On the unsplittable minimal zero-sum sequences over finite cyclic groups of prime order

Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is unsplittable, if there do not exist $g$ in $S$ and $x,y \in G$ such that $g=x+y$ and $Sg^{-1}xy$ is also a minimal zero-sum sequence. In this paper we show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|= \frac{p-1}{2}$, then $S=g^{\frac{p-11}{2}}(\frac{p+3}{2}g)^4(\frac{p-1}{2}g)$ or $g^{\frac{p-7}{2}}(\frac{p+5}{2}g)^2(\frac{p-3}{2}g)$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S| \ge \frac{p-1}{2}$, then \ind(S) \leq 2.Comment: 11 page

Minimal zero-sum sequence of length five over finite cyclic groups of prime power order

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)\cdot\ldots\cdot(x_lg)$ where $g\in G$ and x_1, \ldots, x_l\in[1, \ord(g)], and the index \ind(S) of $S$ is defined to be the minimum of (x_1+\cdots+x_l)/\ord(g) over all possible $g\in G$ such that $\langle g \rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=\langle g\rangle$ is a cyclic group of prime power order $n=p^\mu$ with $p \geq 7$ and $\mu\geq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $\gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then \ind(S)=2 if and only if $S=(mg)(mg)(m\frac{n-1}{2}g)(m\frac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $\gcd(m,n)=1$

The genetic code, algebra of projection operators and problems of inherited biological ensembles

This article is devoted to applications of projection operators to simulate phenomenological properties of the molecular-genetic code system. Oblique projection operators are under consideration, which are connected with matrix representations of the genetic coding system in forms of the Rademacher and Hadamard matrices. Evidences are shown that sums of such projectors give abilities for adequate simulations of ensembles of inherited biological phenomena including ensembles of biological cycles, morphogenetic ensembles of phyllotaxis patterns, mirror-symmetric patterns, etc. For such modeling, the author proposes multidimensional vector spaces, whose subspaces are under a selective control (or coding) by means of a set of matrix operators on base of genetic projectors. Development of genetic biomechanics is discussed. The author proposes and describes special systems of multidimensional numbers under names united-hypercomplex numbers, which attracted his attention when he studied genetic systems and genetic matrices. New rules of long nucleotide sequences are described on the base of the proposed notion of tetra-groups of equivalent oligonucleotides. Described results can be used for developing algebraic biology, bio-technical applications and some other fields of science and technology.Comment: 110 pages,82 figure

On the index-conjecture of length four minimal zero-sum sequences II

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index \ind S of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\hbox{\rm ord}(g)$ over all possible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$ is finite such that $\gcd(|G|,6)=1$, then \ind(S)=1 for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$ is reduced and the (A1) condition is satisfied(see [19]).Comment: arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676 by other author

Zero-sum Subsequences of Length kq over Finite Abelian p-Groups

For a finite abelian group $G$ and a positive integer $k$, let $s_{k}(G)$ denote the smallest integer $\ell\in\mathbb{N}$ such that any sequence $S$ of elements of $G$ of length $|S|\geq\ell$ has a zero-sum subsequence with length $k$. The celebrated Erd\H{o}s-Ginzburg-Ziv theorem determines $s_{n}(C_{n})=2n-1$ for cyclic groups $C_{n}$, while Reiher showed in 2007 that $s_{n}(C_{n}^{2})=4n-3$. In this paper we prove for a $p$-group $G$ with exponent $\exp(G)=q$ the upper bound $s_{kq}(G)\le(k+2d-2)q+3D(G)-3$ whenever $k\geq d$, where $d=\Big\lceil\frac{D(G)}{q}\Big\rceil$ and $p$ is a prime satisfying $p\ge2d+3\Big\lceil\frac{D(G)}{2q}\Big\rceil-3$, where $D(G)$ is the Davenport constant of the finite abelian group $G$. This is the correct order of growth in both $k$ and $d$. As a corollary, we show $s_{kq}(C_{q}^{d})=(k+d)q-d$ whenever $k\geq p+d$ and $2p\geq7d-3$, resolving a case of the conjecture of Gao, Han, Peng, and Sun that $s_{k\exp(G)}(G)=k\exp(G)+D(G)-1$ whenever $k\exp(G)\geq D(G)$. We also obtain a general bound $s_{kn}(C_{n}^{d})\leq9kn$ for $n$ with large prime factors and $k$ sufficiently large. Our methods are inspired by the algebraic method of Kubertin, who proved that $s_{kq}(C_{q}^{d})\leq(k+Cd^{2})q-d$ whenever $k\geq d$ and $q$ is a prime power

On the index-conjecture on the length four minimal zero-sum sequences

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index \ind(S) of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\hbox{\rm ord}(g)$ over all possible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$ is finite such that $\gcd(|G|,6)=1$, then \ind(S)=1 for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$ is reduced and at least one $n_i$ coprime to $|G|$.Comment: International Journal of Number Theory (2013). arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676 by other author

The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts

Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. Families of genetic (4*4)- and (8*8)-matrices show unexpected connections of the genetic system with Walsh functions and Hadamard matrices, which are known in theory of noise-immunity coding, digital communication and digital holography. Dyadic-shift decompositions of such genetic matrices lead to sets of sparse matrices. Each of these sets is closed in relation to multiplication and defines relevant algebra of hypercomplex numbers. It is shown that genetic Hadamard matrices are identical to matrix representations of Hamilton quaternions and its complexification in the case of unit coordinates. The diversity of known dialects of the genetic code is analyzed from the viewpoint of the genetic algebras. An algebraic analogy with Punnett squares for inherited traits is shown. Our results are used in analyzing genetic phenomena. The statement about existence of the geno-logic code in DNA and epigenetics on the base of the spectral logic of systems of Boolean functions is put forward. Our results show promising ways to develop algebraic-logical biology, in particular, in connection with the logic holography on Walsh functions.Comment: 108 pages, 73 figures, added text, added reference

On the structure of zero-sum free set with minimum subset sums in abelian groups

Let $G$ be an additive abelian group and $S\subset G$ a subset. Let $\Sigma(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say $S$ is zero-sum free if $0 \not\in \Sigma(S)$. It was conjectured by R.B.~Eggleton and P.~Erd\"{o}s in 1972 and proved by W.~Gao et. al. in 2008 that $|\Sigma(S)|\geq 19$ provided that $S$ is a zero-sum free subset of an abelian group $G$ with $|S|=6$. In this paper, we determined the structure of zero-sum free set $S$ where $|S|=6$ and $|\Sigma(S)|=19$.Comment: 19 page

On Bousfield's problem for solvable groups of finite Pr\"ufer rank

For a group $G$ and $R=\mathbb Z,\mathbb Z/p,\mathbb Q$ we denote by $\hat G_R$ the $R$-completion of $G.$ We study the map $H_n(G,K)\to H_n(\hat G_R,K),$ where $(R,K)=(\mathbb Z,\mathbb Z/p),(\mathbb Z/p,\mathbb Z/p),(\mathbb Q,\mathbb Q).$ We prove that $H_2(G,K)\to H_2(\hat G_R,K)$ is an epimorphism for a finitely generated solvable group $G$ of finite Pr\"ufer rank. In particular, Bousfield's $HK$-localisation of such groups coincides with the $K$-completion for $K=\mathbb Z/p,\mathbb Q.$ Moreover, we prove that $H_n(G,K)\to H_n(\hat G_R,K)$ is an epimorphism for any $n$ if $G$ is a finitely presented group of the form $G=M\rtimes C,$ where $C$ is the infinite cyclic group and $M$ is a $C$-module

The degeneracy of the genetic code and Hadamard matrices

The matrix form of the presentation of the genetic code is described as the cognitive form to analyze structures of the genetic code. A similar matrix form is utilized in the theory of signal processing. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains 64 triplets. Peculiarities of the degeneracy of the vertebrate mitochondria genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing, spectral analysis, quantum mechanics and quantum computers. A special decomposition of numeric genetic matrices reveals their close relations with a family of 8-dimensional hypercomplex numbers (not Cayley's octonions). Some hypothesis and thoughts are formulated on the basis of these phenomenological facts.Comment: 26 pages; 21 figures; added materials and reference