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Abstract: For \(s,t,u \in {\mathbb{C}}\) , we show rapidly (or globally) convergent series representations of the Tornheim double zeta function T(s, t, u) and (desingularized) symmetric Tornheim double zeta functions. As a corollary, we give a new proof of known results on the values of T(s, s, s) at non-positive integers and the location of the poles of T(s, s, s). Furthermore, we prove that T(s, s, s) can not be written by a polynomial in the form of \(\sum_{k=1}^j c_k \prod_{r=1}^q \zeta^{d_{kr}} (a_{kr} s + b_{kr})\) , where \(a_{kr}, b_{kr}, c_k \in {\mathbb{C}}\) and \(d_{kr} \in {\mathbb{Z}}_{\ge 0}\) . PubDate: 2021-12-11

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Abstract: We show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function \(F \colon I ^2\to I\) is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Aczél [1], Kolmogoroff [18], Nagumo [20] and de Finetti [11]. PubDate: 2021-12-11

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Abstract: The halo conjecture in the theory of differentiation of integrals states that if the maximal operator MB corresponding to a differentiation basis B is of restricted weak type \(\varphi\) , then the basis B differentiates the integrals of functions from the class \(\varphi(L)\) . Using the technique proposed by Antonov and, Sjölin and Soria, which is based on the approximation by simple functions with respect to the integral metrics generated by the truncated maximal convolution operators, for translation invariant bases in locally compact groups of a rather general type, the result has been established that gives an approximation to the conjecture for functions \(\varphi(u)\) close to u, while for the case of \(\varphi(u)=u\) , this implies the validity of the conjecture. The proved theorems generalize the corresponding results obtained by Sjölin and Soria, Moriyón, and Moon. PubDate: 2021-12-11

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Abstract: We consider weakenings of normality in \(\Psi\) -spaces and prove that the existence of an AD family whose \(\Psi\) -space is almost-normal but not normal follows from CH. In contrast, we prove that it is consistent that no MAD family is almost-normal. We also construct a partly-normal not quasi-normal AD family, answering questions of García-Balan and Szeptycki. We finish by showing that the concepts of almost-normal and strongly \(\aleph_0\) -separated AD families are different, even under CH, answering a question of Oliveira-Rodrigues and Santos-Ronchim. PubDate: 2021-12-11

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Abstract: We give necessary and sufficient conditions for a regular sine family of continuous linear set-valued functions associated with a regular cosine family of continuous linear set-valued functions, to be continuous. Then we show that the continuity and Hukuhara differentiability of regular sine and cosine families are equivalent. As an application, we prove the existence and uniqueness of a solution of a second order differential problem. Our results are a much stronger version of results in [5], [6] and [10] that are valid for Banach spaces. PubDate: 2021-12-11

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Abstract: The groups all of whose elements are involutions are characterized by means of two equivalent composite functional equations in two variables. PubDate: 2021-12-03 DOI: 10.1007/s10474-021-01191-1

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Abstract: We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented. PubDate: 2021-12-03 DOI: 10.1007/s10474-021-01187-x

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Abstract: A generalized dual Brunn–Minkowski inequality for the intersection body is established in the framework of the dual Orlicz Brunn–Minkowski theory. This new inequality yields the dual Orlicz Brunn–Minkowski inequality for the dual quermassintegral directly. PubDate: 2021-12-03 DOI: 10.1007/s10474-021-01192-0

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Abstract: Polynomial spaces associated to a convex body C in \((\mathbb{R}^{+})^d\) have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex C. We develop some basic pluripotential theory including notions of C−extremal plurisubharmonic functions VC,K for \(K\subset \mathbb{C}^d\) compact. Using this, we discuss Bernstein−Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting. PubDate: 2021-12-03 DOI: 10.1007/s10474-021-01188-w

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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: 2021-12-01

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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: 2021-12-01

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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: 2021-12-01

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Abstract: We give a construction of a set \(A \subset \mathbb N\) such that any subset \({A' \subset A}\) with \( A' \gg A ^{2/3}\) is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata. PubDate: 2021-12-01

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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: 2021-12-01

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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: 2021-12-01

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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 DOI: 10.1007/s10474-021-01183-1

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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 DOI: 10.1007/s10474-021-01174-2

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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 DOI: 10.1007/s10474-021-01172-4

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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 DOI: 10.1007/s10474-021-01177-z

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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 DOI: 10.1007/s10474-021-01168-0