Authors:Miloš Arsenović; Miroslav Pavlović Pages: 289 - 296 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:Jiří Tomáš Pages: 297 - 316 Abstract: Let M be an m-dimensional manifold and A = D k r /I = R⊕N A a Weil algebra of height r. We prove that any A-covelocity T x A f ∈ T x A *M, x ∈ M is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of T x A M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T A *M ≃ T r *M without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from m ≥ k to all cases of m. We also introduce the space J A (M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J r (M,N). PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0566-15 Issue No:Vol. 67, No. 2 (2017)

Authors:Abdoreza R. Armakan; Mohammed Reza Farhangdoost Pages: 317 - 328 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: 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:Michal Hrbek; Pavel Růžička Pages: 367 - 377 Abstract: A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and regularly weakly based modules over Dedekind domains. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0632-15 Issue No:Vol. 67, No. 2 (2017)

Authors:Nahid Ashrafi; Marjan Sheibani; Huanyin Chen Pages: 417 - 425 Abstract: A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ℕ. We prove that M n (R) is nil clean if and only if R/J(R) is Boolean and M n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ℤ3, B or ℤ3 ⊕ B where B is a Boolean ring, and that M n (R) is weakly nil clean if and only if M n (R) is nil clean for all n ≥ 2. PubDate: 2017-06-01 DOI: 10.21136/cmj.2017.0677-15 Issue No:Vol. 67, No. 2 (2017)

Authors:Wenchang Li; Jingshi Xu Pages: 497 - 513 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: 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: 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: 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:Tamás Milkovszki; Zoltán Muzsnay Pages: 1 - 27 Abstract: A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists. PubDate: 2017-05-12 DOI: 10.21136/cmj.2017.0010-16

Authors:Neha Prabhu Pages: 1 - 17 Abstract: A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k > 1. Building upon a proof by E.M.Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n ≤ x with k prime factors such that a fixed quadratic equation has exactly 2 k solutions modulo n. PubDate: 2017-05-05 DOI: 10.21136/cmj.2017.0712-15

Authors:Dumitru Popa Pages: 1 - 11 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: 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: 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: 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: 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:Kwang-Soon Park Pages: 1 - 22 Abstract: As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h- Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: the integrability of distributions, the geometry of foliations, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic and give some examples of such maps. Finally, we obtain some types of decomposition theorems. PubDate: 2017-04-08 DOI: 10.21136/cmj.2017.0143-16

Authors:Adel Mahmoud Gomaa Pages: 1 - 27 Abstract: We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation (P) $$\dot x\left( t \right) = L\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),{\kern 1pt} t \in \mathbb{R}$$ where L(t): t ∈ R is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([−d, 0],E) be the Banach space of continuous functions from [−d, 0] into E and f d : [a, b] × C([−d, 0],E) → E. Let \(\hat L:{\kern 1pt} \left[ {a,b} \right] \to L\left( E \right)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [−d, 0]. We prove that, under certain conditions, the differential equation with delay (Q) $$\dot x\left( t \right) = \hat L\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right){\kern 1pt} if{\kern 1pt} t \in \left[ {a,b} \right],$$ has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too. PubDate: 2017-04-08 DOI: 10.21136/cmj.2017.0592-15

Authors:Daowei Lu; Shuanhong Wang Pages: 1 - 9 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