A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex...A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder L_n,we determine the exact value of srvc(L_n) for n even. For n odd, upper and lower bounds of srvc(L_n) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.展开更多
A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken ove...A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.展开更多
I show how many connections of Γ?are presently existing from R?to β?as they are being inputted simultaneously through tensor products. I plan to address the Quantum state of this tensor connection st...I show how many connections of Γ?are presently existing from R?to β?as they are being inputted simultaneously through tensor products. I plan to address the Quantum state of this tensor connection step by step throughout the application presented. Also, I will show you how to prove that the connection is true for this tensor connection through its output method using a small bit of tensor calculus and mostly number theory.展开更多
Let E be a vector bundle over a compact Riemannian manifold M. We construct a natural metric on the bundle space E and discuss the relationship between the killing vector fields of E and M. Then we give a proof of the...Let E be a vector bundle over a compact Riemannian manifold M. We construct a natural metric on the bundle space E and discuss the relationship between the killing vector fields of E and M. Then we give a proof of the Bott-Baum-Cheeger Theorem for vector bundle E.展开更多
Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d k(G) for which between any two vertices in G there are at least k internally verte...Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d k(G) for which between any two vertices in G there are at least k internally vertex-disjoint paths of length at most d k(G). For a fixed positive integer d, some conditions to insure d k(G)≤d are given in this paper. In particular, if d≥3 and the sum of degrees of any s (s =2 or 3) nonadjacent vertices is at least n+(s-1)k+1-d, then d k(G)≤d. Furthermore, these conditions are sharp and the upper bound d of k-diameter is best possible.展开更多
The decay number of a connected graph is defined to be the minimum number of the components of the cotree of the graph. Upper bounds of the decay numbers of graphs are obtained according to their edge connectivities. ...The decay number of a connected graph is defined to be the minimum number of the components of the cotree of the graph. Upper bounds of the decay numbers of graphs are obtained according to their edge connectivities. All the bounds in this paper are tight.Moreover, for each integer k between one and the upper bound, there are infinitely many graphs with the decay number k.展开更多
An appropriate optimal number of market segments(ONS)estimation is essential for an enterprise to achieve successful market segmentation,but at present,there is a serious lack of attention to this issue in market segm...An appropriate optimal number of market segments(ONS)estimation is essential for an enterprise to achieve successful market segmentation,but at present,there is a serious lack of attention to this issue in market segmentation.In our study,an independent adaptive ONS estimation method BWCON-NSDK-means++is proposed by integrating a newinternal validity index(IVI)Between-Within-Connectivity(BWCON)and a newstable clustering algorithmNatural-SDK-means++(NSDK-means++)in a novel way.First,to complete the evaluation dimensions of the existing IVIs,we designed a connectivity formula based on the neighbor relationship and proposed the BWCON by integrating the connectivity with other two commonly considered measures of compactness and separation.Then,considering the stability,number of parameters and clustering performance,we proposed the NSDK-means++to participate in the integrationwhere the natural neighbor was used to optimize the initial cluster centers(ICCs)determination strategy in the SDK-means++.At last,to ensure the objectivity of the estimatedONS,we designed a BWCON-based ONS estimation framework that does not require the user to set any parameters in advance and integrated the NSDK-means++into this framework forming a practical ONS estimation tool BWCON-NSDK-means++.The final experimental results showthat the proposed BWCONand NSDK-means++are significantlymore suitable than their respective existing models to participate in the integration for determining theONS,and the proposed BWCON-NSDK-means++is demonstrably superior to the BWCON-KMA,BWCONMBK,BWCON-KM++,BWCON-RKM++,BWCON-SDKM++,BWCON-Single linkage,BWCON-Complete linkage,BWCON-Average linkage and BWCON-Ward linkage in terms of the ONS estimation.Moreover,as an independentmarket segmentation tool,the BWCON-NSDK-means++also outperforms the existing models with respect to the inter-market differentiation and sub-market size.展开更多
In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extende...In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.展开更多
The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature i...The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature in image analysis. In this paper, a very fast algorithm for the detection and localization of the objects and the computation of the Euler number of a binary image is proposed. The proposed algorithm operates in one scan of the image and is based on the Image Block Representation (IBR) scheme. The proposed algorithm is more efficient than conventional pixel based algorithms in terms of execution speed and representation of the extracted information.展开更多
All optical network based on wavelength division multiplexing transmission system with optical cross connect (OXC) is an essential approach for optical commumications.Crosstalk introduced by OXC (specially large scale...All optical network based on wavelength division multiplexing transmission system with optical cross connect (OXC) is an essential approach for optical commumications.Crosstalk introduced by OXC (specially large scale one) is a key limiting factor for its capacity. Optical signal passing through a typical OXC is analyzed in this paper, followed by description of the generation and effect of intraband crosstalk.The power penalties induced by intraband crosstalk versus the number of multiplexed wavelengths M and the number of input fibers N have been given by numerical simulations. The results show that the coherent crosstalk is the most critical limitation on OXC and depends more closely on the number of multiplexed wavelengths M than the number of input fibers N . Crosstalk is suppressed by doubly filtering, fixing optimum decision-threshold and appropriately choosing the number of multiplexed wavelengths M .展开更多
基金Supported by the National Natural Science Foundation of China(11551001,11061027,11261047,11161037,11461054)Supported by the Science Found of Qinghai Province(2016-ZJ-948Q,2014-ZJ-907)
文摘A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder L_n,we determine the exact value of srvc(L_n) for n even. For n odd, upper and lower bounds of srvc(L_n) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.
文摘A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.
文摘I show how many connections of Γ?are presently existing from R?to β?as they are being inputted simultaneously through tensor products. I plan to address the Quantum state of this tensor connection step by step throughout the application presented. Also, I will show you how to prove that the connection is true for this tensor connection through its output method using a small bit of tensor calculus and mostly number theory.
文摘Let E be a vector bundle over a compact Riemannian manifold M. We construct a natural metric on the bundle space E and discuss the relationship between the killing vector fields of E and M. Then we give a proof of the Bott-Baum-Cheeger Theorem for vector bundle E.
文摘Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d k(G) for which between any two vertices in G there are at least k internally vertex-disjoint paths of length at most d k(G). For a fixed positive integer d, some conditions to insure d k(G)≤d are given in this paper. In particular, if d≥3 and the sum of degrees of any s (s =2 or 3) nonadjacent vertices is at least n+(s-1)k+1-d, then d k(G)≤d. Furthermore, these conditions are sharp and the upper bound d of k-diameter is best possible.
基金Supported by the NSFC(10201022)Supported by the NSFCBJ(1012003)
文摘The decay number of a connected graph is defined to be the minimum number of the components of the cotree of the graph. Upper bounds of the decay numbers of graphs are obtained according to their edge connectivities. All the bounds in this paper are tight.Moreover, for each integer k between one and the upper bound, there are infinitely many graphs with the decay number k.
基金supported by the earmarked fund for CARS-29 and the open funds of the Key Laboratory of Viticulture and Enology,Ministry of Agriculture,China.
文摘An appropriate optimal number of market segments(ONS)estimation is essential for an enterprise to achieve successful market segmentation,but at present,there is a serious lack of attention to this issue in market segmentation.In our study,an independent adaptive ONS estimation method BWCON-NSDK-means++is proposed by integrating a newinternal validity index(IVI)Between-Within-Connectivity(BWCON)and a newstable clustering algorithmNatural-SDK-means++(NSDK-means++)in a novel way.First,to complete the evaluation dimensions of the existing IVIs,we designed a connectivity formula based on the neighbor relationship and proposed the BWCON by integrating the connectivity with other two commonly considered measures of compactness and separation.Then,considering the stability,number of parameters and clustering performance,we proposed the NSDK-means++to participate in the integrationwhere the natural neighbor was used to optimize the initial cluster centers(ICCs)determination strategy in the SDK-means++.At last,to ensure the objectivity of the estimatedONS,we designed a BWCON-based ONS estimation framework that does not require the user to set any parameters in advance and integrated the NSDK-means++into this framework forming a practical ONS estimation tool BWCON-NSDK-means++.The final experimental results showthat the proposed BWCONand NSDK-means++are significantlymore suitable than their respective existing models to participate in the integration for determining theONS,and the proposed BWCON-NSDK-means++is demonstrably superior to the BWCON-KMA,BWCONMBK,BWCON-KM++,BWCON-RKM++,BWCON-SDKM++,BWCON-Single linkage,BWCON-Complete linkage,BWCON-Average linkage and BWCON-Ward linkage in terms of the ONS estimation.Moreover,as an independentmarket segmentation tool,the BWCON-NSDK-means++also outperforms the existing models with respect to the inter-market differentiation and sub-market size.
基金supported in part by the National Natural Science Foundation of China(Grant No.12061033)by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region(Grants No.NJZY20119)by the Natural Science Foundation of Inner Mongolia(Grant No.2019MS01007),China.
文摘In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.
文摘The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature in image analysis. In this paper, a very fast algorithm for the detection and localization of the objects and the computation of the Euler number of a binary image is proposed. The proposed algorithm operates in one scan of the image and is based on the Image Block Representation (IBR) scheme. The proposed algorithm is more efficient than conventional pixel based algorithms in terms of execution speed and representation of the extracted information.
文摘All optical network based on wavelength division multiplexing transmission system with optical cross connect (OXC) is an essential approach for optical commumications.Crosstalk introduced by OXC (specially large scale one) is a key limiting factor for its capacity. Optical signal passing through a typical OXC is analyzed in this paper, followed by description of the generation and effect of intraband crosstalk.The power penalties induced by intraband crosstalk versus the number of multiplexed wavelengths M and the number of input fibers N have been given by numerical simulations. The results show that the coherent crosstalk is the most critical limitation on OXC and depends more closely on the number of multiplexed wavelengths M than the number of input fibers N . Crosstalk is suppressed by doubly filtering, fixing optimum decision-threshold and appropriately choosing the number of multiplexed wavelengths M .
