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1. CMB 2015 (vol 58 pp. 271)

 On Domination of Zero-divisor Graphs of Matrix Rings We study domination in zero-divisor graphs of matrix rings over a commutative ring with $1$. Keywords:vector space, linear transformation, zero-divisor graph, domination, local ringCategory:05C69

2. CMB Online first

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

3. CMB 2014 (vol 57 pp. 648)

Tang, Juping; Miao, Long
 On the ${\mathcal F}{\Phi}$-Hypercentre of Finite Groups Let $G$ be a finite group, $\mathcal F$ a class of groups. Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre of $G$ which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A subgroup $H$ is called $\mathcal M$-supplemented in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$ of $H$. The main purpose of this paper is to prove: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that $1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$ is $\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal U}{\Phi}}(G)$. Keywords:${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formationCategories:20D10, 20D20

4. CMB 2014 (vol 58 pp. 110)

Kamalov, F.
 Property T and Amenable Transformation Group $C^*$-algebras It is well known that a discrete group which is both amenable and has Kazhdan's Property T must be finite. In this note we generalize the above statement to the case of transformation groups. We show that if $G$ is a discrete amenable group acting on a compact Hausdorff space $X$, then the transformation group $C^*$-algebra $C^*(X, G)$ has Property T if and only if both $X$ and $G$ are finite. Our approach does not rely on the use of tracial states on $C^*(X, G)$. Keywords:Property T, $C^*$-algebras, transformation group, amenableCategories:46L55, 46L05

5. CMB 2011 (vol 56 pp. 395)

Oancea, D.
 Coessential Abelianization Morphisms in the Category of Groups An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called coessential if every (ordered) generating $n$-tuple of $H$ can be lifted along $\phi$ to a generating $n$-tuple for $G$. We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews--Curtis conjecture. Keywords:coessential epimorphism, Nielsen transformations, Andrew-Curtis transformationsCategories:20F05, 20F99, 20J15

6. CMB 2011 (vol 56 pp. 366)

Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.
 Multiple Solutions for Nonlinear Periodic Problems We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a CarathÃ©odory reaction term $f(t,x)$ that exhibits a $(p-1)$-superlinear growth in $x \in \mathbb{R}$ near $\pm\infty$ and near zero. A special case of the differential operator is the scalar $p$-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative). Keywords:$C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invarianceCategories:34B15, 34B18, 34C25, 58E05

7. CMB 2011 (vol 55 pp. 571)

Miller, A. R.; Paris, R. B.
 A Generalised Kummer-Type Transformation for the ${}_pF_p(x)$ Hypergeometric Function In a recent paper, Miller derived a Kummer-type transformation for the generalised hypergeometric function ${}_pF_p(x)$ when pairs of parameters differ by unity, by means of a reduction formula for a certain KampÃ© de FÃ©riet function. An alternative and simpler derivation of this transformation is obtained here by application of the well-known Kummer transformation for the confluent hypergeometric function corresponding to $p=1$. Keywords:generalised hypergeometric series, Kummer transformationCategories:33C15, 33C20

8. CMB 2011 (vol 55 pp. 297)

Glasner, Eli
 The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G= \operatorname{Aut}(\mu)$ of automorphisms of an atomless standard Borel probability space $(X,\mu)$ is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(\mu)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal. Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functionsCategories:54H11, 22A05, 37B05, 54H20

9. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
 Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$. Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10

10. CMB 2006 (vol 49 pp. 265)

Nicholson, W. K.; Zhou, Y.
 Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in C[x]$, Camillo and Sim\'{o}n called a ring$g(x)$-clean if every element is the sum of a unit and a root of$g(x)$. If$V$is a vector space of countable dimension over a division ring$D,$they showed that$\end {}_{D}V$is$g(x)$-clean provided that$g(x)$has two roots in$C(D)$. If$g(x)=x-x^{2}$this shows that$\end {}_{D}V$is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that$\Mend {}_{R}M$is$g(x)$-clean for any semisimple module$M$over an arbitrary ring$R$provided that$g(x)\in (x-a)(x-b)C[x]$where$a,b\in C$and both$b$and$b-a$are units in$R$. Keywords:Clean rings, linear transformations, endomorphism ringsCategories:16S50, 16E50 11. CMB 2005 (vol 48 pp. 267) Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.  Continuous Adjacency Preserving Maps on Real Matrices It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form$ A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix$R$. The result extends previously known theorems where the map was assumed to be also injective. Keywords:adjacency of matrices, continuous preservers, affine transformationsCategories:15A03, 15A04. 12. CMB 2004 (vol 47 pp. 624) Zhang, Xi  A Compactness Theorem for Yang-Mills Connections In this paper, we consider Yang-Mills connections on a vector bundle$E$over a compact Riemannian manifold$M$of dimension$m> 4$, and we show that any set of Yang-Mills connections with the uniformly bounded$L^{\frac{m}{2}}$-norm of curvature is compact in$C^{\infty}$topology. Keywords:Yang-Mills connection, vector bundle, gauge transformationCategories:58E20, 53C21 13. CMB 2004 (vol 47 pp. 298) Yahaghi, Bamdad R.  Near Triangularizability Implies Triangularizability In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is$\BR$or$\BC$. Keywords:Linear transformation, Compact operator,, Triangularizability, Banach space, Hilbert, spaceCategories:47A15, 47D03, 20M20 14. CMB 2004 (vol 47 pp. 133) Wang, Wei  Embeddability of Some Three-Dimensional Weakly Pseudoconvex${\rm CR}$Structures We prove that a class of perturbations of standard${\rm CR}$structure on the boundary of three-dimensional complex ellipsoid$E_{p,q}$can be realized as hypersurfaces on$\mathbb{C}^2$, which generalizes the result of Burns and Epstein on the embeddability of some perturbations of standard${\rm CR}$structure on$S^3$. Keywords:deformations, embeddability, complex ellipsoidsCategories:32V30, 32G07, 32V35 15. CMB 2001 (vol 44 pp. 337) Vinet, Luc; Zhedanov, Alexei  Spectral Transformations of the Laurent Biorthogonal Polynomials, II. Pastro Polynomials We continue to study the simplest closure conditions for chains of spectral transformations of the Laurent biorthogonal polynomials ($\LBP$). It is shown that the 1-1-periodic$q$-closure condition leads to the$\LBP$introduced by Pastro. We introduce classes of semi-classical and Laguerre-Hahn$\LBP$associated to generic closure conditions of the chain of spectral transformations. Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformationsCategory:33D45 16. CMB 2000 (vol 43 pp. 427) Ivey, Thomas A.  Helices, Hasimoto Surfaces and BÃ¤cklund Transformations Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in$\R^3$that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the B\"acklund transformation for constant torsion curves in$\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in$H^3$or$S^3$leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces. Keywords:surfaces, filament flow, BÃ¤cklund transformationsCategories:53A05, 58F37, 52C42, 58A15 17. CMB 1999 (vol 42 pp. 274) Dădărlat, Marius; Eilers, Søren  The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite, real rank zero$C^\ast$-algebras$E$and$E'$for which there is an isomorphism of ordered groups$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$which is compatible with all the coefficient transformations. The$C^\ast$-algebras$E$and$E'$are not isomorphic since there is no$\Theta$as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra$\OO_\infty$one obtains a pair of non-isomorphic, real rank zero, purely infinite$C^\ast$-algebras with similar properties. Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps,$C^\ast\$-algebras, real rank zero, purely infinite, classificationCategories:46L35, 46L80, 19K14
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