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26. CMB Online first

Sickel, Winfried; Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang
 Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest. Keywords:Besov space, Triebel-Lizorkin space, ball average, CalderÃ³n reproducing formulaCategories:42B25, 46E35, 42B35

27. CMB Online first

Stoyanov, Luchezar
 On Gibbs measures and spectra of Ruelle transfer operators We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the HÃ¶lder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates. Keywords:subshift of finite type, Ruelle transfer operator, Gibbs measureCategories:37A05, 37B10

28. CMB 2016 (vol 60 pp. 165)

Morimoto, Masaharu
 Cokernels of Homomorphisms from Burnside Rings to Inverse Limits Let $G$ be a finite group and let $A(G)$ denote the Burnside ring of $G$. Then an inverse limit $L(G)$ of the groups $A(H)$ for proper subgroups $H$ of $G$ and a homomorphism ${\operatorname{res}}$ from $A(G)$ to $L(G)$ are obtained in a natural way. Let $Q(G)$ denote the cokernel of ${\operatorname{res}}$. For a prime $p$, let $N(p)$ be the minimal normal subgroup of $G$ such that the order of $G/N(p)$ is a power of $p$, possibly $1$. In this paper we prove that $Q(G)$ is isomorphic to the cartesian product of the groups $Q(G/N(p))$, where $p$ ranges over the primes dividing the order of $G$. Keywords:Burnside ring, inverse limit, finite groupCategories:19A22, 57S17

29. CMB Online first

Jensen, Gerd; Pommerenke, Christian
 On the structure of the Schild group in Relativity Theory Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations. The present paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature we associate Lorentz transformations with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups. Keywords:Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroupCategories:22E43, 20H99, 83A05

30. CMB Online first

Shravan Kumar, N.
 Invariant means on a class of von Neumann Algebras related to Ultraspherical Hypergroups II Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means. Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant meanCategories:43A62, 46J10, 43A30, 20N20

31. 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 subgraphCategories:05C12, 05C60, 05C76

32. 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 quotientsCategories:16N60, 16N25

33. 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-orderabilityCategory:20F65

34. 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 spaceCategories:42B20, 42B25

35. 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 coefficientCategories:14M12, 15A22, 05A10

36. 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 groupCategories:05C25, 05E15, 20D10, 20D15

37. CMB 2016 (vol 60 pp. 43)

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

38. 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 stabilityCategory:39B82

39. 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 spacesCategories:46E30, 28A25

40. CMB 2016 (vol 60 pp. 184)

Pathak, Siddhi
 On a Conjecture of Livingston In an attempt to resolve a folklore conjecture of ErdÃ¶s 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 ErdÃ¶s'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 ErdÃ¶s's conjecture for prime $q$. Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbersCategories:11J86, 11J72

41. 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 graphCategories:05C12, 05C35

42. 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 approximationCategories:41A20, 41A25, 41A35

43. 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

44. 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, causticsCategories:58K05, 57R45, 58K60

45. CMB 2016 (vol 59 pp. 776)

Gauthier, Paul M; Sharifi, Fatemeh
 The CarathÃ©odory Reflection Principle and Osgood-CarathÃ©odory Theorem on Riemann Surfaces The Osgood-CarathÃ©odory 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 CarathÃ©odory reflection principle. Keywords:bordered Riemann surface, reflection principle, Osgood-CarathÃ©odoryCategories:30C25, 30F99

46. 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 systemCategories:34K20, 34K06

47. 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 surfaceCategories:30E15, 30F99

48. 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 inequalityCategories:28-XX, 52-XX, 60-XX

49. 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 variationCategories:42B25, 46E35

50. 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 stabilityCategories:05C15, 20C30
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