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.

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 $w$ and $q$, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size $2n$ with even parts congruent to $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ and odd parts at most half the peak.

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 $\mathbb{T}^{m}$.

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems

$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$

$1<p\leq q<N$

$\unicode[STIX]{x1D706}(x)$

$|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$

$\unicode[STIX]{x1D6FF}\in (0,1)$

$\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$

We present the second-order necessary and sufficient conditions for quasiconvex and pseudoconvex functions in terms of their second-order regular subdifferentials.

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

$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)<1+{\displaystyle \frac{c}{2}},\quad z\in \mathbb{D},\end{eqnarray}$$

$c\in (0,3]$

$c\in (0,1]$

$|\unicode[STIX]{x1D6FE}_{n}|$

$n=1,2,3$

$f$

${\mathcal{F}}(c)$

${\mathcal{G}}(c)$

• ∙ If $f\in {\mathcal{F}}(-1/2)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$ for $n\geq 1$, and

  $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$
  where $\text{Li}_{2}(x)$ denotes the dilogarithm function.

• ∙ If $f\in {\mathcal{G}}(c)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$ for $n\geq 1$.

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 ${\mathcal{S}}_{n}$, we consider the ‘affine’ maps $\overline{T}_{a}$ corresponding to $T$ in $\operatorname{GL}(n,\mathbb{Q}_{p})$ and a nonzero $a$ in $\mathbb{Q}_{p}^{n}$ satisfying $\Vert T^{-1}(a)\Vert _{p}<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_{a}$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_{n}$. The dynamics of ‘affine’ maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

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,

$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$

$p^{\prime }$

$n-1$

$m$

$(dn-m)/(n+m)>1$

$n$

$m$

${\sim}n^{-1}$

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.