It is well known that the Two-step Weighted Least-Squares(TWLS) is a widely used method for source localization and sensor position refinement. For this reason, we propose a unified framework of the TWLS method for jo...It is well known that the Two-step Weighted Least-Squares(TWLS) is a widely used method for source localization and sensor position refinement. For this reason, we propose a unified framework of the TWLS method for joint estimation of multiple disjoint sources and sensor locations in this paper. Unlike some existing works, the presented method is based on more general measurement model, and therefore it can be applied to many different localization scenarios.Besides, it does not have the initialization and local convergence problem. The closed-form expression for the covariance matrix of the proposed TWLS estimator is also derived by exploiting the first-order perturbation analysis. Moreover, the estimation accuracy of the TWLS method is shown analytically to achieve the Cramér-Rao Bound(CRB) before the threshold effect takes place. The theoretical analysis is also performed in a common mathematical framework, rather than aiming at some specific signal metrics. Finally, two numerical experiments are performed to support the theoretical development in this paper.展开更多
Single-station passive localization technology avoids the complex time synchronization and information exchange between multiple observatories, and is increasingly important in electronic warfare. Based on a single mo...Single-station passive localization technology avoids the complex time synchronization and information exchange between multiple observatories, and is increasingly important in electronic warfare. Based on a single moving station localization system, a new method with high localization precision and numerical stability is proposed when the measurements from multiple disjoint sources are subject to the same station position and velocity displacement. According to the available measurements including the angle-of-arrival(AOA), time-of-arrival(TOA), and frequency-of-arrival(FOA), the corresponding pseudo linear equations are deduced. Based on this, a structural total least squares(STLS) optimization model is developed and the inverse iteration algorithm is used to obtain the stationary target location. The localization performance of the STLS localization algorithm is derived, and it is strictly proved that the theoretical performance of the STLS method is consistent with that of the constrained total least squares method under first-order error analysis, both of which can achieve the Cramér-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.展开更多
The existence of several non-symmetric balanced incomplete block designs (BIBDs) is still unknown. This is because the non-existence property for non-symmetric BIBDs is still not known and also the existing constructi...The existence of several non-symmetric balanced incomplete block designs (BIBDs) is still unknown. This is because the non-existence property for non-symmetric BIBDs is still not known and also the existing construction methods have not been able to construct these designs despite their design parameters satisfying the necessary conditions for the existence of BIBD. The study aimed to bridge this gap by introducing a new class of non-symmetric BIBDs. The proposed class of BIBDs is constructed through the combination of disjoint symmetric BIBDs. The study was able to determine that the total number of disjoint symmetric BIBDs (n) with parameters (v = b, r = k, λ) that can be obtained from an un-reduced BIBD with parameters (v, k) is given by n = r - λ. When the n symmetric disjoint BIBDs are combined, then a new class of symmetric BIBDs is formed with parameters v<sup>*</sup><sup> </sup>= v, b<sup>*</sup><sup> </sup>= nb, r<sup>*</sup><sup> </sup>= nr, k<sup>*</sup><sup> </sup>= k, λ<sup>*</sup><sup> </sup>= λ, where 2≤ n ≤ r - λ. The study established that the non-existence property of this class of BIBD was that when is not a perfect square then v should be even and when v<sup>*</sup><sup> </sup>is odd then the equation should not have a solution in integers x, y, z which are not all simultaneously zero. In conclusion, the study showed that this construction technique can be used to construct some non-symmetric BIBDs. However, one must first construct the disjoint symmetric BIBDs before one can construct the non-symmetric BIBD. Thus, the disjoint symmetric BIBDs must exist first before the non-symmetric BIBDs exist.展开更多
This research developed a hybrid position-channel network (named PCNet) through incorporating newly designed channel and position attention modules into U-Net to alleviate the crack discontinuity problem in channel an...This research developed a hybrid position-channel network (named PCNet) through incorporating newly designed channel and position attention modules into U-Net to alleviate the crack discontinuity problem in channel and spatial dimensions. In PCNet, the U-Net is used as a baseline to extract informative spatial and channel-wise features from shield tunnel lining crack images. A channel and a position attention module are designed and embedded after each convolution layer of U-Net to model the feature interdependencies in channel and spatial dimensions. These attention modules can make the U-Net adaptively integrate local crack features with their global dependencies. Experiments were conducted utilizing the dataset based on the images from Shanghai metro shield tunnels. The results validate the effectiveness of the designed channel and position attention modules, since they can individually increase balanced accuracy (BA) by 11.25% and 12.95%, intersection over union (IoU) by 10.79% and 11.83%, and F1 score by 9.96% and 10.63%, respectively. In comparison with the state-of-the-art models (i.e. LinkNet, PSPNet, U-Net, PANet, and Mask R–CNN) on the testing dataset, the proposed PCNet outperforms others with an improvement of BA, IoU, and F1 score owing to the implementation of the channel and position attention modules. These evaluation metrics indicate that the proposed PCNet presents refined crack segmentation with improved performance and is a practicable approach to segment shield tunnel lining cracks in field practice.展开更多
Direct soil temperature(ST)measurement is time-consuming and costly;thus,the use of simple and cost-effective machine learning(ML)tools is helpful.In this study,ML approaches,including KStar,instance-based K-nearest l...Direct soil temperature(ST)measurement is time-consuming and costly;thus,the use of simple and cost-effective machine learning(ML)tools is helpful.In this study,ML approaches,including KStar,instance-based K-nearest learning(IBK),and locally weighted learning(LWL),coupled with resampling algorithms of bagging(BA)and dagging(DA)(BA-IBK,BA-KStar,BA-LWL,DA-IBK,DA-KStar,and DA-LWL)were developed and tested for multi-step ahead(3,6,and 9 d ahead)ST forecasting.In addition,a linear regression(LR)model was used as a benchmark to evaluate the results.A dataset was established,with daily ST time-series at 5 and 50 cm soil depths in a farmland as models’output and meteorological data as models’input,including mean(T_(mean)),minimum(Tmin),and maximum(T_(max))air temperatures,evaporation(Eva),sunshine hours(SSH),and solar radiation(SR),which were collected at Isfahan Synoptic Station(Iran)for 13 years(1992–2005).Six different input combination scenarios were selected based on Pearson’s correlation coefficients between inputs and outputs and fed into the models.We used 70%of the data to train the models,with the remaining 30%used for model evaluation via multiple visual and quantitative metrics.Our?ndings showed that T_(mean)was the most effective input variable for ST forecasting in most of the developed models,while in some cases the combinations of variables,including T_(mean)and T_(max)and T_(mean),T_(max),Tmin,Eva,and SSH proved to be the best input combinations.Among the evaluated models,BA-KStar showed greater compatibility,while in most cases,BA-IBK and-LWL provided more accurate results,depending on soil depth.For the 5 cm soil depth,BA-KStar had superior performance(i.e.,Nash-Sutcliffe efficiency(NSE)=0.90,0.87,and 0.85 for 3,6,and 9 d ahead forecasting,respectively);for the 50 cm soil depth,DA-KStar outperformed the other models(i.e.,NSE=0.88,0.89,and 0.89 for 3,6,and 9 d ahead forecasting,respectively).The results con?rmed that all hybrid models had higher prediction capabilities than the LR model.展开更多
This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of ...This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examines the existence of such labelings for disjoint union of plane graphs.展开更多
The algorithm is based on constructing a disjoin kg t set of the minimal paths in a network system.In this paper, cubic notation was used to describe the logic function of a network in a well-balanced state,and then t...The algorithm is based on constructing a disjoin kg t set of the minimal paths in a network system.In this paper, cubic notation was used to describe the logic function of a network in a well-balanced state,and then the sharp-product operation was used to construct the disjoint minimal path set of the network.A computer program has been developed,and when combined with decomposition technology,the reliability of a general lifeline network can be effectively and automatically calculated.展开更多
A consecutive k out of n: F System, with n linearly arranged i. i. d. components, is examined. This paper gives an exact formula for computing the system's reliability directly. Complexity of this algorithm is...A consecutive k out of n: F System, with n linearly arranged i. i. d. components, is examined. This paper gives an exact formula for computing the system's reliability directly. Complexity of this algorithm is an O(n k). Further more, the system's reliability can be computed more quickly and more effectively from this formula than those previously given.展开更多
In Shadow-Fading model, it is difficult to achieve higher Packet Delivery Ratio (PDR) due to the effects of large scale fading. The main goal of this paper is to improve the PDR in Shadow-Fading environment. To achiev...In Shadow-Fading model, it is difficult to achieve higher Packet Delivery Ratio (PDR) due to the effects of large scale fading. The main goal of this paper is to improve the PDR in Shadow-Fading environment. To achieve this objective a protocol has been proposed that discovers backup routes for active sessions. These backup routes salvage the packets of active session in case of active route failure. It is found by research that proactively maintaining backup routes for ac-tive sessions can noticeably improve the PDR in Shadow-Fading environment. This protocol has been implemented with a mechanism of having backup routes and simulations have been conducted by using both node disjoint paths and link disjoint paths. Comparisons have been made between new protocol as well as AODV protocol. Simulation has been carried out using Network Simulator 2 (NS2) and the results show that the proposed QoSAR protocol exhibits higher PDR than AODV protocol in Shadow-fading environment.展开更多
In this article, we present several equivalent conditions ensuring the disjoint supercyclicity of finite weighted pseudo-shifts acting on an arbitrary Banach sequence space.The disjoint supercyclic properties of weigh...In this article, we present several equivalent conditions ensuring the disjoint supercyclicity of finite weighted pseudo-shifts acting on an arbitrary Banach sequence space.The disjoint supercyclic properties of weighted translations on locally compact discrete groups,the direct sums of finite classical weighted backward shifts, and the bilateral backward operator weighted shifts can be viewed as special cases of our main results. Furthermore, we exhibit an interesting fact that any finite bilateral weighted backward shifts on the space ?~2(Z) never satisfy the d-Supercyclicity Criterion by a simple proof.展开更多
The seismic reliability evaluation of lifeline networks has received considerable attention and been widely studied. In this paper, on the basis of an original recursive decomposition algorithm, an improved analytical...The seismic reliability evaluation of lifeline networks has received considerable attention and been widely studied. In this paper, on the basis of an original recursive decomposition algorithm, an improved analytical approach to evaluate the seismic reliability of large lifeline systems is presented. The proposed algorithm takes the shortest path from the source to the sink of a network as decomposition policy. Using the Boolean laws of set operation and the probabilistic operation principal, a recursive decomposition process is constructed in which the disjoint minimal path set and the disjoint minimal cut set are simultaneously enumerated. As the result, a probabilistic inequality can be used to provide results that satisfy a prescribed error bound. During the decomposition process, different from the original recursive decomposition algorithm which only removes edges to simplify the network, the proposed algorithm simplifies the network by merging nodes into sources and removing edges. As a result, the proposed algorithm can obtain simpler networks. Moreover, for a network owning s-independent components in its component set, two network reduction techniques are introduced to speed up the proposed algorithm. A series of case studies, including an actual water distribution network and a large urban gas system, are calculated using the proposed algorithm. The results indicate that the proposed algorithm provides a useful probabilistic analysis method for the seismic reliability evaluation of lifeline networks.展开更多
Let denote the maximum number of disjoint bases in a matroid . For a connected graph , let , where is the cycle matroid of . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs...Let denote the maximum number of disjoint bases in a matroid . For a connected graph , let , where is the cycle matroid of . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid with , elements with the property that have been characterized in terms of matroid invariants such as strength and -partitions. In this paper, we consider matroids with , and determine the minimum of , where is a matroid that contains as a restriction with both and . This minimum is expressed as a function of certain invariants of , as well as a min-max formula. These are applied to imply former results of Haas [3] and of Liu et al. [4].展开更多
Let {E<sub>i</sub>:i∈I}be a family of Archimedean Riesz spaces.The Riesz product space is denoted by Π<sub>i∈I</sub> E<sub>i</sub>.The main result in this paper is the following ...Let {E<sub>i</sub>:i∈I}be a family of Archimedean Riesz spaces.The Riesz product space is denoted by Π<sub>i∈I</sub> E<sub>i</sub>.The main result in this paper is the following conclusion:There exists a completely regular Hausdorff space X such that Π<sub>i∈I</sub> E<sub>i</sub> is Riesz isomorphic to C（X）if and only if for every i ∈ I there exists a completely regular Hausdorff space X<sub>i</sub> such that E<sub>i</sub> is Riesz isomorphic to C（X<sub>i</sub>）.展开更多
The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a gracef...The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a graceful graph for m=j-1 or m≥n＋j,where C_（4n） is a cycle with 4n vertexes,P_m is a path with m＋1 vertexes,and（jC_（4n））∪P_m denotes the disjoint union of j-C_（4n） and P_m.展开更多
基金co-supported by the National Natural Science Foundation of China (Nos. 61201381, 61401513 and 61772548)the China Postdoctoral Science Foundation (No. 2016M592989)+1 种基金the Self-Topic Foundation of Information Engineering University, China (No. 2016600701)the Outstanding Youth Foundation of Information Engineering University, China (No. 2016603201)
文摘It is well known that the Two-step Weighted Least-Squares(TWLS) is a widely used method for source localization and sensor position refinement. For this reason, we propose a unified framework of the TWLS method for joint estimation of multiple disjoint sources and sensor locations in this paper. Unlike some existing works, the presented method is based on more general measurement model, and therefore it can be applied to many different localization scenarios.Besides, it does not have the initialization and local convergence problem. The closed-form expression for the covariance matrix of the proposed TWLS estimator is also derived by exploiting the first-order perturbation analysis. Moreover, the estimation accuracy of the TWLS method is shown analytically to achieve the Cramér-Rao Bound(CRB) before the threshold effect takes place. The theoretical analysis is also performed in a common mathematical framework, rather than aiming at some specific signal metrics. Finally, two numerical experiments are performed to support the theoretical development in this paper.
基金Project supported by the National Natural Science Foundation of China(Nos.61201381,61401513,and 61772548)the China Postdoctoral Science Foundation(No.2016M592989)+1 种基金the Self-Topic Foundation of Information Engineering University,China(No.2016600701)the Outstanding Youth Foundation of Information Engineering University,China(No.2016603201)
文摘Single-station passive localization technology avoids the complex time synchronization and information exchange between multiple observatories, and is increasingly important in electronic warfare. Based on a single moving station localization system, a new method with high localization precision and numerical stability is proposed when the measurements from multiple disjoint sources are subject to the same station position and velocity displacement. According to the available measurements including the angle-of-arrival(AOA), time-of-arrival(TOA), and frequency-of-arrival(FOA), the corresponding pseudo linear equations are deduced. Based on this, a structural total least squares(STLS) optimization model is developed and the inverse iteration algorithm is used to obtain the stationary target location. The localization performance of the STLS localization algorithm is derived, and it is strictly proved that the theoretical performance of the STLS method is consistent with that of the constrained total least squares method under first-order error analysis, both of which can achieve the Cramér-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.
文摘The existence of several non-symmetric balanced incomplete block designs (BIBDs) is still unknown. This is because the non-existence property for non-symmetric BIBDs is still not known and also the existing construction methods have not been able to construct these designs despite their design parameters satisfying the necessary conditions for the existence of BIBD. The study aimed to bridge this gap by introducing a new class of non-symmetric BIBDs. The proposed class of BIBDs is constructed through the combination of disjoint symmetric BIBDs. The study was able to determine that the total number of disjoint symmetric BIBDs (n) with parameters (v = b, r = k, λ) that can be obtained from an un-reduced BIBD with parameters (v, k) is given by n = r - λ. When the n symmetric disjoint BIBDs are combined, then a new class of symmetric BIBDs is formed with parameters v<sup>*</sup><sup> </sup>= v, b<sup>*</sup><sup> </sup>= nb, r<sup>*</sup><sup> </sup>= nr, k<sup>*</sup><sup> </sup>= k, λ<sup>*</sup><sup> </sup>= λ, where 2≤ n ≤ r - λ. The study established that the non-existence property of this class of BIBD was that when is not a perfect square then v should be even and when v<sup>*</sup><sup> </sup>is odd then the equation should not have a solution in integers x, y, z which are not all simultaneously zero. In conclusion, the study showed that this construction technique can be used to construct some non-symmetric BIBDs. However, one must first construct the disjoint symmetric BIBDs before one can construct the non-symmetric BIBD. Thus, the disjoint symmetric BIBDs must exist first before the non-symmetric BIBDs exist.
基金support from the Ministry of Science and Tech-nology of the:People's Republic of China(Grant No.2021 YFB2600804)the Open Research Project Programme of the State Key Labor atory of Interet of Things for Smart City(University of Macao)(Grant No.SKL-IoTSC(UM)-2021-2023/ORPF/A19/2022)the General Research Fund(GRF)project(Grant No.15214722)from Research Grants Council(RGC)of Hong Kong Special Administrative Re gion Government of China are gratefully acknowledged.
文摘This research developed a hybrid position-channel network (named PCNet) through incorporating newly designed channel and position attention modules into U-Net to alleviate the crack discontinuity problem in channel and spatial dimensions. In PCNet, the U-Net is used as a baseline to extract informative spatial and channel-wise features from shield tunnel lining crack images. A channel and a position attention module are designed and embedded after each convolution layer of U-Net to model the feature interdependencies in channel and spatial dimensions. These attention modules can make the U-Net adaptively integrate local crack features with their global dependencies. Experiments were conducted utilizing the dataset based on the images from Shanghai metro shield tunnels. The results validate the effectiveness of the designed channel and position attention modules, since they can individually increase balanced accuracy (BA) by 11.25% and 12.95%, intersection over union (IoU) by 10.79% and 11.83%, and F1 score by 9.96% and 10.63%, respectively. In comparison with the state-of-the-art models (i.e. LinkNet, PSPNet, U-Net, PANet, and Mask R–CNN) on the testing dataset, the proposed PCNet outperforms others with an improvement of BA, IoU, and F1 score owing to the implementation of the channel and position attention modules. These evaluation metrics indicate that the proposed PCNet presents refined crack segmentation with improved performance and is a practicable approach to segment shield tunnel lining cracks in field practice.
文摘Direct soil temperature(ST)measurement is time-consuming and costly;thus,the use of simple and cost-effective machine learning(ML)tools is helpful.In this study,ML approaches,including KStar,instance-based K-nearest learning(IBK),and locally weighted learning(LWL),coupled with resampling algorithms of bagging(BA)and dagging(DA)(BA-IBK,BA-KStar,BA-LWL,DA-IBK,DA-KStar,and DA-LWL)were developed and tested for multi-step ahead(3,6,and 9 d ahead)ST forecasting.In addition,a linear regression(LR)model was used as a benchmark to evaluate the results.A dataset was established,with daily ST time-series at 5 and 50 cm soil depths in a farmland as models’output and meteorological data as models’input,including mean(T_(mean)),minimum(Tmin),and maximum(T_(max))air temperatures,evaporation(Eva),sunshine hours(SSH),and solar radiation(SR),which were collected at Isfahan Synoptic Station(Iran)for 13 years(1992–2005).Six different input combination scenarios were selected based on Pearson’s correlation coefficients between inputs and outputs and fed into the models.We used 70%of the data to train the models,with the remaining 30%used for model evaluation via multiple visual and quantitative metrics.Our?ndings showed that T_(mean)was the most effective input variable for ST forecasting in most of the developed models,while in some cases the combinations of variables,including T_(mean)and T_(max)and T_(mean),T_(max),Tmin,Eva,and SSH proved to be the best input combinations.Among the evaluated models,BA-KStar showed greater compatibility,while in most cases,BA-IBK and-LWL provided more accurate results,depending on soil depth.For the 5 cm soil depth,BA-KStar had superior performance(i.e.,Nash-Sutcliffe efficiency(NSE)=0.90,0.87,and 0.85 for 3,6,and 9 d ahead forecasting,respectively);for the 50 cm soil depth,DA-KStar outperformed the other models(i.e.,NSE=0.88,0.89,and 0.89 for 3,6,and 9 d ahead forecasting,respectively).The results con?rmed that all hybrid models had higher prediction capabilities than the LR model.
文摘This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examines the existence of such labelings for disjoint union of plane graphs.
基金Key Project of Science and Technology from the State Plan Committee.No.101-9914003
文摘The algorithm is based on constructing a disjoin kg t set of the minimal paths in a network system.In this paper, cubic notation was used to describe the logic function of a network in a well-balanced state,and then the sharp-product operation was used to construct the disjoint minimal path set of the network.A computer program has been developed,and when combined with decomposition technology,the reliability of a general lifeline network can be effectively and automatically calculated.
文摘A consecutive k out of n: F System, with n linearly arranged i. i. d. components, is examined. This paper gives an exact formula for computing the system's reliability directly. Complexity of this algorithm is an O(n k). Further more, the system's reliability can be computed more quickly and more effectively from this formula than those previously given.
文摘In Shadow-Fading model, it is difficult to achieve higher Packet Delivery Ratio (PDR) due to the effects of large scale fading. The main goal of this paper is to improve the PDR in Shadow-Fading environment. To achieve this objective a protocol has been proposed that discovers backup routes for active sessions. These backup routes salvage the packets of active session in case of active route failure. It is found by research that proactively maintaining backup routes for ac-tive sessions can noticeably improve the PDR in Shadow-Fading environment. This protocol has been implemented with a mechanism of having backup routes and simulations have been conducted by using both node disjoint paths and link disjoint paths. Comparisons have been made between new protocol as well as AODV protocol. Simulation has been carried out using Network Simulator 2 (NS2) and the results show that the proposed QoSAR protocol exhibits higher PDR than AODV protocol in Shadow-fading environment.
基金supported by the Research Project of Tianjin Municipal Education Commission(2017KJ124)
文摘In this article, we present several equivalent conditions ensuring the disjoint supercyclicity of finite weighted pseudo-shifts acting on an arbitrary Banach sequence space.The disjoint supercyclic properties of weighted translations on locally compact discrete groups,the direct sums of finite classical weighted backward shifts, and the bilateral backward operator weighted shifts can be viewed as special cases of our main results. Furthermore, we exhibit an interesting fact that any finite bilateral weighted backward shifts on the space ?~2(Z) never satisfy the d-Supercyclicity Criterion by a simple proof.
基金Natural Science Funds for the Innovative Research Group of China Under Grant No.50621062
文摘The seismic reliability evaluation of lifeline networks has received considerable attention and been widely studied. In this paper, on the basis of an original recursive decomposition algorithm, an improved analytical approach to evaluate the seismic reliability of large lifeline systems is presented. The proposed algorithm takes the shortest path from the source to the sink of a network as decomposition policy. Using the Boolean laws of set operation and the probabilistic operation principal, a recursive decomposition process is constructed in which the disjoint minimal path set and the disjoint minimal cut set are simultaneously enumerated. As the result, a probabilistic inequality can be used to provide results that satisfy a prescribed error bound. During the decomposition process, different from the original recursive decomposition algorithm which only removes edges to simplify the network, the proposed algorithm simplifies the network by merging nodes into sources and removing edges. As a result, the proposed algorithm can obtain simpler networks. Moreover, for a network owning s-independent components in its component set, two network reduction techniques are introduced to speed up the proposed algorithm. A series of case studies, including an actual water distribution network and a large urban gas system, are calculated using the proposed algorithm. The results indicate that the proposed algorithm provides a useful probabilistic analysis method for the seismic reliability evaluation of lifeline networks.
文摘Let denote the maximum number of disjoint bases in a matroid . For a connected graph , let , where is the cycle matroid of . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid with , elements with the property that have been characterized in terms of matroid invariants such as strength and -partitions. In this paper, we consider matroids with , and determine the minimum of , where is a matroid that contains as a restriction with both and . This minimum is expressed as a function of certain invariants of , as well as a min-max formula. These are applied to imply former results of Haas [3] and of Liu et al. [4].
基金Supported by the National Natural Science Foundation of China
文摘Let {E<sub>i</sub>:i∈I}be a family of Archimedean Riesz spaces.The Riesz product space is denoted by Π<sub>i∈I</sub> E<sub>i</sub>.The main result in this paper is the following conclusion:There exists a completely regular Hausdorff space X such that Π<sub>i∈I</sub> E<sub>i</sub> is Riesz isomorphic to C（X）if and only if for every i ∈ I there exists a completely regular Hausdorff space X<sub>i</sub> such that E<sub>i</sub> is Riesz isomorphic to C（X<sub>i</sub>）.
文摘The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a graceful graph for m=j-1 or m≥n＋j,where C_（4n） is a cycle with 4n vertexes,P_m is a path with m＋1 vertexes,and（jC_（4n））∪P_m denotes the disjoint union of j-C_（4n） and P_m.
