Convergence rates of Neumann problems for Stokes systems

In quantitative homogenization of the Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, this paper studies the convergence rates of the velocity in $L^2$ and $H^1$ as well as those of the pressure term in $L^2$, without any smoothness assumptions on the coefficients.     

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between ?-quasi-cyclic codes over a finite field ${\mathbb{F}}_q$ and linear codes over a ring $R={\mathbb{F}}_q[Y]/(Y^m?1)$. Using this correspondence, we prove that every ?-quasi-cyclic self-dual code of length m? over a finite field ${\mathbb{F}}_q$ can be obtained by the building-up construction, provided that $\mathrm{char}({\mathbb{F}}_q)=2$ or $q\equiv 1(\mathrm{mod}4)$, m is a prime p, and q is a primitive element of ${\mathbb{F}}_p$. We determine possible weight enumerators of a binary ?-quasi-cyclic self-dual code of length p? (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ?-quasi-cyclic codes of length 3?) optimal self-dual codes of lengths $30,36,42,48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40,20,12]$ code over ${\mathbb{F}}_3$ and a new 6-quasi-cyclic self-dual $[30,15,10]$ code over ${\mathbb{F}}_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28,14,9]$ code over ${\mathbb{F}}_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over ${\mathbb{F}}_4$.     

Existence of steady subsonic Euler flows through infinitely long periodic nozzles

In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles which are periodic in $x_1$-direction with the period L. It is shown that when the variation of Bernoulli function at some given section is small and mass flux is in a suitable regime, there exists a unique global subsonic flow in the nozzle. Furthermore, the flow is also periodic in $x_1$-direction with the period L. If, in particular, the Bernoulli function is a constant, we also get the existence of subsonic-sonic flows when the mass flux takes the critical value.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].     

Bordered surfaces in the 3-sphere with maximum symmetry

We consider orientation-preserving actions of finite groups G on pairs $(S^3,\mathrm{\Sigma })$, where Σ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus $g>1$ the maximum order of such a G for all embeddings of a surface of genus g, and classified the corresponding embeddings.In the present paper we obtain analogous results for the case of bordered surfaces Σ (i.e. with non-empty boundary, orientable or not). Now the genus g gets replaced by the algebraic genus α of Σ (the rank of its free fundamental group); for each $\alpha >1$ we determine the maximum order $m_{\alpha }$ of an action of G, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into $S^3$. For example, the maximal possibility $12(\alpha ?1)$ is obtained for the finitely many values $\alpha =2,3,4,5,9,11,25,97,121$ and 241.     

On the number of unlabeled vertices in edge-friendly labelings of graphs

Let $G$ be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$, and $f$ be a $0?1$ labeling of $E\left(G\right)$ so that the absolute difference in the number of edges labeled $1$ and $0$ is no more than one. Call such a labeling $f$ edge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident to more edges labeled $1$ than $0$, are labeled $1$, and vertices which are incident to more edges labeled $0$ than $1$, are labeled $0$. Vertices that are incident to an equal number of edges of both labels we call unlabeled. Call a procedure on a labeled graph a label switching algorithm if it consists of pairwise switches of labels. Given an edge-friendly labeling of $K_n$, we show a label switching algorithm producing an edge-friendly relabeling of $K_n$ such that all the vertices are labeled. We call such a labeling opinionated.     

Counting cubic extensions with given quadratic resolvent

Given a number field k and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of k whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha })$, for an explicit $\alpha <1$.     

Two timescales asymptotes of the weakly compressible Stokes system

We establish a two timescale asymptotics of the weakly compressible Stokes system which has dissipation of order $\epsilon >0$. For any $L^2$ initial data, over time scale of order 1, the solutions of the weakly compressible Stokes system converge strongly to those of the acoustic system as $\epsilon \to 0$. Over time scale of order $1/\epsilon$, the limit system is the incompressible Stokes system with the initial data projected on the incompressible mode. For the periodic domain, the convergence is weak due to the fast oscillation generated by acoustic waves. For the Navier-slip boundary condition with the reciprocal of slip length being square of the Knudsen number, the acoustic waves are damped by the viscous boundary layer, and consequently the strong convergence is justified.     

Diffuse reflection diameter and radius for convex-quadrilateralizable polygons

In this paper we study the diffuse reflection diameter and diffuse reflection radius problems for convex-quadrilateralizable polygons. In the usual model of diffuse reflection, a light ray incident at a point on the reflecting surface is reflected from that point in all possible inward directions. A ray reflected from a polygonal edge may graze that reflecting edge but an incident ray cannot graze the reflecting edge. The diffuse reflection diameter of a simple polygon $P$ is the minimum number of diffuse reflections that may be needed in the worst case to illuminate any target point $t$ from any point light source $s$ inside $P$. We show that the diameter is upper bounded by $\frac{3n?10}{4}$ in our usual model of diffuse reflection for convex-quadrilateralizable polygons. To the best of our knowledge, this is the first upper bound on diffuse reflection diameter within a fraction of $n$ for such a class of polygons. We also show that the diffuse reflection radius of a convex-quadrilateralizable simple polygon with $n$ vertices is at most $\frac{3n?10}{8}$ under the usual model of diffuse reflection. In other words, there exists a point inside such a polygon from which $\frac{3n?10}{8}$ usual diffuse reflections are always sufficient to illuminate the entire polygon. In order to establish these bounds for the usual model, we first show that the diameter and radius are $\frac{n?4}{2}$ and $?\frac{n?4}{4}?$ respectively, for the same class of polygons for a relaxed model of diffuse reflections; in the relaxed model an incident ray is permitted to graze a reflecting edge before turning and reflecting off the same edge at any interior point on that edge. We also show that the worst-case diameter and radius lower bounds of $\frac{n?4}{2}$ and $?\frac{n?4}{4}?$ respectively, are sometimes attained in the usual model, as well as in the relaxed model of diffuse reflection.