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Abstract: Abstract We prove that for any ring R of Krull dimension not greater than 1 and n ⩾ 3, the group En (R[X, X−1]) acts transitively on Umn(R[X, X−1]). In particular, we obtain that for any ring R with Krull dimension not greater than 1, all finitely generated stably free modules over R[X, X−1] are free. All the obtained results are proved constructively. PubDate: 2022-10-21

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Abstract: Abstract Let [·] be the floor function. In this paper, we prove by asymptotic formula that when \(1 < c < {{3441} \over {2539}}\) , then every sufficiently large positive integer N can be represented in the form $$N = \left[ {p_1^c} \right] + \left[ {p_2^c} \right] + \left[ {p_3^c} \right] + \left[ {p_4^c} \right] + \left[ {p_5^c} \right],$$ where p1, p2, p3, p4, p5 are primes such that p1 = x2 + y2 + 1. PubDate: 2022-10-20

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Abstract: Abstract We extend the notions of quasi-monomial groups and almost monomial groups in the framework of supercharacter theories, and we study their connection with Artin’s conjecture regarding the holomorphy of Artin L-functions. PubDate: 2022-10-19

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Abstract: Abstract We consider the Keller-Segel-Navier-Stokes system $$\left\{ {\begin{array}{*{20}{c}} {{n_t} + {\bf{u}} \cdot \nabla n = \Delta n - \nabla \cdot (n\nabla v),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \Omega ,\;t > 0,} \\ {{v_t} + {\bf{u}} \cdot \nabla v = \Delta v - v + w,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \Omega ,\;t > 0,} \\ {{w_t} + {\bf{u}} \cdot \nabla w = \Delta w - w + n,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \Omega ,\;t > 0,} \\ {{{\bf{u}}_t} + ({\bf{u}} \cdot \nabla ){\bf{u}} = \Delta {\bf{u}} + \nabla P + n\nabla \phi ,\;\nabla \cdot {\bf{u}} = 0,}&{x \in \Omega ,\;t > 0,} \end{array}} \right.$$ which is considered in bounded domain Ω ⊂ ℝN (N ∈ {2, 3}) with smooth boundary, where \(\phi \in {C^{1 + \delta }}\left( {\overline \Omega } \right)\) with δ ∈ (0, 1). We show that if the initial data \({\left\ {{n_0}} \right\ _{{L^{N/2}}\left( \Omega \right)}}\) , \({\left\ {\nabla {v_0}} \right\ _{{L^N}\left( \Omega \right)}}\) , \({\left\ {\nabla {w_0}} \right\ _{{L^N}\left( \Omega \right)}}\) and \({\left\ {{{\bf{u}}_0}} \right\ _{{L^N}\left( \Omega \right)}}\) is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state ( \({\bar n_0},{\bar n_0},{\bar n_0},0\) ) exponentially with \({\bar n_0}: = \left( {1/\left \Omega \right } \right)\int_\Omega {{n_0}\left( x \right){\rm{d}}x} \) . PubDate: 2022-10-12

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Abstract: Abstract For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary. PubDate: 2022-10-07

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Abstract: Abstract For a complex character χ of a finite group G, it is known that the product \({\rm{sh}}\left( \chi \right) = \prod\limits_{l \in L\left( \chi \right)} {\left( {\chi \left( 1 \right) - l} \right)} \) is a multiple of ∣G∣, where L(χ) is the image of χ on G − {1} The character χ is said to be a sharp character of type L if L = L(χ) and sh(χ) = ∣G∣. If the principal character of G is not an irreducible constituent of χ, then the character χ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups G with normalized sharp characters of type {−1, 0, 2}. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12. PubDate: 2022-10-06

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Abstract: Abstract We prove that if the average number of Sylow subgroups of a finite group is less than \({{41} \over 5}\) and not equal to \({{29} \over 4}\) , then G is solvable or G/F(G) ≅ A5. In particular, if the average number of Sylow subgroups of a finite group is \({{29} \over 4}\) , then G/N ≅ A5, where N is the largest normal solvable subgroup of G. This generalizes an earlier result by Moretó et al. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2021.0131-21

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Abstract: Abstract We study sums and products in a field. Let F be a field with ch(F) ≠ 2, where ch(F) is the characteristic of F. For any integer k ⩾ 4, we show that any x ∈ F can be written as a1 + … + ak with a1, …, ak ∈ F and a1… ak = 1, and that for any α ∈ F {0} we can write every x ∈ F as a1 … ak with a1, …, ak ∈ F and a1 + … + ak = α. We also prove that for any x ∈ F and k ∈ {2, 3, …} there are a1, …, a2k ∈ F such that a1 + … + a2k = x = a1 … a2k. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2021.0184-21

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Abstract: Abstract Let a = (a1,a2, …, an) be a nonincreasing sequence of positive real numbers. Denote by S = {1, 2, …, n} the index set and by Jk = {I = {r1, r2, …, rk}, 1 ⩽ r1 < r2 < … < rk ⩽ n} the set of all subsets of S of cardinality k, 1 ⩽ k ⩽ n − 1. In addition, denote by \({a_I} = {a_{{r_1}}} + {a_{{r_2}}} + \ldots + {a_{{r_k}}},\,\,1 \leqslant k \leqslant n - 1,\,\,1 \leqslant {r_1} < {r_2} < \ldots < {r_k} \leqslant n\) , the sum of k arbitrary elements of sequence a, where \({a_{{I_1}}} = {a_1} + {a_2} + \ldots + {a_k}\) and \({a_{{I_n}}} = {a_{n - k + 1}} + {a_{n - k + 2}} + \ldots + {a_n}\) . We consider bounds of the quantities \(R{S_k}(a) = {a_{{I_1}}}/{a_{{I_n}}},\,\,L{S_k}(a) = {a_{{I_1}}} - {a_{{I_n}}}\) and \({S_{k,\alpha}}(a) = \sum\limits_{I \in {J_k}} {a_I^\alpha}\) in terms of \(A = \sum\limits_{i = 1}^n {{a_i}} \) and \(B = \sum\limits_{i = 1}^n {a_i^2} \) . Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0155-21

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Abstract: Abstract Let χ be a semibrick in an extriangulated category. If χ is a τ-semibrick, then the Auslander-Reiten quiver \(\Gamma ({\cal F}({\cal X}))\) of the filtration subcategory \({\cal F}({\cal X})\) generated by χ is \({\mathbb{ZA}_\infty}\) . If χ = {Xi} i=1 t is a τ-cycle semibrick, then \(\Gamma ({\cal F}({\cal X}))\) is \({\mathbb{ZA}_\infty}/{\tau _{\mathbb{A}}}\) . PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0145-21

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Abstract: Abstract We systematically investigate the expressions and congruences for both a one-parameter family {Gn(x)} as well as a two-parameter family {Gn(r, m)} of sequences. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0224-21

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Abstract: Abstract We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f(U) is a dense set of ℂ. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0021-21

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Abstract: Abstract Let G be a group generated by a set of finite order elements. We prove that any bicrossed product Hm,d(q) ⋈ k[G] between the generalized Taft algebra Hm,d(q) and group algebra k[G] is actually the smash product Hm,d(q)♯k[G]. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of G. As an application, the classification of \({H_{m,d}}(q)\bowtie k[{C_{{n_1}}} \times {C_{{n_2}}}]\) is completely presented by generators and relations, where Cn denotes the n-cyclic group. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0176-21

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Abstract: Abstract We propose a lower bound sequence for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse, in terms of an S-type eigenvalues inclusion set and inequality scaling techniques. In addition, it is proved that the lower bound sequence converges. Several numerical experiments are given to demonstrate that the lower bound sequence is sharper than some existing ones in most cases. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2021.0092-21

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Abstract: Abstract The irregularity of a graph G = (V, E) is defined as the sum of imbalances ∣du − dv∣ over all edges uv ∈ E, where du denotes the degree of the vertex u in G. This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with n vertices and p pendant vertices (1 ⩽ p ⩽ n − 1), and characterize the corresponding extremal graphs. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0125-21

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Abstract: Abstract We give a concrete description of complex symmetric monomial Toeplitz operators \({T_{{z^p}{{\bar z}^q}}}\) on the weighted Bergman space A2(Ω), where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for \({T_{{z^p}{{\bar z}^q}}}\) to be complex symmetric. When p,q ∈ ℕ2, we prove that \({T_{{z^p}{{\bar z}^q}}}\) is complex symmetric on A2(Ω) if and only if p1 = q2 and P2 = q1. Moreover, we completely characterize when monomial Toeplitz operators \({T_{{z^p}{{\bar z}^q}}}\) on \({A^2}\left( {{\mathbb{D}_n}} \right)\) are JU-symmetric with the n × n symmetric unitary matrix U. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0210-21

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Abstract: Abstract Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space \(H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})\) . Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on \(H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})\) of operators in mixed Journé’s class. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0115-21

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Abstract: Abstract We introduce and study the concepts of weak n-injective and weak n-flat modules in terms of super finitely presented modules whose projective dimension is at most n, which generalize the n-FP-injective and n-flat modules. We show that the class of all weak n-injective R-modules is injectively resolving, whereas that of weak n-flat right R-modules is projectively resolving and the class of weak n-injective (or weak n-flat) modules together with its left (or right) orthogonal class forms a hereditary (or perfect hereditary) cotorsion theory. PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0225-21

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Abstract: Abstract We first describe the Sekine quantum groups \({{\cal A}_k}\) (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of \({{\cal A}_k}\) and describe their representation rings \(r({{\cal A}_k})\) . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of \(r({{\cal A}_k})\) . PubDate: 2022-10-01 DOI: 10.21136/CMJ.2022.0112-21

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Abstract: Abstract Given a graph G, let f(G) denote the maximum number of edges in a bipartite subgraph of G. Given a fixed graph H and a positive integer m, let f(m, H) denote the minimum possible cardinality of f(G), as G ranges over all graphs on m edges that contain no copy of H. In this paper we prove that \(f\left( {m,{\theta _{k,s}}} \right) \geqslant {1 \over 2}m + \Omega \left( {{m^{\left( {2k + 1} \right)/\left( {2k + 2} \right)}}} \right)\) , which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write K k ′ and K t,s ′ for the subdivisions of Kk and Kt,s. We show that \(f\left( {m,K_k^\prime } \right) \geqslant {1 \over 2}m + \Omega \left( {{m^{\left( {5k - 8} \right)/\left( {6k - 10} \right)}}} \right)\) and \(f\left( {m,K_{t,s}^\prime } \right) \geqslant {1 \over 2}m + \Omega \left( {{m^{\left( {5t - 1} \right)/\left( {6t - 2} \right)}}} \right)\) , improving a result of Q. Zeng, J. Hou. We also give lower bounds on wheel-free graphs. All of these contribute to a conjecture of N. Alon, B. Bollobás, M. Krivelevich, B. Sudakov (2003). PubDate: 2022-06-03 DOI: 10.21136/CMJ.2022.0302-20