It is well known that a totally disconnected compact metric space without isolated points is a Cantor set.In this note me give a simple proof of this theorem.
This paper discusses the total irredundance relations between the graph G and its clone-contraction graph H, that is, let H be the clone-contraction graph of G and v1,v2,...,vk be all contraction vertices ofH. IfS is ...This paper discusses the total irredundance relations between the graph G and its clone-contraction graph H, that is, let H be the clone-contraction graph of G and v1,v2,...,vk be all contraction vertices ofH. IfS is a maximal total irredundant set of H such that A = S ∩ {V1,V2,…,Vk} contains as few vertices as possible, then S＇= S-A is the maximal total irredundant set of G. Furthermore, we obtain the bound of the total irredundance A（G） number： irt ≤△（G）/2△（G）＋1 n, which n is the order of graph G, and △（G） is maximum degree in G.展开更多
This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. In a previous paper of the authors, a complete result on the multiple blow-up rates was obtain...This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. In a previous paper of the authors, a complete result on the multiple blow-up rates was obtained. In the present paper, we continue to consider the blow-up sets to the system via a complete classification for the nonlinear parameters. That is the discussion on single point versus total blow-up of the solutions. It is mentioned that due to the influence of the localized sources, there is some substantial difficulty to be overcomed there to deal with the single point blow-up of the solutions.展开更多
Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.
This paper deals with the Gevreg-hypoellipticity for a class of totally characteristic operators with the elliptic condition and the discrete boundary spectrum condition respectively.
The question associated with total domination on the queen’s graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that...The question associated with total domination on the queen’s graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that every square of an n × n board is attacked? Beginning in 2005 with Amirabadi, Burchett, and Hedetniemi [2] [3], work on this problem, and two other related problems, has seen progress. Bounds have been given for the values of all three domination parameters on the queen’s graph. In this paper, formations of queens are given that provide new bounds for the values of total, paired, and connected domination on the queen’s graph, denoted , , and respectively. For any n × n board size, the new bound of is arrived at, along with the separate bounds of , for with , and , for with .展开更多
We have introduced the total domination polynomial for any simple non isolated graph G in [7] and is defined by Dt（G, x） = ∑in=yt（G） dr（G, i） x＇, where dr（G, i） is the cardinality of total dominating sets of...We have introduced the total domination polynomial for any simple non isolated graph G in [7] and is defined by Dt（G, x） = ∑in=yt（G） dr（G, i） x＇, where dr（G, i） is the cardinality of total dominating sets of G of size i, and yt（G） is the total domination number of G. In [7] We have obtained some properties of Dt（G, x） and its coefficients. Also, we have calculated the total domination polynomials of complete graph, complete bipartite graph, join of two graphs and a graph consisting of disjoint components. In this paper, we presented for any two isomorphic graphs the total domination polynomials are same, but the converse is not true. Also, we proved that for any n vertex transitive graph of order n and for any v ∈ V（G）, dt（G, i） = 7 dt（V）（G, i）, 1 〈 i 〈 n. And, for any k-regular graph of order n, dr（G, i） = （7）, i 〉 n-k and d,（G, n-k） = （kn） - n. We have calculated the total domination polynomial of Petersen graph D,（P, x） = 10X4 ＋ 72x5 ＋ 140x6 ＋ 110x7 ＋ 45x8 ＋ [ 0x9 ＋ x10. Also, for any two vertices u and v of a k-regular graph Hwith N（u） ≠ N（v） and if Dr（G, x） = Dt（ H, x ）, then G is also a k-regular graph.展开更多
In intensity modulated radiation treatment (IMRT) planning, the use of asymmetrically collimated fields that are placed on central axis or its off-set is mostly required. Output is the main topic discussed today for e...In intensity modulated radiation treatment (IMRT) planning, the use of asymmetrically collimated fields that are placed on central axis or its off-set is mostly required. Output is the main topic discussed today for extremely small and/or severe irregularly shaped fields. The air scatter data are involved directly or indirectly in obtaining the output. Despite the fact that extensive data have been published in many studies to provide a guide on the magnitude of output factor for clinical accelerators, there are very few data reviewed about output factor in-air or phantom for off-set fields. This study was aimed to investigate the impact of these conditions for small fields. This study was conducted in Elekta Synergy linear accelerator which produces 6 MV X-ray energy. The in-air output factor (Sc) has been measured by CC04 ion chamber with brass-alloy “build-up” cap and Dose-1 electrometer, and the total output (Scp) measurements were carried on at dose maximum depth in phantom by the same chamber and Thermoluminescence dosimeter (TLD) for 1 - 10 cm2 fields. The all measurements at center of isocenter and off-set fields at three directions (X2, Y1, Diagonal) were done. By decreasing field size from 10 to 2 cm2 at isocenter, the Sc value using CC04 was decreased to 5.4% and Scp using CC04 and TLD to 14.5% and 11% respectively. By increasing off-set value, the Sc and Scp values were increased in all directions comparing to central fields. The maximum increase was obtained in Y1 direction for Sc and Scp. TLD results for Scp is slightly higher than CC04. The dosimetric properties of small fields and their off-set should be evaluated and modelled appropriately in the treatment planning system to ensure accurate dose calculation in Intensity Modulated Radiation Treatment.展开更多
文摘It is well known that a totally disconnected compact metric space without isolated points is a Cantor set.In this note me give a simple proof of this theorem.
基金Supported by the National Natural Science Foundation of China (10571071,10371048)
文摘This paper discusses the total irredundance relations between the graph G and its clone-contraction graph H, that is, let H be the clone-contraction graph of G and v1,v2,...,vk be all contraction vertices ofH. IfS is a maximal total irredundant set of H such that A = S ∩ {V1,V2,…,Vk} contains as few vertices as possible, then S＇= S-A is the maximal total irredundant set of G. Furthermore, we obtain the bound of the total irredundance A（G） number： irt ≤△（G）/2△（G）＋1 n, which n is the order of graph G, and △（G） is maximum degree in G.
基金China Postdoctoral Science Foundation(20110490409)Science Foundation(L2010146)of Liaoning Education Department
文摘This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. In a previous paper of the authors, a complete result on the multiple blow-up rates was obtained. In the present paper, we continue to consider the blow-up sets to the system via a complete classification for the nonlinear parameters. That is the discussion on single point versus total blow-up of the solutions. It is mentioned that due to the influence of the localized sources, there is some substantial difficulty to be overcomed there to deal with the single point blow-up of the solutions.
文摘Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.
基金Supported by National Natural Science Foundation of P.R.China
文摘This paper deals with the Gevreg-hypoellipticity for a class of totally characteristic operators with the elliptic condition and the discrete boundary spectrum condition respectively.
文摘The question associated with total domination on the queen’s graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that every square of an n × n board is attacked? Beginning in 2005 with Amirabadi, Burchett, and Hedetniemi [2] [3], work on this problem, and two other related problems, has seen progress. Bounds have been given for the values of all three domination parameters on the queen’s graph. In this paper, formations of queens are given that provide new bounds for the values of total, paired, and connected domination on the queen’s graph, denoted , , and respectively. For any n × n board size, the new bound of is arrived at, along with the separate bounds of , for with , and , for with .
文摘We have introduced the total domination polynomial for any simple non isolated graph G in [7] and is defined by Dt（G, x） = ∑in=yt（G） dr（G, i） x＇, where dr（G, i） is the cardinality of total dominating sets of G of size i, and yt（G） is the total domination number of G. In [7] We have obtained some properties of Dt（G, x） and its coefficients. Also, we have calculated the total domination polynomials of complete graph, complete bipartite graph, join of two graphs and a graph consisting of disjoint components. In this paper, we presented for any two isomorphic graphs the total domination polynomials are same, but the converse is not true. Also, we proved that for any n vertex transitive graph of order n and for any v ∈ V（G）, dt（G, i） = 7 dt（V）（G, i）, 1 〈 i 〈 n. And, for any k-regular graph of order n, dr（G, i） = （7）, i 〉 n-k and d,（G, n-k） = （kn） - n. We have calculated the total domination polynomial of Petersen graph D,（P, x） = 10X4 ＋ 72x5 ＋ 140x6 ＋ 110x7 ＋ 45x8 ＋ [ 0x9 ＋ x10. Also, for any two vertices u and v of a k-regular graph Hwith N（u） ≠ N（v） and if Dr（G, x） = Dt（ H, x ）, then G is also a k-regular graph.
文摘In intensity modulated radiation treatment (IMRT) planning, the use of asymmetrically collimated fields that are placed on central axis or its off-set is mostly required. Output is the main topic discussed today for extremely small and/or severe irregularly shaped fields. The air scatter data are involved directly or indirectly in obtaining the output. Despite the fact that extensive data have been published in many studies to provide a guide on the magnitude of output factor for clinical accelerators, there are very few data reviewed about output factor in-air or phantom for off-set fields. This study was aimed to investigate the impact of these conditions for small fields. This study was conducted in Elekta Synergy linear accelerator which produces 6 MV X-ray energy. The in-air output factor (Sc) has been measured by CC04 ion chamber with brass-alloy “build-up” cap and Dose-1 electrometer, and the total output (Scp) measurements were carried on at dose maximum depth in phantom by the same chamber and Thermoluminescence dosimeter (TLD) for 1 - 10 cm2 fields. The all measurements at center of isocenter and off-set fields at three directions (X2, Y1, Diagonal) were done. By decreasing field size from 10 to 2 cm2 at isocenter, the Sc value using CC04 was decreased to 5.4% and Scp using CC04 and TLD to 14.5% and 11% respectively. By increasing off-set value, the Sc and Scp values were increased in all directions comparing to central fields. The maximum increase was obtained in Y1 direction for Sc and Scp. TLD results for Scp is slightly higher than CC04. The dosimetric properties of small fields and their off-set should be evaluated and modelled appropriately in the treatment planning system to ensure accurate dose calculation in Intensity Modulated Radiation Treatment.
