Kato–Milne cohomology and polynomial forms

Given a prime number p, a field F with $\mathrm{char}(F)=p$ and a positive integer n, we study the class-preserving modifications of Kato–Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p-regular forms. A p-regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order $p?1$ partial derivatives vanish simultaneously. We define a ${\stackrel{?}{C}}_{p,m}$ field to be a field over which every p-regular form of dimension greater than $p^m$ is isotropic. The main results are that for a ${\stackrel{?}{C}}_{p,m}$ field F, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m?1}?1$ and for any $n??(m?1){\log }_2?(p)?+1$, $H_p^{n+1}(F)=0$.     

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 ${\mathbb{F}}_4$.     

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups $\mathrm{S}\mathrm{U}(n)$ for $n\ge 6$, $\mathrm{S}\mathrm{O}(n)$ for $n\ge 7$, $\mathrm{S}\mathrm{p}(n)$ for $n\ge 3$, ${\mathrm{E}}_6,{\mathrm{E}}_7,{\mathrm{E}}_8$, and ${\mathrm{F}}_4$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.     

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

On the denominators of harmonic numbers

Let $H_n$ be the n-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers n such that $v_n$ is divisible by the least common multiple of $1,2,?,?n^{1/4}?$ has density one. In particular, for any positive integer m, the set of positive integers n such that $v_n$ is divisible by m has density one.     

On a discrete bilinear singular operator

We prove that for a large class of functions P and Q, the discrete bilinear operator $T_{P,Q}(f,g)(n)={\sum }_{m\in \mathbb{Z}?\{0\}}f(n?P(m))g(n?Q(m))\frac{1}{m}$ is bounded from $l^2\times l^2$ into $l^{1+?,\infty }$ for any $?\in (0,1]$.     

Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.