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Journal of the Australian Mathematical Society
Journal Prestige (SJR): 0.46  Citation Impact (citeScore): 1 Number of Followers: 0  Subscription journalISSN (Print) 1446-7887 - ISSN (Online) 1446-8107 Published by Cambridge University Press  [389 journals]
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- JAZ volume 109 Issue 2 Cover and Front matter
- PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000673
Issue No: Vol. 109, No. 2 (2020)
- PubDate: 2020-10-01T00:00:00.000Z
- JAZ volume 109 Issue 2 Cover and Back matter
- PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000685
Issue No: Vol. 109, No. 2 (2020)
- PubDate: 2020-10-01T00:00:00.000Z
- THE THICKNESS OF SCHUBERT CELLS AS INCIDENCE STRUCTURES
- Authors: JOHN BAMBERG; ARUN RAM, JON XU
Pages: 145 - 156
Abstract: This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000363
Issue No: Vol. 109, No. 2 (2020)
- Authors: JOHN BAMBERG; ARUN RAM, JON XU
- RANK GENERATING FUNCTIONS FOR ODD-BALANCED UNIMODAL SEQUENCES, QUANTUM
JACOBI FORMS, AND MOCK JACOBI FORMS- Authors: MICHAEL BARNETT; AMANDA FOLSOM, WILLIAM J. WESLEY
Pages: 157 - 175
Abstract: Let
$\unicode[STIX]{x1D707}(m,n)$ (respectively, 
$\unicode[STIX]{x1D702}(m,n)$ ) denote the number of odd-balanced unimodal sequences of size 
$2n$ and rank 
$m$ with even parts congruent to 
$2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ (respectively, 
$0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ ) and odd parts at most half the peak. We prove that two-variable generating functions for 
$\unicode[STIX]{x1D707}(m,n)$ and 
$\unicode[STIX]{x1D702}(m,n)$ are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single 
$C^{\infty }$ function in 
$\mathbb{R}\times \mathbb{R}$ to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000405
Issue No: Vol. 109, No. 2 (2020)
- Authors: MICHAEL BARNETT; AMANDA FOLSOM, WILLIAM J. WESLEY
- BOCHNER–RIESZ MEANS ON BLOCK-SOBOLEV SPACES IN COMPACT LIE GROUP
- Authors: JIECHENG CHEN; DASHAN FAN, FAYOU ZHAO
Pages: 176 - 192
Abstract: On a compact Lie group
$G$ of dimension 
$n$ , we study the Bochner–Riesz mean 
$S_{R}^{\unicode[STIX]{x1D6FC}}(f)$ of the Fourier series for a function 
$f$ . At the critical index 
$\unicode[STIX]{x1D6FC}=(n-1)/2$ , we obtain the convergence rate for 
$S_{R}^{(n-1)/2}(f)$ when 
$f$ is a function in the block-Sobolev space. The main theorems extend some known results on the 
$m$ -torus 
$\mathbb{T}^{m}$ .
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000430
Issue No: Vol. 109, No. 2 (2020)
- Authors: JIECHENG CHEN; DASHAN FAN, FAYOU ZHAO
COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS- Authors: J. C. DE ALBUQUERQUE; JOÃO MARCOS DO Ó, EDCARLOS D. SILVA
Pages: 193 - 216
Abstract: We study the existence of positive ground state solutions for the following class of
$(p,q)$ -Laplacian coupled systems 
$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where 
$1
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000260
Issue No: Vol. 109, No. 2 (2020)
- Authors: J. C. DE ALBUQUERQUE; JOÃO MARCOS DO Ó, EDCARLOS D. SILVA
- CHARACTERIZATIONS OF QUASICONVEX AND PSEUDOCONVEX FUNCTIONS BY THEIR
SECOND-ORDER REGULAR SUBDIFFERENTIALS- Authors: MOHAMMAD TAGHI NADI; JAFAR ZAFARANI
Pages: 217 - 229
Abstract: We present the second-order necessary and sufficient conditions for quasiconvex and pseudoconvex functions in terms of their second-order regular subdifferentials.
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000090
Issue No: Vol. 109, No. 2 (2020)
- Authors: MOHAMMAD TAGHI NADI; JAFAR ZAFARANI
- LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND
CONVEX FUNCTIONS- Authors: SAMINATHAN PONNUSAMY; NAVNEET LAL SHARMA, KARL-JOACHIM WIRTHS
Pages: 230 - 249
Abstract: Let
${\mathcal{S}}$ be the family of analytic and univalent functions 
$f$ in the unit disk 
$\mathbb{D}$ with the normalization 
$f(0)=f^{\prime }(0)-1=0$ , and let 
$\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$ denote the logarithmic coefficients of 
$f\in {\mathcal{S}}$ . In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families 
${\mathcal{F}}(c)$ and 
${\mathcal{G}}(c)$ of functions 
$f\in {\mathcal{S}}$ defined by 
$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000065
Issue No: Vol. 109, No. 2 (2020)
- Authors: SAMINATHAN PONNUSAMY; NAVNEET LAL SHARMA, KARL-JOACHIM WIRTHS
- Authors: RIDDHI SHAH; ALOK KUMAR YADAV
Pages: 250 - 261
Abstract: Consider the action of
$\operatorname{GL}(n,\mathbb{Q}_{p})$ on the 
$p$ -adic unit sphere 
${\mathcal{S}}_{n}$ arising from the linear action on 
$\mathbb{Q}_{p}^{n}\setminus \{0\}$ . We show that for the action of a semigroup 
$\mathfrak{S}$ of 
$\operatorname{GL}(n,\mathbb{Q}_{p})$ on 
${\mathcal{S}}_{n}$ , the following are equivalent: (1) 
$\mathfrak{S}$ acts distally on 
${\mathcal{S}}_{n}$ ; (2) the closure of the image of 
$\mathfrak{S}$ in 
$\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000272
Issue No: Vol. 109, No. 2 (2020)
- Authors: RIDDHI SHAH; ALOK KUMAR YADAV
- A STABILITY VERSION OF THE GAUSS–LUCAS THEOREM AND APPLICATIONS
- Authors: STEFAN STEINERBERGER
Pages: 262 - 269
Abstract: Let
$p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, 
$p^{\prime }(z)=0$ , are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that 
$p$ has 
$n+m$ roots, where 
$n$ are inside the unit disk, 
$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$ then 
$p^{\prime }$ has 
$n-1$ roots inside the unit disk and 
$m$ roots at distance at least 
$(dn-m)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for 
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000284
Issue No: Vol. 109, No. 2 (2020)
- Authors: STEFAN STEINERBERGER
- PLURIPOLAR SETS, REAL SUBMANIFOLDS AND PSEUDOHOLOMORPHIC DISCS
- Authors: ALEXANDRE SUKHOV
Pages: 270 - 288
Abstract: We prove that a compact subset of full measure on a generic submanifold of an almost complex manifold is not a pluripolar set. Several related results on boundary behavior of plurisubharmonic functions are established. Our approach is based on gluing a family of complex discs to a generic manifold along a boundary arc and could admit further applications.
PubDate: 2020-10-01T00:00:00.000Z
DOI: 10.1017/S1446788719000119
Issue No: Vol. 109, No. 2 (2020)
- Authors: ALEXANDRE SUKHOV
