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1. CMB Online first

Haase, Christian; Hofmann, Jan
 Convex-normal (pairs of) polytopes In 2012 Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between $k$- and $(k+1)$-convex-normality (for $k\geq 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement of the normal fan of $Q$. If every edge $e_P$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap \mathbb{Z}^d = (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$. Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopesCategories:52B20, 14M25, 90C10

2. CMB 2016 (vol 59 pp. 564)

Li, Boyu
 Normal Extensions of Representations of Abelian Semigroups A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation on an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extend a result due to Athavale to representations on abelian lattice ordered semigroups. Keywords:subnormal operator, normal extension, regular dilation, lattice ordered semigroupCategories:47B20, 47A20, 47D03

3. CMB 2016 (vol 59 pp. 461)

Ara, Pere; O'Meara, Kevin C.
 The Nilpotent Regular Element Problem We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular. Keywords:nilpotent element, von Neumann regular element, unit-regular, Bergman's normal formCategories:16E50, 16U99, 16S10, 16S15

4. CMB 2016 (vol 59 pp. 575)

Li, Jifu; Hu, Zhiguang; Deng, Shaoqiang
 Cohomogeneity One Randers Metrics An action of a Lie group $G$ on a smooth manifold $M$ is called cohomogeneity one if the orbit space $M/G$ is of dimension $1$. A Finsler metric $F$ on $M$ is called invariant if $F$ is invariant under the action of $G$. In this paper, we study invariant Randers metrics on cohomogeneity one manifolds. We first give a sufficient and necessary condition for the existence of invariant Randers metrics on cohomogeneity one manifolds. Then we obtain some results on invariant Killing vector fields on the cohomogeneity one manifolds and use that to deduce some sufficient and necessary condition for a cohomogeneity one Randers metric to be Einstein. Keywords:cohomogeneity one actions, normal geodesics, invariant vector fields, Randers metricsCategories:53C30, 53C60

5. CMB 2015 (vol 58 pp. 799)

Kong, Qingjun; Guo, Xiuyun
 On $s$-semipermutable or $s$-quasinormally Embedded Subgroups of Finite Groups Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow $p$-subgroup $G_{p}$ of $G$ with $(p,|H|)=1$; $H$ is said to be $s$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-quasinormal subgroup of $G$. We fix in every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$ satisfying $1\lt |D|\lt |P|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either $s$-semipermutable or $s$-quasinormally embedded in $G$. Some recent results are generalized and unified. Keywords:$s$-semipermutable subgroup, $s$-quasinormally embedded subgroup, saturated formation.Categories:20D10, 20D20

6. CMB 2014 (vol 58 pp. 9)

Chavan, Sameer
 Irreducible Tuples Without the Boundary Property We examine spectral behavior of irreducible tuples which do not admit boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $(T_1, \dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots + T^*_mT_m$. We use this simple observation to ensure boundary property for an irreducible, essentially normal joint $q$-isometry provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}[z_1, \dots, z_m]$-modules (of which the Drury-Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal. Keywords:boundary representations, subnormal, joint p-isometryCategories:47A13, 46E22

7. CMB 2014 (vol 58 pp. 160)

Pollack, Paul; Vandehey, Joseph
 Some Normal Numbers Generated by Arithmetic Functions Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number $0. f(1) f(2) f(3) \dots$ obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and ErdÅs in 1946 to prove the $10$-normality of $0.235711131719\ldots$. Keywords:normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's numberCategories:11K16, 11A63, 11N25, 11N37

8. CMB Online first

Pollack, Paul; Vandehey, Joseph
 Some normal numbers generated by arithmetic functions Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number $0. f(1) f(2) f(3) \dots$ obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and ErdÅs in 1946 to prove the $10$-normality of $0.235711131719\ldots$. Keywords:normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's numberCategories:11K16, 11A63, 11N25, 11N37

9. CMB 2014 (vol 57 pp. 579)

Larson, Paul; Tall, Franklin D.
 On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of $\omega_1$ is hereditarily paracompact. Keywords:locally compact, hereditarily normal, paracompact, Axiom R, PFA$^{++}$Categories:54D35, 54D15, 54D20, 54D45, 03E65, 03E35

10. CMB 2013 (vol 56 pp. 745)

Fu, Xiaoye; Gabardo, Jean-Pierre
 Dimension Functions of Self-Affine Scaling Sets In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$. Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension functionCategory:42C40

11. CMB 2011 (vol 56 pp. 459)

Athavale, Ameer; Patil, Pramod
 On Certain Multivariable Subnormal Weighted Shifts and their Duals To every subnormal $m$-variable weighted shift $S$ (with bounded positive weights) corresponds a positive Reinhardt measure $\mu$ supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for $m \geq 2$, the dimensions of the $1$-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual ${\tilde S}$ are different if a certain radial function happens to be integrable with respect to $\mu$ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to ${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual. Keywords:subnormal, Reinhardt, Betti numbersCategory:47B20

12. CMB 2011 (vol 55 pp. 368)

Nie, Zhaohu
 The Secondary Chern-Euler Class for a General Submanifold We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern. Keywords:secondary Chern-Euler class, normal sphere bundle, Euler characteristic, index, non-isolated singularities, blow-upCategory:57R20

13. CMB 2011 (vol 54 pp. 249)

Dattori da Silva, Paulo L.
 A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic. Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$)Categories:35A01, 58Jxx

14. CMB 2010 (vol 54 pp. 21)

 Generalized D-symmetric Operators II Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L }(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$. This note will initiate a study on the class of pairs $(A,B)$ such that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal R}(\delta_{A^{\ast}B^{\ast}})}$. Keywords:generalized derivation, adjoint, D-symmetric operator, normal operatorCategories:47B47, 47B10, 47A30

15. CMB 2008 (vol 51 pp. 508)

Cavicchioli, Alberto; Spaggiari, Fulvia
 A Result in Surgery Theory We study the topological $4$-dimensional surgery problem for a closed connected orientable topological $4$-manifold $X$ with vanishing second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has one end and $F(r)$ is the free group of rank $r\ge 1$. Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups. Keywords:four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly mapCategories:57N65, 57R67, 57Q10

16. CMB 2008 (vol 51 pp. 261)

Neeb, Karl-Hermann
 On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups An $n$-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field. Keywords:quantum torus, normal form, automorphisms of quantum toriCategory:16S35

17. CMB 2005 (vol 48 pp. 195)

Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
 On Suslinian Continua A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most $\omega_1$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set. Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finiteCategories:54F15, 54D15, 54F50

18. CMB 2001 (vol 44 pp. 323)

Schuman, Bertrand
 Une classe d'hamiltoniens polynomiaux isochrones Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S]. Keywords:Hamiltonian system, normal forms, resonance, linearizationCategories:34C20, 58F05, 58F22, 58F30
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