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Abstract: Abstract Let F/k be a finite abelian extension of number fields with k imaginary quadratic. Let OF be the ring of integers of F and n ⩾ 2 a rational integer. We construct a submodule in the higher odd-degree algebraic K-groups of OF using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of F, which is the cardinal of the finite algebraic K-group K2n−2(OF). PubDate: 2023-11-07

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Abstract: Abstract The main purpose of the present paper is to study representations of BiHom-Hopf algebras. We first introduce the notion of BiHom-Hopf algebras, and then discuss BiHom-type modules, Yetter-Dinfeld modules and Drinfeld doubles with parameters. We get some new n-monoidal categories via the category of BiHom-(co)modules and the category of BiHom-Yetter-Drinfeld modules. Finally, we obtain a center construction type theorem on BiHom-Hopf algebras. PubDate: 2023-11-03

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Abstract: Abstract We improve a few results related to Huppert’s ϱ-σ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups. PubDate: 2023-11-02

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Abstract: Abstract The aim is to investigate the behaviour of (homomorphic images of) periodic linear groups which are factorized by mutually permutable subgroups. Mutually permutable subgroups have been extensively investigated in the finite case by several authors, among which, for our purposes, we only cite J. C. Beidleman and H. Heineken (2005). In a previous paper of ours (see M. Ferrara, M. Trombetti (2022)) we have been able to generalize the first main result of J. C. Beidleman, H. Heineken (2005) to periodic linear groups (showing that the commutator subgroups and the intersection of mutually permutable subgroups are subnormal subgroups of the whole group), and, in this paper, we completely generalize all other main results of J. C. Beidleman, H. Heineken (2005) to (homomorphic images of) periodic linear groups. PubDate: 2023-11-02

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Abstract: Abstract We characterize clean elements of \({\cal R}(L)\) and show that \(\alpha \in {\cal R}(L)\) is clean if and only if there exists a clopen sublocale U in L such that \({\mathfrak{c}_L}({\rm{coz}}(\alpha - {\bf{1}})) \subseteq U \subseteq {_L}({\rm{coz}}(\alpha))\) . Also, we prove that \({\cal R}(L)\) is clean if and only if \({\cal R}(L)\) has a clean prime ideal. Then, according to the results about \({\cal R}(L)\) , we immediately get results about \({{\cal C}_c}(L)\) . PubDate: 2023-10-27

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Abstract: Abstract A graph is 1-planar if it can be drawn in the Euclidean plane so that each edge is crossed by at most one other edge. A 1-planar graph on n vertices is optimal if it has 4n − 8 edges. We prove that 1-planar graphs with girth at least 6 are (1,1,1,1)-colorable (in the sense that each of the four color classes induces a subgraph of maximum degree one). Inspired by the decomposition of 1-planar graphs, we conjecture that every 1-planar graph is (2,2,2,0,0)-colorable. PubDate: 2023-10-18

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Abstract: Abstract Let R be a fusion ring and Rℂ:= R ⊗ℤ ℂ be the corresponding fusion algebra. We first show that the algebra Rℂ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, Rℂ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra R(PSL(2, q)):= r(PSL(2, q)) ⊗ℤ ℂ up to isomorphism, where r(PSL(2, q)) is the interpolated fusion ring with even q ⩾ 2. PubDate: 2023-10-13

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Abstract: Abstract We study the chemotaxis system with singular sensitivity and logistic-type source: ut = Δu − χ∇ · (u∇v/v) + ru − μuk, 0 = Δv − v + u under the non-flux boundary conditions in a smooth bounded domain Ω ⊂ ℝn, χ, r, μ > 0, k > 1 and n ⩾ 1. It is shown with k ∈ (1, 2) that the system possesses a global generalized solution for n ⩾ 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1. PubDate: 2023-10-13

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Abstract: Abstract It is proved that every pair of sufficiently large odd integers can be represented by a pair of equations, each containing two squares of primes, two cubes of primes, two fourth powers of primes and 105 powers of 2. PubDate: 2023-10-12

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Abstract: Abstract A Coburn theorem says that a nonzero Toeplitz operator on the Hardy space is one-to-one or its adjoint operator is one-to-one. We study the corresponding problem for certain Toeplitz operators on the Bergman space. PubDate: 2023-10-12

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Abstract: Abstract For any square-free positive integer m ≡ 10 (mod 16) with m ⩾ 26, we prove that the class number of the real cyclotomic field ℚ(ζ4m +ζ 4m −1 ) is greater than 1, where ζ4m is a primitive 4mth root of unity. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0364-22

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Abstract: Abstract Let p be an odd prime, and let a be an integer not divisible by p. When m is a positive integer with p ≡ 1 (mod 2m) and 2 is an mth power residue modulo p, we determine the value of the product \(\prod\limits_{k \in {R_m}(p)} {(1 + \tan (\pi ak/p))} \) , where $${R_m}(p) = \{0 < k < p:k \in \mathbb{Z}\,\,{\rm{is}}\,\,{\rm{an}}\,\,m{\rm{th}}\,\,{\rm{power}}\,\,{\rm{residue}}\,\,{\rm{modulo}}\,p\} .$$ In particular, if p = x2 + 64y2 with x, y ∈ \(\mathbb{Z}\) , then $$\prod\limits_{k \in {R_4}(p)} {\left({1 + \tan \,\pi {{ak} \over p}} \right) = {{(- 1)}^y}{{(- 2)}^{(p - 1)/8}}.}$$ PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0395-22

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Abstract: Abstract We consider k-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number α > 1 of finite type τ < ∞ and any constant ε > 0, we can show that where Qk is the set of positive k-free integers and the implied constant depends only on α, ε, k and β. This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type . PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0304-22

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Abstract: Abstract We prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted Lp and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of nontrivial weights in the optimal region satisfying the conditions required. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0222-22

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Abstract: Abstract A majority coloring of a digraph D with k colors is an assignment π: V(D) → {1, 2, …, k} such that for every v ∈ V(D) we have π(w) = π(v) for at most half of all out-neighbors w ∈ N+(v). A digraph D is majority k-choosable if for any assignment of lists of colors of size k to the vertices, there is a majority coloring of D from these lists. We prove that if U(D) is a 1-planar graph without a 4-cycle, then D is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0170-22

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Abstract: Abstract We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function an with the periodic Bernoulli polynomial weight \(\overline{B}_{x}(nx)\) and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order 0 or 1 gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order 1 of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order 1 subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer’s Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0322-22

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Abstract: Abstract Let k be a fixed integer. We study the asymptotic formula of R(H,r,k), which is the number of positive integer solutions 1 ⩽ x, y, z ⩽ H such that the polynomial x2 + y2 + z2 + k is r-free. We obtained the asymptotic formula of R(H, r, k) for all r ⩾ 2. Our result is new even in the case r = 2. We proved that R(H, 2, k) = ckH3 + O(H9/4+ε), where ck > 0 is a constant depending on k. This improves upon the error term O(H7/3+ε) obtained by G.-L. Zhou, Y. Ding (2022). PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0394-22

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Abstract: Abstract We investigate the representation theory of the positively based algebra Am,d, which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that Am,d is of finite representative type if d ⩽ 4, of tame type if d = 5, and of wild type if d ⩾ 6. In the case when d ⩽ 4, all indecomposable representations of Am,d are constructed. Furthermore, their right cell representations as well as left cell representations of Am,d are described. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0254-22

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Abstract: Abstract We introduce equivariant formal deformation theory of associative algebra morphisms. We also present an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal deformation theory of associative algebra morphisms. We discuss some examples of equivariant deformations and use the Maurer-Cartan equation to characterize equivariant deformations. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0171-22

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Abstract: Abstract We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards. PubDate: 2023-10-01 DOI: 10.21136/CMJ.2023.0358-22