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1. CMB Online first

Rosales, Leobardo
Generalizing Hopf's boundary point lemma
We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with Hölder continuous top-order coefficients and lower-order coefficients in a Morrey space.

Keywords:partial differential equation, divergence form, Hopf boundary point lemma
Categories:35B50, 35A07

2. CMB Online first

Karimianpour, Camelia
Branching Rules for $n$-fold Covering Groups of $\mathrm{SL}_2$ over a Non-Archimedean Local Field
Let $\mathtt{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $\mathtt{G}$ to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

Keywords:local field, covering group, representation, Hilbert symbol, $\mathsf{K}$-type

3. CMB Online first

Hong, Kyungpyo; Oh, Seungsang
Bounds on multiple self-avoiding polygons
A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problem of this study, we consider multiple self-avoiding polygons in a confined region, as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds of the number $p_{m \times n}$ of distinct multiple self-avoiding polygons in the $m \times n$ rectangular grid on the square lattice. For $m=2$, $p_{2 \times n} = 2^{n-1}-1$. And, for integers $m,n \geq 3$, $$2^{m+n-3} \left(\tfrac{17}{10} \right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \ 2^{m+n-3} \left(\tfrac{31}{16} \right)^{(m-2)(n-2)}.$$

Keywords:ring polymer, self-avoiding polygon
Categories:57M25, 82B20, 82B41, 82D60

4. CMB Online first

Awonusika, Richard; Taheri, Ali
A spectral identity on Jacobi polynomials and its analytic implications
The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq j\leq \ell$, $\alpha,\beta\gt -1$) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt -1$) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented.

Keywords:Jacobi coefficient, Laplace-Beltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial
Categories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15

5. CMB Online first

Dang, Pei; Liu, Hua; Qian, Tao
Hilbert Transformation and Representation of the $ax+b$ Group
In this paper we study the Hilbert transformations over $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the form $ \lambda I+\eta H, $ where $I$ and $H$ are the identity operator and Hilbert transformation respectively, and $\lambda,\eta$ are complex numbers. In the related literature this result was proved through first invoking the boundedness result of the operator, proved though a big machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is Gelfand-Naimark's representation of the ax+b group. Furthermore we prove a similar result on the unit circle. Although there does not exist a group like ax+b on the unit circle, we construct a semigroup to play the same symmetry role for the Hilbert transformations over the circle $L^2(\mathbb{T}).$

Keywords:singular integral, Hilbert transform, the $ax+b$ group
Categories:30E25, 44A15, 42A50

6. CMB Online first

Maier, Helmut; Rassias, Michael Th.
On the size of an expression in the Nyman-Beurling-Báez-Duarte criterion for the Riemann Hypothesis
A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty \left|1-\zeta A_N \left(\frac{1}{2}+it \right) \right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where the infimum is over all Dirichlet polynomials $$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we investigate $d_N^2$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line.

Keywords:Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion
Categories:30C15, 11M26

7. CMB Online first

Reichstein, Zinovy B.
On a property of real plane curves of even degree
F. Cukierman asked whether or not for every smooth real plane curve $X \subset \mathbb{P}^2$ of even degree $d \geqslant 2$ there exists a real line $L \subset \mathbb{P}^2$ such $X \cap L$ has no real points. We show that the answer is ``yes" if $d = 2$ or $4$ and ``no" if $n \geqslant 6$.

Keywords:real algebraic geometry, plane curve, maximizer function, bitangent
Categories:14P05, 14H50

8. CMB Online first

Nemirovski, Stefan; Shafikov, Rasul Gazimovich
Uniformization and Steinness
It is shown that the unit ball in $\mathbb{C}^n$ is the only complex manifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

Keywords:Stein manifold, covering, spherical domain
Categories:32T15, 32Q30

9. CMB Online first

Roche, Alan; Vinroot, C. Ryan
A factorization result for classical and similitude groups
For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well known result of Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.

Keywords:classical group, similitude group, involution, $p$-adic group, dual of representation
Categories:20G15, 22E50

10. CMB Online first

Lee, Hao
Irregular Weight one points with $D_{4}$ Image
Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

Keywords:weight one points, irregular, dihedral image, generalized eigenform, eigencurve, tangent space,
Categories:11F33, 11F80

11. CMB Online first

Wang, Zhenjian
On Deformations of Nodal Hypersurfaces
We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

Keywords:nodal hypersurface, deformation, Torelli theorem
Categories:32S35, 14C30, 14D07, 32S25

12. CMB Online first

Koskivirta, Jean-Stefan
Normalization of closed Ekedahl-Oort strata
We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Keywords:Ekedahl-Oort stratification, Shimura variety
Categories:14K10, 20G40, 11G18

13. CMB Online first

Medini, Andrea; van Mill, Jan; Zdomskyy, Lyubomyr S.
Infinite powers and Cohen reals
We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^\omega$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.

Keywords:infinite power, zero-dimensional, first-countable, homogeneous, Cohen real, h-homogeneous, rigid
Categories:03E35, 54B10, 54G20

14. CMB Online first

Marković, Marijan
Differential-free characterisation of smooth mappings with controlled growth
In this paper we give some generalizations and improvements of the Pavlović result on the Holland-Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in $\mathbf{R}^m$.

Keywords:Bloch type space, Lipschitz type space, Holland-Walsh characterisation, hyperbolic distance, analytic function, Mobius transform
Categories:32A18, 30D45

15. CMB Online first

Li, Junfeng; Yu, Haixia
Oscillatory Hyper-Hilbert Transform Associated with Plane Curves
In this paper, the bounded properties of oscillatory hyper-Hilbert transform along certain plane curves $\gamma(t)$ $$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(x-t,y-\gamma(t))e^{ i t^{-\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$ were studied. For a general curves, these operators are bounded in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their boundedness in $L^p(\mathbb{R}^{2})$ were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta-3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.

Keywords:oscillatory hyper-Hilbert transform, oscillatory integral
Categories:42B20, 42B35

16. CMB Online first

Loeffler, David
A note on $p$-adic Rankin-Selberg $L$-functions
We prove an interpolation formula for the values of certain $p$-adic Rankin-Selberg $L$-functions associated to non-ordinary modular forms.

Keywords:$p$-adic $L$-function, Iwasawa theory
Categories:11F85, 11F67, 11G40, 14G35

17. CMB Online first

He, Yubo; Qin, Dongdong; Tang, Xianhua
Ground state and multiple solutions for Kirchhoff type equations with critical exponent
In this paper, we consider the following critical Kirchhoff type equation: \begin{align*} \left\{ \begin{array}{lll} - \left(a+b\int_{\Omega}|\nabla u|^2 \right)\Delta u=Q(x)|u|^4u + \lambda |u|^{q-1}u,~~\mbox{in}~~\Omega, \\ u=0,\quad \text{on}\quad \partial \Omega, \end{array} \right. \end{align*} By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\lt q\lt 5$. The relation between the number of maxima of $Q$ and the number of positive solutions for the problem is also investigated.

Keywords:Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state
Categories:35J20, 35J60, 35J25

18. CMB Online first

Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan
Remarks on inner functions and optimal approximants
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.

Keywords:inner function, reproducing Kernel Hilbert Space, operator-theoretic function theory
Categories:46E22, 30J05

19. CMB Online first

Bu, Shangquan; Cai, Gang
Periodic solutions of second order degenerate differential equations with delay in Banach spaces
We give necessary and sufficient conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate differential equation with finite delays: $(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where $A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from $L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$.

Keywords:second order degenerate differential equation, Fourier multiplier theorem, well-posedness, Lebesgue-Bochner space, Besov space
Categories:34G10, 34K30, 43A15, 47D06

20. CMB 2017 (vol 60 pp. 705)

Benelkourchi, Slimane
Envelope Approach to Degenerate Complex Monge-Ampère Equations on Compact Kähler Manifolds
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

Keywords:degenerate complex Monge-Ampère equation, compact Kähler manifold, big cohomology, plurisubharmonic function
Categories:32W20, 32Q25, 32U05

21. 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

22. CMB Online first

Yukita, Tomoshige
Growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers
In this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra with at least one dihedral angle of the form $\frac{\pi}{k}$ for an integer $k\geq{7}$. Combining a classical result by Parry with a previous result of ours, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers.

Keywords:Coxeter group, growth function, growth rate, Perron number
Categories:20F55, 20F65

23. 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

24. CMB Online first

Jeong, Imsoon; de Dios Pérez, Juan; Suh, Young Jin; Woo, Changhwa
Lie derivatives and Ricci tensor on real hypersurfaces in complex two-plane Grassmannians
On a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ we have the Lie derivation ${\mathcal L}$ and a differential operator of order one associated to the generalized Tanaka-Webster connection $\widehat {\mathcal L} ^{(k)}$. We give a classification of real hypersurfaces $M$ on $G_2({\mathbb C}^{m+2})$ satisfying $\widehat {\mathcal L} ^{(k)}_{\xi}S={\mathcal L}_{\xi}S$, where $\xi$ is the Reeb vector field on $M$ and $S$ the Ricci tensor of $M$.

Keywords:real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, shape operator, Ricci tensor, Lie derivation
Categories:53C40, 53C15

25. 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
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