• East Asian mathematical journal
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• v.35 no.1
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• pp.9-15
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• 2019

Kaavya introduce an involution on the set of partitions with crank 0 and studied the number of partitions of n which are invariant under Kaavya's involution. If a partition ${\lambda}$ with crank 0 is invariant under her involution, we say ${\lambda}$ is a self-conjugate partition with crank 0. We prove that the number of such partitions of n is equal to the number of partitions with rank 0 which are invariant under the usual partition conjugation. We also study arithmetic properties of such partitions and their q-theoretic implication.

• Korean Journal of Mathematics
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• v.26 no.2
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• pp.167-174
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• 2018

By acting the dihedral group $D_k$ on the set of k-tuple multi-partitions, we introduce k-gonal partitions for all positive integers k. We give generating functions for these new partition functions and investigate their arithmetic properties.

• Journal of the Korean Institute of Telematics and Electronics
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• v.15 no.1
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• pp.12-18
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• 1978

From the state table of sequential machines, Roomers) obtained minimal k co-mapping chain(k-CC) and proposed an algorithm obtaining binary partitions which were seed partitions of S-shift registers. By comparing and processing bits simply, this paper obtained two different algorithms more efficient than that of Roome's for obtaining such binary partitions and defined the concept of the triple pair of the basis partitions. By using the concept, given set of basis partitions was reduced to the set containing elements of the triple pair only and the algorithm became quite simple.

• ETRI Journal
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• v.36 no.4
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• pp.635-642
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• 2014

Three-dimensional integrated circuits (3D ICs) implement heterogeneous systems in the same platform by stacking several planar chips vertically with through-silicon via (TSV) technology. 3D ICs have some advantages, including shorter interconnect lengths, higher integration density, and improved performance. Thermal-aware design would enhance the reliability and performance of the interconnects and devices. In this paper, we propose thermal-aware floorplanning with min-cut die partitioning for 3D ICs. The proposed min-cut die partition methodology minimizes the number of connections between partitions based on the min-cut theorem and minimizes the number of TSVs by considering a complementary set from the set of connections between two partitions when assigning the partitions to dies. Also, thermal-aware floorplanning methodology ensures a more even power distribution in the dies and reduces the peak temperature of the chip. The simulation results show that the proposed methodologies reduced the number of TSVs and the peak temperature effectively while also reducing the run-time.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.367-382
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• 2011

We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$ into $\mathbb{R}^3$.

• Kyungpook Mathematical Journal
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• v.56 no.3
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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.

• Journal of Industrial Technology
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• v.28 no.B
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• pp.209-214
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• 2008

Set-valued attributes appear in many applications to model complex objects occurring in the real world. One of the most important operations on set-valued attributes is the set join, because it provides a various method to express complex queries. Currently proposed set join algorithms are based on block nested loop join in which inverted files are partitioned horizontally into blocks. Evaluating these joins are expensive because they generate intermediate partial results severely and finally obtain the final results after merging partial results. In this paper, we present an efficient processing of set join algorithm. We propose a new set join algorithm that vertically partitions inverted files into blocks, where each block fits in memory, and performs block nested loop join without producing intermediate results. Our experiments show that the vertical bitmap nested set join algorithm outperforms previously proposed set join algorithms.

• Journal of the Korean Statistical Society
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• v.13 no.2
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• pp.114-120
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• 1984

A parallel flats fraction for the $3^n$ design is defined as union of flats ${t}At=c_i(mod 3)}, i=1,2,\cdots, f$ and is symbolically written as At=C where A is rank r. The A matrix partitions the effects into n+1 alias sets where $u=(3^{n-r}-1)/2. For each alias set the f flats produce an ACPM from which a detection matrix is constructed. The set of all possible parallel flats fraction C can be partitioned into equivalence classes. In this paper, we develop some properties of a detection matrix and C.

• Korean Journal of Mathematics
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• v.14 no.1
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• pp.71-77
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• 2006

We find the ${\epsilon}$-fuzzy equivalence relation generated by the union of two ${\epsilon}$-fuzzy equivalence relations on a set, find the ${\epsilon}$-fuzzy equivalence relation generated by a fuzzy relation on a set, and find sufficient conditions for the composition ${\mu}{\circ}{\nu}$ of two ${\epsilon}$-fuzzy equivalence relations ${\mu}$ and ${\nu}$ to be the ${\epsilon}$-fuzzy equivalence relation generated by ${\mu}{\cup}{\nu}$. Also we study fuzzy partitions of ${\epsilon}$-fuzzy equivalence relations.

• Journal of the Korean Statistical Society
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• v.9 no.1
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• pp.1-12
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• 1980

A parallel flats faraction for the $3^n$ factorial experiment is symbolically written as $At = C(r\timesf)$ where $A(r\timesn)$ is of rank r. The A-matrix partitions the nonnegligible effects into $(3^{n-r}-1)/2+1$ alias sets. The $U_i$ effects in the i-th alias set are related pairwise by elements from $S_3$, the symmetric group on three symbols. For each alias set the f flats produce an $f \times u_i$ alias componet permutation matrices (ACPM) with elements from $S_3$. All the information concerning the relationships among levels of the effects is contained in the ACPM.