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1. 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, unirationalityCategories:14J28, 14J15, 14M20

2. 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 stabilityCategories:05C15, 20C30

3. CMB Online first

Liao, Fanghui; Liu, Zongguang
 Some Properties of Triebel-Lizorkin and Besov Spaces Associated with Zygmund Dilations In this paper, using CalderÃ³n'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, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theoremCategories:42B20, 42B35

4. 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 PrÃ¼fer $v$-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 ringCategories:13A15, 13F05, 13F25

5. 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

6. 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 ringsCategories:15A09, 15A23

7. 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) formulaCategories:20G05, 43A85, 43A32, 43A40

8. 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 operatorCategories:53C15, 53B25

9. 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 systemCategories:34K20, 34K06

10. 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 functionsCategories:14P20, 58A07

11. 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 retractionCategories:53C23, 47H20, 54E40, 58D07

12. 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

13. 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 tensorCategories:53C40, 53C15

14. 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 groupsCategories:57M25, 57M27, 06F15

15. CMB Online first

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

16. CMB Online first

 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

17. CMB Online first

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

18. CMB Online first

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

19. 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

20. CMB Online first

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

21. CMB Online first

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

22. CMB Online first

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

23. CMB Online first

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

24. CMB Online first

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

25. 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
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