Journal of Combinatorial Theory Series A最新文献

英文中文

On locally n × n grid graphs 在局部 n×n 网格图上

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-26 DOI: 10.1016/j.jcta.2024.105957

Carmen Amarra , Wei Jin , Cheryl E. Praeger

We investigate locally

n \times n

grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on n vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2n by previous work of Blokhuis and Brouwer. We show that if each pair is joined by at least

2 (n - 1)

such paths then the diameter is at most 3 and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally

n \times n

grid for odd prime powers n, and apply these results to locally

5 \times 5

grid graphs to obtain a classification for the case where either all μ-graphs (induced subgraphs on the set of common neighbours of two vertices at distance two) have order at least 8 or all μ-graphs have order c for some constant c.

我们研究的是局部 n×n 网格图，即任何顶点的邻域都是 n 个顶点上两个完整图的笛卡尔乘积的图。我们考虑的是这些图的子类，其中距离为 2 的每对顶点都有足够多的长度为 2 的路径相连。根据 Blokhuis 和 Brouwer 以前的研究，已知此类路径的数量最多为 2n。我们证明，如果每对图都至少有 2(n-1) 条这样的路径连接，那么直径最多为 3，并且我们给出了图的阶数的严格上限。我们证明，符合这一上限的图是完整图的距离规则反顶盖。我们展示了此类图的一个无穷族，它们在奇素数 n 的情况下是局部 n×n 网格图，并将这些结果应用于局部 5×5 网格图，从而获得了一种分类，即所有 μ 图（距离为 2 的两个顶点的公共相邻集合上的诱导子图）的阶数至少为 8，或者所有 μ 图的阶数为某个常数 c。

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引用次数: 0

On power monoids and their automorphisms 论幂单子及其自动形态

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-25 DOI: 10.1016/j.jcta.2024.105961

Salvatore Tringali, Weihao Yan

Endowed with the binary operation of set addition, the family

P_{fin, 0} (N)

of all finite subsets of

N

containing 0 forms a monoid, with the singleton {0} as its neutral element.

We show that the only non-trivial automorphism of

P_{fin, 0} (N)

is the involution

X \mapsto \max X - X

. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.

我们证明，Pfin,0(N)的唯一非难自变量是内卷X↦maxX-X。证明利用了加法数论的思想，通过对我们所说的有限整数集合的拳数维度（即其结合为集合本身的最小（离散）区间数）的非常规归纳来进行。

引用次数: 0

On non-empty cross-t-intersecting families 关于非空交叉相交族

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-24 DOI: 10.1016/j.jcta.2024.105960

Anshui Li , Huajun Zhang

Let

A_{1}, A_{2}, \dots, A_{m}

be families of k-element subsets of a n-element set. We call them cross-t-intersecting if

| A_{i} \cap A_{j} | \geq t

for any

A_{i} \in A_{i}

and

A_{j} \in A_{j}

with

i \neq j

. In this paper we will prove that, for

n \geq 2 k - t + 1

, if

A_{1}, A_{2}, \dots, A_{m}

are non-empty cross-t-intersecting families, then

\sum_{1 \leq i \leq m} | A_{i} | \leq \max {(\begin{matrix} n \\ k \end{matrix}) - \sum_{1 \leq i \leq t - 1} (\begin{matrix} k \\ i \end{matrix}) (\begin{matrix} n - k \\ k - i \end{matrix}) + m - 1, m M (n, k, t)},

where

M (n, k, t)

is the size of the maximum t-intersecting family of

(\begin{matrix} [n] \\ k \end{matrix})

. Moreover, the extremal families attaining the upper bound are characterized.

设 A1,A2,...,Am 是 n 元素集合的 k 元素子集族。对于任意 Ai∈Ai 和 Aj∈Aj 且 i≠j 的情况，如果|Ai∩Aj|≥t，我们称它们为交叉-t-交集。本文将证明，对于 n≥2k-t+1，如果 A1，A2，...，Am 是非空跨 t 交集族，则∑1≤i≤m|Ai|≤max{(nk)-∑1≤i≤t-1(ki)(n-kk-i)+m-1,mM(n,k,t)}，其中 M(n,k,t) 是 ([n]k) 的最大 t 交集族的大小。此外，还描述了达到上限的极值族的特征。

引用次数: 0

Avoiding intersections of given size in finite affine spaces AG(n,2) 在有限仿射空间 AG(n,2) 中避免给定大小的交集

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-24 DOI: 10.1016/j.jcta.2024.105959

Benedek Kovács , Zoltán Lóránt Nagy

We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size

m \in [0, 2^{n}]

of the n-dimensional binary affine space

AG (n, 2)

. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all m-element point sets in the n-dimensional affine space.

We also show constructions of point sets for which the intersection sizes with k-dimensional affine subspaces take values from a set of a small size compared to

2^{k}

. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.

Keywords: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.

我们研究了 k 维仿射子空间与 n 维二元仿射空间 AG(n,2) 大小为 m∈[0,2n] 的点集的交集大小集。按照厄尔多斯、富雷迪、罗斯柴尔德和 T. 索斯的主题，我们部分确定了 k 维仿射子空间中的哪些局部密度在 n 维仿射空间的所有 m 元素点集中是不可避免的。这些都是由仿射子空间和所谓的子空间规避集建立起来的。同时，我们改进了关于子空间逃避集的已知上界，并应用了关于数字的规范带符号数字（CSD）表示的结果。关键词：不可避免、仿射子空间、逃避集、随机方法、规范带符号数字系统。

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引用次数: 0

A rank two Leonard pair in Terwilliger algebras of Doob graphs Doob 图的特威里格代数中的二阶伦纳德对

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-23 DOI: 10.1016/j.jcta.2024.105958

John Vincent S. Morales

Let

Γ = Γ (n, m)

denote the Doob graph formed by the Cartesian product of the nth Cartesian power of the Shrikhande graph and the mth Cartesian power of the complete graph on four vertices. Let

T = T (x)

denote the Terwilliger algebra of Γ with respect to a fixed vertex x of Γ and let W denote an arbitrary non-thin irreducible T-module in the standard module of Γ. In (Morales and Palma, 2021 [25]), it was shown that there exists a Lie algebra embedding π from the special orthogonal algebra

{so}_{4}

into T and that W is an irreducible

π ({so}_{4})

-module. In this paper, we consider two Cartan subalgebras

h, \tilde{h}

{so}_{4}

such that

h, \tilde{h}

generate

{so}_{4}

. Using the embedding

π : {so}_{4} \to T

, we show that

π (h)

and

π (\tilde{h})

act on W as a rank two Leonard pair. We also obtain several direct sum decompositions of W akin to how split decompositions are obtained from Leonard pairs of rank one.

让Γ=Γ(n,m) 表示由四个顶点上的 Shrikhande 图的第 n 个笛卡尔幂和完整图的第 m 个笛卡尔幂的笛卡尔乘积形成的 Doob 图。让 T=T(x) 表示关于 Γ 的固定顶点 x 的 Γ 的特尔维利格代数，让 W 表示 Γ 的标准模块中的任意非薄不可还原 T 模块。莫拉莱斯和帕尔马，2021 [25]）中证明，存在一个从特殊正交代数 so4 到 T 的列代数嵌入 π，并且 W 是一个不可还原的 π(so4)- 模块。在本文中，我们考虑 so4 的两个 Cartan 子代数 h,h˜，使得 h,h˜ 产生 so4。利用嵌入π:so4→T，我们证明π(h)和π(h˜)作为秩二伦纳德对作用于 W。我们还得到了 W 的几个直接和分解，类似于从一阶伦纳德对得到分裂分解的方法。

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引用次数: 0

Covering the set of p-elements in finite groups by proper subgroups 用适当的子群覆盖有限群中 p 元素的集合

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-20 DOI: 10.1016/j.jcta.2024.105954

Attila Maróti , Juan Martínez , Alexander Moretó

Let p be a prime and let G be a finite group which is generated by the set $G_{p}$ of its p-elements. We show that if G is solvable and not a p-group, then the minimal number $σ_{p} (G)$ of proper subgroups of G whose union contains $G_{p}$ is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p-solvable groups G, we always have $σ_{p} (G) \geq p + 1$ . We study the case of alternating and symmetric groups G in detail.

设 p 是素数，G 是有限群，由其 p 元素集 Gp 生成。我们证明，如果 G 是可解而非 p 群，那么其联合包含 Gp 的 G 的适当子群的最小数目 σp(G) 等于比其联合是 G 的 G 的适当子群的最小数目少 1。对于 p 可解群 G，我们总是有 σp(G)≥p+1。我们将详细研究交替群和对称群 G 的情况。

引用次数: 0

Proofs of some conjectures of Merca on truncated series involving the Rogers-Ramanujan functions 梅尔卡关于涉及罗杰斯-拉马努扬函数的截断数列的一些猜想的证明

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-19 DOI: 10.1016/j.jcta.2024.105956

Yongqiang Chen, Olivia X.M. Yao

In 2012, Andrews and Merca investigated the truncated version of the Euler pentagonal number theorem. Their work has opened up a new study on truncated theta series and has inspired several mathematicians to work on the topic. In 2019, Merca studied the Rogers-Ramanujan functions and posed three groups of conjectures on truncated series involving the Rogers-Ramanujan functions. In this paper, we present a uniform method to prove the three groups of conjectures given by Merca based on a result due to Pólya and Szegö.

2012 年，安德鲁斯和梅尔卡研究了欧拉五边形数定理的截断版本。他们的研究开启了截断θ级数的新研究，并激发了多位数学家对这一课题的研究。2019 年，Merca 研究了 Rogers-Ramanujan 函数，并就涉及 Rogers-Ramanujan 函数的截断数列提出了三组猜想。在本文中，我们根据 Pólya 和 Szegö 的一个结果，提出了证明 Merca 提出的三组猜想的统一方法。

引用次数: 0

On the proportion of metric matroids whose Jacobians have nontrivial p-torsion 关于雅各布有非三角 p 扭转的公因子矩阵的比例

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-16 DOI: 10.1016/j.jcta.2024.105953

Sergio Ricardo Zapata Ceballos

We study the proportion of metric matroids whose Jacobians have nontrivial p-torsion. We establish a correspondence between these Jacobians and the $F_{p}$ -rational points on configuration hypersurfaces, thereby relating their proportions. By counting points over finite fields, we prove that the proportion of these Jacobians is asymptotically equivalent to $1 / p$ .

我们研究了雅各布具有非难 p 扭转的度量矩阵的比例。我们在这些雅各布与配置超曲面上的 Fp 有理点之间建立了对应关系，从而将它们的比例联系起来。通过计算有限域上的点，我们证明这些雅各布的比例在渐近上等同于 1/p。

引用次数: 0

Approximate generalized Steiner systems and near-optimal constant weight codes 近似广义斯泰纳系统和近优恒定权重码

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-11 DOI: 10.1016/j.jcta.2024.105955

Miao Liu , Chong Shangguan

Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for all fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds.

Let $A_{q} (n, d, w)$ denote the maximum size of q-ary CWCs of length n with constant weight w and minimum distance d. One of our main results shows that for all fixed $q, w$ and odd d, one has $\lim_{n \to \infty} \frac{A_{q} (n, d, w)}{(\begin{matrix} n \\ t \end{matrix})} = \frac{{(q - 1)}^{t}}{(\begin{matrix} w \\ t \end{matrix})}$ , where $t = \frac{2 w - d + 1}{2}$ . This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion, and can be viewed as a counterpart of a celebrated result of Rödl on the existence of near-optimal Steiner systems. Note that prior to our work, very little is known about $A_{q} (n, d, w)$ for $q \geq 3$ . A similar result is proved for the maximum size of CCCs.

We provide different proofs for our two main results, based on two strengthenings of the well-known Frankl-Rödl-Pippenger theorem on the existence of near-optimal matchings in hypergraphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcourt-Postle, and Glock-Joos-Kim-Kühn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations.

We also present several intriguing open questions for future research.

恒重码（CWC）和恒组成码（CCC）是组合学和编码理论近六十年来广泛研究的两类重要编码。本文证明，对于所有固定奇数距离，存在近似达到经典约翰逊型上界的近优 CWC 和 CCC。让 Aq(n,d,w) 表示长度为 n、权重为 w、距离为 d 的 q-ary CWCs 的最大尺寸。我们的一个主要结果表明，对于所有固定的 q、w 和奇数 d，都有 limn→∞Aq(n,d,w)(nt)=(q-1)t(wt)，其中 t=2w-d+12。这意味着最初由埃齐昂提出的近优广义斯坦纳系统的存在，可以看作是罗德尔关于近优斯坦纳系统存在的著名结果的对应物。请注意，在我们的研究之前，人们对 q≥3 时的 Aq(n,d,w) 知之甚少。我们基于著名的弗兰克尔-罗德尔-皮彭格（Frankl-Rödl-Pippenger）超图中近优匹配存在性定理的两个加强版，为我们的两个主要结果提供了不同的证明：第一个证明基于卡恩（Kahn）对上述定理的线性规划变式，第二个证明基于德尔库特-波斯特尔（Delcourt-Postle）和格洛克-朱斯-金-金-利切夫（Glock-Joos-Kim-Kühn-Lichev）最近关于避免某些禁止配置的近优匹配存在性的独立工作。我们还为未来研究提出了几个引人入胜的开放性问题。

{"title":"Approximate generalized Steiner systems and near-optimal constant weight codes","authors":"Miao Liu , Chong Shangguan","doi":"10.1016/j.jcta.2024.105955","DOIUrl":"10.1016/j.jcta.2024.105955","url":null,"abstract":"<div>Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for all fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds.Let <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>w</mi><mo>)</mo></math> denote the maximum size of q-ary CWCs of length n with constant weight w and minimum distance d. One of our main results shows that for all fixed <math><mi>q</mi><mo>,</mo><mi>w</mi></math> and odd d, one has <math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>t</mi></mrow></msup></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>w</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow></mfrac></math>, where <math><mi>t</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>w</mi><mo>−</mo><mi>d</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math>. This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion, and can be viewed as a counterpart of a celebrated result of Rödl on the existence of near-optimal Steiner systems. Note that prior to our work, very little is known about <math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>w</mi><mo>)</mo></math> for <math><mi>q</mi><mo>≥</mo><mn>3</mn></math>. A similar result is proved for the maximum size of CCCs.We provide different proofs for our two main results, based on two strengthenings of the well-known Frankl-Rödl-Pippenger theorem on the existence of near-optimal matchings in hypergraphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcourt-Postle, and Glock-Joos-Kim-Kühn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations.We also present several intriguing open questions for future research.</div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105955"},"PeriodicalIF":0.9,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000943/pdfft?md5=65eb96db9a426be78f5105ffe48c2ece&pid=1-s2.0-S0097316524000943-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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A note on tournament m-semiregular representations of finite groups 关于有限群 m-semiregular 代表锦标赛的说明

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-09-04 DOI: 10.1016/j.jcta.2024.105952

Jia-Li Du

For a positive integer m, a group G is said to admit a tournament m-semiregular representation (TmSR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to G and acts semiregularly on the vertex set of Γ with m orbits. It is easy to see that every finite group of even order does not admit a TmSR for any positive integer m. The T1SR is the well-known tournament regular representation (TRR for short). In 1970s, Babai and Imrich proved that every finite group of odd order admits a TRR except for $Z_{3}^{2}$ , and every group (finite or infinite) without element of order 2 having an independent generating set admits a T2SR in (1979) [3]. Later, Godsil correct the result by showing that the only finite groups of odd order without a TRR are $Z_{3}^{2}$ and $Z_{3}^{3}$ by a probabilistic approach in (1986) [11]. In this note, it is shown that every finite group of odd order has a TmSR for every $m \geq 2$ .

对于正整数 m，如果存在一个锦标赛 Γ，使得 Γ 的自变群与 G 同构，并以 m 个轨道半规则地作用于 Γ 的顶点集，则称群 G 接受锦标赛 m 半规则表示（简称 TmSR）。不难看出，对于任意正整数 m，每个偶数阶有限群都不存在 TmSR。20 世纪 70 年代，Babai 和 Imrich 在（1979）[3] 中证明了除了 Z32 之外，每个奇阶有限群都有一个 TRR，而每个无 2 阶元素且有独立生成集的群（有限或无限）都有一个 T2SR。后来，Godsil 在 (1986) [11] 中用概率方法证明了唯一没有 TRR 的奇阶有限群是 Z32 和 Z33，从而纠正了这一结果。在本注中，我们证明了每一个奇阶有限群对于每一个 m≥2 都有一个 TmSR。

{"title":"A note on tournament m-semiregular representations of finite groups","authors":"Jia-Li Du","doi":"10.1016/j.jcta.2024.105952","DOIUrl":"10.1016/j.jcta.2024.105952","url":null,"abstract":"<div>For a positive integer m, a group G is said to admit a tournament m-semiregular representation (TmSR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to G and acts semiregularly on the vertex set of Γ with m orbits. It is easy to see that every finite group of even order does not admit a TmSR for any positive integer m. The T1SR is the well-known tournament regular representation (TRR for short). In 1970s, Babai and Imrich proved that every finite group of odd order admits a TRR except for <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math>, and every group (finite or infinite) without element of order 2 having an independent generating set admits a T2SR in (1979) [3]. Later, Godsil correct the result by showing that the only finite groups of odd order without a TRR are <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math> and <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></msubsup></math> by a probabilistic approach in (1986) [11]. In this note, it is shown that every finite group of odd order has a TmSR for every <math><mi>m</mi><mo>≥</mo><mn>2</mn></math>.</div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105952"},"PeriodicalIF":0.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000918/pdfft?md5=9f9703a561ce567e377942546fcc91e2&pid=1-s2.0-S0097316524000918-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142137277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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