Set Of Minimal Elements Research Articles

On the minimal elements in conjugacy classes of the complex reflection group G(r,1,n)

In this paper we study the minimal length elements in conjugacy classes (abbreviated as minimal elements) of the complex reflection group G(r,1,n). We set up some connections between the set of minimal elements in different conjugacy classes of G(r,1,n), which yields an recursive way to produce all minimal elements. For each w∈G(r,1,n), we give a necessary and sufficient condition under which w is minimal in terms of its standard components in the standard decomposition of w. We construct an explicit example which shows that the standard basis elements labeled by two different minimal elements in the same conjugacy class may have different images in the cocenter of the non-degenerate cyclotomic Hecke algebra of type G(r,1,n), which disproves an earlier speculation. Finally, we show that the standard basis elements labeled by some minimal elements form an integral basis for the cocenter of the degenerate cyclotomic Hecke algebra of type G(r,1,n).

Interior Operators Generated by Ideals in Complete Domains

This article presents some properties of a special class of interior operators generated by ideals. The mathematical framework is given by complete domains, namely complete posets in which the set of minimal elements is a basis. The first part of the paper presents some preliminary results; in the second part we present the novel interior operator denoted by G(i,I), an operator built starting from an interior operator i and an ideal I. Various properties of this operator are presented; in particular, the connection between the properties of the ideal I and the properties of the operator G(i,I). Two such properties (denoted by Pi and Qi) are extensively analyzed and characterized. Additionally, some characterizations for compact elements are presented.

Random optimization on random sets

Random sets and random preorders naturally appear in financial market modeling with transaction costs. In this paper, we introduce and study a concept of essential minimum for a family of vector-valued random variables, as a set of minimal elements with respect to some random preorder. We provide some conditions under which the essential minimum is not empty and we present two applications in optimisation for mathematical finance and economics.

Boxicity and Poset Dimension

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times\cdots\times [a_k,b_k]$. The boxicity of $G$, box$(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes; i.e., each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\mathcal{P}=(S,P)$ be a poset, where $S$ is the ground set and $P$ is a reflexive, antisymmetric and transitive binary relation on $S$. The dimension of $\mathcal{P}$, $\dim(\mathcal{P})$, is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_{\mathcal{P}}$ be the underlying comparability graph of $\mathcal{P}$; i.e., $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\mathcal{P}$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\mathcal{P}$, box$(G_{\mathcal{P}})/(\chi(G_{\mathcal{P}})-1) \le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$, where $\chi(G_{\mathcal{P}})$ is the chromatic number of $G_{\mathcal{P}}$ and $\chi(G_{\mathcal{P}})\ne1$. It immediately follows that if $\mathcal{P}$ is a height-$2$ poset, then box$(G_{\mathcal{P}})\le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$ since the underlying comparability graph of a height-$2$ poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the extended double cover of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that, corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\mathcal{P}_c$ be the natural height-$2$ poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\mbox{box}(G)}{2} \le \dim(\mathcal{P}_c) \le 2\mbox{box}(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\mathcal{P})\le 2\mbox{box}(G_\mathcal{P})$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\mathcal{P})\le 2\mbox{ tree width }(G_{\mathcal{P}})+4$, since boxicity of any graph is known to be at most its tree width $+\; 2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $\Delta$ is $O(\Delta\log^2\Delta)$, which is an improvement over the best-known upper bound of $\Delta^2+2$. (2) There exist graphs with boxicity $\Omega(\Delta\log\Delta)$. This disproves a conjecture that the boxicity of a graph is $O(\Delta)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon>0$ unless $NP=ZPP$.

Partial Latin Squares Having a Santilli’s Autotopism in their Autotopism Groups

An autotopism Θ = (α, β, γ) of a partial Latin square P = (pi, j ) with symbols in N = {0, 1, …, n − 1} is said to be a Santilli’s autotopism of P if there exist jalpha;, jβ,jγ ∈ N such that α(i) = pi, jα , β(i) = pi, jβ and γ(i) = pi, jγ , for all i ∈ N. In this paper, we are interested in the set SPLS (Θ) of partial Latin squares having Θ as a Santilli’s autotopism. Specifically, the cardinality of the set of minimal elements of SPLS(Θ) is obtained and a classification of Latin squares Santilli’s isotopisms of order ≤ 5 fixing at least one symbol is given.

Compact metric spaces as minimal-limit sets in domains of bottomed sequences

Every compact metric space $X$ is homeomorphically embedded in an $\omega$-algebraic domain $D$ as the set of minimal limit (that is, non-finite) elements. Moreover, $X$ is a retract of the set $L(D)$ of all limit elements of $D$. Such a domain $D$ can be chosen so that it has property M and finite-branching, and the height of $L(D)$ is equal to the small inductive dimension of $X$. We also show that the small inductive dimension of $L(D)$ as a topological space is equal to the height of $L(D)$ for domains with property M. These results give a characterisation of the dimension of a space $X$ as the minimal height of $L(D)$ in which $X$ is embedded as the set of minimal elements. The domain in which we embed an $n$-dimensional compact metric space $X$ ($n \leq \infinity$) has a concrete structure in that it consists of finite/infinite sequences in $\{0,1,\bot\}$ with at most $n$ copies of $\bot$.

New steps in the amazing world of sequences and schedules

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timep i,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan. We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules. Furthermore, we investigate the problemO∥C max on an operation set with spanning tree structure.

Bounds on the minimum distance of the duals of extended BCH codes over $$\mathbb{F}_p $$

LetC be an extended cyclic code of lengthp m over $$\mathbb{F}_p $$ . The border ofC is the set of minimal elements (according to a partial order on [0,p m −1]) of the complement of the defining-set ofC. We show that an affine-invariant code whose border consists of only one cyclotomic coset is the dual of an extended BCH code if, and only if, this border is the cyclotomic coset, sayF(t, i), ofp t −1−i, with 1 ≦t ≦ m and 0 ≦i < p−1. We then study such privileged codes. We first make precize which duals of extendedBCH codes they are. Next, we show that Weil's bound in this context gives an explicit formula; that is, the couple (t, i) fully determines the value of the Weil bound for the code with borderF(t, i). In the case where this value is negative, we use the Roos method to bound the minimum distance, greatly improving the BCH bound.