Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$ , $\mathbb{F}_{5}$ , and $\mathbb{F}_{7}$ . In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.     

On the field of definition of p-torsion points on elliptic curves over the rationals

Let $S_\mathbb Q (d)$ be the set of primes $p$ for which there exists a number field $K$ of degree $\le d$ and an elliptic curve $E/\mathbb Q $ , such that the order of the torsion subgroup of $E(K)$ is divisible by $p$ . In this article we give bounds for the primes in the set $S_\mathbb Q (d)$ . In particular, we show that, if $p\ge 11$ , $p\ne 13,37$ , and $p\in S_\mathbb Q (d)$ , then $p\le 2d+1$ . Moreover, we determine $S_\mathbb Q (d)$ for all $d\le 42$ , and give a conjectural formula for all $d\ge 1$ . If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large $d$ . Under further assumptions on the non-cuspidal points on modular curves that parametrize those $j$ -invariants associated to Cartan subgroups, the formula is valid for all $d\ge 1$ .     

Weighted Hardy operators in the local generalized vanishing Morrey spaces

In this paper we study $p\rightarrow q$ -boundedness of the multi-dimensional Hardy type operators in the vanishing local generalized Morrey spaces $V\mathcal L ^{p,\varphi }_\mathrm{{loc}}(\mathbb R ^n,w)$ defined by an almost increasing function $\varphi (r)$ and radial type weight $w(|x|)$ . We obtain sufficient conditions, in terms of some integral inequalities imposed on $\varphi $ and $w$ , for such a boundedness. In the case where the function $\varphi (r)$ and the weight are power functions, these conditions are also necessary.     

Compact composition operators acting between weighted Bergman spaces of the unit ball

In this paper, we study a composition operator ${C_{\varphi}}$ on the weighted Bergman space ${A_{\alpha}^p(B)}$ of the unit ball B in ${{\mathbb{C}}^N}$ . Under a natural condition we estimate the essential norm of ${C_{\varphi}}$ . As a consequence of this estimate, we also give a function-theoretic characterization of ${\varphi}$ that induces a compact composition operator on ${A_{\alpha}^p(B)}$ .     

On Suborbital Graphs for the Extended Modular Group {\hat{\Gamma}}

In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$      

Lattice-free sets,multi-branch split disjunctions,and mixed-integer programming

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734, 2008). By analyzing $n$ -dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with $n$ integer variables is a $t$ -branch split cut. We use this result to give a finite cutting-plane algorithm to solve mixed-integer programs. We also show that the minimum value $t$ , for which all facets of polyhedral mixed-integer sets with $n$ integer variables can be generated as $t$ -branch split cuts, grows exponentially with $n$ . In particular, when $n=3$ , we observe that not all facet-defining inequalities are 6-branch split cuts.     

Representation Formulas of Curves in a Two- and Three-Dimensional Lightlike Cone

In this paper we give the representation formulas for spacelike curves in two-dimensional lightlike cone ${{\mathbb Q}^2}$ and three-dimensional lightlike cone ${{\mathbb Q}^3}$ . Using these formulas we discuss the properties and structures of cone curves in ${{\mathbb Q}^2}$ and ${{\mathbb Q}^3}$ . Some examples are also given.     

Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

Characterization of some simple groups by the multiplicity pattern

Let $G$ be a finite group and let ${\mathrm{Irr}}(G)$ denote the set of all complex irreducible characters of $G.$ Let ${\mathrm{cd}}(G)$ be the set of all character degrees of $G.$ For each positive integer $d,$ the multiplicity of $d$ in $G$ is defined to be the number of irreducible characters of $G$ having the same degree $d.$ The multiplicity pattern ${\mathrm{mp}}(G)$ is the vector whose first coordinate is $|G:G^{\prime }|$ and for $i\ge 1,$ the $(i+1)$ th-coordinate of ${\mathrm{mp}}(G)$ is the multiplicity of the $i$ th-smallest nontrivial character degree of $G.$ In this paper, we show that every nonabelian simple group with at most $7$ distinct character degrees is uniquely determined by the multiplicity pattern.     

Optimal relationships between L^p-norms for the Hardy operator and its dual

We obtain sharp two-sided inequalities between $L^p$ -norms $(1<p<\infty )$ of functions $\textit{Hf}$ and $H^*f$ , where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty ).$ In an equivalent form, it gives sharp constants in the two-sided relationships between $L^p$ -norms of functions $H\varphi -\varphi $ and $\varphi $ , where $\varphi $ is a nonnegative nonincreasing function on $(0,+\infty )$ with $\varphi (+\infty )=0.$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $p=2k \,\,(k\in \mathbb N )$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $p\ge 2$ , and gives a sharp version of this result for $1<p<2$ .     

\mathfrak D -parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians

In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C }^{m+2})$ with $\mathfrak D $ -parallel normal Jacobi operator ${\bar{R}}_N$ and $\mathfrak D $ -parallel structure Jacobi operator $R_{\xi }$ if the distribution $\mathfrak D $ or $\mathfrak D ^{\bot }$ component of the Reeb vector field is invariant by the shape operator, respectively.     

On the W^{2,1+\varepsilon }-estimates for the Monge-Ampère equation and related real analysis

We build upon the techniques introduced by De Philippis and Figalli regarding $W^{2,1+\varepsilon }$ bounds for the Monge-Ampère operator, to improve the recent $A_\infty $ estimates for $\Vert D^2 \varphi \Vert $ to $A_2$ ones. Also, we prove a $(1,2)-$ Poincaré inequality and weak $(q,p)-$ Poincaré inequalities associated to the Monge-Ampère quasi-metric structure. In turn, these Poincaré inequalities are used to prove Harnack’s inequality for non-negative solutions to the linearized Monge-Ampère under minimal geometric assumptions.     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.