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Abstract: For any positive integer m, let \(\mathbb{Z}_{m}\) be the set of residue classes modulo m. For \(A\subseteq \mathbb{Z}_{m}\) and \(\overline{n}\in \mathbb{Z}_{m}\) , let the representation function \(R_{A}(\overline{n})\) denote the number of solutions of the equation \(\overline{n}=\overline{a}+\overline{a'}\) with unordered pairs \((\overline{a}, \overline{a'})\in A \times A\) . We characterize the partitions of \(\mathbb{Z}_{2p}\) with \(A\cup B=\mathbb{Z}_{2p}\) and \( A\cap B =2\) such that \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in\mathbb{Z}_{2p}\) , where p is an odd prime. PubDate: 2021-10-01

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Abstract: For a real biquadratic field, we denote by \(\lambda\) , \(\mu\) and \(\nu\) the Iwasawa invariants of cyclotomic \(\mathbb{Z}_{2}\) -extension of \(k\) . We give certain families of real biquadratic fields \(k\) such that \(\mu=0\) . PubDate: 2021-10-01

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Abstract: Recently, we have proved that the rectangular pointwise Lipschitz regularity of a continuous function on the unit square is directly related with the local suprema of the coefficients of the function in the tensor product Faber–Schauder basis. In this paper, we provide print dimension information on the distribution at all bi-scales of these local suprema. We apply our results for self-affine functions associated to the Schauder product function and a particular type of Sierpinski carpets. PubDate: 2021-10-01

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Abstract: The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice \(\mathbb{MON}\) of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in \(\mathbb{MON}\) is established. These results play a crucial role in the complete description of all cancellable elements of the lattice \(\mathbb{MON}\) . It turns out that there are precisely five such elements. PubDate: 2021-10-01

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Abstract: Let \(\sigma\) be a partition of the set of all primes \(\mathbb{P}\) . Let G be a finite group and \(\mathfrak{F}\) be a Fitting class of finite groups. In the theory of formations of finite soluble groups, a well known result of Bryce and Cossey is: a local formation \(\mathfrak{F}\) is a Fitting class if and only if every value of the canonical formation function F of \(\mathfrak{F}\) is a Fitting class. In this paper, we give the dual theory of the result of Bryce and Cossey. We proved that an \(\sigma\) -local Fitting class \(\mathfrak{F}\) is a formation if and only if every value of the canonical \(\sigma\) -local \(H_{\sigma}\) -function of \(\mathfrak{F}\) is a formation. PubDate: 2021-10-01

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Abstract: We find the solutions \(f,g,h \colon S \to H\) of the following extension of Pexider's functional equation $$\underset{\lambda\in K}{\sum}f(x+\lambda y)=g(x)+h(y),\quad x,y\in S,$$ where (S,+) is an abelian semigroup, K is a finite subgroup of the automorphism group of S and (H,+) is an abelian group. PubDate: 2021-10-01

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Abstract: We deal with the combinatorial principle Weak Diamond. We prove that, if it holds for a given cardinal, we can get this principle with more than two colours or some relevant ideal is not too saturated. Then we point out a model theoretic consequence of Weak Diamond. PubDate: 2021-10-01

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Abstract: Let \(c \geq 2\) be any fixed real number. Matomäki [4] inverstigated the set of \(A > 1\) such that the integer part of \( A^{c^k}\) is a prime number for every \(k \in \mathbb{N}\) . She proved that the set is uncountable, nowhere dense, and has Lebesgue measure 0. In this article, we show that the set has Hausdorff dimension 1. PubDate: 2021-10-01

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Abstract: An edge-ordered graph is a graph with a total ordering of its edges. A path \(P=v_1v_2\ldots v_k\) in an edge-ordered graph is called increasing if \((v_iv_{i+1}) < (v_{i+1}v_{i+2})\) for all \(i = 1,\ldots,k-2\) ; and it is called decreasing if \((v_iv_{i+1}) > (v_{i+1}v_{i+2})\) for all \(i = 1,\ldots,k-2\) . We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing. Let G be a graph. In a straight-line drawing D of G, its vertices are drawn as different points in the plane and its edges are straight line segments. Let \(\overline{\alpha}(G)\) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length \(\overline{\alpha}(G)\) . Let \(\overline{\tau}(G)\) be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of \(\overline{\tau}(G)\) edges. In this paper we show that \(\overline \alpha(K_n) = \Omega(\log\log n)\) , \(\overline \alpha(K_n) = O(\log n), \overline \tau(K_n) = \Omega(\log\log \log n)\) and \(\overline \tau(K_n) = O(\sqrt{n \log n})\) . PubDate: 2021-10-01

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Abstract: Let \((X, d, \mu)\) be a space of homogeneous type. Let \(T_{1}, T_{2}\) be singular integral operators with nonsmooth kernels. By establishing sparse domination, we obtain quantitative boundedness for the composite operator \(T_{1}T_{2}\) on \(L^{p}(X, \omega)\) with \(\omega\in A_{p}\) and the weighted endpoint estimate for the composite operator \(T_{1}T_{2}\) . PubDate: 2021-10-01

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Abstract: Let \(\sigma =\{\sigma_i i\in I\}\) be some partition of the set of all primes \(\mathbb{P}\) and G be a finite group. A group is said to be \(\sigma\) -primary if it is a finite \(\sigma_{i}\) -group for some i. A subgroup A of G is said to be \({\sigma}\) -subnormal in G if there is a subgroup chain \(A=A_{0} \leq A_{1} \leq \cdots \leq A_{t}=G\) such that either \(A_{i-1}\trianglelefteq A_{i}\) or \(A_{i}/(A_{i-1})_{A_{i}}\) is \(\sigma\) -primary for all \(i=1, \ldots , t\) . A subgroup S of G is m- \(\sigma\) -permutable in G if \(S=\langle M, B \rangle\) for some modular subgroup M and \(\sigma\) -permutable subgroup B of G. We say that a subgroup H of G is m- \(\sigma\) -embedded in G if there exist an m- \(\sigma\) -permutable subgroup S and a \(\sigma\) -subnormal subgroup T of G such that \(H^G=HT\) and \(H\cap T\leq S\leq H\) , where \(H^G = \langle H^x x \in G \rangle\) is the normal closure of H in G. In this paper, we study the properties of m- \(\sigma\) -embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized. PubDate: 2021-10-01

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Abstract: We discuss recent developments in the following important areas of Alfréd Rényi’s research interest: axiomatization of quantitative dependence measures, qualitative independence in combinatorics, conditional qualitative independence in statistics/data science and in measure theory/probability theory, and finally, prime gaps that are responsible for Rényi’s early career reputation. Most authors of this paper are main contributors to the new developments. PubDate: 2021-10-01

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Abstract: Answering a question of E. K. van Douwen and W. F. Pfeffer [1], we prove that the countable power of the Sorgenfrey line is a D-space. To establish this result we use a method of proof which we call reverse induction. This method allows to establish certain properties of a product \(\prod_{{i}=0}^{\infty}{X}_{i}\) by making a kind of ``reverse induction step'' from \(\prod_{{i}={n}+1}^{\infty}{X}_{i} {\rm to} \prod_{{i}={n}}^{\infty}{X}_{i}\) for an arbitrary natural n. PubDate: 2021-10-01

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Abstract: We get a version of the colouring property Pr1 proving \(\Pr_1(\lambda,\lambda,\lambda,\partial)\) always when \(\lambda= \partial^+,\partial \) are regular cardinals and some stationary subset of \(\lambda\) consisting of ordinals of cofinality \(< \partial\) do not reflect in any ordinal \(< \lambda\) . PubDate: 2021-10-01

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Abstract: The classes of semicommutative, PS I, 2-primal, weakly-2-primal, NI, NR and Dedekind-finite rings form a strictly increasing chain of classes of rings. Examples have been provided by several authors to show most of the inclusions are strict. In this paper it is shown that every inclusion is strict. Furthermore, in most instances examples with the additional properties of being reflexive and/or abelian are constructed. In particular, an abelian nonreflexive NI non-weakly-2-primal ring is given, which shows that the classes of NI and weakly-2-primal rings differ; a fact that has been previously claimed without proof. PubDate: 2021-10-01

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Abstract: We extend Morley’s trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space. We will show that the Morley tetrahedron of an isosceles tetrahedron is also isosceles tetrahedron. Furthermore, by the formula for distance in barycentric coordinate, we introduce and prove a general theorem on an isosceles tetrahedron. PubDate: 2021-09-03

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Abstract: Let \(p \ge 1\) . We give upper and lower bounds for $$\begin{aligned} {M}_{p}(N): = \bigg \Vert \mathop {\mathrm{sup}}\limits _{0 \le t \le 1} \bigg {\sum _{n=1}^N} {e}(nx + n^{2}t)\bigg \bigg \Vert _{L^{p}[0,1]}^{p} \end{aligned}$$ that are of the same order of magnitude. PubDate: 2021-09-03

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Abstract: A subgroup H of a group G is said to be an \(IC\mathrm{\Phi}\) -subgroup of G if \(H \cap [H,G] \le \mathrm{\Phi}(H)\) . We prove the p-nilpotency of a finite group G under the assumption that certain p-subgroups of G are \(IC\mathrm{\Phi}\) -subgroups of G. Our main result generalizes and extends some recent work of Gao and Li [1]. PubDate: 2021-09-03

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Abstract: Using the method of undetermined coefficients and the theory of Pellian equation, we show that there exist infinitely many isosceles Heron triangles whose sides are certain polynomial values. PubDate: 2021-09-03

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Abstract: The cosine addition formula on a semigroup S is the functional equation \(g(xy) = g(x)g(y) - f(x)f(y)\) for all \(x,y \in S\) . We find its general solution for \(g,f \colon S \to \mathbb{C}\) , using the recently found general solution of the sine addition formula \(f(xy) = f(x)g(y) + g(x)f(y)\) on semigroups. A simpler proof of this latter result is also included, with some details added to the solution. We also solve the cosine subtraction formula \(g(x\sigma(y)) = g(x)g(y) + f(x)f(y)\) on monoids, where \(\sigma\) is an automorphic involution. The solutions of these functional equations are described mostly in terms of additive and multiplicative functions, but for some semigroups there exist points where f and/or g can take arbitrary values. The continuous solutions on topological semigroups are also found. PubDate: 2021-09-03