• Journal of applied mathematics & informatics
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• v.37 no.3_4
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• pp.163-176
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• 2019

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of those pebbles at an adjacent vertex. The t-pebbling number, $f_t(G)$, of a connected graph G, is the smallest positive integer such that from every placement of $f_t(G)$ pebbles, t pebbles can be moved to any specified vertex by a sequence of pebbling moves. A graph G has the 2t-pebbling property if for any distribution with more than $2f_t(G)$ - q pebbles, where q is the number of vertices with at least one pebble, it is possible, using the sequence of pebbling moves, to put 2t pebbles on any vertex. In this paper, we determine the t-pebbling number for the middle graph of a complete binary tree $M(B_h)$ and we show that the middle graph of a complete binary tree $M(B_h)$ satisfies the 2t-pebbling property.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.403-408
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• 2006

A pebbling move on a connected graph G is taking two pebbles off of one vertex and placing one of them on an adjacent vertex. The domination cover pebbling number ${\psi}(G)$ is the minimum number of pebbles required so that any initial configuration of pebbles can be transformed by a sequence of pebbling moves so that the set of vertices that contain pebbles forms a domination set of G. We determine the domination cover pebbling number for fans, fuses, and pseudo-star.

• Journal of the Korean Ceramic Society
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• v.50 no.5
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• pp.333-340
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• 2013

To fabricate spherical lithium titanate ($Li_2TiO_3$) pebbles which are used for a breeder material in fusion reactor, titanium oxide ($TiO_2$) granules were used as a starting material. The granules were pre-sintered, and then aqueous lithium nitrate solution infiltrated into the granules at vacuum condition. The granules were crystallized to $Li_2TiO_3$ after sintering under the control of process parameters. In this study, the concentration of lithium in the solution, as well as the number of penetration times and sintering temperature affected the final crystallite phase and the microstructure of the pebbles. In particular, the sphericity and size of the pebbles were effectively controlled by a technical rolling process. The useful spherical $Li_2TiO_3$ pebbles which have 10~20% porosity and 60~120 N compressive strength were obtained through the sintering at $1000{\sim}1100^{\circ}C$ in the multi-times infiltration process with 50 wt% solution. The physical properties of pebbles such as density, porosity and strength, can be controlled by a selection of $TiO_2$ powders and control of processing parameters. It can be thought that the lithium penetration method is a useful method for the fabrication of mass product of spherical $Li_2TiO_3$ pebbles.

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.121-134
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• 2008

A pebbling step on a graph consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The covering cover pebbling number of a graph is the smallest number of pebbles, such that, however the pebbles are initially placed on the vertices of the graph, after a sequence of pebbling moves, the set of vertices with pebbles forms a covering of G. In this paper we find the covering cover pebbling number of n-cube and diameter two graphs. Finally we give an upperbound for the covering cover pebbling number of graphs of diameter d.

• Nuclear Engineering and Technology
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• v.55 no.8
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• pp.2966-2976
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• 2023

A tritium breeder is a lithium-based material capable of producing tritium. Many researchers designing nuclear-fusion energy are studying tritium production using pebbles, which are solid-type breeders. The sphericity and size of the pebbles are critical in obtaining pebbles with good tritium-breeding efficiency. Furthermore, tritium-release efficiency can be increased by using pebbles with appropriate porosities. Promising raw materials for tritium-breeding materials include Li₄SiO₄ and Li₂TiO₃. Li₄SiO₄ has a higher lithium density than Li₂TiO₃ and exhibits excellent tritium-breeding efficiency. However, it has the disadvantage of being easily decomposed during the Li₄SiO₄-green-pebble sintering process because of its low structural stability at high temperatures and high lithium density. In this study, we attempted to determine the optimal conditions for manufacturing Li₄SiO₄ pebbles using the droplet-freeze-drying method. The optimal Li₄SiO₄ slurry conditions and sintering temperatures were determined. The optimal Li₄SiO₄ slurry-fabrication conditions were 3 wt% polyvinyl alcohol and 75 wt% Li₄SiO₄ based on the deionized-water weight content. The sintering temperature at which Li₄SiO₄ did not decompose and exhibited the optimum porosity of 10.8% was 900 ℃.

• The Journal of Engineering Geology
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• v.19 no.3
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• pp.373-387
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• 2009

Many cultural heritages with various periods including the 'Three Kingdoms Period' were discovered in Chulmamyeon, Busan. Detailed geomorphological and geological investigation was performed due to archaeological importance of the study area. Aerial photographic analysis for geomorphology and analysis of vertical sedimentary sections to interpret the sedimentary characteristics were carried out. Many pebbles with specific features were discovered during the investigation. The characteristics and origin of the pebbles are analysed based on their size, rock type and shape. The study area is located in a relatively low and gentle slope compared with neighboring regions. The strong N-S trending lineaments may influence the topographic features and the distribution of wells discovered in this area. The exposed sections during excavation are divided into several layers depending on colors and materials included. Some of the layers may be developed not by nature but by human activities. The similarity of the pebbles indicates that these pebbles may artificially be collected or produced for a specific purpose. This study shows that geomorphology and geology can be important in location and development of archaeological sites. The analysis and interpretation for the origin of the pebbles indicates that geological studies can be helpful to interpretation of archaeological studies.

• The Pure and Applied Mathematics
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• v.9 no.1
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• pp.57-61
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• 2002

Let G be a connected graph. A pebbling move on a graph G is the movement of taking two pebbles off from a vertex and placing one of them onto an adjacent vertex. The pebbling number f(G) of a connected graph G is the least n such that any distribution of n pebbles on the vertices of G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. In this paper, the pebbling numbers of the compositions of two graphs are computed.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.7-14
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• 2001

Let G be a connected graph. A pebbling move on a graph G is taking two pebbles off one vertex and placing one of them on an adjacent vertex. The pebbling number of a connected graph G, f(G), is the least n such that any distribution of n pebbles on the vertices of G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. In this paper, the pebbling numbers of the lexicographic products of some graphs are computed.

• Geomechanics and Engineering
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• v.4 no.4
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• pp.281-294
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• 2012

Filtration using granular media such as quarried sand, anthracite and granular activated carbon is a well-known technique used in both water and wastewater treatment. A relatively new pre-filtration method called pebble matrix filtration (PMF) technology has been proved effective in treating high turbidity water during heavy rain periods that occur in many parts of the world. Sand and pebbles are the principal filter media used in PMF laboratory and pilot field trials conducted in the UK, Papua New Guinea and Serbia. However during first full-scale trials at a water treatment plant in Sri Lanka in 2008, problems were encountered in sourcing the required uniform size and shape of pebbles due to cost, scarcity and Government regulations on pebble dredging. As an alternative to pebbles, hand-made clay pebbles (balls) were fired in a kiln and their performance evaluated for the sustainability of the PMF system. These clay balls within a filter bed are subjected to stresses due to self-weight and overburden, therefore, it is important that clay balls should be able to withstand these stresses in water saturated conditions. In this paper, experimentally determined physical properties including compression failure load (Uniaxial Compressive Strength) and tensile strength at failure (theoretical) of hand-made clay balls are described. Hand-made clay balls fired between the kiln temperatures of $875^{\circ}C$ to $960^{\circ}C$ gave failure loads of between 3.0 kN and 7.1 kN. In another test when clay balls were fired to $1250^{\circ}C$ the failure load was 35.0 kN compared to natural Scottish cobbles with an average failure load of 29.5 kN. The uniaxial compressive strength of clay balls obtained by experiment has been presented in terms of the tensile yield stress of clay balls. Based on the effective stress principle in soil mechanics, a method for the estimation of maximum theoretical load on clay balls used as filter media is proposed and compared with experimental failure loads.

• Economic and Environmental Geology
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• v.18 no.2
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• pp.177-184
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• 1985

The carbonate flat pebble conglomerates (CFPC) are interbedded as lenticular bed in the greenish rhythmite of the upper part of $Hwaj{\check{o}}l$ Formation, $Jos{\check{o}}n$ Supergroup. Pebbles are composed mainly of lime-mudstone with small amounts of bioclasts and silt-sized subangular quartz grains. The matrix among pebbles is composed mainly of sparry calcite with relatively much amounts of bioclasts, silt-sized subangular quartz grains and authigenic pyrite crystals or grains. The sparry calcite of the matrix seems to be the results of neomorphism of skeletal sands and bioclasts. The pebbles are well rounded and no plastic deformations are found. Some pebbles show the outer rim of glauconite. CFPC are not associated with any other intertidal features such as stromatolites, flaser bedding and channel structures. Also any features indicative of subaerial exposure such as dessication cracks, fenestrae and so on are not found in the bed. The sedimentological features of CFPC suggest that the following conditions appear to have been necessary for the formation of CFPC : 1) episodic deposition of thin, permeable calcareous beds separated argillaceous beds; 2) preservation of these beds near the sediment-water interface where they could become rapidly cemented; 3) erosion and redeposition of the partially lithified beds by storms or other exceptional erosional events. Eventually storm erosion and redeposition together represent only one of several critical conditions in the genesis of CFPC. The CFPC are very common in Cambrian and lower Ordovician formations, and become very rare in the younger carbonate formations. The expansion of infauna after Ordovician Period eliminated the widespread potential for rapid submarine cementation which is one of the critical factors to form CFPC.