{"title":"Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator","authors":"Sara Iman, Mohammad-Reza Jahed-Motlagh","doi":"10.1016/j.amc.2024.129068","DOIUrl":"10.1016/j.amc.2024.129068","url":null,"abstract":"<div><div>This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-cospectral equienergetic trees of diameter at most four","authors":"Fenjin Liu , Ke Su , Wei Wang , Hao Zhang","doi":"10.1016/j.amc.2024.129104","DOIUrl":"10.1016/j.amc.2024.129104","url":null,"abstract":"<div><div>No general method differing from computer search has been discovered for finding noncospectral equienergetic trees. We first utilize techniques of spectral graph theory and Diophantine Equation to analyze the energy of two families of short-diameter trees. Seven infinite families noncospectral equienergetic trees are obtained of which six pairs of them have equal number vertices. This contributes to an open problem posed by Li, Shi and Gutman for constructing noncospectral equienergetic trees.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Perfect matching cover indices of generalized rotation snarks","authors":"Wenjuan Zhou, Rong-Xia Hao, Yilun Luo","doi":"10.1016/j.amc.2024.129101","DOIUrl":"10.1016/j.amc.2024.129101","url":null,"abstract":"<div><div>Let Γ be a 3-regular bridgeless graph and <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> be the perfect matching cover index of Γ. It is conjectured by Berge that <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>5</mn></math></span>. Esperet and Mazzuoccolo (2013) <span><span>[4]</span></span> proved that deciding whether Γ satisfies <span><math><mi>τ</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>4</mn></math></span> is NP-complete. Máčajová and Škoviera (2021) <span><span>[13]</span></span> gave a family <span><math><mi>R</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><mi>k</mi><mo>≥</mo><mn>4</mn><mo>}</mo></math></span> of rotation snarks. We construct a family <span><math><mrow><mi>FR</mi></mrow><mo>=</mo><mo>{</mo><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span> of generalized rotation snarks. In this paper, we show that each <span><math><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> satisfies <span><math><mi>τ</mi><mo>(</mo><mi>F</mi><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn></math></span>. As a corollary, each <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> satisfies <span><math><mi>τ</mi><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Graphs with burning number three","authors":"Yinkui Li , Guiyu Shi , Xiaoxiao Qin","doi":"10.1016/j.amc.2024.129100","DOIUrl":"10.1016/j.amc.2024.129100","url":null,"abstract":"<div><div>Recently, Bonato et al. proposed the concept of the burning number of a graph to measure the speed of contagion spread on a network. They pointed out that the burning number of graph <em>G</em> is <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> and showed that the graph <em>G</em> with burning number 2 if and only if <em>G</em> has maximum degree <span><math><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>1</mn></math></span> or <span><math><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>2</mn></math></span>. Consider the problem of finding the maximum degree of a graph is solvable in polynomial time, graphs with burning number 2 can be recognized in polynomial time and thus the lower bound of burning number be improved to 3. In this paper, we characterize graphs with burning number 3 in terms of maximum degree and diameter.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Disjoint path covers of star graphs","authors":"Hongwei Qiao, Jixiang Meng","doi":"10.1016/j.amc.2024.129098","DOIUrl":"10.1016/j.amc.2024.129098","url":null,"abstract":"<div><div>Given a graph <em>G</em>, let <em>S</em> and <em>T</em> be two vertex-disjoint subsets of equal size <em>k</em> of <em>G</em>. A <em>k</em>-disjoint path cover of <em>G</em> corresponding to <em>S</em> and <em>T</em> is the union of <em>k</em> vertex-disjoint paths among <em>S</em> and <em>T</em> that spans <em>G</em>. If every vertex of <em>S</em> should be joined to a prescribed vertex in <em>T</em>, it is defined to be paired, otherwise it is unpaired. Let <span><math><mi>S</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be a star graph with bipartition <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. Let <span><math><mi>S</mi><mo>⊆</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><mi>T</mi><mo>⊆</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> be two vertex subsets of equal size <em>k</em>. It is shown in this paper that <span><math><mi>S</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> admits an unpaired <em>k</em>-disjoint path cover between <em>S</em> and <em>T</em>, where <span><math><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. In view of the degree of <span><math><mi>S</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, this result is optimal.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Junwei Wang , Weili Xiong , Feng Ding , Yihong Zhou , Erfu Yang
{"title":"Parameter estimation method for separable fractional-order Hammerstein nonlinear systems based on the on-line measurements","authors":"Junwei Wang , Weili Xiong , Feng Ding , Yihong Zhou , Erfu Yang","doi":"10.1016/j.amc.2024.129102","DOIUrl":"10.1016/j.amc.2024.129102","url":null,"abstract":"<div><div>This paper investigates the problem of parameter estimation for fractional-order Hammerstein nonlinear systems. To handle the identification difficulty of the parameters of the system and the order, the maximum likelihood and hierarchical identification principles are combined to derive a maximum likelihood gradient-based iterative algorithm. Moreover, to achieve the higher estimation accuracy, the multi-innovation identification theory is introduced, based on which the residual can be formulated as a linear combination of the innovation. Then, a multi-innovation maximum likelihood gradient-based iterative algorithm is proposed, which further improves the innovation utilization. Meanwhile, the computational cost of the proposed algorithm is assessed through the use of flops, which is less than those of its peers. Finally, the convergence analysis and simulation examples demonstrate the efficacy and robustness of the proposed algorithms.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Estimation of parameters and valuation of options written on multiple assets described by uncertain fractional differential equations","authors":"Yue Xin , Yi Zhang , Idin Noorani , Farshid Mehrdoust , Jinwu Gao","doi":"10.1016/j.amc.2024.129109","DOIUrl":"10.1016/j.amc.2024.129109","url":null,"abstract":"<div><div>This study suggests the pricing problems of options dependent on multiple assets, spread, basket, and quanto options when the asset dynamics are described by the uncertain fractional differential equation. The solutions of these option prices are analytically provided and the algorithms related to each one of these derivatives are designed. For the first time, we apply the minimum cover method to estimate the parameters of the uncertain fractional differential equations based on the real data related to the stock prices of some markets. Through the uncertain hypothesis test, we demonstrate that the estimated uncertain fractional differential equations can successfully fit the observed data. We then experimentally show that the <em>α</em>-paths obtained by the estimated uncertain fractional differential equations favorably cover the sample data. Finally, some numerical experiments based on the uncertain fractional differential equation estimated by the minimum cover method are accomplished to confirm the achievement of the presented results.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Theory of Hermite and Laguerre Bessel function from the umbral point of view","authors":"M. Artioli , G. Dattoli , U. Zainab","doi":"10.1016/j.amc.2024.129103","DOIUrl":"10.1016/j.amc.2024.129103","url":null,"abstract":"<div><div>The theoretical underpinnings of hybrid families of special functions are examined through an umbral reformulation. Our discussion encompasses diverse families of Bessel-type functions and special polynomials, all situated within a unifying umbral-algebraic formalism. The method presented capitalizes on recent advancements in the formal treatment of higher transcendental functions, enabling novel and intriguing generalizations.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Distance (signless) Laplacian spectra and energies of two classes of cyclic polyomino chains","authors":"Yonghong Zhang , Ligong Wang","doi":"10.1016/j.amc.2024.129099","DOIUrl":"10.1016/j.amc.2024.129099","url":null,"abstract":"<div><div>Let <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>T</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the distance matrix and the diagonal matrix of vertex transmissions of a graph <em>G</em>, respectively. The distance Laplacian matrix and the distance signless Laplacian matrix of <em>G</em> are defined as <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>L</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>T</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>Q</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>T</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, respectively. In this paper, we consider the distance Laplacian spectra and the distance signless Laplacian spectra of the linear cyclic polyomino chain <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the Möbius cyclic polyomino chain <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By utilizing the properties of circulant matrices, we give the characteristic polynomials and the eigenvalues for the distance Laplacian matrices and the distance signless Laplacian matrices of the graphs <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively. Furthermore, we provide the exactly values of the distance Laplacian energy and the distance signless Laplacian energy of the graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and the upper bounds on the distance Laplacian energy and the distance signless Laplacian energy of the graph <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Contagion probability in linear threshold model","authors":"Ying Ying Keng , Kiam Heong Kwa","doi":"10.1016/j.amc.2024.129090","DOIUrl":"10.1016/j.amc.2024.129090","url":null,"abstract":"<div><div>We study a linear threshold model on a simple undirected connected network <em>G</em> where each non-seed becomes active if and only if the proportion of its active neighbors exceeds its adoption threshold. Each threshold function <span><math><mi>ϕ</mi><mo>:</mo><mi>V</mi><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> is viewed as a point <span><math><mo>(</mo><mi>ϕ</mi><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>ϕ</mi><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> in the <em>n</em>-cube <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where <span><math><mi>V</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> is the set of nodes in <em>G</em>. We define <em>ϕ</em> as a contagious point of a subset <em>S</em> of nodes if it can induce full contagion from <em>S</em>. Consequently, the volume of the set of contagious points of <em>S</em> in <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> represents the probability of full contagion from <em>S</em> when the adoption threshold of each node is independently and uniformly distributed in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, which we term the contagion probability of <em>S</em> and denote by <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math></span>. We derive an explicit formula for <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math></span>, showing that <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math></span> is determined by how likely <em>S</em> can produce full contagion exclusively through each spanning tree of the quotient graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> of <em>G</em> in which <em>S</em> is treated as a single node. Besides, we compare <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math></span> with the contagion threshold of <em>S</em>, which is denoted by <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> and is the probability of full contagion from <em>S</em> when all nodes share a common adoption threshold <em>q</em> chosen uniformly at random from <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. We show that the presence of a cycle in <span><math><msub><mrow><mi>G</mi>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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