totally even character χ, up to the dim ℓ-th power of the Gauss sum related to χ and an element in the field generated by the values of χℓ and χ over Q. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward. PubDate: 2017-04-08 DOI: 10.21136/cmj.2017.0134-16

Authors:Shaban Khidr; Osama Abdelkader Pages: 1 - 9 Abstract: Abstract Let D be a C d q-convex intersection, d ≥ 2, 0≤ q ≤ n − 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, C k -estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \) -equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n − q ≤ s ≤ n. In addition, we solve the \(\bar \partial \) -equation with a support condition in C k -spaces. More precisely, we prove that for a \(\bar \partial \) -closed form f in \(C_{0{,_q}}^k\left( {X\backslash D,E} \right),{\kern 1pt} 1 \leqslant q \leqslant n - 2,{\kern 1pt} n \geqslant 3\) , with compact support and for ε with 0 < ε < 1 there exists a form u in \(C_{0{,_{q - 1}}}^k\left( {X\backslash D,E} \right)\) with compact support such that \(\bar \partial u = f{\kern 1pt} in{\kern 1pt} X\backslash \bar D\) . Applications are given for a separation theorem of Andreotti-Vesentini type in C k -setting and for the solvability of the \(\bar \partial \) -equation for currents. PubDate: 2017-04-08 DOI: 10.21136/cmj.2017.0039-16

Authors:Kamal Paykan Pages: 1 - 7 Abstract: Abstract A ring R is called a right PS-ring if its socle, Soc(R R ), is projective. Nicholson and Watters have shown that if R is a right PS-ring, then so are the polynomial ring R[x] and power series ring R[[x]]. In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R[[x −1; α, δ]] and the skew polynomial ring R[x; α, δ], where R is an associative ring equipped with an automorphism α and an α-derivation δ. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided. PubDate: 2017-04-08 DOI: 10.21136/cmj.2017.0672-15

Authors:Azam Babai; Zeinab Akhlaghi Pages: 1 - 11 Abstract: Abstract Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G) and m k (G) be the number of elements of order k in G. Let nse(G) = {m k (G): k ∈ ω(G)}. Assume r is a prime number and let G be a group such that nse(G) = nse(S r ), where S r is the symmetric group of degree r. In this paper we prove that G ≅ S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components. PubDate: 2017-04-08 DOI: 10.21136/cmj.2017.0700-15

Authors:Daowei Lu; Shuanhong Wang Pages: 1 - 9 Abstract: Abstract Let H be a finite-dimensional bialgebra. In this paper, we prove that the category LR(H) of Yetter-Drinfeld-Long bimodules, introduced by F.Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category \(\begin{array}{*{20}{c}} {H \otimes H*} \\ {H \otimes H*} \end{array}YD\) over the tensor product bialgebra \(H \otimes H*\) as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results. PubDate: 2017-03-03 DOI: 10.21136/cmj.2017.0666-15