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Abstract: In this paper we study the $C^1$-regularity of solutions of one-dimensional variational obstacle problems in $W^{1,1}$ when the obstacles are $C^{1,\sigma}$ and the Lagrangian is locally H\"older continuous and globally elliptic. In this framework, we prove that the solutions of one-dimensional variational obstacle problems are $C^1$ for all boundary data if and only if the value function is Lipschitz continuous at all boundary data.
received 18.08.2017, accepted 07.11.2018 (31 pages)Abstract: Using $\lambda$ operations, we give some results on the kernel of the natural map from the monoid algebra $\Z R$ of a commutative ring $R$ to the ring of $S$-Witt vectors of $R$. As a byproduct we obtain a very natural interpretation of a power series used by Dwork in his proof of the rationality of zeta functions for varieties over finite fields.
received 18.05.2018, accepted 13.10.2018 (10 pages)Abstract: Since 1882 it is known that the so-called {\it Schur}'s quartic contains exactly 64 lines. However, it has not yet been established what is the maximum number of pairwise disjoint lines that it can have. The aim of our work is to show in an elementary and self-contained way that the maximum number of pairwise disjoint lines in {\it Schur}'s quartic is 16 (without using {\it Nikulins}'s Theorem (\cite{Nikulin}) or {\it Miyaoka}'s upper bound (\cite{Miyaoka})).
received 20.09.2017, accepted 03.10.2018 (11 pages)Abstract: Let $G$ be a finite group and $A\leq \operatorname{Aut}(G)$. The index $\left|G\,:\, C_G(A)\right|$ is called the index of $A$ in $G$ and is denoted by $\operatorname{Ind}_G(A)$. In this paper, we study the influence of $\operatorname{Ind}_G(A)$ on the structure of $G$ and prove that $\left[G,\, A\right]$ is solvable in case where $A$ is cyclic, $\operatorname{Ind}_G(A)$ is squarefree and the orders of $G$ and $A$ are coprime. Moreover, for arbitrary $A\leq \operatorname{Aut}(G)$ whose order is coprime to the order of $G$, we show that when $[G,A]$ is solvable, the Fitting height of $[G,A]$ is bounded above by the number of primes (counted with multiplicities) dividing $\operatorname{Ind}_G(A)$ and this bound is best possible.
received 30.05.2012, accepted 25.09.2018 (7 pages)Abstract: Weak and q-weak variational measures defined by Brian S.~Thomson [{\em On {\sl VBG} functions and the Denjoy--Khintchine integral}, Real Analysis Exchange, {\bf41}(1) (2015/16), 173--226] are shown to coincide with variational measures resulting from Riemann definitions of some wide Denjoy type integrals. This fact is applied in characterizations of these integrals, via absolute continuity of weak and q-weak measures. In related results, it is discussed if these Riemann definitions can be essentially simplified. The paper is a follow-up to a paper by the author published some time ago in the Rendiconti [{\em On Riemann-type definition for the wide Denjoy integral}, Rendiconti del Seminario Matematico della Universit\`a di Padova, {\bf126} (2011), 175--200].
received 19.03.2018, accepted 25.09.2018 (28 pages)Abstract: (No abstract)
received 20.04.2018, accepted 25.09.2018 (3 pages)Abstract: We introduce a notion of {\em pure-minimality} for chain complexes of modules and show that it coincides with (homotopic) minimality in standard settings, while being a more useful notion for complexes of flat modules. As applications, we characterize von Neumann regular rings and left perfect rings.
received 15.06.2018, accepted 25.09.2018 (18 pages)Abstract: A self-normalizing subgroup is always self-centralizing, but the converse is not necessarily true. Given a finite group $G$, we denote by $w(G)$ the number of all self-centralizing subgroups of $G$ which are not self-normalizing. We observe that $w(G) = 0$ if and only if $G$ is abelian, and that if $G$ is nonabelian nilpotent then $w(G)\geq 3$. We also prove that if $w(G)\leq 20$ then $G$ is solvable. Finally, we provide structural information in the case when $w(G)\leq 3$.
received 25.06.2018, accepted 25.09.2018 (13 pages)Abstract: In this paper, we prove that any Cohen-Macaulay monomial ideal generated ÿby at most five elements is clean.
received 13.10.2017, accepted 21.06.2018 (15 pages)Abstract: We introduce and study the derived moduli stack $\mathbb{Symp}(X,n)$ of $n$-shifted symplectic structures on a given derived stack $X$, as introduced in [8]. In particular, under reasonable assumptions on $X$, we prove that $\mathbb{Symp}(X,n)$ carries a canonical quadratic form, in the sense of [14]. This generalizes a classical result of Fricke and Habermann (see [3]), which was established in the $C^{\infty}$-setting, to the broader context of derived algebraic geometry, thus proving a conjecture stated in [14].
received 26.09.2017, accepted 15.06.2018 (22 pages)Abstract: The torsion anomalous conjecture states that for any variety $V$ in an abelian variety there are only finitely many maximal $V$-torsion anomalous varieties. We prove this conjecture for $V$ of codimension $2$ in a product $E^N$ of an elliptic curve $E$ without CM, complementing previous results for $E$ with CM. We also give an effective upper bound for the normalized height of these maximal $V$-torsion anomalous varieties.
received 30.01.2017, accepted 11.05.2018 (13 pages)Abstract: Given a prime number $p$ and $x$ an element of the Tate field $\mathbb{C}_p$, the main goal of the present paper is to provide an explicit generating set, which is given by the trace function of $x$ and all its derivatives, for the $\mathbb{C}_p$-Banach algebra of the Galois equivariant Krasner analytic functions defined on the complement in $\mathbb{P}^1(\mathbb{C}_p)$ of the orbit of $x$ with values in $\mathbb{C}_p$.
received 26.09.2017, accepted 13.03.2018 (14 pages)Abstract: In this study we first define the concept of Nadler type contraction in the setting of $H$-cone $b$-metric space with respect to cone $b$-metric spaces over Banach algebras. Next we prove the Banach contraction principle for such contractions by means of the notion of spectral radius and a solid cone in underlying Banach algebra. Finally we observe that the main result achieved in this work extends and generalizes the well known results associated with contractions of Nadler type.
received 05.02.2018, accepted 12.03.2018 (9 pages)Abstract: The algebra of noncommutative differential forms has been defined by A. Connes in }$\left[ 4\right] .$ Using this algebra, M. Karoubi has defined cyclic homology and Hochschild homology groups (see. $\left[ 13\right] $). These groups are related to the algebraic K-theory. The purpose of this paper is to provide the noncommutative differential forms algebra with the structure of Gerstenhaber-Voronov algebras.
received 13.06.2017, accepted 01.03.2018 (14 pages)Abstract: If $\{\gamma^{s+1}G\}$ and $\{\zeta_s(G)\}$ denote respectively the lower and upper central series of the group $G$, $s\ge 0$ an integer, and if $\gamma^{s+1}G/(\gamma^{s+1}G\cap\zeta_s(G))$ is polycyclic (resp. polycyclic-by-finite) for some $s$, then we prove that $G/\zeta_{2s}(G)$ is polycyclic (resp. polycyclic-by-finite). The corresponding result with polycyclic replaced by finite was proved in 2009 by G. A. Fern ndez-Alcober and M. Morigi. We also present an alternative approach to the latter.
received 25.08.2017, accepted 23.02.2018 (10 pages)Abstract: We address in this paper the issue of singularities of ultrahyperfunctions. Following the Carmichael's approach for ultrahyperfunctions, we study the relation between the singular spectrum of a class of tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. In passing, a simple version of the celebrated edge of the wedge theorem for this setting is derived from the integral representation without using cohomology.
received 18.08.2017, accepted 09.02.2018 (14 pages)Abstract: A finite group $G$ is said to be \textsl{twisted cyclic} if there exist $\phi \in \text{Aut}(G)$ and $x \in G$ such that $G=\{(x^i)\phi^j\ :\ i,j \in \mathbb{Z}\}$. In this note, we classify all groups satisfying this property and determine that, if a finite group $G$ is twisted cyclic, then $G$ is isomorphic to $\mathbb{Z}_{p^n}$, $\mathbb{Z}_p\times \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$, $Q_8$, $\mathbb{Z}_{p^n}\times \mathbb{Z}_{p^n}$ or direct products of these groups for some prime $p$ and some $n\in \mathbb{Z}^{+}$.
received 03.01.2018, accepted 09.02.2018 (9 pages)Abstract: \textit{Nunke's Problem} asks when the torsion product of two abelian $p$-groups is a direct sum of countable reduced groups. In previous work the author gave a complete answer to this question when the groups involved have countable length. In this paper a complete answer is given in the case of groups of uncountable length, at least in any set-theoretic universe in which $2^{\aleph_1}=\aleph_2$.
received 18.01.2017, accepted 03.03.2017 (18 pages)