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26. CMB 2016 (vol 60 pp. 26)

Azimi, Ali; Farrokhi Derakhshandeh Ghouchan, Mohammad
Self $2$-distance Graphs
All finite simple self $2$-distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self $2$-distance graph.

Keywords:distance graph, regular graph, forbidden subgraph
Categories:05C12, 05C60, 05C76

27. CMB Online first

Eroǧlu, Münevver Pınar; Argaç, Nurcan
On Identities with Composition of Generalized Derivations
Let $R$ be a prime ring with extended centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that $dim_{C}(RC) \gt 4$, $f(X_{1},\dots,X_{n})$ a multilinear polynomial over $C$ which is not central-valued on $R$ and $f(R)$ the set of all evaluations of the multilinear polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u \in C$ for all $u\in f(R)$ then one of the following conditions holds: (I) there exists $a\in Q$ such that $a^2=0$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$; (II) there exists $a\in Q$ such that $0\neq a^2\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$; (III) $char(R)=2$ and one of the following holds: (i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$ and $a^{2}=b^{2}\in C$; (ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$ is central-valued on $R$; (iii) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and $a^2+d(a)=0$; (iv) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$, $a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$. Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$.

Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotients
Categories:16N60, 16N25

28. CMB Online first

Louder, Larsen; Wilton, Henry
Stackings and the $W$-cycles conjecture
We prove Wise's $W$-cycles conjecture: Consider a compact graph $\Gamma'$ immersing into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma'$. Unless $\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of $\Gamma'$. As a corollary, any finitely generated subgroup of a one-relator group has finitely generated Schur multiplier.

Keywords:free groups, one-relator groups, right-orderability
Category:20F65

29. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces
In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey space
Categories:42B20, 42B25

30. CMB Online first

Reichstein, Zinovy; Vistoli, Angelo
On the dimension of the locus of determinantal hypersurfaces
The characteristic polynomial $P_A(x_0, \dots, x_r)$ of an $r$-tuple $A := (A_1, \dots, A_r)$ of $n \times n$-matrices is defined as \[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r A_r) \, . \] We show that if $r \geqslant 3$ and $A := (A_1, \dots, A_r)$ is an $r$-tuple of $n \times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples $A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$ is irreducible of dimension $(r-1)n^2 + 1$.

Keywords:determinantal hypersurface, matrix invariant, $q$-binomial coefficient
Categories:14M12, 15A22, 05A10

31. CMB 2016 (vol 60 pp. 12)

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 group
Categories:05C25, 05E15, 20D10, 20D15

32. CMB 2016 (vol 60 pp. 43)

Bouchemakh, Isma; Fatma, Kaci
On the Dual König 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 König property of $\mathcal{H}(P)$ in two subclasses of N-free class.

Keywords:poset, interval, N-free, hypergraph, König property, dual König property
Category:05C65

33. CMB 2016 (vol 60 pp. 95)

Choi, Chang-Kwon; 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 stability
Category:39B82

34. CMB 2016 (vol 60 pp. 104)

Diestel, Geoff
An Extension of Nikishin's Factorization Theorem
A Nikishin-Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.

Keywords:factorization, type, cotype, Banach spaces
Categories:46E30, 28A25

35. CMB 2016 (vol 60 pp. 197)

Tang, Zikai; Deng, Hanyuan
Degree Kirchhoff Index of Bicyclic Graphs
Let $G$ be a connected graph with vertex set $V(G)$. The degree Kirchhoff index of $G$ is defined as $S'(G) =\sum_{\{u,v\}\subseteq V(G)}d(u)d(v)R(u,v)$, where $d(u)$ is the degree of vertex $u$, and $R(u, v)$ denotes the resistance distance between vertices $u$ and $v$. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoff index among all $n$-vertex bicyclic graphs with exactly two cycles.

Keywords:degree Kirchhoff index, resistance distance, bicyclic graph, extremal graph
Categories:05C12, 05C35

36. CMB 2016 (vol 60 pp. 184)

Pathak, Siddhi
On a Conjecture of Livingston
In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{ -1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settle Erdös's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients will be needed to settle Erdös's conjecture for prime $q$.

Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbers
Categories:11J86, 11J72

37. CMB Online first

Xu, Xu; Zhu, Laiyi
Rational function operators from Poisson integrals
In this paper, we construct two classes of rational function operators by using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

Keywords:rational function operators, Poisson integrals, convergence rate, uniform approximation, mean approximation
Categories:41A20, 41A25, 41A35

38. CMB 2016 (vol 60 pp. 173)

Oubbi, Lahbib
On Ulam Stability of a Functional Equation in Banach Modules
Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping. For any rational number $r \ne 2$, C. Baak, D. H. Boo, and Th. M. Rassias have proved the Hyers-Ulam stability of the following functional equation: \begin{align*} r f \left(\frac{\sum_{j=1}^d x_j}{r} \right) & + \sum_{\substack{i(j) \in \{0,1\} \\ \sum_{j=1}^d i(j)=\ell}} r f \left( \frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r} \right) = (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d f(x_j) \end{align*} where $d$ and $\ell$ are positive integers so that $1 \lt \ell \lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$, $p, q \in \mathbb{N}$ with $p \le q$. In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers-Ulam stable. We thus extend and generalize Baak et al.'s result. Different questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered.

Keywords:linear functional equation, Hyers-Ulam stability, Banach modules, C*-algebra homomorphisms.
Categories:39A30, 39B10, 39A06, 46Hxx

39. CMB 2016 (vol 59 pp. 806)

Izumiya, Shyuichi
Geometric Interpretation of Lagrangian Equivalence
As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics and wave front propagations are revealed.

Keywords:wave front propagations, big wave fronts, graph-like Legendrian unfoldings, caustics
Categories:58K05, 57R45, 58K60

40. CMB 2016 (vol 59 pp. 776)

Gauthier, Paul M; Sharifi, Fatemeh
The Carathéodory Reflection Principle and Osgood-Carathéodory Theorem on Riemann Surfaces
The Osgood-Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

Keywords:bordered Riemann surface, reflection principle, Osgood-Carathéodory
Categories:30C25, 30F99

41. CMB 2016 (vol 59 pp. 849)

Nah, Kyeongah; Röst, Gergely
Stability Threshold for Scalar Linear Periodic Delay Differential Equations
We prove that for the linear scalar delay differential equation $$ \dot{x}(t) = - a(t)x(t) + b(t)x(t-1) $$ with non-negative periodic coefficients of period $P\gt 0$, the stability threshold for the trivial solution is $r:=\int_{0}^{P} \left(b(t)-a(t) \right)\mathrm{d}t=0,$ assuming that $b(t+1)-a(t)$ does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that in general, $r=0$ is not a stability threshold.

Keywords:delay differential equation, stability, periodic system
Categories:34K20, 34K06

42. CMB Online first

Gauthier, Paul M; Sharifi, Fatemeh
Luzin-type holomorphic approximation on closed subsets of open Riemann surfaces
It is known that if $E$ is a closed subset of an open Riemann surface $R$ and $f$ is a holomorphic function on a neighbourhood of $E,$ then it is ``usually" not possible to approximate $f$ uniformly by functions holomorphic on all of $R.$ We show, however, that for every open Riemann surface $R$ and every closed subset $E\subset R,$ there is closed subset $F\subset E,$ which approximates $E$ extremely well, such that every function holomorphic on $F$ can be approximated much better than uniformly by functions holomorphic on $R$.

Keywords:Carleman approximation, tangential approximation, Myrberg surface
Categories:30E15, 30F99

43. CMB Online first

Werner, Elisabeth; Ye, Deping
Mixed $f$-divergence for multiple pairs of measures
In this paper, the concept of the classical $f$-divergence for a pair of measures is extended to the mixed $f$-divergence for multiple pairs of measures. The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric inequality for the mixed $f$-divergence are proved.

Keywords:Alexandrov-Fenchel inequality, $f$-dissimilarity, $f$-divergence, isoperimetric inequality
Categories:28-XX, 52-XX, 60-XX

44. CMB Online first

Liu, Feng; Wu, Huoxiong
Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation
Categories:42B25, 46E35

45. CMB 2016 (vol 60 pp. 154)

Liu, Ye
On Chromatic Functors and Stable Partitions of Graphs
The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated to a finite graph restricted to the category $\mathrm{FI}$ of finite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb.

Keywords:chromatic functor, stable partition, representation stability
Categories:05C15, 20C30

46. CMB 2016 (vol 59 pp. 813)

Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of Their generalized Tanaka-Webster Lie Derivative
On a real hypersurface $M$ in a non-flat complex space form there exist the Levi-Civita and the k-th generalized Tanaka-Webster connections. The aim of the present paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operator with respect to the Levi-Civita connections coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in directions of any vecro field orthogonal to the structure vector field.

Keywords:$k$-th generalized Tanaka-Webster connection, non-flat complex space form, real hypersurface, Lie derivative, structure Jacobi operator
Categories:53C15, 53B25

47. 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

48. 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

49. 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

50. 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
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