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51. CMB Online first

Chen, Jianlong; Patricio, Pedro; Zhang, Yulin; Zhu, Huihui
Characterizations and representations of core and dual core inverses
In this paper, double commutativity and the reverse order law for the core inverse are considered. Then, new characterizations of the Moore-Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

Keywords:regularities, group inverses, Moore-Penrose inverses, core inverses, dual core inverses, Dedekind-finite rings
Categories:15A09, 15A23

52. CMB Online first

Karzhemanov, Ilya
On polarized K3 surfaces of genus 33
We prove that the moduli space of smooth primitively polarized $\mathrm{K3}$ surfaces of genus $33$ is unirational.

Keywords:K3 surface, moduli space, unirationality
Categories:14J28, 14J15, 14M20

53. CMB 2016 (vol 60 pp. 63)

Chang, Gyu Whan
Power Series Rings Over Prüfer $v$-multiplication Domains, II
Let $D$ be an integral domain, $X^1(D)$ be the set of height-one prime ideals of $D$, $\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be two disjoint nonempty sets of indeterminates over $D$, $D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and $D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type power series ring over $D[\{X_{\beta}\}]$. Assume that $D$ is a Prüfer $v$-multiplication domain (P$v$MD) in which each proper integral $t$-ideal has only finitely many minimal prime ideals (e.g., $t$-SFT P$v$MDs, valuation domains, rings of Krull type). Among other things, we show that if $X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$, then ${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D - \{0\}}$ is a Krull domain. We also prove that if $D$ is a $t$-SFT P$v$MD, then the complete integral closure of $D$ is a Krull domain and ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every height-one maximal $t$-ideal $M$ of $D$.

Keywords:Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ring
Categories:13A15, 13F05, 13F25

54. CMB 2016 (vol 59 pp. 834)

Liao, Fanghui; Liu, Zongguang
Some Properties of Triebel-Lizorkin and Besov Spaces Associated with Zygmund Dilations
In this paper, using Calderón's reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel-Lizorkin and Besov spaces associated with Zygmund dilations.

Keywords:Triebel-Lizorkin and Besov spaces, Riesz potential, Calderón's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theorem
Categories:42B20, 42B35

55. CMB 2016 (vol 60 pp. 111)

Ghaani Farashahi, Arash
Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$.

Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula
Categories:20G05, 43A85, 43A32, 43A40

56. CMB 2016 (vol 59 pp. 760)

Fichou, Goulwen; Quarez, Ronan; Shiota, Masahiro
Artin Approximation Compatible with a Change of Variables
We propose a version of the classical Artin approximation which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a Nash equation by a Nash solution in a compatible way with a given Nash change of variables. This result is closely related to the so-called nested Artin approximation and becomes false in the analytic setting. We provide local and global versions of this approximation in real and complex geometry together with an application to the Right-Left equivalence of Nash maps.

Keywords:Artin approximation, global case, Nash functions
Categories:14P20, 58A07

57. 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 retraction
Categories:53C23, 47H20, 54E40, 58D07

58. 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 tensor
Categories:53C40, 53C15

59. 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 groups
Categories:57M25, 57M27, 06F15

60. 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 Grassmannian
Categories:55N91, 19L47

61. 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, $\mathsf{L}^2$-summands
Categories:46B20, 46C05

62. CMB 2016 (vol 59 pp. 652)

Su, Huadong
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 ring
Categories:05C25, 16U60, 05C12

63. 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 methods
Categories:35J60, 35J92, 46E30, 49R05

64. 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 group
Categories:13A50, 20F55

65. 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 filtrations
Categories:32S35, 32S22, 14H50

66. 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 martingales
Categories:42C40, 42C15, 43A70, 11S85

67. CMB 2016 (vol 59 pp. 734)

Dimassi, Mouez
Semi-classical Asymptotics for Schrödinger Operator with Oscillating Decaying Potential
We study the distribution of the discrete spectrum of the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$.

Keywords:periodic Schrödinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift function
Categories:81Q10, 35P20, 47A55, 47N50, 81Q15

68. 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, isometries
Categories:42A45, 42A50, 41A44

69. 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 form
Categories:16E50, 16U99, 16S10, 16S15

70. CMB 2016 (vol 60 pp. 146)

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 continuation
Categories:31B20, 31B05

71. 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 metrics
Categories:53C30, 53C60

72. 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 dimension
Categories:13M99, 05E40

73. 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 sums
Categories:11S80, 11G15, 11R18

74. 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 nonlinearities

75. 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 function
Categories:47A16, 54B20, 43A15
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