# Rendiconti del Seminario Matematico della Università di PadovaThe Mathematical Journal of the University of Padua

ISSN 0041-8994
e-ISSN: 2240-2926

## Forthcoming Articles

(Last update: February 7, 2018)

The following articles will soon be published in this journal. Their publication status can be :

• online first – final version available (without page numbers nor DOI);
• accepted – manuscript available.

To view the abstract, or download the full paper in PDF[1], click on the article's title.

## Online first

•

Abstract: See the paper itself.

## Accepted Papers

• ### Finite groups with the pp-embedding property

Institute of Mathematics
University of Bialystok
K. Ciolkowskiego 1M
15-245 Bialystok
Poland

Abstract: A subset $X$ of a finite group $G$ is called g-independent if there is no proper subset $Y$ of $X$ such that $\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle.$ The group $G$ has the embedding property if every g-independent subset of $G$ can be embedded in a minimal generating set of $G$. If $X$ is a set of prime power order elements, then we say that $G$ has the pp-embedding property. In this note we classify all finite solvable groups with the pp-embedding property. Moreover we prove that this class is equal to the class of finite solvable groups with the embedding property.

received 26.10.2017, accepted 19.01.2018 (11 pages)
• ### Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

Department of Applied Mathematics
Northwestern Polytechnical University
Xian
China

Abstract: See the article

received 18.03.2017, accepted 09.12.2017 (32 pages)
• ### On the generalized $\sigma$-Fitting subgroup of finite groups

Hu, Huang

School of Mathematics and Statistics
Jiangsu Normal University
Xuzhou, 221116
P. R. China

Skiba

Department of Mathematics
Francisk Skorina Gomel State University
Gomel 246019
Belarus


Abstract: see the article

received 26.02.2017, accepted 29.11.2017 (15 pages)
• ### On the monotonicity of Hilbert functions

Department of Mathematics
Indian Institute of Technology Bombay
Powai
Mumbai 400 076
India


Abstract: In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ (with respect to $\mathfrak{m}$) is non-decreasing. Examples include
(1) Complete intersections $A = Q/(f,g)$ where $(Q,\mathfrak{n})$ is regular local of dimension three and $f \in \mathfrak{n}^2 \setminus \mathfrak{n}^3$.
(2) One dimensional Cohen-Macaulay quotients of a two dimensional Cohen-Macaulay local ring with pseudo-rational singularity.

received 16.02.2017, accepted 09.11.2017 (7 pages)
• ### Zeros of irreducible characters of metabelian $p$-groups

2 Amner Road
London SW11 6AA
UK

Abstract: We show that if $\chi$ is an irreducible complex character of a metabelian $p$ - group $P,$ where $p$ is an odd prime, and if $x\in P$ satisfies $\chi(x)\neq 0$, then the order of $x$ divides $|P|/\chi(1)^2.$

received 21.06.2017, accepted 09.11.2017 (9 pages)
• ### Cohen-Macaulayness and sequentially Cohen-Macaulayness of monomial ideals

Department of Mathematics
Tabriz
Iran

Department of Mathematics
Razi University
Kermanshah
Iran


Abstract: In this paper, we give a characterization for Cohen--Macaulay rings $R/I$ where $I\subset R=K[y_1, \dots, y_n]$ is a monomial ideal which satisfies $\rm{bigsize}\, I=\rm{size}\, I$. Next, we let $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a polynomial ring and $I\subset S$ a monomial ideal. We study the sequentially Cohen-Macaulayness of $S/I$ with respect to $Q=(y_1,\ldots,y_n)$. Moreover, if $I\subset R$ is a monomial ideal such that the associated prime ideals of $I$ are in pairwise disjoint sets of variables, a classification of $R/I$ to be sequentially Cohen-Macaulay is given. Finally, we compute $\rm{grade}\,(Q, M)$ where $M$ is a sequentially Cohen-Macaulay $S$-module with respect to $Q$.

received 02.04.2017, accepted 23.10.2017 (14 pages)
• ### On generalized $\Pi$-property of subgroups of finite groups

School of Mathematics
Jilin University
No. 2699 Qianjin Street
Changchun
China

Abstract: In this note, we extend the concept of $\Pi$-property of subgroups of finite groups and generalize some recent results. In particular, we generalize the main results of Li et al. [8] and Miao et al. [9].

received 28.08.2017, accepted 23.10.2017 (17 pages)
• ### On the uniqueness of the factorization of power digraphs modulo n

Department of Mathematics
North-Eastern Hill University
Shillong-793022
India

Department of Basic Sciences & Social Sciences
North-Eastern Hill University
Shillong-793022
India

Abstract: For each pair of integers $n=\prod_{i=1}^{r}p_i^{e_i}$ and $k \geq 2$, a digraph $G(n,k)$ is one with vertex set $\{0,1,\ldots ,n-1\}$ and for which there exists a directed edge from $x$ to $y$ if $x^k \equiv y \pmod n$. Using the Chinese Remainder Theorem, the digraph $G(n,k)$ can be written as a direct product of digraphs $G(p_i^{e_i},k)$ for all $i$ such that $1 \leq i \leq r$. A fundamental constituent $G_{P}^{*}(n,k)$, where $P \subseteq Q=\{p_1,p_2,\ldots,p_r\}$, is a subdigraph of $G(n,k)$ induced on the set of vertices which are multiples of $\prod_{{p_i} \in P}p_i$ and are relatively prime to all primes $p_j \in Q \smallsetminus P$. In this paper, we investigate the uniqueness of the factorization of trees attached to cycle vertices of the type $0$, $1$, and $(1,0)$, and in general, the uniqueness of $G(n,k)$. Moreover, we provide a necessary and sufficient condition for the isomorphism of the fundamental constituents $G_{P}^{*}(n,k_1)$ and $G_{P}^{*}(n,k_2)$ of $G(n,k_1)$ and $G(n,k_2)$ respectively for $k_1 \neq k_2$.

received 01.06.2017, accepted 20.10.2017 (28 pages)
• ### Semisimple groups that are quasi-split over a tamely ramified extension

Université Claude Bernard Lyon 1
CNRS UMR 5208
Institut Camille Jordan
43 Blvd. du 11 novembre 1918
F-69622 Villeurbane cedex
France

Abstract: Let $K$ be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple $K$-group $G$ is quasi-split if and only if it quasi-splits after a finite tamely ramified extension of $K$.

received 07.07.2017, accepted 16.10.2017 (9 pages)
• ### Complex manifolds as families of homotopy algebras

Institut de Mathématiques de Toulouse
UMR5219, Université de Toulouse
CNRS, UPS IMT
F-31062 Toulouse Cedex 9
France

Abstract: We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex structure on a smooth manifold corresponds in this way to a family of algebras indexed by the points of the manifold.

received 24.03.2017, accepted 25.09.2017 (38 pages)
• ### A brief journey through extensions of rational groups

University of Stralsund
Zur Schwedenschanze 15
18435 Stralsund
Germany


Abstract: Let $A$ and $B$ be rational groups, i.e. torsion-free groups of rank-1 and thus subgroups of the rational numbers. This paper gives a short overview of the structure of $\text{Ext}\,(A,B)$ especially considering some interesting classes of torsion-free pairs.

received 06.03.2017, accepted 20.09.2017 (5 pages)
• ### Algebraic cycles on a very special EPW sextic

Institut de Recherche Mathématique Avancée
CNRS _ Université de Strasbourg
7 Rue René Descartes
67084 Strasbourg CEDEX
France

Abstract: Motivated by the Beauville--Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten--Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperk\"ahler fourfolds.

received 22.06.2016, accepted 28.08.2017 (27 pages)
• ### Algèbres de distributions et ${\cal D}$-modules arithmétiques

IRMA
Université de Strasbourg
7, rue René Descartes
67084 Strasbourg cedex
France

IRMAR
Université de Rennes 1
Campus de Beaulieu
35042 Rennes cedex
France

Abstract: Let $p$ be a prime number, $V$ a complete discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over $Spec \,V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level $m$, generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra over a finite extension $K$ of ${\bf Q}_p$ . We then show that these distribution algebras can be identified with invariant arithmetic differential operators over $G$, and prove a coherence result when the ramification index of $K$ is $received 13.10.2015, accepted 07.07.2017 (66 pages) download: manuscript (778K) • ### Countable recognizability and residual properties of groups Dipartimento di Matematica e Applicazioni Università di Napoli Federico II Complesso Universitario Monte S. Angelo Via Cintia, Napoli Italy Abstract: A class of groups {\mgoth X} is said to be countably recognizable if a group belongs to~{\mgoth X} whenever all its countable subgroups lie in {\mgoth X}. It is proved here that the class of groups whose subgroups are closed in the profinite topology is countably recognizable. Moreover, countably detectable properties of the finite residual of a group are studied. received 17.01.2017, accepted 29.05.2017 (11 pages) download: manuscript (306K) • ### A result on a singular Cauchy problem with a radial point revisited using microdifferential calculus Department of Mathematics Edificio Mario Laserna Cra 1 Este No 19A 40 Bogota Colombia Universidad de los Andes Colombia Abstract: We study the ramified Cauchy problem for a linear PDE with a radial point using the theory of microdifferential operators. received 23.09.2016, accepted 11.05.2017 (20 pages) download: manuscript (447K) • ### Mostow s fibration for canonical embeddings of compact homogeneous$CR$manifolds Dipartimento di Matematica e Fisica III Università di Roma Largo San Leonardo Murialdo, 1 00146 Roma Italy Abstract: We define a class of compact homogeneous$CR$manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy-Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings. received 17.12.2016, accepted 09.05.2017 (30 pages) download: manuscript (278K) • ### Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries RUDN University S. M. Nikol’skii Institute 6 Miklukho-Maklay St Moscow, 117198 Russia  Abstract: We introduce a class of Morrey-type spaces$M^\lambda_{p,q}$, which includes classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem. This theorem is then applied to obtaining the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operator. received 05.10.2016, accepted 26.04.2017 (20 pages) download: manuscript (173K) • ### Motifs et adjoints Institut de Mathématiques de Jussieu - PRG Case 247 4 place Jussieu 75252 Paris Cedex 05 France Abstract: We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrasted with the more classical situation where one deals with a finite type morphism of schemes. Among various applications, one is a construction of a Tate-\v Safarevi\v c motive" attached to an abelian variety over a function field. We also deduce a possible approach to \text{Bloch's} conjecture on surfaces, by reduction to curves. received 18.11.2015, accepted 22.03.2017 (44 pages) download: manuscript (702K) • ### Definable categories and$\mathbb{T}$-motives Dipartimento di Matematica "F. Enriques" Università degli Studi di Milano Via C. Saldini, 50 I-20133 Milano Italy School of Mathematics University of Manchester Oxford Road Manchester M13 9PL UK Abstract: Making use of Freyd's free abelian category on a preadditive category we show that if$T:D\rightarrow \mathcal{A}$is a representation of a quiver$D$in an abelian category$\mathcal{A}$then there is an abelian category$\mathcal{A} (T)$, a faithful exact functor$F_T: \mathcal{A} (T) \to \mathcal{A}$and an induced representation$\tilde T: D \to \mathcal{A} (T)$such that$F_T\tilde T= T$universally. We then can show that$\mathbb{T}$-motives as well as Nori's motives are given by a certain category of functors on definable categories. received 03.10.2016, accepted 15.03.2017 (18 pages) download: manuscript (395K) • ### Nunke’s problem for abelian$p$-groups of uncountable length Department of Mathematics Whitman College Walla Walla, WA 99362 USA Abstract: \textit{Nunke's Problem} asks when the torsion product of two abelian$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) download: manuscript (292K) • ### The isoperimetric problem in Grushin space$\mathbb{R}^{h+1}$with density$|x|^p$School of Science Nanjing University of Science and Technology Nanjing 210094 P. R. China Abstract: In this paper we study the isoperimetric problem in a class of$x$-spherically symmetric sets in the Grushin space$\mathbb{R}^{h+1}$with density$|x|^p$,$p>-h+1$. First we prove the existence of weighted isoperimetric sets. Then we deduce that, up to a vertical translation, a dilation and a negligible set, the weighted isoperimetric set is only of the form$\big\{(x,y)\in \mathbb{R}^{h+1}:|y|<\int_{\arcsin|x|}^{\frac{\pi}{2}}\sin^{\alpha+1}(t)dt,\hspace{2mm}|x|<1\big\}$. received 31.01.2017, accepted 24.02.2017 (17 pages) download: manuscript (423K) • ### Cyclic subgroup commutativity degrees of finite groups Faculty of Mathematics Al. I. Cuza University Iasi Romania Abstract: In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group$G$. This quantity measures the probability of two random cyclic subgroups of$G$commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented. received 13.12.2016, accepted 21.02.2017 (15 pages) download: manuscript (311K) • ### Weak local-global compatibility and ordinary representations Instytut Matematyczny PAN Ul. Sniadeckich 8 00-656 Warsawa Poland Abstract: We introduce a general formalism with minimal requirements under which we are able to prove the pro-modular Fontaine-Mazur conjecture. We verify it in the ordinary case using the recent construction of Breuil and Herzig. received 04.07.2016, accepted 13.02.2017 (22 pages) download: manuscript (436K) • ### Some sufficient conditions for p-nilpotence of a finite group School of Mathematics and Statistics Jiangsu Normal University Xuzhou, 221116 China lcw2000@126.com Abstract: In this paper, we give some new characterizations of finite$p$-nilpotent groups by using the notion of$\mathcal{H}C$-subgroups and extend several recent results. received 28.09.2016, accepted 17.01.2017 (8 pages) download: manuscript (98K) • ### On an open problem about$\pi$-quasi-$\mathcal{F}$-groups Department of Mathematics University of Science and Technology of China Hefei, 230026 P.R. China Department of Mathematics University of Science and Technology of China Hefei, 230026 P.R. China School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou, 22 1116 P.R. China Abstract: Let$\mathfrak{F}$be a class of finite groups,$p$a prime and$\pi$a set of some primes. A finite group$G$is called a$p$-quasi-$\mathfrak{F}$-group (respectively, by$\pi$-quasi-$\mathfrak{F}$-group) provided that for every$\mathfrak{F}$-eccentric$G$-chief factor$H/K$of order divisible by$p$(respectively, by at least one prime in$\pi$), the automorphisms of$H/K$induced by all elements of$G$are inner. In this paper, we obtain the characterizations of$p$-quasi-$\mathfrak{F}$-groups and$\pi$-quasi-$\mathfrak{F}$-groups, which give a positive answer to an open problem in the book [3]. received 29.07.2016, accepted 04.11.2016 (8 pages) download: manuscript (336K) • ### On$W$-$S$-permutable subgroups of finite groups Department of Mathematics Shanghai University Shanghai 200444 P. R. China Abstract: A subgroup$H$of a finite group$G$is said to be$W$-$S$-permutable in$G$if there is a subgroup$K$of$G$such that$G=HK$and$H\cap K$is a nearly$S$-permutable subgroup of$G$. In this article, we analyse the structure of a finite group$G$by using the properties of$W$-$S$-permutable subgroups and obtain some new characterizations of finite$p$-nilpotent groups and finite supersolvable groups. Some known results are generalized. received 20.04.2016, accepted 06.10.2016 (12 pages) download: manuscript (355) • ### On$H_\sigma$-permutably embedded subgroups of finite groups Guo Zhang Department of Mathematics University of Science and Technology of China Hefei 230026 P. R. China Skiba Sinitsa Department of Mathematics Francisk Skorina Gomel State University Gomel 246019 Belarus  Abstract: Let$G$be a finite group. Let$\sigma =\{\sigma_{i} | i\in I\}$be a partition of the set of all primes$\Bbb{P}$and$n$an integer. We write$\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$,$\sigma (G) =\sigma (|G|)$. A set$ {\cal H}$of subgroups of$G$is said to be a complete Hall$\sigma $-set of$G$if every member of${\cal H}\setminus \{1\}$is a Hall$\sigma _{i}$-subgroup of$G$for some$\sigma _{i}$and${\cal H}$contains exact one Hall$\sigma _{i}$-subgroup of$G$for every$\sigma _{i}\in \sigma (G)$. A subgroup$A$of$G$is called: (i) a$\sigma$-Hall subgroup of$G$if$\sigma (A) \cap \sigma (|G:A|)=\emptyset$; (ii)${\sigma}$-permutable in$G$if$G$possesses a complete Hall$\sigma$-set${\cal H}$such that$AH^{x}=H^{x}A$for all$H\in {\cal H}$and all$x\in G$. We say that a subgroup$A$of$G$is$H_{\sigma}$-permutably embedded in$G$if$A$is a${\sigma}$-Hall subgroup of some${\sigma}$-permutable subgroup of$G$. We study finite groups$G$having an$H_{\sigma}$-permutably embedded subgroup of order$|A|$for each subgroup$A$of$G$. Some known results are generalized. received 07.06.2016, accepted 18.08.2016 (13 pages) download: manuscript (187K) • ### Test modules for flatness Yasar University Department of Mathematics 35100, Izmir Turkey Bitlis Eren University Department of Mathematics 13000, Bitlis Turkey  Abstract: A right$R$-module$M$is said to be a test module for flatness (shortly: an f-test module) provided for each left$R$-module$N$,$Tor(M,N)=0$implies$N$is flat.$f$-test modules are flat version of the Whitehead test modules for injectivity defined by Trlifaj. In this paper the properties of f-test modules are investigated and are used to characterize various families of rings. The structure of a ring over which every (finitely generated) right$R$-module is flat or f-test is investigated. Abelian groups that are Whitehead test modules for injectivity or f-test are characterized. received 22.02.2015, accepted 14.06.2016 (15 pages) download: manuscript (129K) • ### On the piecewise approximation of bi-Lipschitz curves Department of Mathematics University of Erlangen Cauerstrasse 11 90158 Erlangen Germany Abstract: In this paper we deal with the task of uniformly approximating an$L$-biLipschitz curve by means of piecewise linear ones. This is rather simple if one is satisfied to have approximating functions which are$L'$-biLipschitz, for instance this was already done with$L'=4L$in [3, Lemma~5.5]. The main result of this paper is to do the same with$L'=L+\varepsilon$(which is of course the best possible result); in the end, we generalize the result to the case of closed curves. received 12.05.2015, accepted 31.05.2016 (30 pages) download: manuscript (408K) • ### Configurations of points on degenerate varieties and properness of moduli spaces Department of Mathematics Box 1917 Brown University Providence, RI, 02912 U.S.A SISSA Via Bonomea 265 34136 Trieste Italy  Abstract: see the paper itself received 19.09.2014, accepted 01.04.2016 (16 pages) download: manuscript (341K) • ### A comparison of logarithmic overconvergent de Rham-Witt and log-crystalline cohomology for projective smooth varieties with normal crossing divisor University of Exeter Mathematics Exeter, EX4 4QF Devon UK Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld Germany Abstract: In this note we derive for a smooth projective variety$X$with normal crossing divisor$Z$an integral comparison between the log-crystalline cohomology of the associated log-scheme and the logarithmic overconvergent de Rham-Witt cohomology defined by Matsuue. This extends our previous result that in the absence of a divisor$Z$the crystalline cohomology and overconvergent de Rham-Witt cohomology are canonically isomorphic. received 07.09.2015, accepted 14.03.2016 (6 pages) download: manuscript (221K) • ### Primary group rings Institute of Mathematical Sciences Faculty of Science University of Malaya 50603 Kuala Lumpur Malaysia Abstract: Let$R$be an associative ring with identity and let$J(R)$denote the Jacobson radical of$R$. We say that$R$is primary if$R/J(R)$is simple Artinian and$J(R)$is nilpotent. In this paper we obtain necessary and sufficient conditions for the group ring$RG$, where$G$is a nontrivial abelian group, to be primary. received 25.08.2015, accepted 23.02.2016 (5 pages) download: manuscript (89K) • ### A construction of a uniform continuous minimizing movement associated with a singular functional Department of Mathematics College of Humanities and Sciences Nihon University 3-25-40, Sakurajosui, Setagaya Tokyio 156-8550 Japan Abstract: A minimizing movement is constructed associated with a singular functional introduced by Alt and Caffarelli in order to study a free boundary problem. The main purpose of the present research is to construct a minimizing movement, which is uniformly continuous with respect to both time and space variables. The strategy is to regularize the singular term of time discretized functionals, and then to pass to the limit in the regularization parameter in the sense of$\mit\Gamma$-convergence keeping the time discretization parameter fixed. received 30.06.2015, accepted 04.02.2016 (27 pages) download: manuscript (192K) • ### Existence of stationary solutions for some integro-differential equations with superdiffusion Department of Mathematics University of Toronto Ontario, M5S 2E4 Canada Institute Camille Jordan UMR 5208 CNRS University Lyon 1 Villeurbane, 69622 France Abstract: The work deals with the existence of solutions of an integro-differential equation arising in population dynamics in the case of anomalous diffusion. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for non-Fredholm elliptic operators in unbounded domains are being used. received 20.07.2015, accepted 28.01.2016 (16 pages) download: manuscript (329K) • ### A new characterization of some families of finite simple groups Department of pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University Tehran Iran Abstract: Let$G$be a finite group. A vanishing element of$G$is an element$g\in G$such that$\chi(g)=0$for some irreducible complex character$\chi$of$G$. Denote by${\rm Vo}(G)$the set of the orders of vanishing elements of$G$. In this paper, we prove that if$G$is a finite group such that${\rm Vo}(G)={\rm Vo}(M)$and$|G|=|M|$, then$G\cong M$, where$M$is a sporadic simple group, an alternating group, a projective special linear group$L_2(p)$, where$p$is an odd prime or a finite simple$K_{n}$-group, where$n\in \{3,4\}$. These results confirm the conjecture posed in [17] for the simple groups under study. received 12.02.2015, accepted 18.01.2016 (16 pages) download: manuscript (164K) • ### A harmonic mean inequality for the digamma function and related results Morsbacher Str, 10 51545 Waldbröl Germany Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF UK  Abstract: We present some inequalities and a concavity property of the digamma function$\psi=\Gamma'/\Gamma$, where$\Gamma$denotes Euler's gamma function. In particular, we offer a new characterization of Euler's constant$\gamma=0.57721...$. We prove that$-\gamma$is the minimum of the harmonic mean of$\psi(x)$and$\psi(1/x)$for$x>0$. received 19.08.2015, accepted 11.01.2016 (6 pages) download: manuscript (256K) • ### Wavelet transform of Beurling-Björck type ultradistributions DST Centre for Interdisciplinary Mathematical Sciences Faculty of Science Banaras Hindu University Varanasi - 221 005 India Abstract: Wavelet transform of a distribution in$\mathscr{M}_{\omega}^{'}$involving wavelet of infraexponential decay (subexponential decay) is studied. An inversion formula is obtained which is valid in the weak topology of$\mathcal{D}^{'}$. A discussion on extension of the results to ultradistribution space of compact support is also given. received 21.08.2015, accepted 11.01.2016 (11 pages) download: manuscript (151K) • ### Strong local-global compatibility in the$p$-adic Langlands program for$U(2)$Mathematical Institute, University of Oxford Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG England Department of Mathematics, UCSD 9500 Gilman Dr. #0112 La Jolla, CA 92093-0112 USA Abstract: For certain mod$p$Galois representations$\bar{\rho}$, arising from modular forms on definite unitary groups in two variables, we express the$\bar{\rho}$-part of completed cohomology$\hat{H}_{\bar{\rho}}^0$(away from$\Sigma=\Sigma_p\cup \Sigma_0$) as a tensor product$\Pi_p\otimes \Pi_{\Sigma_0}$. Here$\Pi_p$is attached to the universal deformation$\rho^{univ}$via the$p$-adic local Langlands correspondence for$\text{GL}_2(\mathbb{Q}_p)$, and$\Pi_{\Sigma_0}$is given by the local Langlands correspondence in families, of Emerton and Helm. received 09.05.2015, accepted 20.11.2015 (17 pages) download: manuscript (448K) • ### Coverings of commutators in profinite groups Department of Mathematics University of Brasilia Brasilia-DF 70910-900 Brazil Abstract: Let$w$be a group-word. Suppose that the set of all$w$-values in a profinite group$G$is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup$w(G)$depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall's problem on conciseness of group-words. received 29.10.2015, accepted 19.11.2015 (19 pages) download: manuscript (320K) • ### Weak local-global compatibility in the p-adic Langlands program for U(2) Mathematical Institute, University of Oxford Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG England chojecki@maths.ox.ac.uk Department of Mathematics, UCSD 9500 Gilman Dr. #0112 La Jolla, CA 92093-0112 USA csorensen@ucsd.edu  Abstract: We study the completed cohomology$\hat{H}^0$of a definite unitary group$G$in two variables associated with a CM-extension$\mathcal{K}/F$. When the prime$p$splits, we prove that (under technical asumptions) the$p$-adic local Langlands correspondence for$\text{GL}_2(\Bbb{Q}_p)$occurs in$\hat{H}^0$. As an application, we obtain a result towards the Fontaine-Mazur conjecture over$\mathcal{K}$: if$x$is a point on the eigenvariety such that$\rho_x$is geometric (and satisfying additional hypotheses which we suppress), then$x$must be a classical point. Thus, not only is$\rho_x$modular, but the weight of$x$defines an accessible refinement. This follows from a recent result of Colmez (which describes the locally analytic vectors in$p$-adic unitary principal series), knowing that$\rho_x$admits a triangulation compatible with the weight. received 09.05.2015, accepted 25.10.2015 (28 pages) download: manuscript (512K) • ### Cyclic non-$S$-permutable subgroups and non-normal maximal subgroups Faculty of Mathematical Science Department of Pure Mathematics Shahrekord University P. O. Box 115 Shahrekord Iran Abstract: A finite group$G$is said to be a$T$-group (resp.$PT$-group,$PST$-group) if normality (resp. permutability,$S$-permutability) is a transitive relation. Ballester-Bolinches et al. gave some new characterizations of the soluble$T$-,$PT$- and$PST$-groups. A finite group$G$is called a$T_c$-group (resp.${PT}_c$-group,${PST}_c$-group) if each cyclic subnormal subgroup is normal (resp. permutable,$S$-permutable) in$G$. The present work defines the${NNM}_c$-,${PNM}_c$-, and${SNM}_c$-groups and presents new characterizations of the wider classes of soluble$T_c$-,${PT}_c$-, and${PST}_c$-groups. received 23.02.2015, accepted 23.07.2015 (7 pages) download: manuscript (83K) • ### Example of minimizer of the average-distance problem with non closed set of corners Department of Mathematical Sciences Carnegie Mellon University Pittsburg, PA, 15213 United States Abstract: The average-distance problem, in the penalized formulation, involves minimizing \begin{equation*} E_\mu^\lambda(\Sigma):=\int_{{\mathbb R}^d} \inf_{y\in\Sigma} |x-y|\operatorname{d}\mu(x)+\lambda{\mathcal H}^1(\Sigma), \end{equation*} among {compact, connected sets$\Sigma$, where${\mathcal H}^1$denotes the 1-Hausdorff measure},$d\geq 2$,$\mu$is a given measure and$\lambda$a given parameter. Regularity of minimizers is a delicate problem. It is known that even if$\mu$is absolutely continuous with respect to Lebesgue measure,$C^1$regularity does not hold in general. An interesting question is whether the set of corners, i.e. points where$C^1$regularity does not hold, is closed. The aim of this paper is to provide an example of minimizer whose set of corners is not closed, with reference measure$\mu\$ absolutely continuous with respect to Lebesgue measure.

received 10.02.2015, accepted 06.07.2015 (27 pages)