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1. CMB Online first

Li, Ji; Wick, Brett D.
Weak Factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms
This paper provides a constructive proof of the weak factorization of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms.

Keywords:Hardy space, BMO space, multilinear Riesz transform, weak factorization
Categories:42B35, 42B20

2. CMB Online first

Franz, Matthias
Symmetric products of equivariantly formal spaces
Let \(X\) be a CW complex with a continuous action of a topological group \(G\). We show that if \(X\) is equivariantly formal for singular cohomology with coefficients in some field \(\Bbbk\), then so are all symmetric products of \(X\) and in fact all its \(\Gamma\)-products. In particular, symmetric products of quasi-projective M-varieties are again M-varieties. This generalizes a result by Biswas and D'Mello about symmetric products of M-curves. We also discuss several related questions.

Keywords:symmetric product, equivariant formality, maximal variety, Gamma product
Categories:55N91, 55S15, 14P25

3. CMB Online first

Zhang, Tao; Zhou, Chunqin
Classification of solutions for harmonic functions with Neumann boundary value
In this paper, we classify all solutions of \[ \left\{ \begin{array}{rcll} -\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+}, \\ \dfrac{\partial u}{\partial t}&=&-c|x|^{\beta}e^{u} \quad &\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\}, \\ \end{array} \right. \] with the finite conditions \[ \int_{\partial \mathbb{R}^{2}_{+}}|x|^{\beta}e^{u}ds \lt C, \qquad \sup\limits_{\overline{\mathbb{R}^{2}_{+}}}{u(x)}\lt C. \] Here, $c$ is a positive number and $\beta \gt -1$.

Keywords:Neumann problem, singular coefficient, classification of solutions
Categories:35A05, 35J65

4. CMB 2017 (vol 60 pp. 402)

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 mean
Categories:43A62, 46J10, 43A30, 20N20

5. CMB Online first

Wang, Li-an Daniel
A Multiplier Theorem on Anisotropic Hardy Spaces
We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss.

Keywords:anisotropic Hardy space, multiplier, Fourier transform
Categories:42B30, 42B25, 42B35

6. CMB Online first

Alhasanat, Ahmad; Ou, Chunhua
Periodic steady-state solutions of a liquid film model via a classical method
In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness, as well as the asymptotic formula, of the periodic solutions.

Keywords:film flow, classical methods, asymptotic analysis
Categories:34E05, 34E10, 34E15

7. CMB Online first

Bichon, Julien; Kyed, David; Raum, Sven
Higher $\ell^2$-Betti numbers of universal quantum groups
We calculate all $\ell^2$-Betti numbers of the universal discrete Kac quantum groups $\hat{\mathrm U}^+_n$ as well as their half-liberated counterparts $\hat{\mathrm U}^*_n$.

Keywords:$\ell^2$-Betti number, free unitary quantum group, half-liberated unitary quantum group, free product formula, extension
Categories:16T05, 46L65, 20G42

8. CMB Online first

Józiak, Paweł
Remarks on Hopf images and quantum permutation groups $S_n^+$
Motivated by a question of A. Skalski and P.M. Sołtan (2016) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

Keywords:Hopf image, quantum permutation group, compact quantum group
Categories:20G42, 81R50, 46L89, 16W35

9. CMB Online first

Gupta, Purvi
A real-analytic nonpolynomially convex isotropic torus with no attached discs
We show by means of an example in $\mathbb C^3$ that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e, polynomial convexity.

Keywords:polynomial hull, isotropic submanifold, holomorphic disc
Categories:32V40, 32E20, 53D12

10. CMB Online first

Takahashi, Tomokuni
Projective plane bundles over an elliptic curve
We calculate the dimension of cohomology groups for the holomorphic tangent bundles of each isomorphism class of the projective plane bundle over an elliptic curve. As an application, we construct the families of projective plane bundles, and prove that the families are effectively parametrized and complete.

Keywords:projective plane bundle, vector bundle, elliptic curve, deformation, Kodaira-Spencer map
Categories:14J10, 14J30, 14D15

11. CMB Online first

Haralampidou, Marina; Oudadess, Mohamed; Palacios, Lourdes; Signoret, Carlos
A characterization of $C^{\ast}$-normed algebras via positive functionals
We give a characterization of $C^{\ast}$-normed algebras, among certain involutive normed ones. This is done through the existence of enough specific positive functionals. The same question is also examined in some non normed (topological) algebras.

Keywords:$C^{\ast}$-normed algebra, $C^*$-algebra, (pre-)locally $C^*$-algebra, pre-$C^*$-bornological algebra, positive functional, locally uniformly $A$-convex algebra, perfect locally $m$-convex algebra, $C^*$-(resp. $^*$-) subnormable algebra
Categories:46H05, 46K05

12. CMB Online first

Zhang, Guo-Bao; Tian, Ge
Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models
In this paper, we study a two-component Lotka-Volterra competition system on an one-dimensional spatial lattice. By the method of the comparison principle together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as $j+ct \rightarrow -\infty$, where $j\in\mathbb{Z}$, $t\gt 0$, but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H. Wu.

Keywords:lattice dynamical system, competition model, traveling wavefront, stability
Categories:34A33, 34K20, 92D25

13. CMB Online first

Sebbar, Abdellah; Al-Shbeil, Isra
Elliptic Zeta functions and equivariant functions
In this paper we establish a close connection between three notions attached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Keywords:modular form, equivariant function, elliptic zeta function
Categories:11F12, 35Q15, 32L10

14. CMB Online first

Tran, Anh T.; Yamaguchi, Yoshikazu
The asymptotics of the higher dimensional Reidemeister torsion for exceptional surgeries along twist knots
We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible $\operatorname{SL}_2(\mathbb{C})$-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coefficients in the higher dimensional Reidemeister torsion explicitly.

Keywords:Reidemeister torsion, graph manifold, asymptotic behavior, exceptional surgery
Categories:57M27, 57M50

15. CMB Online first

Saito, Hiroki; Tanaka, Hitoshi
The Fefferman-Stein type inequalities for strong and directional maximal operators in the plane
The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with an additional composition of the Hardy-Littlewood maximal operator in the plane.

Keywords:directional maximal operator, Fefferman-Stein type inequality, Hardy-Littlewood maximal operator, strong maximal operator
Categories:42B25, 42B35

16. CMB Online first

Kalaj, David; Vujadinović, Djordjije
Gradient of solution of the Poisson equation in the unit ball and related operators
In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

Keywords:Möbius transformation, Poisson equation, Newtonian potential, Cauchy transform, Bessel function
Categories:35J05, 47G10

17. 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, subgroup
Categories:22E43, 20H99, 83A05

18. CMB Online first

Koşan, Tamer; Sahinkaya, Serap; Zhou, Yiqiang
Additive maps on units of rings
Let $R$ be a ring. A map $f: R\rightarrow R$ is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$ of $R$. Here a map $f: R\rightarrow R$ is called unit-additive if $f(u+v)=f(u)+f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of Xu, Pei and Yi showing that, for any field $F$, every unit-additive map of ${\mathbb M}_n(F)$ is additive for all $n\ge 2$, this paper is about the question when every unit-additive map of a ring is additive. It is proved that every unit-additive map of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong \mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${\mathbb M}_n(R)$ is additive for all $n\ge 2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unit-homomorphic if $f(uv)=f(u)f(v)$ for all units $u,v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Keywords:additive map, unit, 2-sum property, semilocal ring, exchange ring with primitive factors Artinian
Categories:15A99, 16U60, 16L30

19. CMB Online first

Maican, Mario
Moduli of space sheaves with Hilbert polynomial $4m+1$
We investigate the moduli space of sheaves supported on space curves of degree $4$ and having Euler characteristic $1$. We give an elementary proof of the fact that this moduli space consists of three irreducible components.

Keywords:moduli of sheaves, semi-stable sheaves
Categories:14D20, 14D22

20. CMB Online first

Llibre, Jaume; Valls, Claudia
Global phase portraits for the Abel quadratic polynomial differential equations of second kind with $Z_2$-symmetries
We provide normal forms and the global phase portraits on the Poincaré disk for all Abel quadratic polynomial differential equations of the second kind with $\mathbb Z_2$-symmetries.

Keywords:Abel polynomial differential system of the second kind, vector field, phase portrait
Categories:37J35, 37K10

21. CMB Online first

Bu, Shangquan; Cai, Gang
Hölder continuous solutions of degenerate differential equations with finite delay
Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.

Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, Hölder continuous function space
Categories:34N05, 34G10, 47D06, 47A10, 34K30

22. CMB Online first

Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian
On subcritically Stein fillable 5-manifolds
We make some elementary observations concerning subcritically Stein fillable contact structures on $5$-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the $5$-sphere the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected $5$-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

Keywords:subcritically Stein fillable, 5-manifold, almost contact structure, thickening
Categories:53D35, 32Q28, 57M20, 57Q10, 57R17

23. CMB Online first

Ha, Pham Hoang; Kawakami, Yu
A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Keywords:minimal surface, Gauss map, unicity theorem
Categories:53A10, 30D35, 53C42

24. 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

25. CMB Online first

Gilligan, Bruce
Levi's problem for pseudoconvex homogeneous manifolds
Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.

Keywords:complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator
Categories:32M10, 32U10, 32A10, 32Q28
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