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1. 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

2. CMB Online first

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 group
Categories:19A22, 57S17

3. CMB Online first

Gurbuz, 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

4. CMB Online first

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

5. CMB Online first

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

6. CMB Online first

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

7. CMB Online first

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

8. CMB Online first

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

9. CMB Online first

Deng, Hanyuan; Tang, Zikai
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

10. 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

11. CMB Online first

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

12. 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

13. 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

14. CMB Online first

Kaimakamis, George; Panagiotidou, Konstantina; de Dios Perez, Juan
A classification of three-dimensional real hypersurfaces in non-flat complex space forms in terms of their generalized Tanaka-Webster Lie derivatives
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

15. CMB Online first

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

16. CMB Online first

Chang, Gyu Whan
Power series rings over Prufer $v$-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

17. 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

18. 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

19. CMB Online first

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

20. CMB Online first

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

21. CMB Online first

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

22. CMB Online first

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

23. 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 stability
Category:39B82

24. CMB Online first

de Dios Pérez, Juan; 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

25. CMB Online first

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
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