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26. CMB 2016 (vol 59 pp. 673)

Bačák, Miroslav; Kovalev, Leonid V.
 Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive. Keywords:finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retractionCategories:53C23, 47H20, 54E40, 58D07

27. CMB Online first

Chung, Jaeyoung; Ju, Yumin; Rassias, John
 Cubic functional equations on restricted domains of Lebesgue measure zero Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to Y$. We prove the Ulam-Hyers stability theorem for the cubic functional equation \begin{align*} f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)=0 \end{align*} in restricted domains. As an application we consider a measure zero stability problem of the inequality \begin{align*} \|f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)\|\le \epsilon \end{align*} for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure 0. Keywords:Baire category theorem, cubic functional equation, first category, Lebesgue measure, Ulam-Hyers stabilityCategory:39B82

28. CMB 2016 (vol 59 pp. 721)

Pérez, Juan de Dios; Lee, Hyunjin; Suh, Young Jin; Woo, Changhwa
 Real Hypersurfaces in Complex Two-plane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce the notion of generalized Tanaka-Webster (in shortly, GTW) Reeb parallel Ricci tensor for Hopf hypersurface $M$ in $G_2({\mathbb C}^{m+2})$. Next, we give a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ with GTW Reeb parallel Ricci tensor. Keywords:Complex two-plane Grassmannian, real hypersurface, Hopf hypersurface, generalized Tanaka-Webster connection, parallelism, Reeb parallelism, Ricci tensorCategories:53C40, 53C15

29. CMB 2016 (vol 59 pp. 472)

Clay, Adam; Desmarais, Colin; Naylor, Patrick
 Testing Bi-orderability of Knot Groups We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 499 of the corresponding knot groups. With our methods we are able to deal with 191 more. Keywords:knots, fundamental groups, orderable groupsCategories:57M25, 57M27, 06F15

30. CMB 2016 (vol 59 pp. 483)

Crooks, Peter; Holden, Tyler
 Generalized Equivariant Cohomology and Stratifications For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group. Keywords:equivariant cohomology theory, stratification, affine GrassmannianCategories:55N91, 19L47

31. CMB 2016 (vol 59 pp. 769)

García-Pacheco, Francisco Javier; Hill, Justin R.
 Geometric Characterizations of Hilbert Spaces We study some geometric properties related to the set $\Pi_X:= \{ (x,x^* )\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^* (x )=1 \}$ obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $(h,k )\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space. Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$-summandsCategories:46B20, 46C05

32. CMB 2016 (vol 59 pp. 652)

 On the Diameter of Unitary Cayley Graphs of Rings The unitary Cayley graph of a ring $R$, denoted $\Gamma(R)$, is the simple graph defined on all elements of $R$, and where two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $R$. The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved that for each integer $n\geq1$, there exists a ring $R$ such that ${\rm diam}(\Gamma(R))=n$. We also show that ${\rm diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with ${\rm diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively. Keywords:unitary Cayley graph, diameter, $k$-good, unit sum number, self-injective ringCategories:05C25, 16U60, 05C12

33. CMB 2016 (vol 59 pp. 606)

Mihăilescu, Mihai; Moroşanu, Gheorghe
 Eigenvalues of $-\Delta_p -\Delta_q$ Under Neumann Boundary Condition The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from $\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $(\lambda_1, +\infty )$ plus an isolated point $\lambda =0$. This comprehensive result is strongly related to our framework which is complementary to the well-known case $p=q\neq 2$ for which a full description of the set of eigenvalues is still unavailable. Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methodsCategories:35J60, 35J92, 46E30, 49R05

34. CMB 2016 (vol 59 pp. 617)

Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei
 Canonical Systems of Basic Invariants for Unitary Reflection Groups It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group. Keywords:basic invariant, invariant theory, finite unitary reflection groupCategories:13A50, 20F55

35. CMB Online first

Bouchemakh, Isma; Fatma, Kaci
 On the dual KÃ¶nig property of the order-interval hypergraph of two classes of N-free posets Let $P$ be a finite N-free poset. We consider the hypergraph $\mathcal{H}(P)$ whose vertices are the elements of $P$ and whose edges are the maximal intervals of $P$. We study the dual KÃ¶nig property of $\mathcal{H}(P)$ in two subclasses of N-free class. Keywords:poset, interval, N-free, hypergraph, KÃ¶nig property, dual KÃ¶nig propertyCategory:05C65

36. CMB 2016 (vol 59 pp. 449)

Abdallah, Nancy
 On Hodge Theory of Singular Plane Curves The dimensions of the graded quotients of the cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$ with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on $H^2(U,\mathbb C)$. Keywords:plane curves, Hodge and pole order filtrationsCategories:32S35, 32S22, 14H50

37. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
 Characterization of Low-pass Filters on Local Fields of Positive Characteristic In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated to the scaling function for multiresolution analysis of $L^2(K)$. We use probability and martingale methods to provide such a characterization. Keywords:multiresolution analysis, local field, low-pass filter, scaling function, probability, conditional probability and martingalesCategories:42C40, 42C15, 43A70, 11S85

38. CMB 2016 (vol 59 pp. 734)

Dimassi, Mouez
 Semi-classical Asymptotics for SchrÃ¶dinger Operator with Oscillating Decaying Potential We study the distribution of the discrete spectrum of the SchrÃ¶dinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$. Keywords:periodic SchrÃ¶dinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift functionCategories:81Q10, 35P20, 47A55, 47N50, 81Q15

39. CMB 2016 (vol 59 pp. 497)

De Carli, Laura; Samad, Gohin Shaikh
 One-parameter Groups of Operators and Discrete Hilbert Transforms We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators in $\ell^2$. Keywords:discrete Hilbert transform, groups of operators, isometriesCategories:42A45, 42A50, 41A44

40. 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

41. CMB Online first

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
 Co-Maximal Graphs of Subgroups of Groups Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite. Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable groupCategories:05C25, 05E15, 20D10, 20D15

42. CMB Online first

Khavinson, Dmitry; Lundberg, Erik; Render, Hermann
 The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$ the inhomogeneous difference equation $h ( t+1,y) -h (t,y) =g ( t,y)$ has an entire harmonic solution $h$. Keywords:reflection principle, entire harmonic function, analytic continuationCategories:31B20, 31B05

43. 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

44. CMB 2016 (vol 59 pp. 748)

Dolžan, David
 The Metric Dimension of the Total Graph of a Finite Commutative Ring We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields. Keywords:total graph, finite ring, metric dimensionCategories:13M99, 05E40

45. CMB 2016 (vol 59 pp. 624)

Otsubo, Noriyuki
 Homology of the Fermat Tower and Universal Measures for Jacobi Sums We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a similar manner to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve. Keywords:Fermat curves, Ihara-Anderson theory, Jacobi sumsCategories:11S80, 11G15, 11R18

46. CMB 2016 (vol 59 pp. 693)

Chen, Chung-Chuan
 Recurrence of Cosine Operator Functions on Groups In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent. Keywords:topologically multiple recurrence, recurrence, topological transitivity, hypercyclicity, cosine operator functionCategories:47A16, 54B20, 43A15

47. CMB 2016 (vol 59 pp. 417)

Song, Hongxue; Chen, Caisheng; Yan, Qinglun
 Existence of Multiple Solutions for a $p$-Laplacian System in $\textbf{R}^{N}$ with Sign-changing Weight Functions In this paper, we consider the quasi-linear elliptic problem \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$ are continuous functions which change sign in $\mathbb{R}^{N}$. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem. Keywords:Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearitiesCategory:35J66

48. CMB 2016 (vol 59 pp. 234)

Beardon, Alan F.
 Non-discrete Frieze Groups The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practical way. In addition, this discussion sheds light on why there are only seven conjugacy classes in the classical case. Keywords:frieze groups, isometry groupsCategories:51M04, 51N30, 20E45

49. CMB 2016 (vol 59 pp. 326)

Jiang, Chunlan; Shi, Rui
 On the Uniqueness of Jordan Canonical Form Decompositions of Operators by $K$-theoretical Data In this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in $\mathcal {L}(\mathcal {H})$. A complete criterion for similarity for this class of operators in terms of $K$-theory for Banach algebras is given. Keywords:strongly irreducible operator, similarity invariant, reduction theory of von Neumann algebras, $K$-theoryCategories:47A15, 47C15, 47A65

50. CMB 2016 (vol 59 pp. 508)

De Nicola, Antonio; Yudin, Ivan
 Generalized Goldberg Formula In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally KÃ¤hler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra. Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasi-Sasakian manifoldCategories:53C25, 53D35
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