UTCC Scholarhttps://scholar.utcc.ac.thThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 10 Apr 2021 22:05:35 GMT2021-04-10T22:05:35Z5081Combinatorial aspects of the generalized Euler's totienthttps://scholar.utcc.ac.th/handle/6626976254/3588Title: Combinatorial aspects of the generalized Euler's totient
Authors: Pabhapote, N.; Laohakosol, V.
Abstract: A generalized Euler's totient is defined as a Dirichlet convolution of a power function and a product of the SouriauHsuMbius function with a completely multiplicative function. Two combinatorial aspects of the generalized Euler's totient, namely, its connections to other totients and its relations with counting formulae, are investigated.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/35882010-01-01T00:00:00ZDistributive property of completely multiplicative functionshttps://scholar.utcc.ac.th/handle/6626976254/3593Title: Distributive property of completely multiplicative functions
Authors: Pabhapote, N.; Laohakosol, V.
Abstract: Dirichlet product of any two arithmetic functions. Taking a generalized Möbius function as an element in this product, our first objective is to investigate, through the concepts of discriminative and partially discriminative products, under which conditions such distributive property yields a necessary and sufficient condition for complete multiplicativity. Our second objective is to derive an extension of the IvićHaukkanencharacterization of completely multiplicative functions through the concept of semidiscriminative product.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/35932010-01-01T00:00:00ZGDDs with two associate classes and with three groups of sizes 1, n, n and λ1 &t; λ2https://scholar.utcc.ac.th/handle/6626976254/3462Title: GDDs with two associate classes and with three groups of sizes 1, n, n and λ1 &t; λ2
Authors: Lapchinda, W.; Punnim, N.; Pabhapote, N.
Abstract: A group divisible design GDD(v = 1 + n + n, 3, 3, λ1, λ2) is an ordered pair (V, B) where V is an (1 + n + n)setof symbols and B is a collection of 3subsets(called blocks) of V satisfying the following properties: the (1 + n + n)set is divided into 3 groups of sizes 1, n and n; each pair of symbols from the same group occurs in exactly λ1 blocks in B; and each pair of symbols fromdifferent groups occurs in exactly λ2 blocks in B. Let λ1, λ2 be positive integers. Then the spectrum of λ1, λ2, denoted by Spec(λ1, λ2), is defined by Spec(λ1, λ2) = {n ∈ ℕ : a GDD(v = 1 + n + n, 3, 3, λ1, λ2) exists}. We found in [10] the spectrum Spec(λ1, λ2) provided that λ1 ≥ λ2 in all situations. We find in this paper Spec(λ1, λ2) when λ1 &t; λ2 in all situations.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/34622013-01-01T00:00:00ZGroup divisible designs with two associate classes and with two unequal groupshttps://scholar.utcc.ac.th/handle/6626976254/3507Title: Group divisible designs with two associate classes and with two unequal groups
Authors: Pabhapote, N.
Abstract: A group divisible design GDD(m, n; 3, λ 1, λ 2) is an ordered triple (V,G,B), where V is a m + nset of symbols, G is a partition of V into 2 sets of sizes m, n, each set being called group, and B is a collection of 3subsets(called blocks) of V , such that each pair of symbols from the same group occurs in exactly λ 1 blocks; and each pair of symbols from different groups occurs in exactly λ 2 blocks. In this paper, we find necessary and sufficient conditions for the existence of a GDD(m, n; 3, λ 1, λ 2) with λ 1 ≥ λ 2.
Sun, 01 Jan 2012 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/35072012-01-01T00:00:00ZGroup divisible designs with two associate classes and λ1 λ2 = 1https://scholar.utcc.ac.th/handle/6626976254/3622Title: Group divisible designs with two associate classes and λ1 λ2 = 1
Authors: Lapchinda, W.; Pabhapote, N.
Abstract: A group divisible design GDD(v, g, k, λ1, λ2) is a collection of ksubsets (called blocks) of a vsetof symbols where: the vset is divided into g groups; each pair of symbols from the same group occurs in exactly λ1 blocks; and each pair symbols from different groups occurs in exactly λ2 blocks. Pairs of symbols occurring in the same group are known to statisticians as firstassociates, and pairs occurring in different groups are called second associates. The existence of such GDDs has been of interest over the years, going back to at least the work of Bose and Shimamoto in 1952 who began classifying such designs. Recently, such an existence problem when g = 2 was solved in the case where the groups have the same size and the blocks have size 3. In this paper, we continue to focus on blocks of size 3, solving the problem when the required designs having twogroups of unequal sizes and λ1 λ2 = 1 and prove that the conditions are sufficient.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/36222010-01-01T00:00:00ZGroup divisible designs with two associate classes and λ 2 = 4https://scholar.utcc.ac.th/handle/6626976254/3540Title: Group divisible designs with two associate classes and λ 2 = 4
Authors: Uiyyasathian, C.; Pabhapote, N.
Abstract: A group divisible design GDD(v = v 1 + v 2 + · · · + v g,g, k, λ 1, λ 2) is an ordered triple (V, G, B), where V is a vset of symbols, G is a partition of V into g sets of size v 1, v 2,⋯ v g, each set being called group, and B is a collection of ksubsets (called blocks) ofV, such that each pair of symbols from the same group occurs in exactly λ 1 blocks; and each pair of symbols from different groups occurs in exactly λ 2 blocks. Here, we focus on an existence problem of GDDs with two associate classes or when g = 2, and with blocks of size 3, when the required designs have two groups of unequal sizes and λ 2 = 4. We obtain the necessary conditions and prove that these conditions are sufficient.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/35402011-01-01T00:00:00ZGroup divisible designs with two associate classes and λ2 =1https://scholar.utcc.ac.th/handle/6626976254/3550Title: Group divisible designs with two associate classes and λ2 =1
Authors: Pabhapote, N.; Punnim, N.
Abstract: The original classiffcation of PBIBDs defined group divisible designs GDD(ν= ν1 + ν2 +⋯+ νg,g,k, λ1, λ2) with λ1 ≠ 0. In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with λ2 =1.
https://scholar.utcc.ac.th/handle/6626976254/3550Group divisible designs with two associate classes and λ2 = 3https://scholar.utcc.ac.th/handle/6626976254/3553Title: Group divisible designs with two associate classes and λ2 = 3
Authors: Chaiyasena, A.; Pabhapote, N.
Abstract: Necessary and sufficient conditions for the existence of Group divisible designs with two groups of unequal sizes and block size tree with λ2 = 3, λ1 ≥ 3 are here considered. We find that the necessary conditions, derived from graph theoretic conditions, are sufficient as well. We present some constructions to prove sufficiency.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholar.utcc.ac.th/handle/6626976254/35532011-01-01T00:00:00Z