• 대한수학회지
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• 제51권5호
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• pp.919-940
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• 2014

We conjecture that if $k{\geq}2$ is an integer and G is a graph of order n with minimum degree at least (n+2k)/2, then for any k independent edges $e_1$, ${\cdots}$, $e_k$ in G and for any integer partition $n=n_1+{\cdots}+n_k$ with $n_i{\geq}4(1{\leq}i{\leq}k)$, G has k disjoint cycles $C_1$, ${\cdots}$, $C_k$ of orders $n_1$, ${\cdots}$, $n_k$, respectively, such that $C_i$ passes through $e_i$ for all $1{\leq}i{\leq}k$. We show that this conjecture is true for the case k = 2. The minimum degree condition is sharp in general.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.657-668
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• 2016

Given a partition ${\lambda}=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_l)$ of a positive integer n, let Tab(${\lambda}$, k) be the set of all tabloids of shape ${\lambda}$ whose weights range over the set of all k-compositions of n and ${\mathcal{OP}}^k_{\lambda}_{rev}$ the set of all ordered partitions into k blocks of the multiset $\{1^{{\lambda}_l}2^{{\lambda}_{l-1}}{\cdots}l^{{\lambda}_1}\}$. In [2], Butler introduced an inversion-like statistic on Tab(${\lambda}$, k) to show that the rank-selected $M{\ddot{o}}bius$ invariant arising from the subgroup lattice of a finite abelian p-group of type ${\lambda}$ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(${\lambda}$, k) and ${\mathcal{OP}}^k_{\hat{\lambda}}$. When k = 2, we also introduce a major-like statistic on Tab(${\lambda}$, 2) and study its connection to the inversion statistic due to Butler.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.155-166
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• 2023

In 2018, Andrews introduced the partition functions ${\mathfrak{EO}}$(n) and ${\bar{\mathfrak{EO}}}$(n). The first of these denotes the number of partitions of n in which every even part is less than each odd part, and the second counts the number of partitions enumerated by the first in which only the largest even part appears an odd number of times. In 2021, Pore and Fathima introduced a new partition function ${\mathfrak{EO}}_e$(n) which counts the number of partitions of n which are enumerated by ${\bar{\mathfrak{EO}}}$(n) together with the partitions enumerated by ${\bar{\mathfrak{EO}}}$(n) where all parts are odd and the number of parts is even. They also proved some particular congruences for ${\bar{\mathfrak{EO}}}$(n) and ${\mathfrak{EO}}_e$(n). In this paper, we establish infinitely many families of congruences modulo 2, 4, 5 and 8 for ${\bar{\mathfrak{EO}}}$(n) and modulo 4 for ${\mathfrak{EO}}_e$(n). For example, if p ≥ 5 is a prime with Legendre symbol $({\frac{-3}{p}})=-1$, then for all integers n ≥ 0 and α ≥ 0, we have ${\bar{\mathfrak{EO}}}(8{\cdot}p^{2{\alpha}+1}(pn+j)+{\frac{19{\cdot}p^{2{\alpha}+2}-1}{3}}){\equiv}0$ (mod 8); 1 ≤ j ≤ (p - 1).

• ETRI Journal
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• 제45권5호
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• pp.874-886
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• 2023

A joint resource-optimization scheme is investigated for nonorthogonal multiple access (NOMA)-enhanced scalable video coding (SVC) multicast in unmanned aerial vehicle (UAV)-assisted radio-access networks (RANs). This scheme allows a ground base station and UAVs to simultaneously multicast successive video layers in SVC with successive interference cancellation in NOMA. A video quality-maximization problem is formulated as a mixed-integer nonlinear programming problem to determine the UAV deployment and association, RAN spectrum allocation for multicast groups, and UAV transmit power. The optimization problem is decoupled into the UAV deployment-association, spectrum-partition, and UAV transmit-power-control subproblems. A heuristic strategy is designed to determine the UAV deployment and association patterns. An upgraded knapsack algorithm is developed to solve spectrum partition, followed by fast UAV power fine-tuning to further boost the performance. The simulation results confirm that the proposed scheme improves the average peak signal-to-noise ratio, aggregate videoreception rate, and spectrum utilization over various baselines.

• 대한산업공학회지
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• 제10권1호
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• pp.3-10
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• 1984

A materials acquisition planning (MAP) problem that involves the determination of how much to order of a number of different items from a number of different suppliers is considered. This particular problem is modelled as a nonlinear mixed integer programming problem. A solution procedure based upon the partition of variables is developed to handle the MAP problem. This solution procedure utilizes a modified Hooke-Jeeves Pattern Search procedure along with a linear programming simplex algorithm. An example problem is presented and the results of applying the suggested solution procedure to this problem are reported.

• 대한수학회보
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• 제52권5호
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• pp.1683-1709
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• 2015

Let (${\Omega}$, S) be an association scheme where ${\Omega}$ is a non-empty finite set and S is a partition of ${\Omega}{\times}{\Omega}$. For a positive integer k we say that (${\Omega}$, S) is k-equivalenced if each non-diagonal element of S has valency k. In this paper we focus on 4-equivalenced association schemes, and prove that they are transitive.

• 한국경영과학회지
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• 제15권2호
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• pp.1-15
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• 1990

A mathematical definition of the cluster is suggested. A nonlinear 0-1 integer programming formulation for the multi-dimensional entity clustering problem is developed. A heuristic method named MDEC (Multi-Dimensional Entity Clustering) using centroids and the binary partition is developed and the numerical examples are shown. This method has an advantage of providing bottle-neck entity informations.

• East Asian mathematical journal
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• 제28권1호
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• pp.101-107
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• 2012

For each dimension exceeds 1, determining the number of multi-dimensional partitions of a positive integer is an open question in combinatorial number theory. For n ${\leq}$ 14 and d ${\geq}$ 1 we derive a formula for the function ${\wp}_d(n)$ where ${\wp}_d(n)$ denotes the number of partitions of n arranged on a d-dimensional space. We also give an another definition of the d-dimensional partitions using the union of finite number of divisor sets of integers.

• 대한수학회보
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• 제59권5호
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• pp.1279-1287
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• 2022

A partition of n is complete if every positive integer from 1 to n can be represented by the sum of its parts. The concept of complete partitions has been extended in several ways. In this paper, we consider the number of k-relaxed r-complete partitions of n and the number of double-complete partitions of n.

• 대한수학회지
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• 제59권5호
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• pp.945-962
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• 2022

For a positive integer ℓ, $\bar{A}_{\ell}(n)$ denotes the number of over-partitions of n into parts not divisible by ℓ. In this article, we find certain Ramanujan-type congruences for $\bar{A}_{r{\ell}}(n)$, when r ∈ {8, 9} and we deduce infinite families of congruences for them. Furthermore, we also obtain Ramanujan-type congruences for $\bar{A}_{13}(n)$ by using an algorithm developed by Radu and Sellers [15].