An LRHTS（v） （or LARHTS（v）） is a collection of {（X, Bi） ： 1≤i ≤ 4（v - 2）}, where X is a v-set, each （X, Bi） is a resolvable （or almost resolvable） HTS（v）, and all Bis form a partition of all cycle tr...An LRHTS（v） （or LARHTS（v）） is a collection of {（X, Bi） ： 1≤i ≤ 4（v - 2）}, where X is a v-set, each （X, Bi） is a resolvable （or almost resolvable） HTS（v）, and all Bis form a partition of all cycle triples and transitive triples on X. An OLRHTS（v） （or OLARHTS（v）） is a collection {（Y／{y}, Ay^j） ： y ∈ Y,j = 0, 1, 2, 3}, where Y is a （v ＋ 1）-set, each （Y／{y}, Ay^j） is a resolvable （or almost resolvable） HTS（v）, and all Ay^js form a partition of all cycle and transitive triples on Y. In this paper, we establish some directed and recursive constructions for LRHTS（v）, LARHTS（v）, OLRHTS（v）, OLARHTS（v） and give some new results.展开更多
A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） ...A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） （X,B） is called pure and denoted by PMTS（v） if （x, y, z） ∈ B implies （z, y, x） ∈B. A large set of MTS（v）s （LMTS（v）） is a collection of v - 2 pairwise disjoint MTS（v）s on a v-set. A self-converse large set of PMTS（v）s, denoted by LPMTS＊ （v）, is an LMTS（v） containing [ v-2/2] converse pairs of PMTS（v）s. In this paper, some results about the existence and non-existence for LPMTS＊ （v） are obtained.展开更多
There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have al...There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have already been solved. For the first three types of triangles, their large sets have already been solved, and their overlarge sets have been investigated. In this paper, we establish the spectrum of LTi（v,λ）, OLTi（v）（i = 1, 2）, and give the existence of LT3（v, λ） and OLT3（v, λ） with λ even.展开更多
An LPDTS(υ)is a collection of 3(υ-2)disjoint pure directed triple systems on the same set ofυelements.It is showed in Tian's doctoral thesis that there exists an LPDTS(υ)forυ≡0,4(mod 6),υ≥4.In this paper,w...An LPDTS(υ)is a collection of 3(υ-2)disjoint pure directed triple systems on the same set ofυelements.It is showed in Tian's doctoral thesis that there exists an LPDTS(υ)forυ≡0,4(mod 6),υ≥4.In this paper,we establish the existence of an LPDTS(υ)forυ≡1,3(mod 6),υ>3.Thus the spectrum for LPDTS(υ)is completely determined to be the set{υ:υ≡0,1(mod 3),υ≥4}.展开更多
In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a trip...In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.展开更多
A family(X,B1),(X,B2),...,(X,Bq)of q STS(v)s is aλ-fold large set of STS(v)and denoted by LSTSλ(v)if every 3-subset of X is contained in exactlyλSTS(v)s of the collection.It is indecomposable and denoted by IDLSTS...A family(X,B1),(X,B2),...,(X,Bq)of q STS(v)s is aλ-fold large set of STS(v)and denoted by LSTSλ(v)if every 3-subset of X is contained in exactlyλSTS(v)s of the collection.It is indecomposable and denoted by IDLSTSλ(v)if there exists no LSTSλ(v)contained in the collection for anyλ【λ.In 1995,Griggs and Rosa posed a problem:For which values ofλ】1 and orders v≡1,3(mod 6)do there exist IDLSTSλ(v)?In this paper,we use partitionable candelabra systems(PCSs)and holeyλ-fold large set of STS(v)(HLSTSλ(v))as auxiliary designs to establish a recursive construction for IDLSTSλ(v)and show that there exists an IDLSTSλ(v)forλ=2,3,4 and v≡1,3(mod 6).展开更多
A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to...A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.展开更多
A k-cycle system of order v with index A, denoted by CS（v, k, λ）, is a collection A of k-cycles （blocks） of Kv such that each edge in Kv appears in exactly λ blocks of A. A large set of CS（v, k, λ）s is a part...A k-cycle system of order v with index A, denoted by CS（v, k, λ）, is a collection A of k-cycles （blocks） of Kv such that each edge in Kv appears in exactly λ blocks of A. A large set of CS（v, k, λ）s is a partition of the set of all k-cycles of Kv into CS（v, k, λ）s, and is denoted by LCS（v, k, λ）. A （v - 1）-cycle in K, is called almost Hamilton. The completion of the existence problem for LCS（v, v- 1,λ） depends only on one case： all v ≥ 4 for λ=2. In this paper, it is shown that there exists an LCS（v,v - 1,2） for all v ≡ 2 （mod 4）, v ≥ 6.展开更多
A hybrid triple system of order v, briefly by HTS （v）, is a pair （X,/3） where X is a v-set and /3 is a collection of cyclic and transitive triples （called blocks） on X such that every ordered pair of X belongs t...A hybrid triple system of order v, briefly by HTS （v）, is a pair （X,/3） where X is a v-set and /3 is a collection of cyclic and transitive triples （called blocks） on X such that every ordered pair of X belongs to exactly one block of/3. An HTS （v） is called pure and denoted by PHTS （v） if one element of the block set {（x,y,z）, （z,y, ss）, （z,sc,y）, （y,x,z）, （y,z,x）, （x,z,y）, （x,y,z）, （z,y,x）} is contained in 13 then the others will not be contained in/3. A self-converse large set of disjoint PHTS （v）s, denoted by LPHTS＊（v）, is a collection of 4（v - 2） disjoint PHTS （v）s which contains exactly （v - 2）/2 converse octads of PHTS （v）s. In this paper, some results about the existence for LPHTS＊ （v） are obtained.展开更多
In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a...In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.展开更多
A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposab...A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposable and denoted by IDLSTSx（v） if there does not exist an LSTSx, （v） contained in the collection for any λ 〈 λ. In this paper, we show that for λ = 5, 6, there is an IDLSTSλ（v） for v ≡ 1 or 3 （rood 6） with the exception IDLSTS6（7）.展开更多
The Phlaythong large iron deposit in Shampasak of southern Laos,is located in the Kon Tum microblock （Fig.1A）,central-southern part of the Indo-China block,and the geographic coordinate of the central mining area is...The Phlaythong large iron deposit in Shampasak of southern Laos,is located in the Kon Tum microblock （Fig.1A）,central-southern part of the Indo-China block,and the geographic coordinate of the central mining area is 14°43′04″ N and 106°07′02″ E.展开更多
Many classical clustering algorithms do good jobs on their prerequisite but do not scale well when being applied to deal with very large data sets(VLDS).In this work,a novel division and partition clustering method(DP...Many classical clustering algorithms do good jobs on their prerequisite but do not scale well when being applied to deal with very large data sets(VLDS).In this work,a novel division and partition clustering method(DP) was proposed to solve the problem.DP cut the source data set into data blocks,and extracted the eigenvector for each data block to form the local feature set.The local feature set was used in the second round of the characteristics polymerization process for the source data to find the global eigenvector.Ultimately according to the global eigenvector,the data set was assigned by criterion of minimum distance.The experimental results show that it is more robust than the conventional clusterings.Characteristics of not sensitive to data dimensions,distribution and number of nature clustering make it have a wide range of applications in clustering VLDS.展开更多
In this paper,it is shown that for a minimal system(X,G),if H is a normal subgroup of G with finite index n,then X can be decomposed into n components of closed sets such that each component is minimal under H-action....In this paper,it is shown that for a minimal system(X,G),if H is a normal subgroup of G with finite index n,then X can be decomposed into n components of closed sets such that each component is minimal under H-action.Meanwhile,we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms,the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension,extending a previous result by Glasscock,Koutsogiannis and Richter.展开更多
Artificial intelligence(AI)is based on complex artificial neural networks,characterized by layered network architecture,parallel processing of large data sets and iterative algorithms for processing large data sets.AI...Artificial intelligence(AI)is based on complex artificial neural networks,characterized by layered network architecture,parallel processing of large data sets and iterative algorithms for processing large data sets.AI-assisted screening studies have demonstrated non-inferior diagnostic performance,reduced human workload by up to 70%,and reduced recall rates by 25% compared to human double reading.Natural language models promise high accuracy in advising on breast cancer prevention(80%),guiding tumor boards for personalized treatment decisions(50-70%),and planning oncoplastic or radiotherapy treatment for standard cases(72%),but AI sometimes produces errors and fails in complex cases.The main technical advantage of AI is that it can perform routine tasks faster and with fewer errors than humans.This is relevant for scheduling,summarizing reports,recording services for billing and quality assurance.The main concerns in healthcare are the quality of training data,the stability of AI systems,cybersecurity,liability and transparency.Currently,human experts still outperform AI in most areas.AI self-correcting algorithms and the alignment of AI constructed goals with human ethics are imperative to prevent patient harm.The ability of AI to uncover hidden patterns in multi-omics,immune regulation and tumor defense,as well as to develop new drugs,will advance the integrative fight against breast cancer.展开更多
In this article, we establish the existence of an LHMTS（mv） for v ≡ 2 （mod 6） and m≡ 3 （mod 6）. Thus there exists an LHMTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3） except possibly for v=6, m≡ 1, 5 （mo...In this article, we establish the existence of an LHMTS（mv） for v ≡ 2 （mod 6） and m≡ 3 （mod 6）. Thus there exists an LHMTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3） except possibly for v=6, m≡ 1, 5 （mod 6） and m≠1. In the similar way, the existence of LHDTS（mv） is completely determined, i.e., there exists an LHDTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3）.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11471096)
文摘An LRHTS（v） （or LARHTS（v）） is a collection of {（X, Bi） ： 1≤i ≤ 4（v - 2）}, where X is a v-set, each （X, Bi） is a resolvable （or almost resolvable） HTS（v）, and all Bis form a partition of all cycle triples and transitive triples on X. An OLRHTS（v） （or OLARHTS（v）） is a collection {（Y／{y}, Ay^j） ： y ∈ Y,j = 0, 1, 2, 3}, where Y is a （v ＋ 1）-set, each （Y／{y}, Ay^j） is a resolvable （or almost resolvable） HTS（v）, and all Ay^js form a partition of all cycle and transitive triples on Y. In this paper, we establish some directed and recursive constructions for LRHTS（v）, LARHTS（v）, OLRHTS（v）, OLARHTS（v） and give some new results.
基金Supported by National Natural Science Foundation of China (Grant No.10771051)
文摘A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） （X,B） is called pure and denoted by PMTS（v） if （x, y, z） ∈ B implies （z, y, x） ∈B. A large set of MTS（v）s （LMTS（v）） is a collection of v - 2 pairwise disjoint MTS（v）s on a v-set. A self-converse large set of PMTS（v）s, denoted by LPMTS＊ （v）, is an LMTS（v） containing [ v-2/2] converse pairs of PMTS（v）s. In this paper, some results about the existence and non-existence for LPMTS＊ （v） are obtained.
基金Supported by the National Natural Science Foundation of China(No.10371031) Doctor fund of Hebei Normal University
文摘There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have already been solved. For the first three types of triangles, their large sets have already been solved, and their overlarge sets have been investigated. In this paper, we establish the spectrum of LTi（v,λ）, OLTi（v）（i = 1, 2）, and give the existence of LT3（v, λ） and OLT3（v, λ） with λ even.
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10371002)the Tianyuan Mathematics Foundation of the National Natural Science Foundation of China(Grant No.10526032)the Natural Science Foundation of Universities of Jiangsu Province(Grant No.05KJB110111).
文摘An LPDTS(υ)is a collection of 3(υ-2)disjoint pure directed triple systems on the same set ofυelements.It is showed in Tian's doctoral thesis that there exists an LPDTS(υ)forυ≡0,4(mod 6),υ≥4.In this paper,we establish the existence of an LPDTS(υ)forυ≡1,3(mod 6),υ>3.Thus the spectrum for LPDTS(υ)is completely determined to be the set{υ:υ≡0,1(mod 3),υ≥4}.
文摘In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.
基金supported by National Natural Science Foundation of China(Grant Nos.10971051,10701060,10831002)Qing Lan Project of Jiangsu Province,China
文摘A family(X,B1),(X,B2),...,(X,Bq)of q STS(v)s is aλ-fold large set of STS(v)and denoted by LSTSλ(v)if every 3-subset of X is contained in exactlyλSTS(v)s of the collection.It is indecomposable and denoted by IDLSTSλ(v)if there exists no LSTSλ(v)contained in the collection for anyλ【λ.In 1995,Griggs and Rosa posed a problem:For which values ofλ】1 and orders v≡1,3(mod 6)do there exist IDLSTSλ(v)?In this paper,we use partitionable candelabra systems(PCSs)and holeyλ-fold large set of STS(v)(HLSTSλ(v))as auxiliary designs to establish a recursive construction for IDLSTSλ(v)and show that there exists an IDLSTSλ(v)forλ=2,3,4 and v≡1,3(mod 6).
基金Supported by National Natural Science Foundation of China (Grant Nos. 10771013 and 10831002)
文摘A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.
基金Supported in part by the National Natural Science Foundation of China(No.10901051,11201143)the Fundamental Research Funds for the Central Universities(No.2016MS66)the Co-construction Project of Bejing Municipal Commission of Education
文摘A k-cycle system of order v with index A, denoted by CS（v, k, λ）, is a collection A of k-cycles （blocks） of Kv such that each edge in Kv appears in exactly λ blocks of A. A large set of CS（v, k, λ）s is a partition of the set of all k-cycles of Kv into CS（v, k, λ）s, and is denoted by LCS（v, k, λ）. A （v - 1）-cycle in K, is called almost Hamilton. The completion of the existence problem for LCS（v, v- 1,λ） depends only on one case： all v ≥ 4 for λ=2. In this paper, it is shown that there exists an LCS（v,v - 1,2） for all v ≡ 2 （mod 4）, v ≥ 6.
基金Supported by Tianyuan Mathematics Foundation of National Natural Science Foundation of China(No.11126285)
文摘A hybrid triple system of order v, briefly by HTS （v）, is a pair （X,/3） where X is a v-set and /3 is a collection of cyclic and transitive triples （called blocks） on X such that every ordered pair of X belongs to exactly one block of/3. An HTS （v） is called pure and denoted by PHTS （v） if one element of the block set {（x,y,z）, （z,y, ss）, （z,sc,y）, （y,x,z）, （y,z,x）, （x,z,y）, （x,y,z）, （z,y,x）} is contained in 13 then the others will not be contained in/3. A self-converse large set of disjoint PHTS （v）s, denoted by LPHTS＊（v）, is a collection of 4（v - 2） disjoint PHTS （v）s which contains exactly （v - 2）/2 converse octads of PHTS （v）s. In this paper, some results about the existence for LPHTS＊ （v） are obtained.
基金the National Natural Science Foundation of China(No.10671055)Natural Science Foundation of Hebei(No.A2007000230)
文摘In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971051 and 11071056)
文摘A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposable and denoted by IDLSTSx（v） if there does not exist an LSTSx, （v） contained in the collection for any λ 〈 λ. In this paper, we show that for λ = 5, 6, there is an IDLSTSλ（v） for v ≡ 1 or 3 （rood 6） with the exception IDLSTS6（7）.
基金financially supported by the Special fund for Foreign Mineral Resources Risk Exploration (Grant No.Sichuan Financial Investment (2010)331)China Geological Survey (Grant No.12120114012501)
文摘The Phlaythong large iron deposit in Shampasak of southern Laos,is located in the Kon Tum microblock （Fig.1A）,central-southern part of the Indo-China block,and the geographic coordinate of the central mining area is 14°43′04″ N and 106°07′02″ E.
基金Projects(60903082,60975042)supported by the National Natural Science Foundation of ChinaProject(20070217043)supported by the Research Fund for the Doctoral Program of Higher Education of China
文摘Many classical clustering algorithms do good jobs on their prerequisite but do not scale well when being applied to deal with very large data sets(VLDS).In this work,a novel division and partition clustering method(DP) was proposed to solve the problem.DP cut the source data set into data blocks,and extracted the eigenvector for each data block to form the local feature set.The local feature set was used in the second round of the characteristics polymerization process for the source data to find the global eigenvector.Ultimately according to the global eigenvector,the data set was assigned by criterion of minimum distance.The experimental results show that it is more robust than the conventional clusterings.Characteristics of not sensitive to data dimensions,distribution and number of nature clustering make it have a wide range of applications in clustering VLDS.
文摘In this paper,it is shown that for a minimal system(X,G),if H is a normal subgroup of G with finite index n,then X can be decomposed into n components of closed sets such that each component is minimal under H-action.Meanwhile,we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms,the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension,extending a previous result by Glasscock,Koutsogiannis and Richter.
文摘Artificial intelligence(AI)is based on complex artificial neural networks,characterized by layered network architecture,parallel processing of large data sets and iterative algorithms for processing large data sets.AI-assisted screening studies have demonstrated non-inferior diagnostic performance,reduced human workload by up to 70%,and reduced recall rates by 25% compared to human double reading.Natural language models promise high accuracy in advising on breast cancer prevention(80%),guiding tumor boards for personalized treatment decisions(50-70%),and planning oncoplastic or radiotherapy treatment for standard cases(72%),but AI sometimes produces errors and fails in complex cases.The main technical advantage of AI is that it can perform routine tasks faster and with fewer errors than humans.This is relevant for scheduling,summarizing reports,recording services for billing and quality assurance.The main concerns in healthcare are the quality of training data,the stability of AI systems,cybersecurity,liability and transparency.Currently,human experts still outperform AI in most areas.AI self-correcting algorithms and the alignment of AI constructed goals with human ethics are imperative to prevent patient harm.The ability of AI to uncover hidden patterns in multi-omics,immune regulation and tumor defense,as well as to develop new drugs,will advance the integrative fight against breast cancer.
基金Supported by National Natural Science Foundation of China(Grant Nos.11471096 and 11771119)
文摘In this article, we establish the existence of an LHMTS（mv） for v ≡ 2 （mod 6） and m≡ 3 （mod 6）. Thus there exists an LHMTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3） except possibly for v=6, m≡ 1, 5 （mod 6） and m≠1. In the similar way, the existence of LHDTS（mv） is completely determined, i.e., there exists an LHDTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3）.
