Pub Date : 2024-11-19DOI: 10.1016/j.amc.2024.129187
Arnab Mapui, Santwana Mukhopadhyay
The present work deals with the problem of prescribed-time control of non-linear impulsive systems consisting of external perturbations. Lyapunov-like sufficient conditions for prescribed-time and practical prescribed-time stability are provided for vanishing and non-vanishing perturbations, respectively. Depending on the user's requirements, some sequences of stabilizing impulses are constructed in this regard. It is shown that the systems consisting of vanishing-type disturbances can attain prescribed-time stabilization at the origin. On the other hand, in the presence of non-vanishing disturbances, which can consist of bounded or unbounded disturbances, the trajectories of the system can enter a stable region only within the prescribed time. Moreover, the stable region is independent of the impulsive strength and the prescribed-time. The efficacy of the proposed results is provided through various examples and their numerical simulations.
{"title":"Lyapunov-like conditions for prescribed-time stability of perturbed impulsive systems","authors":"Arnab Mapui, Santwana Mukhopadhyay","doi":"10.1016/j.amc.2024.129187","DOIUrl":"10.1016/j.amc.2024.129187","url":null,"abstract":"<div><div>The present work deals with the problem of prescribed-time control of non-linear impulsive systems consisting of external perturbations. Lyapunov-like sufficient conditions for prescribed-time and practical prescribed-time stability are provided for vanishing and non-vanishing perturbations, respectively. Depending on the user's requirements, some sequences of stabilizing impulses are constructed in this regard. It is shown that the systems consisting of vanishing-type disturbances can attain prescribed-time stabilization at the origin. On the other hand, in the presence of non-vanishing disturbances, which can consist of bounded or <strong><em>unbounded</em></strong> disturbances, the trajectories of the system can enter a stable region only within the prescribed time. Moreover, the stable region is independent of the impulsive strength and the prescribed-time. The efficacy of the proposed results is provided through various examples and their numerical simulations.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129187"},"PeriodicalIF":3.5,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696447","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}
Pub Date : 2024-11-19DOI: 10.1016/j.amc.2024.129197
Mei-Li Wang , Rong-Xia Hao , Jou-Ming Chang , Sejeong Bang
A graph G is 2-edge Hamiltonian connected if for any edge set with , has a Hamiltonian cycle containing all edges of , where is the graph obtained from G by including all edges of . The problem of determining whether a graph is 2-edge Hamiltonian connected is challenging, as it is known to be NP-complete. This property is stronger than Hamiltonian connectedness, which indicates the existence of a Hamiltonian path between every pair of vertices in a graph. This paper first provides a characterization and a sufficiency for 2-edge Hamiltonian connectedness. Through this, we shed light on the fact that many well-known networks are 2-edge Hamiltonian connected, including BCube data center networks and some variations of hypercubes, and so on. Additionally, we demonstrate that DCell data center networks and Cartesian product graphs containing almost all generalized hypercubes are 2-edge Hamiltonian connected.
如果对于任意边集 E⊆{uv:u,v∈V(G),u≠v}（|E|≤2），G∪E 有一个包含 E 的所有边的哈密顿循环，其中 G∪E 是由 G 通过包含 E 的所有边得到的图，则图 G 是 2 边哈密顿连通的。这一特性比哈密顿连通性更强，后者表示图中每对顶点之间都存在一条哈密顿路径。本文首先给出了 2 边哈密尔顿连通性的特征和充分性。由此，我们揭示了许多著名的网络都具有 2 边哈密顿连通性，包括 BCube 数据中心网络和超立方体的一些变体等。此外，我们还证明了 DCell 数据中心网络和包含几乎所有广义超立方体的笛卡尔积图都是 2 边哈密顿连通的。
{"title":"2-Edge Hamiltonian connectedness: Characterization and results in data center networks","authors":"Mei-Li Wang , Rong-Xia Hao , Jou-Ming Chang , Sejeong Bang","doi":"10.1016/j.amc.2024.129197","DOIUrl":"10.1016/j.amc.2024.129197","url":null,"abstract":"<div><div>A graph <em>G</em> is 2-edge Hamiltonian connected if for any edge set <span><math><mi>E</mi><mo>⊆</mo><mo>{</mo><mi>u</mi><mi>v</mi><mo>:</mo><mspace></mspace><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>u</mi><mo>≠</mo><mi>v</mi><mo>}</mo></math></span> with <span><math><mo>|</mo><mi>E</mi><mo>|</mo><mo>≤</mo><mn>2</mn></math></span>, <span><math><mi>G</mi><mo>∪</mo><mi>E</mi></math></span> has a Hamiltonian cycle containing all edges of <span><math><mi>E</mi></math></span>, where <span><math><mi>G</mi><mo>∪</mo><mi>E</mi></math></span> is the graph obtained from <em>G</em> by including all edges of <span><math><mi>E</mi></math></span>. The problem of determining whether a graph is 2-edge Hamiltonian connected is challenging, as it is known to be NP-complete. This property is stronger than Hamiltonian connectedness, which indicates the existence of a Hamiltonian path between every pair of vertices in a graph. This paper first provides a characterization and a sufficiency for 2-edge Hamiltonian connectedness. Through this, we shed light on the fact that many well-known networks are 2-edge Hamiltonian connected, including BCube data center networks and some variations of hypercubes, and so on. Additionally, we demonstrate that DCell data center networks and Cartesian product graphs containing almost all generalized hypercubes are 2-edge Hamiltonian connected.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129197"},"PeriodicalIF":3.5,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696446","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}
Pub Date : 2024-11-19DOI: 10.1016/j.amc.2024.129186
Donna Quanjie Dou , Lisa Hui Sun
Gamma-positivity is one of the basic properties that may be possessed by polynomials with symmetric coefficients, which directly implies that they are unimodal. It originates from the study of Eulerian polynomials by Foata and Schützenberger. Then, the alternatingly gamma-positivity for symmetric polynomials was defined by Sagan and Tirrell. Later, Ma et al. further introduced the notions of bi-gamma-positive and alternatingly bi-gamma-positive for a polynomial which correspond to that both of the polynomials in the symmetric decomposition of are gamma-positive and alternatingly gamma-positive, respectively. In this paper we establish the alternatingly bi-gamma-positivity of the Boros–Moll polynomials by verifying both polynomials in the symmetric decomposition of their reciprocals are unimodal and alternatingly gamma-positive. It confirms a conjecture proposed by Ma, Qi, Yeh and Yeh.
{"title":"A conjecture on Boros-Moll polynomials due to Ma, Qi, Yeh and Yeh","authors":"Donna Quanjie Dou , Lisa Hui Sun","doi":"10.1016/j.amc.2024.129186","DOIUrl":"10.1016/j.amc.2024.129186","url":null,"abstract":"<div><div>Gamma-positivity is one of the basic properties that may be possessed by polynomials with symmetric coefficients, which directly implies that they are unimodal. It originates from the study of Eulerian polynomials by Foata and Schützenberger. Then, the alternatingly gamma-positivity for symmetric polynomials was defined by Sagan and Tirrell. Later, Ma et al. further introduced the notions of <em>bi-gamma-positive</em> and <em>alternatingly bi-gamma-positive</em> for a polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> which correspond to that both of the polynomials in the symmetric decomposition of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are gamma-positive and alternatingly gamma-positive, respectively. In this paper we establish the alternatingly bi-gamma-positivity of the Boros–Moll polynomials by verifying both polynomials in the symmetric decomposition of their reciprocals are unimodal and alternatingly gamma-positive. It confirms a conjecture proposed by Ma, Qi, Yeh and Yeh.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129186"},"PeriodicalIF":3.5,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696448","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}
Pub Date : 2024-11-18DOI: 10.1016/j.amc.2024.129202
Jiahao Zhang , Changxiang He , Rongquan Feng
As a generalization of strongly regular graphs, van Dam and Omidi [8] introduced the concept of strongly walk-regular graphs. A graph is called strongly ℓ-walk-regular if the number of walks of length ℓ from a vertex to another vertex depends only on whether the two vertices are adjacent, not adjacent, or identical. They proved that this class of graphs falls into several subclasses including connected regular graphs with four eigenvalues, which are called genuine strongly ℓ-walk-regular. In this paper, we prove that the least eigenvalue of a connected genuine strongly 3-walk-regular graph is no more than −2 and characterize all graphs reaching this upper bound.
{"title":"On the least eigenvalue of genuine strongly 3-walk-regular graphs","authors":"Jiahao Zhang , Changxiang He , Rongquan Feng","doi":"10.1016/j.amc.2024.129202","DOIUrl":"10.1016/j.amc.2024.129202","url":null,"abstract":"<div><div>As a generalization of strongly regular graphs, van Dam and Omidi <span><span>[8]</span></span> introduced the concept of strongly walk-regular graphs. A graph is called strongly <em>ℓ</em>-walk-regular if the number of walks of length <em>ℓ</em> from a vertex to another vertex depends only on whether the two vertices are adjacent, not adjacent, or identical. They proved that this class of graphs falls into several subclasses including connected regular graphs with four eigenvalues, which are called genuine strongly <em>ℓ</em>-walk-regular. In this paper, we prove that the least eigenvalue of a connected genuine strongly 3-walk-regular graph is no more than −2 and characterize all graphs reaching this upper bound.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129202"},"PeriodicalIF":3.5,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696451","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}
The total σ-irregularity is given by , where indicates the degree of a vertex z within the graph G. It is known that the graphs maximizing -irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to , where and . We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.
σt(G)=∑{u,v}⊆V(G)(dG(u)-dG(v))2，其中 dG(z) 表示图 G 中顶点 z 的度数。由于人们通常希望具有尽可能多不同度数的图能达到最大不规则度，因此我们将此不变量修改为 σtf(n)(G)=∑{u,v}⊆V(G)|dG(u)-dG(v)|f(n), 其中 n=|V(G)| 且 f(n)>0.我们研究上述修正在什么条件下能获得反规则图的最大值。我们考虑了一般图、树图和化学图，并随结果提出了一些问题和猜想。
{"title":"Extremizing antiregular graphs by modifying total σ-irregularity","authors":"Martin Knor , Riste Škrekovski , Slobodan Filipovski , Darko Dimitrov","doi":"10.1016/j.amc.2024.129199","DOIUrl":"10.1016/j.amc.2024.129199","url":null,"abstract":"<div><div>The total <em>σ</em>-irregularity is given by <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><msup><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span> indicates the degree of a vertex <em>z</em> within the graph <em>G</em>. It is known that the graphs maximizing <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to <span><math><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>|</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span>, where <span><math><mi>n</mi><mo>=</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span>. We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129199"},"PeriodicalIF":3.5,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696454","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}
Pub Date : 2024-11-18DOI: 10.1016/j.amc.2024.129201
Peng-Li Zhang , Xiao-Dong Zhang
The incidence -spectral radius of a k-uniform hypergraph G with n vertices and m edges is defined as the spectral radius of the incidence -tensor , where R is the incidence matrix of G, and is an order k dimension m identity tensor. Since the -entry of is involved in the number of edges in G containing vertices simultaneously, more structural properties of G from the entry of than other commonly used tensors associated with hypergraphs may be discovered. In this paper, we present several bounds on the incidence -spectral radius of G in terms of degree sequences, which are better than some known results in some cases.
具有 n 个顶点和 m 条边的 k-Uniform 超图 G 的入射 Q 谱半径定义为入射 Q 张量 Q⁎:=RIRT 的谱半径，其中 R 是 G 的入射矩阵，I 是一个 k 维 m 次方的标识张量。由于 Q⁎的 (i1,i2,...,ik) 项涉及到 G 中同时包含顶点 i1,i2,...,ik 的边的数量，因此从 Q⁎ 项中可以发现 G 的结构属性比其他与超图相关的常用张量更多。在本文中，我们以度序列为单位提出了 G 的入射 Q 谱半径的几个约束，这些约束在某些情况下优于一些已知结果。
{"title":"Bounds for the incidence Q-spectral radius of uniform hypergraphs","authors":"Peng-Li Zhang , Xiao-Dong Zhang","doi":"10.1016/j.amc.2024.129201","DOIUrl":"10.1016/j.amc.2024.129201","url":null,"abstract":"<div><div>The incidence <span><math><mi>Q</mi></math></span>-spectral radius of a <em>k</em>-uniform hypergraph <em>G</em> with <em>n</em> vertices and <em>m</em> edges is defined as the spectral radius of the incidence <span><math><mi>Q</mi></math></span>-tensor <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>:</mo><mo>=</mo><mi>R</mi><mi>I</mi><msup><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msup></math></span>, where <em>R</em> is the incidence matrix of <em>G</em>, and <span><math><mi>I</mi></math></span> is an order <em>k</em> dimension <em>m</em> identity tensor. Since the <span><math><mo>(</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>-entry of <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is involved in the number of edges in <em>G</em> containing vertices <span><math><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> simultaneously, more structural properties of <em>G</em> from the entry of <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> than other commonly used tensors associated with hypergraphs may be discovered. In this paper, we present several bounds on the incidence <span><math><mi>Q</mi></math></span>-spectral radius of <em>G</em> in terms of degree sequences, which are better than some known results in some cases.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129201"},"PeriodicalIF":3.5,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696453","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}
Pub Date : 2024-11-12DOI: 10.1016/j.amc.2024.129160
Yayu Yang, Mingzu Zhang, Jixiang Meng
<div><div>The interconnection network between the storage system and the multi-core computing system is the bridge for communication of enormous amounts of data access and storage, which is the critical factor in affecting the performance of high-performance computing systems. By enforcing additional restrictions on the definition of <em>R</em>-structure and <em>R</em>-substructure connectivities to satisfy that each remaining vertex has not less than <em>r</em> neighbors, we can dynamically assess the cardinality of the separated component to meet the above conditions under structure faulty, thereby enhancing the evaluation of the fault tolerance and reliability of high-performance computing systems. Let <em>R</em> be a connected subgraph of a connected graph <em>G</em>. The <em>r</em>-restricted <em>R</em>-structure connectivity <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>;</mo><mi>R</mi><mo>)</mo></math></span> (resp. <em>r</em>-restricted <em>R</em>-substructure connectivity <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>;</mo><mi>R</mi><mo>)</mo></math></span>) of <em>G</em> is the minimum cardinality of a set of subgraphs <span><math><mi>F</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></math></span> such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is isomorphic to <em>R</em> (resp. <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a connected subgraph of <em>R</em>) for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></math></span>, and <span><math><mi>G</mi><mo>−</mo><mi>F</mi></math></span> is disconnected with the minimum degree of each component being at least <em>r</em>. Note that <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>;</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> reduces to <em>r</em>-restricted connectivity <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also called <em>r</em>-good neighbor connectivity). In this paper, we focus on <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>;</mo><mi>R</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>;</mo><mi>R</mi><mo>)</mo></math></span> for the <em>L</em>-ary <em>n</em>-dimensional hamming graph
存储系统和多核计算系统之间的互连网络是海量数据访问和存储的通信桥梁，是影响高性能计算系统性能的关键因素。通过对 R-结构和 R-子结构连通性的定义进行额外限制，满足每个剩余顶点的邻居不少于 r 个，我们就可以动态评估分离组件的明细度，以满足结构故障下的上述条件，从而提高对高性能计算系统容错性和可靠性的评估。设 R 是连通图 G 的一个连通子图。G 的 r 限制 R 结构连通性 κr(G;R)（或 r 限制 R 子结构连通性 κrs(G;R)）是 Fi 与 R 同构（或 Fi 是连通子图 F={F1,F2,...,Fm}）的子图集合的最小卡片度。请注意，κr(G;K1) 简化为 r 限制连通性 κr(G)（也称为 r 好邻居连通性）。在本文中，我们主要研究 Lary n 维汉明图 KLn 的 κr(KLn;R) 和 κrs(KLn;R) ，其中 R∈{K1,K1,1,KL1} 。对于 0≤r≤n-3、n≥3 和 L≥3，我们确定了 KLn 的（L-1）r-好邻居连通性，即 κ(L-1)r(KLn)=(L-1)(n-r)Lr，以及 KLn 在 PMC 模型和 MM* 模型下的（L-1）r-好邻居可诊断性，即 t(L-1)r(KLn)=[(L-1)(n-r)-1]Lr-1。同时，我们还推导出 1≤r≤n-3, n≥4 时，κ(L-1)r(KLn;K1,1)=κ(L-1)rs(KLn;K1,1)=12(L-1)Lr(n-r)。此外，我们还给出了 n≥3 时的 κ2(KLn;KL1) (resp. κ2s(KLn;KL1))上限，并证明它对于三元 n 立方体 K3n 是尖锐的。具体地说，当 n≥3 时，κ2(K3n;K31)=κ2s(K3n;K31)=2(n-1)。
{"title":"Fault tolerance assessment for hamming graphs based on r-restricted R-structure(substructure) fault pattern","authors":"Yayu Yang, Mingzu Zhang, Jixiang Meng","doi":"10.1016/j.amc.2024.129160","DOIUrl":"10.1016/j.amc.2024.129160","url":null,"abstract":"<div><div>The interconnection network between the storage system and the multi-core computing system is the bridge for communication of enormous amounts of data access and storage, which is the critical factor in affecting the performance of high-performance computing systems. By enforcing additional restrictions on the definition of <em>R</em>-structure and <em>R</em>-substructure connectivities to satisfy that each remaining vertex has not less than <em>r</em> neighbors, we can dynamically assess the cardinality of the separated component to meet the above conditions under structure faulty, thereby enhancing the evaluation of the fault tolerance and reliability of high-performance computing systems. Let <em>R</em> be a connected subgraph of a connected graph <em>G</em>. The <em>r</em>-restricted <em>R</em>-structure connectivity <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>;</mo><mi>R</mi><mo>)</mo></math></span> (resp. <em>r</em>-restricted <em>R</em>-substructure connectivity <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>;</mo><mi>R</mi><mo>)</mo></math></span>) of <em>G</em> is the minimum cardinality of a set of subgraphs <span><math><mi>F</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></math></span> such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is isomorphic to <em>R</em> (resp. <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a connected subgraph of <em>R</em>) for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></math></span>, and <span><math><mi>G</mi><mo>−</mo><mi>F</mi></math></span> is disconnected with the minimum degree of each component being at least <em>r</em>. Note that <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>;</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> reduces to <em>r</em>-restricted connectivity <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also called <em>r</em>-good neighbor connectivity). In this paper, we focus on <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>;</mo><mi>R</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>;</mo><mi>R</mi><mo>)</mo></math></span> for the <em>L</em>-ary <em>n</em>-dimensional hamming graph ","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"489 ","pages":"Article 129160"},"PeriodicalIF":3.5,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661265","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}
Pub Date : 2024-11-12DOI: 10.1016/j.amc.2024.129149
Yang Yang , Shuocong Geng , Dong Yue , Sergey Gorbachev , Iakov Korovin
An event-triggered formation control strategy is proposed for a multi-agent system (MAS) suffered from unknown disturbances. In identifier-critic-actor neural networks (NNs), the strategy only needs to calculate the negative gradient of an approximated Hamilton-Jacobi-Bellman (HJB) equation, instead of the gradient descent method associated with Bellman residual errors. This simplified method removes the requirement for a complicated gradient calculation process of residual square of HJB equation. The weights in critic-actor NNs only update as the triggered condition is violated, and the computational burdens caused by frequent updates are thus reduced. Without dynamics information in prior, a disturbance observer is also constructed to approximate disturbances in an MAS. From stability analysis, it is proven that all signals are bounded. Two numerical examples are illustrated to verify that the proposed control strategy is effective.
本文针对遭受未知干扰的多代理系统（MAS）提出了一种事件触发的编队控制策略。在识别器-批判者-行动者神经网络（NNs）中，该策略只需计算近似汉密尔顿-雅各比-贝尔曼（HJB）方程的负梯度，而无需采用与贝尔曼残差相关的梯度下降法。这种简化方法消除了对 HJB 方程残差平方的复杂梯度计算过程的要求。批判-行为网络中的权重只在触发条件被违反时更新，因此减少了频繁更新带来的计算负担。在没有先验动态信息的情况下，还构建了一个扰动观测器来近似 MAS 中的扰动。稳定性分析证明，所有信号都是有界的。两个数值示例验证了所提出的控制策略是有效的。
{"title":"Event-triggered approximately optimized formation control of multi-agent systems with unknown disturbances via simplified reinforcement learning","authors":"Yang Yang , Shuocong Geng , Dong Yue , Sergey Gorbachev , Iakov Korovin","doi":"10.1016/j.amc.2024.129149","DOIUrl":"10.1016/j.amc.2024.129149","url":null,"abstract":"<div><div>An event-triggered formation control strategy is proposed for a multi-agent system (MAS) suffered from unknown disturbances. In identifier-critic-actor neural networks (NNs), the strategy only needs to calculate the negative gradient of an approximated Hamilton-Jacobi-Bellman (HJB) equation, instead of the gradient descent method associated with Bellman residual errors. This simplified method removes the requirement for a complicated gradient calculation process of residual square of HJB equation. The weights in critic-actor NNs only update as the triggered condition is violated, and the computational burdens caused by frequent updates are thus reduced. Without dynamics information in prior, a disturbance observer is also constructed to approximate disturbances in an MAS. From stability analysis, it is proven that all signals are bounded. Two numerical examples are illustrated to verify that the proposed control strategy is effective.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"489 ","pages":"Article 129149"},"PeriodicalIF":3.5,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661266","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}
Pub Date : 2024-11-12DOI: 10.1016/j.amc.2024.129176
Xuelong Liu, Guoju Ye, Wei Liu, Fangfang Shi
This work aims to solve a fuzzy initial value problem for fractional difference equations and to study a class of discrete fractional cobweb models with fuzzy data under the Caputo granular difference operator. Based on relative-distance-measure fuzzy interval arithmetic, we first present several new concepts for fuzzy functions in the field of fuzzy discrete fractional calculus, such as the forward granular difference operator, Riemann-Liouville fractional granular sum, Riemann-Liouville and Caputo granular differences. The composition rules and Leibniz laws used to solve a fuzzy initial value problem for fractional difference equations are also presented. As applications, we obtain the solutions of fuzzy discrete Caputo fractional cobweb models, provide conditions for the convergence of the solution to the equilibrium value, and discuss different cases of how the trajectory of the granular solution converges to the equilibrium value. The developed results are also illustrated through several numerical examples.
{"title":"Fuzzy discrete fractional granular calculus and its application to fractional cobweb models","authors":"Xuelong Liu, Guoju Ye, Wei Liu, Fangfang Shi","doi":"10.1016/j.amc.2024.129176","DOIUrl":"10.1016/j.amc.2024.129176","url":null,"abstract":"<div><div>This work aims to solve a fuzzy initial value problem for fractional difference equations and to study a class of discrete fractional cobweb models with fuzzy data under the Caputo granular difference operator. Based on relative-distance-measure fuzzy interval arithmetic, we first present several new concepts for fuzzy functions in the field of fuzzy discrete fractional calculus, such as the forward granular difference operator, Riemann-Liouville fractional granular sum, Riemann-Liouville and Caputo granular differences. The composition rules and Leibniz laws used to solve a fuzzy initial value problem for fractional difference equations are also presented. As applications, we obtain the solutions of fuzzy discrete Caputo fractional cobweb models, provide conditions for the convergence of the solution to the equilibrium value, and discuss different cases of how the trajectory of the granular solution converges to the equilibrium value. The developed results are also illustrated through several numerical examples.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"489 ","pages":"Article 129176"},"PeriodicalIF":3.5,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661263","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}
Pub Date : 2024-11-12DOI: 10.1016/j.amc.2024.129174
Jie Liu , Driss Boutat , Da-Yan Liu , Xue-Feng Zhang
This research note establishes a specific framework for transforming nonlinear multi-input and multi-output diffeomorphism systems into extended observable normal forms without using differential geometry techniques. For this purpose, the nonlinear MIMO systems whose nonlinear terms do not need to be Lipschitz, are proposed. First, a change of coordinates is designed to eliminate the square items and coupled items for each nonlinear dynamical subsystem. Second, coupled auxiliary dynamics are constructed to transform the nonlinear multi-input and multi-output diffeomorphism systems into extended observable normal forms such that the finite-time and robust step-by-step sliding mode observer can be applied. Then, the state variables for the considered nonlinear dynamical systems are estimated by using the inverse of the transformations. Finally, the validity of the proposed design methods is verified by two numerical examples.
本研究论文建立了一个特定框架，无需使用微分几何技术，即可将非线性多输入和多输出差分系统转化为扩展的可观测正则表达式。为此，本文提出了非线性 MIMO 系统，其非线性项无需为 Lipschitz。首先，设计了一种坐标变化，以消除每个非线性动力学子系统的平方项和耦合项。其次，构建耦合辅助动力学，将非线性多输入和多输出差分系统转化为扩展的可观测正态形式，从而应用有限时间和鲁棒逐步滑模观测器。然后，利用逆变换估计所考虑的非线性动力系统的状态变量。最后，通过两个数值示例验证了所提设计方法的有效性。
{"title":"Nonlinear MIMO observable normal forms with output injection and output diffeomorphism","authors":"Jie Liu , Driss Boutat , Da-Yan Liu , Xue-Feng Zhang","doi":"10.1016/j.amc.2024.129174","DOIUrl":"10.1016/j.amc.2024.129174","url":null,"abstract":"<div><div>This research note establishes a specific framework for transforming nonlinear multi-input and multi-output diffeomorphism systems into extended observable normal forms without using differential geometry techniques. For this purpose, the nonlinear MIMO systems whose nonlinear terms do not need to be Lipschitz, are proposed. First, a change of coordinates is designed to eliminate the square items and coupled items for each nonlinear dynamical subsystem. Second, coupled auxiliary dynamics are constructed to transform the nonlinear multi-input and multi-output diffeomorphism systems into extended observable normal forms such that the finite-time and robust step-by-step sliding mode observer can be applied. Then, the state variables for the considered nonlinear dynamical systems are estimated by using the inverse of the transformations. Finally, the validity of the proposed design methods is verified by two numerical examples.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"489 ","pages":"Article 129174"},"PeriodicalIF":3.5,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661264","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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