Sang-Mok Kim

Irreducible Width 2 Posets of Linear Discrepancy 3

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x) − hL(y)| ≤ k, where hL(x) is the height of x in L. Tannenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem for characterizing the posets of linear discrepancy 2. Howard et al. (Order 24:139–153, 2007) showed that this problem is equivalent to finding all posets of linear discrepancy 3 such that the removal of any point reduces the linear discrepancy. In this paper we determine all of these minimal posets of linear discrepancy 3 that have width 2. We do so by showing that, when removing a specific maximal point in a minimal linear discrepancy 3 poset, there is a unique linear extension that witnesses linear discrepancy 2.

The Linear Discrepancy of a Fuzzy Poset

In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X × X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the ‘less than’ function, G is the ‘greater than’ function, and I is the ‘incomparable to’ function. Using this approach, we are able to define a special class of fuzzy posets, and define the ‘skeleton’ of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)|f(x)?f(y)| for f ∈ F and x, y ∈ X with I(x, y) > ½, where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.

POSETS ADMITTING THE LINEARITY OF ISOMETRIES

Abstract. In this paper, we deal with a characterization of the posetswith the property that every poset isometry of F nq fixing the origin is alinear map. We say such a poset to be admitting the linearity of isome-tries. We show that a poset P admits the linearity of isometries over F nq if and only if P is a disjoint sum of chains of cardinality 2 or 1 whenq= 2, or P is an anti-chain otherwise. 1. IntroductionLet F q be a finite field with q elements, and F nq the vector space of n-tuplesover F q . In 1995, Brualdi et al. [1] introduced a non-Hamming metric on F n which is associated to an arbitrary poset on [n] = {1,2,...,n}. It is calleda poset metric. The poset metric spaces have been extensively studied in[1, 3, 4, 5, 8]. We briefly introduce basic notions for poset metric on F nq .Let P = ([n],≤) be a poset on [n] of coordinate positions of vectors on F nq .A subset I of P is called an order ideal (or a down-set) if x ∈ I and y ≤ ximply y ∈ I. For an arbitrary subset A of P, we denote by hAi the smallestorder ideal of P containing A. The P-weight of a vector u = (u

Estimation for the Variation of the Concentration of Greenhouse Gases with Modified Shannon Entropy

Entropy is a measure of disorder or uncertainty. This terminology is qualitatively used in the understanding of its correlation to pollution in the environmental area. In this research, three different entropies were defined and characterized in order to quantify the qualitative entropy previously used in the environmental science. We are dealing with newly defined distinct entropies E1, E2, and E3 originated from Shannon entropy in the information theory, reflecting concentration of three major green house gases CO2, N2O and CH4 represented as the probability variables. First, E1 is to evaluate the total amount of entropy from concentration difference of each green house gas with respect to three periods, due to industrial revolution, post-industrial revolution, and information revolution, respectively. Next, E2 is to evaluate the entropy reflecting the increasing of the logarithm base along with the accumulated time unit. Lastly, E3 is to evaluate the entropy with a fixed logarithm base by 2 depending on the time. Analytical results are as follows. E1 shows the degree of prediction reliability with respect to variation of green house gases. As E1 increased, the concentration variation becomes stabilized, so that it follows from linear correlation. E2 is a valid indicator for the mutual comparison of those green house gases. Although E3 locally varies within specific periods, it eventually follows a logarithmic curve like a similar pattern observed in thermodynamic entropy.

SOME HOMOGENEITY CLASSES OF POSETS OF HEIGHT 2

In this paper, we nd the inclusion relation among four cate- gories of posets, i.e., ideal-homogeneous, tower-homogeneous, quasi-com- plement-preserved, and complement-preserved posets.

INCLUSION AND EXCLUSION FOR FINITELY MANY TYPES OF PROPERTIES

Inclusion and exclusion is used in many papers to count certain objects exactly or asymptotically. Also it is used to derive the Bonferroni inequalities in probabilistic area (6). Inclusion and exclusion on flnitely many types of properties is flrst used in R. Meyer (7) in probability form and flrst used in the paper of McKay, Palmer, Read and Robinson (8) as a form of counting version of inclusion and exclusion on two types of properties. In this paper, we provide a proof for inclusion and exclusion on flnitely many types of properties in counting version. As an example, the asymptotic number of general cubic graphs via inclusion and exclusion formula is given for this generalization.

THE LINEAR DISCREPANCY OF 3 £ 3 £ 3

is the meaningful smallest product of three chains of each size 2n+1 since is a 1-element poset. The linear discrepancy of the product of three chains is found as . But the case of the product of three chains is not known yet. In this paper, we determine ld as a case to determine the linear discrepancy of the product of three chains of each size 2n + 1.

Sets of type-(1,n) in symmetric designs for λ≥3

A set of type-(m,n)S is a set of points of a design with the property that each block of the design meets either m points or n points of S. The notions of type and of parameters of a k-set (there called characters) were introduced for the first time by Tallini Scafati in [M. Tallini Scafati, {k,n}-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri. Note I and Note II, Rend. Accad. Naz. Lincei 40 (8) (1996) 812-818 (1020-1025)]. If m=1, S gives rise to a subdesign of the design. Under weaker conditions for the order of each symmetric design, the parameters of sets of type-(1,n) in projective planes were characterised by G. Tallini and the biplane case was dealt with by S. Kim, by solving the corresponding Diophantine equation for each case, separately. In this paper, we first characterise the parameters of sets of type-(1,n) in the triplane with more generalised order conditions than prime power order. Next, we generalise the result on triplanes to arbitrary symmetric designs for @l>=3. As results, a non-existence condition for special parameter sets and a characterisation of parameters for the existence, restricted by some derived bounds, are given.

ON NUMBER OF WAYS TO SHELL THE k-DIMENSIONAL TREES

Which spheres are shellable?[2]. We present one of them which is the k-tree with n-labeled vertices. We found that the number of ways to shell the k-dimensional trees on n-labeled vertices is