Abstract We consider the sequence {ξn,t}t≥1 of controlled critical branching processes with immigration, where n = 1, 2, … is an integer parameter limiting the population size. It is shown that for n → ∞ the stationary distributions of considered branching processes normalized by n $begin{array}{} sqrt{n} end{array} $ converge to the distribution of a random variable whose square has a gamma distribution.
摘要 我们考虑了有移民的受控临界分支过程序列{ξn,t}t≥1，其中n = 1, 2, ...是限制种群规模的整数参数。研究表明，对于 n → ∞，所考虑的分支过程的静态分布以 n $begin{array}{} 归一化。sqrt{n}end{array} $ 收敛到其平方具有伽马分布的随机变量的分布。
{"title":"Limit theorem for stationary distribution of a critical controlled branching process with immigration","authors":"Vladimir I. Vinokurov","doi":"10.1515/dma-2023-0030","DOIUrl":"https://doi.org/10.1515/dma-2023-0030","url":null,"abstract":"Abstract We consider the sequence {ξn,t}t≥1 of controlled critical branching processes with immigration, where n = 1, 2, … is an integer parameter limiting the population size. It is shown that for n → ∞ the stationary distributions of considered branching processes normalized by n $begin{array}{} sqrt{n} end{array} $ converge to the distribution of a random variable whose square has a gamma distribution.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"6 1","pages":"325 - 337"},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139328386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the problem of testing the hypothesis that the tested sequence is a sequence of independent random variables that take values 1 and –1 with equal probability. To solve this problem, the Discrete Fourier Transform (spectral) test of the NIST package uses the statistic TFourier, the exact limiting distribution of which is unknown. In this paper a new statistic is proposed and its limiting distribution is established. This new statistic is a slight modification of TFourier. A hypothesis about the limit distribution of TFourier is formulated, which is confirmed by numerical experiments presented by Pareschi F., Rovatti R. and Setti G.
摘要 我们考虑的问题是测试这样一个假设，即被测序列是一个独立随机变量序列，其取值 1 和 -1 的概率相等。为了解决这个问题，NIST 软件包中的离散傅立叶变换（频谱）检验使用了统计量 TFourier，但其确切的极限分布是未知的。本文提出了一种新的统计量，并确定了其极限分布。这一新统计量是对 TFourier 统计量的轻微修改。Pareschi F., Rovatti R. 和 Setti G. 提出的数值实验证实了这一假设。
{"title":"Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence","authors":"M. P. Savelov","doi":"10.1515/dma-2023-0029","DOIUrl":"https://doi.org/10.1515/dma-2023-0029","url":null,"abstract":"Abstract We consider the problem of testing the hypothesis that the tested sequence is a sequence of independent random variables that take values 1 and –1 with equal probability. To solve this problem, the Discrete Fourier Transform (spectral) test of the NIST package uses the statistic TFourier, the exact limiting distribution of which is unknown. In this paper a new statistic is proposed and its limiting distribution is established. This new statistic is a slight modification of TFourier. A hypothesis about the limit distribution of TFourier is formulated, which is confirmed by numerical experiments presented by Pareschi F., Rovatti R. and Setti G.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"7 1","pages":"317 - 323"},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139330293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The class of nonabelian 2-groups H with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group H, and put forward criteria of their bijectivity. For the generalized group of quaternions of order 2m, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.
摘要 索引为 2 的循环子群的非阿贝尔 2 群 H 类包括二面群、广义四元组、半二面群和模态最大循环群，它们在离散数学和密码学中有着广泛的应用。我们介绍了群 H 上的片断四元数变换，并提出了它们的双射性标准。对于阶数为 2m 的广义四元数群，我们在所考虑的片断-夸西亚芬变换类中获得了正交变换、完全变换及其左类似物的完整分类。我们还评估了它们的心数。
{"title":"Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions","authors":"B. Pogorelov, M. Pudovkina","doi":"10.1515/dma-2023-0028","DOIUrl":"https://doi.org/10.1515/dma-2023-0028","url":null,"abstract":"Abstract The class of nonabelian 2-groups H with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group H, and put forward criteria of their bijectivity. For the generalized group of quaternions of order 2m, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"5 1","pages":"299 - 316"},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139330860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function |x|; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [x]. $begin{array}{} displaystyle [sqrt{x}]. end{array}$ For the set of such recursive sequences, an algorithmically unsolvable problem is indicated.
{"title":"On polynomial-modular recursive sequences","authors":"S. Marchenkov","doi":"10.1515/dma-2023-0027","DOIUrl":"https://doi.org/10.1515/dma-2023-0027","url":null,"abstract":"Abstract We consider recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function |x|; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [x]. $begin{array}{} displaystyle [sqrt{x}]. end{array}$ For the set of such recursive sequences, an algorithmically unsolvable problem is indicated.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"25 1","pages":"293 - 298"},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139328931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For a nonempty set ω of primes, V. A. Vedernikov had constructed ω-fibred formations of groups via function methods. We study lattice properties of ω-fibred formations of finite groups with direction δ satisfying the condition δ0 ≤ δ. The lattice ωδFθ of all ω-fibred formations with direction δ and θ-valued ω-satellite is shown to be algebraic under the condition that the lattice of formations θ is algebraic. As a corollary, the lattices ωδF, ωδFτ, τωδF, ωδnF of ω-fibred formations of groups are shown to be algebraic.
摘要 对于素数的非空集ω，V. A. Vedernikov 通过函数方法构造了群ω-纤维形式。我们研究了满足δ0 ≤ δ条件的有限群方向为δ的ω纤维网格的网格性质。在网格的网格θ是代数的条件下，所有方向为δ的ω纤维网格和θ值ω卫星的网格ωδFθ被证明是代数的。作为一个推论，ωδF，ωδFτ，τωδF，ωδnF 的ω-纤维形式群的网格被证明是代数的。
{"title":"On algebraicity of lattices of ω-fibred formations of finite groups","authors":"Serafim P. Maksakov, M. Sorokina","doi":"10.1515/dma-2023-0026","DOIUrl":"https://doi.org/10.1515/dma-2023-0026","url":null,"abstract":"Abstract For a nonempty set ω of primes, V. A. Vedernikov had constructed ω-fibred formations of groups via function methods. We study lattice properties of ω-fibred formations of finite groups with direction δ satisfying the condition δ0 ≤ δ. The lattice ωδFθ of all ω-fibred formations with direction δ and θ-valued ω-satellite is shown to be algebraic under the condition that the lattice of formations θ is algebraic. As a corollary, the lattices ωδF, ωδFτ, τωδF, ωδnF of ω-fibred formations of groups are shown to be algebraic.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"67 1","pages":"283 - 291"},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139330317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let Γ be a diameter 3 distance-regular graph with a strongly regular graph Γ3, where Γ3 is the graph whose vertex set coincides with the vertex set of the graph Γ and two vertices are adjacent whenever they are at distance 3 in the graph Γ. Computing the parameters of Γ3 by the intersection array of the graph Γ is considered as the direct problem. Recovering the intersection array of the graph Γ by the parameters of Γ3 is referred to as the inverse problem. The inverse problem for Γ3 has been solved earlier by A. A. Makhnev and M. S. Nirova. In the case where Γ3 is a pseudo-geometric graph of a net, a series of admissible intersection arrays has been obtained: {c2(u2 − m2) + 2c2m − c2 − 1, c2(u2 − m2), (c2 − 1)(u2 − m2) + 2c2m − c2; 1, c2, u2 − m2} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov). The cases c2 = 1 and c2 = 2 have been examined by A. A. Makhnev, M. P. Golubyatnikov and A. A. Makhnev, M. S. Nirova, respectively. In this paper in the class of graphs with the intersection arrays {mn − 1, (m − 1)(n + 1)}, {n − m + 1}; 1, 1, (m − 1)(n + 1)} all admissible intersection arrays for {3 ≤ m ≤ 13} are found: {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} and {350,336,15; 1, 1,336}. It is demonstrated that graphs with the intersection arrays {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} and {90,84,7; 1, 1,84} do not exist.
摘要 假设Γ是一个直径为 3 的距离正则图，它有一个强正则图Γ3，其中Γ3 是顶点集与图Γ的顶点集重合的图，并且只要两个顶点在图Γ中的距离为 3，它们就是相邻的。通过图 Γ 的交点阵列计算 Γ3 的参数被视为直接问题。通过 Γ3 的参数恢复图 Γ 的交点阵列称为逆问题。A. A. Makhnev 和 M. S. Nirova 早先已经解决了 Γ3 的逆问题。在 Γ3 是一个网的伪几何图形的情况下，得到了一系列可接受的交点阵列：{c2(u2 - m2) + 2c2m - c2 - 1, c2(u2 - m2), (c2 - 1)(u2 - m2) + 2c2m - c2; 1, c2, u2 - m2} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov)。A. A. Makhnev, M. P. Golubyatnikov 和 A. A. Makhnev, M. S. Nirova 分别研究了 c2 = 1 和 c2 = 2 的情况。在本文中，在具有交集阵列 {mn - 1, (m - 1)(n + 1)}, {n - m + 1}; 1, 1, (m - 1)(n + 1)} 的一类图形中，发现了 {3 ≤ m ≤ 13} 的所有可容许交集阵列：{20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} 和 {350,336,15; 1, 1,336}.事实证明，不存在交集数组为 {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} 和 {90,84,7; 1, 1,84} 的图形。
{"title":"On small distance-regular graphs with the intersection arrays {mn − 1, (m − 1)(n + 1), n − m + 1; 1, 1, (m − 1)(n + 1)}","authors":"A. Makhnev, M. P. Golubyatnikov","doi":"10.1515/dma-2023-0025","DOIUrl":"https://doi.org/10.1515/dma-2023-0025","url":null,"abstract":"Abstract Let Γ be a diameter 3 distance-regular graph with a strongly regular graph Γ3, where Γ3 is the graph whose vertex set coincides with the vertex set of the graph Γ and two vertices are adjacent whenever they are at distance 3 in the graph Γ. Computing the parameters of Γ3 by the intersection array of the graph Γ is considered as the direct problem. Recovering the intersection array of the graph Γ by the parameters of Γ3 is referred to as the inverse problem. The inverse problem for Γ3 has been solved earlier by A. A. Makhnev and M. S. Nirova. In the case where Γ3 is a pseudo-geometric graph of a net, a series of admissible intersection arrays has been obtained: {c2(u2 − m2) + 2c2m − c2 − 1, c2(u2 − m2), (c2 − 1)(u2 − m2) + 2c2m − c2; 1, c2, u2 − m2} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov). The cases c2 = 1 and c2 = 2 have been examined by A. A. Makhnev, M. P. Golubyatnikov and A. A. Makhnev, M. S. Nirova, respectively. In this paper in the class of graphs with the intersection arrays {mn − 1, (m − 1)(n + 1)}, {n − m + 1}; 1, 1, (m − 1)(n + 1)} all admissible intersection arrays for {3 ≤ m ≤ 13} are found: {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} and {350,336,15; 1, 1,336}. It is demonstrated that graphs with the intersection arrays {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} and {90,84,7; 1, 1,84} do not exist.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"218 1","pages":"273 - 281"},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139331533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity.
{"title":"Nonlinearity of functions over finite fields","authors":"V. G. Ryabov","doi":"10.1515/dma-2023-0021","DOIUrl":"https://doi.org/10.1515/dma-2023-0021","url":null,"abstract":"Abstract The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"231 - 246"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48978494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We prove that any Boolean function in n variables can be modeled by a testable Boolean circuit with two additional inputs in the basis “conjunction, oblique conjunction, disjunction, negation” so that the circuit admits a complete diagnostic test of the length at most 2n + 3 with respect to stuck-at faults of the type 1 at gate outputs.
{"title":"Short complete diagnostic tests for circuits with two additional inputs in some basis","authors":"K. A. Popkov","doi":"10.1515/dma-2023-0020","DOIUrl":"https://doi.org/10.1515/dma-2023-0020","url":null,"abstract":"Abstract We prove that any Boolean function in n variables can be modeled by a testable Boolean circuit with two additional inputs in the basis “conjunction, oblique conjunction, disjunction, negation” so that the circuit admits a complete diagnostic test of the length at most 2n + 3 with respect to stuck-at faults of the type 1 at gate outputs.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"219 - 230"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41403232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider predicates on a finite set that are invariant with respect to an affine operation fG, where G is some Abelian group. Such predicates are said to be multiaffine for the group G. Special attention is paid to predicates that are affine for a group Gq of addition modulo q=ps, where p is a prime number and s=1. We establish the predicate multiaffinity criterion for a group Gq. Then we introduce disjunctive normal forms (DNF) for predicates on a finite set and obtain properties of DNFs of predicates that are multiaffine for a group Gq. Finally we show how these properties can be used to design a polynomial algorithm that decides satisfiability of a system of predicates which are multiaffine for a group Gq, if predicates are specified by DNF.
{"title":"On properties of multiaffine predicates on a finite set","authors":"S. Selezneva","doi":"10.1515/dma-2023-0023","DOIUrl":"https://doi.org/10.1515/dma-2023-0023","url":null,"abstract":"Abstract We consider predicates on a finite set that are invariant with respect to an affine operation fG, where G is some Abelian group. Such predicates are said to be multiaffine for the group G. Special attention is paid to predicates that are affine for a group Gq of addition modulo q=ps, where p is a prime number and s=1. We establish the predicate multiaffinity criterion for a group Gq. Then we introduce disjunctive normal forms (DNF) for predicates on a finite set and obtain properties of DNFs of predicates that are multiaffine for a group Gq. Finally we show how these properties can be used to design a polynomial algorithm that decides satisfiability of a system of predicates which are multiaffine for a group Gq, if predicates are specified by DNF.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"259 - 267"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47853471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For sequences of independent random variables having a Bernoulli distribution the joint distribution and the limit joint distribution of statistics of three tests of the NIST statistical package («Monobit Test», «Frequency Test within a Block», and «Cumulative Sums Test») are obtained. In the case when two blocks are used in «Frequency Test within a Block», pairwise covariance of these statistics is given.
{"title":"The limit joint distributions of statistics of three tests of the NIST package","authors":"Maksim P. Savelov","doi":"10.1515/dma-2023-0022","DOIUrl":"https://doi.org/10.1515/dma-2023-0022","url":null,"abstract":"Abstract For sequences of independent random variables having a Bernoulli distribution the joint distribution and the limit joint distribution of statistics of three tests of the NIST statistical package («Monobit Test», «Frequency Test within a Block», and «Cumulative Sums Test») are obtained. In the case when two blocks are used in «Frequency Test within a Block», pairwise covariance of these statistics is given.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"247 - 257"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43618843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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