Computing Dixmier Invariants and Some Geometric Configurations of Quartic Curves with 2 Involutions

In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1902.0291

Computing Moment Maps of Hypersurfaces using MAXIMA

We use Maxima to compute the moment matrices of hypersurfaces. After that, we compute the Hilbert-Mumford numerical criterion for plane cubics and plane quartics, and give the stability of these curves

Equivariant mirror symmetry for the weighted projective line

In this paper, we establish equivariant mirror symmetry for the weighted projective line. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [{\it Geometry \& Topology} 24:2049-2092, 2017]. More precisely, we prove the equivalence of the $R$-matrices for A-model and B-model at large radius limit, and establish isomorphism for $R$-matrices for general radius. We further demonstrate that the graph sum of higher genus cases are the same for both models, hence establish equivariant mirror symmetry for the weighted projective line.Comment: 15 pages, 2 figure

Kinesthetic imagery: does it exist and how can we measure it?

[Introduction]: The emergence of sport psychology has influenced how athletes train and compete. Increasingly, coaches and athletes are incorporating mental as well as physical skills into their training programs and competition routines. Imagery is one such mental skill. To develop an imagery program tailored to the athlete three pieces of information are vital: the imagery ability of the athlete; the effect of imagery on performance; and the motive for using imagery. This paper explores measurement of the imagery ability of the athlete. Specifically, the aim was to create new and valid measures of kinaesthetic imagery and examine the relationship these measures share with existing measures of imagery

Patterns in words of ordered set partitions

An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(\sigma)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $\sigma$ and obtained the formula for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $3$. In this paper, we define a new pattern avoidance for the set $\mathcal{OP}_{n,k}$, called $\mathcal{WOP}_{n,k}(\sigma)$, which includes the questions proposed by Godbole, Goyt, Herdan and Pudwell. We obtain formulas for $|\mathcal{WOP}_{n,k}(\sigma)|$ combinatorially for any $\sigma$ of length $3$. We also define 3 kinds of descent statistics on ordered set partitions and study the distribution of the descent statistics on $\mathcal{WOP}_{n,k}(\sigma)$ for $\sigma$ of length $3$.Comment: 42 pages, 16 figure

Classical pattern distributions in Sn(132)\mathcal{S}_{n}(132) and Sn(123)\mathcal{S}_{n}(123)

Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations.Comment: 23 pages, 5 fugure

Quadrant marked mesh patterns in 123-avoiding permutations

Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions

Almost Automorphically and Almost Periodically Forced Circle Flows of Almost Periodic Parabolic Equations on S^1

We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$. The structure of the minimal set $M$ is thoroughly investigated under the assumption that the center space $V^c(M)$ associated with $M$ is no more than $2$-dimensional. Such situation naturally occurs while, for instance, $M$ is hyperbolic or uniquely ergodic. It is shown in this paper that $M$ is a $1$-cover of the hull $H(f)$ provided that $M$ is hyperbolic (equivalently, ${\rm dim}V^c(M)=0$). If ${\rm dim}V^c(M)=1$ (resp. ${\rm dim}V^c(M)=2$ with ${\rm dim}V^u(M)$ being odd), then either $M$ is an almost $1$-cover of $H(f)$ and topologically conjugate to a minimal flow in $\mathbb{R}\times H(f)$; or $M$ can be (resp. residually) embedded into an almost periodically (resp. almost automorphically) forced circle-flow $S^1\times H(f)$. When $f(t,u,u_x)=f(t,u,-u_x)$ (which includes the case $f=f(t,u)$), it is proved that any minimal set $M$ is an almost $1$-cover of $H(f)$. In particular, any hyperbolic minimal set $M$ is a $1$-cover of $H(f)$. Furthermore, if ${\rm dim}V^c(M)=1$, then $M$ is either a $1$-cover of $H(f)$ or is topologically conjugate to a minimal flow in $\mathbb{R}\times H(f)$. For the general spatially-dependent nonlinearity $f=f(t,x,u,u_{x})$, we show that any stable or linearly stable minimal invariant set $M$ is residually embedded into $\mathbb{R}^2\times H(f)$.Comment: 49 page

Optimal bilinear control of nonlinear Schr\"{o}dinger equations with singular potentials

In this paper, we consider an optimal bilinear control problem for the nonlinear Schr\"{o}dinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our results generalize the ones in \cite{Sp} in several aspects.Comment: 14 page

Structure of ω\omega-limit Sets for Almost-periodic Parabolic Equations on S1S^1 with Reflection Symmetry

The structure of the $\omega$-limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$ and satisfies $f(t,u,u_x)=f(t,u,-u_x)$. We show that any $\omega$-limit set $\Omega$ contains at most two minimal sets. Moreover, any hyperbolic $\omega$-limit set $\Omega$ is a spatially-homogeneous $1$-cover of hull $H(f)$. When $\dim V^c(\Omega)=1$ ($V^c(\Omega)$ is the center space associated with $\Omega$), it is proved that either $\Omega$ is a spatially-homogeneous, or $\Omega$ is a spatially-inhomogeneous $1$-cover of $H(f)$.Comment: Accepted by J.Diff.Eqns. arXiv admin note: text overlap with arXiv:1507.0170