Abstract: Suppose G is linear group. If G has characteristic zero, we prove that G is (polycyclic-by-finite)-Engel if and only if G has a normal series < 1 > = T0 ≤ T1 ≤ ··· ≤ Ts = T ≤ G with s and the index (G:T) finite and each Ti/Ti–1 either polycyclic-by-finite, or G-hypercentral with [Ti, T] ≤ Ti–1, or G-hypercentral, abelian and Chernikov. This is much more complex than the positive characteristic case where G is (polycyclic-by-finite)-Engel if and only if G is (polycyclic-by-finite)-by-hypercentral. PubDate: 2019-03-19

Abstract: In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier–Legendre series expansions, arXiv:1710.03221, 2017), about the interplay between hypergeometric functions and Fourier–Legendre ( \(\text {FL}\) ) series expansions. In the section “Hypergeometric series related to \(\pi ,\pi ^2\) and the lemniscate constant”, through the FL-expansion of \([x(1-x)]^\mu \) (with \(\mu +1\in \frac{1}{4}{\mathbb {N}}\) ) we prove that all the hypergeometric series $$\begin{aligned}&\sum _{n\ge 0}\frac{(-1)^n(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\\&\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2 \end{aligned}$$ return rational multiples of \(\frac{1}{\pi },\frac{1}{\pi ^2}\) or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of \(\frac{\log x}{\sqrt{x}}\) and related functions, we show that in many cases the hypergeometric \(\phantom {}_{p+1} F_{p}(\ldots , z)\) function evaluated at \(z=\pm 1\) can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$\begin{aligned} \sum _{n\ge 0}\frac{1}{(2n+1)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2,\quad \sum _{n\ge 0}\frac{1}{(2n+1)^3}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2. \end{aligned}$$ In the section “Twisted hypergeometric series” we show that the conversion of some \(\phantom {}_{p+1} F_{p}(\ldots ,\pm 1)\) values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form \(\sum _{n\ge 0} a_n b_n\) where \(a_n\) is a Stirling number of the first kind and \(\sum _{n\ge 0}b_n z^n = \phantom {}_{p+1} F_{p}(\ldots ;z)\) . PubDate: 2019-03-18

Abstract: We deal with complete surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in the hyperbolic space \({\mathbb {H}}^{2+p}\) . Supposing that such a surface \(M^{2}\) satisfies a linear Weingarten condition of the type \(K=aH+b\) for some appropriate real constants a and b, where H and K denote the mean and Gaussian curvatures, respectively, we show that \(M^{2}\) must be either totally umbilical or isometric to one of the following flat surfaces: \({\mathbb {S}}^{1}(r)\times {\mathbb {R}}\) , \(\mathbb S^{1}(r)\times {\mathbb {S}}^1(\sqrt{{\tilde{r}}^2-r^2})\) or \(\mathbb S^{1}(r)\times {\mathbb {H}}^1(-\sqrt{{\tilde{r}}^2+r^2})\) . Furthermore, we obtain a version of the classical Liebmann’s Theorem (Liebmann in Math Phys Klasse 44–55, 1899) showing that the only compact (without boundary) surfaces having positive constant Gaussian curvature, immersed with flat normal bundle in the hyperbolic space \({\mathbb {H}}^{2+p}\) , are the totally umbilical round spheres. PubDate: 2019-02-20

Abstract: In this paper we study the fractional Fourier transformation on the space S of Schwartz test functions, study some of its properties, and establish a two sided inverse for it. Also, we establish a convolution theorem for the fractional Fourier transform. We use duality to define fractional Fourier transform of tempered distributions. We define fractional convolution of a function and a tempered distribution and fractional convolution of tempered distributions, and show continuity of the convolution operators involved. PubDate: 2019-02-14

Abstract: It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class group of Krull monoids. We prove that a C-monoid is seminormal if and only if its reduced class semigroup is a union of groups. Based on this characterization we establish a criterion (in terms of the class semigroup) when seminormal C-monoids are half-factorial. PubDate: 2019-02-09

Abstract: We associate a Banach algebra \(\mathbf A _\Phi \) to each bounded bilinear map \(\Phi : \mathrm X\times \mathrm Y\rightarrow \mathrm {Z}\) on Banach spaces, and we find out that this Banach algebra can be useful for many purposes in the theory of Banach algebras. For example, the construction of \(\mathbf A _\Phi \) enables us to provide many simple examples of Banach algebras with different topological centers, which are neither Arens regular nor either left or right strongly Arens irregular. It also gives examples of Banach algebras which are not n-weakly amenable for each natural number n. We also find out that the dual of \(\mathbf A _\Phi \) does not enjoy the factorization property of any level. PubDate: 2019-02-05

Abstract: We propose a generalization of Ledet’s conjecture, which predicts the essential dimension of cyclic p-groups in characteristic p, for finite commutative unipotent group schemes. And we present some evidence for this conjecture and discuss some consequences. PubDate: 2019-02-05

Abstract: For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or \(\mathbb {R}^N\) . PubDate: 2019-01-19

Abstract: Let \((X, *, 0)\) be a d-algebra and \(\alpha , \beta \) are endomorphisms on X. Motivated by some results on derivations, \((\alpha ,\beta )\) -derivation in rings, and the generalizations of BCK and BCI-algebras, in this paper, we introduce the notion of \((\alpha ,\beta )\) -derivations on d-algebras, construct several examples and investigate some simple and important results. PubDate: 2019-01-03

Abstract: In this paper, we introduce a notion of dependency between subsets of an arbitrary fixed non-empty set \(\Omega \) . To be more detailed, we introduce a preorder \(\leftarrow \) on the power set \(\mathcal {P}(\Omega )\) having the further property that \(B \leftarrow A\) if and only if \(\{b\} \leftarrow A\) for any \(b \in B\) . We shall argue that this relation generalizes well-studied notions of dependence occurring in such fields as linear algebra, topology, and combinatorics. Furthermore, we show that this relation is characterized by two set operators whose fixed points have interesting geometric and order-theoretic properties. After giving some some elementary results about such a dependency relation, we provide some specific examples taken from graph theory. An interesting property we will provide consists of the possibility to characterize partial orders on a finite lattice in terms of a suitable dependency relation. Finally, we introduce and analyze some specific classes of dependency relations, namely attractive and anti-attractive dependency relations. PubDate: 2019-01-02

Abstract: The main purpose of this paper is to investigate the character amenability of semigroup algebras. In this regard, the new concept character amenability modulo an ideal of Banach algebras are introduced. For a large class of semigroups such as E-inversive E-semigroup and eventually inverse semigroups, it is shown that the semigroup S is amenable if and only if the semigroup algebra \(l^1(S)\) is character amenable modulo an ideal. Some characterizations of character amenability modulo an ideal of Banach algebras are studied and interesting examples are presented. PubDate: 2019-01-01

Abstract: Let G be a group with identity e and let R be a commutative G-graded ring. In this paper, we will investigate commutative graded rings which satisfy the condition \((*)\) . We say that a graded ring R satisfy the condition \((*)\) if P is a graded prime ideal of R and if \(\{I_{\alpha }\}_{\alpha \in \Delta }\) is a family of graded ideals of R, then P contains \(\cap _{\alpha \in \Delta }I_{\alpha }\) only if P contains some \(I_{\alpha }\) . PubDate: 2018-12-01

Abstract: A theorem on the solutions of the problem \(U'(w)={\gamma }F(U(w),w), U(w_1)=u_1,\ U(w_2)=u_2\) is applied for finding the functional solutions of the system of partial differential equations $$\begin{aligned} {\nabla }\cdot (a(u,w){\nabla }u)= & {} 0,\quad u=u_1{\ \hbox {on}\quad {{\Gamma }}_1},\quad u=u_2{\ \hbox {on}\quad {{\Gamma }}_2},\quad \frac{{\partial }u}{{\partial }n}=0{\ \hbox {on}\quad {{\Gamma }}_3} \\ {\nabla }\cdot (b(u,w){\nabla }w)= & {} 0,\quad w=w_1{\ \hbox {on}\quad {{\Gamma }}_1},\quad w=w_2{\ \hbox {on}\quad {{\Gamma }}_2},\quad \frac{{\partial }w}{{\partial }n}=0{\ \hbox {on}\quad {{\Gamma }}_3}. \end{aligned}$$ The problem of existence and uniqueness of solutions is considered. PubDate: 2018-12-01

Abstract: We consider the equation 1 $$\begin{aligned} - (r(x)y'(x))'+q(x)y(x)=f(x),\quad x\in {\mathbb {R}} \end{aligned}$$ where \(f\in L_1({\mathbb {R}}) \) and 2 $$\begin{aligned}&r >0,\quad q\ge 0,\quad 1/r\in L_1^{\mathrm{loc}}({\mathbb {R}}),\quad q\in L_1^{\mathrm{loc}}({\mathbb {R}}),\end{aligned}$$ 3 $$\begin{aligned}&\lim _{ d \rightarrow \infty }\int _{x-d}^x\frac{dt}{r(t)}\cdot \int _{x-d}^x q(t)dt=\infty . \end{aligned}$$ By a solution of (1), we mean any function y, absolutely continuous in \({\mathbb {R}}\) together with \(ry'\) , which satisfies (1) almost everywhere in \({\mathbb {R}}.\) Under conditions (2) and (3), we give a criterion for correct solvability of (1) in the space \(L_1({\mathbb {R}})\) . PubDate: 2018-12-01

Abstract: Consider a sequence \((a_n)_{n\ge 1}\) of complex numbers such that for some positive number p we have \(\lim _{n\rightarrow \infty }\frac{1}{n^p}\left( \sum _{k=1}^na_k\right) =\ell \in \mathbb {C}\) . We prove that, under some mild conditions on the sequence (positivity or absolute boundedness in the mean, etc.), we have $$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{n^{q}}\sum _{k=1}^nk^{q-p}a_kf\left( \frac{k}{n}\right) = p\ell \int _0^1x^{q-1}f(x)dx \end{aligned}$$ for all \(q>0\) and all Riemann integrable functions \(f:[0,1]\rightarrow \mathbb {C}\) . Some applications to probability theory and to number theory are also discussed. PubDate: 2018-12-01

Authors:Kathryn E. Hare; Jimmy He Abstract: Let G be a compact, connected simple Lie group and \(\mathfrak {g}\) its Lie algebra. It is known that if \(\mu \) is any G-invariant measure supported on an adjoint orbit in \(\mathfrak {g}\) , then for each integer k, the k-fold convolution product of \(\mu \) with itself is either singular or in \( L^{2}\) . This was originally proven by computations that depended on the Lie type of \(\mathfrak {g}\) , as well as properties of the measure. In this note, we observe that the validity of this dichotomy is a direct consequence of the Duistermaat–Heckman theorem from symplectic geometry and that, in fact, any convolution product of (even distinct) orbital measures is either singular or in \(L^{2+\varepsilon }\) for some \(\varepsilon >0\) . An abstract transference result is given to show that the \(L^{2}\) -singular dichotomy holds for certain of the G-invariant measures supported on conjugacy classes in G. PubDate: 2018-01-11 DOI: 10.1007/s40574-017-0154-9

Authors:Maurizio Brunetti; Adriana Ciampella Abstract: Let p be an odd prime. In this paper we determine the group of length-preserving automorphisms for the ordinary Steenrod algebra A(p) and for \({\mathcal {B}}(p)\) , the algebra of cohomology operations for the cohomology of cocommutative \(\mathbb {F}_p\) -Hopf algebras. Contrarily to the \(p=2\) case, no length-preserving strict monomorphism turns out to exist. PubDate: 2018-01-10 DOI: 10.1007/s40574-017-0155-8

Authors:Rasoul Akrami; Ali Reza Janfada; Mohammad Reza Miri Abstract: Let \(\mathcal {M}\) be a Hilbert \(C^*\) -module. Set \( x =\big <x,x\big >^\frac{1}{2}\) for every \(x\in \mathcal {M}\) and \(\mathcal {C}=\overline{\{\big <x,y\big >:x,y\in \mathcal {M}\}}\) . We prove that \( xa = x a \) for every \(x\in \mathcal {M}\) and \(a\in \mathcal {C}\) if and only if \(\mathcal {C}\) is a commutative \(C^*\) -algebra. PubDate: 2017-12-26 DOI: 10.1007/s40574-017-0152-y