When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the PC, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional PCs with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. This correspondence is extended to a larger class of one-dimensional models that include imperfect interfaces, array of resonators, or dispersive media. Numerical simulations are proposed to illustrate the theoretical findings.
当半无限声子晶体（PC）接触时，它们的边界可能存在局部模式。核心问题通常是预测它们的存在并确定其稳定性。随着拓扑绝缘体领域的迅速发展，人们开发出了强大的工具来解决这些问题。特别是，当应用于具有镜像对称性的一维系统时，体-界对应关系声称界面模式的存在是由 PC 的体特性计算出的拓扑不变量给出的，这确保了强大的稳定性。这种一维体-边界对应关系已在各种研究中得到证实。最近的尝试利用了表面阻抗的概念，依赖于传递矩阵的分析计算。在本研究中，我们证明了所有具有镜像对称性的一维 PC 的表面阻抗随频率的单调演化。这一结果使我们建立了更强版本的体界对应关系，不仅保证了拓扑保护界面态的存在性，而且保证了其唯一性。这一对应关系被扩展到包括不完美界面、谐振器阵列或色散介质在内的更大一类一维模型。我们提出了数值模拟来说明理论发现。
{"title":"Surface impedance and topologically protected interface modes in one-dimensional phononic crystals","authors":"A. Coutant, B. Lombard","doi":"10.1098/rspa.2023.0533","DOIUrl":"https://doi.org/10.1098/rspa.2023.0533","url":null,"abstract":"When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the PC, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional PCs with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. This correspondence is extended to a larger class of one-dimensional models that include imperfect interfaces, array of resonators, or dispersive media. Numerical simulations are proposed to illustrate the theoretical findings.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"48 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.
{"title":"Improved estimates for the number of non-negative integer matrices with given row and column sums","authors":"Maximilian Jerdee, Alec Kirkley, M. E. J. Newman","doi":"10.1098/rspa.2023.0470","DOIUrl":"https://doi.org/10.1098/rspa.2023.0470","url":null,"abstract":"The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"48 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Daniel J. VandenHeuvel, Pascal R. Buenzli, Matthew J. Simpson
Mathematical modelling of biological population dynamics often involves proposing high-fidelity discrete agent-based models that capture stochasticity and individual-level processes. These models are often considered in conjunction with an approximate coarse-grained differential equation that captures population-level features only. These coarse-grained models are only accurate in certain asymptotic parameter regimes, such as enforcing that the time scale of individual motility far exceeds the time scale of birth/death processes. When these coarse-grained models are accurate, the discrete model still abides by conservation laws at the microscopic level, which implies that there is some macroscopic conservation law that can describe the macroscopic dynamics. In this work, we introduce an equation learning framework to find accurate coarse-grained models when standard continuum limit approaches are inaccurate. We demonstrate our approach using a discrete mechanical model of epithelial tissues, considering a series of four case studies that consider problems with and without free boundaries, and with and without proliferation, illustrating how we can learn macroscopic equations describing mechanical relaxation, cell proliferation, and the equation governing the dynamics of the free boundary of the tissue. While our presentation focuses on this biological application, our approach is more broadly applicable across a range of scenarios where discrete models are approximated by approximate continuum-limit descriptions.
{"title":"Pushing coarse-grained models beyond the continuum limit using equation learning","authors":"Daniel J. VandenHeuvel, Pascal R. Buenzli, Matthew J. Simpson","doi":"10.1098/rspa.2023.0619","DOIUrl":"https://doi.org/10.1098/rspa.2023.0619","url":null,"abstract":"Mathematical modelling of biological population dynamics often involves proposing high-fidelity discrete agent-based models that capture stochasticity and individual-level processes. These models are often considered in conjunction with an approximate coarse-grained differential equation that captures population-level features only. These coarse-grained models are only accurate in certain asymptotic parameter regimes, such as enforcing that the time scale of individual motility far exceeds the time scale of birth/death processes. When these coarse-grained models are accurate, the discrete model still abides by conservation laws at the microscopic level, which implies that there is some macroscopic conservation law that can describe the macroscopic dynamics. In this work, we introduce an equation learning framework to find accurate coarse-grained models when standard continuum limit approaches are inaccurate. We demonstrate our approach using a discrete mechanical model of epithelial tissues, considering a series of four case studies that consider problems with and without free boundaries, and with and without proliferation, illustrating how we can learn macroscopic equations describing mechanical relaxation, cell proliferation, and the equation governing the dynamics of the free boundary of the tissue. While our presentation focuses on this biological application, our approach is more broadly applicable across a range of scenarios where discrete models are approximated by approximate continuum-limit descriptions.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"22 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Long-wave asymptotic approximations are developed for two-dimensional acoustic waves along rigid ducts. The waves are scattered by obstacles, constrictions, bulges and/or bends. Matched asymptotic expansions are used, requiring the calculation of blockage coefficients, which are defined in terms of the solution of related potential-flow problems. The emphasis is on estimating reflection and transmission coefficients, correct to first order in the ratio of the waveguide width to the wavelength. Detailed results are given for sharp bends of arbitrary angle, including right-angled bends and hairpin bends. Applications to multiple scattering by labyrinthine structures are also made.
{"title":"Going round the bend: reflection and transmission of long waves by waveguide corners and labyrinths","authors":"P. A. Martin","doi":"10.1098/rspa.2023.0635","DOIUrl":"https://doi.org/10.1098/rspa.2023.0635","url":null,"abstract":"Long-wave asymptotic approximations are developed for two-dimensional acoustic waves along rigid ducts. The waves are scattered by obstacles, constrictions, bulges and/or bends. Matched asymptotic expansions are used, requiring the calculation of blockage coefficients, which are defined in terms of the solution of related potential-flow problems. The emphasis is on estimating reflection and transmission coefficients, correct to first order in the ratio of the waveguide width to the wavelength. Detailed results are given for sharp bends of arbitrary angle, including right-angled bends and hairpin bends. Applications to multiple scattering by labyrinthine structures are also made.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"14 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
H. Tanaka, S. Yanagihara, K. Shiomi, T. Kuroda, Y. Oku
Soft-hard matter friction is a long-standing tribology problem that remains unclarified, requiring engineers to empirically predict the wear life. To clarify this issue, this study examines the transient running-in regime of rubber friction on a hard rough substrate and models the temporal wear progression using the spectrum curves of surface roughness for both materials. Performing a series of friction tests and three-dimensional surface-height measurements, the time-dependent behaviours of the power spectral densities (PSDs) are divided into two phases, namely the initial non-steady and long-term steady phases. The detailed spectral analyses of worn rubber surfaces in the initial phase lead to a blended PSD function between self-affine and K -correlation surface models, consisting of one variable (the Hurst exponent) that is saturated by the substrate self-affinity. Supported by the Greenwood–Williamson theory concerning rough contact mechanics, the volumetric estimate with the blended PSD function is used to assess the volume rate of wear debris in the steady phase, which is validated experimentally. These findings not only improve the wear predictions of soft materials from the initial measurements of worn surfaces but also help clarify the constrained multiscale mechanism of wear.
软硬物质摩擦是一个长期存在的摩擦学问题，至今仍未得到澄清，需要工程师根据经验预测磨损寿命。为了澄清这一问题，本研究探讨了橡胶在硬质粗糙基体上的瞬态磨合机制，并利用两种材料的表面粗糙度频谱曲线建立了时间磨损进展模型。通过一系列摩擦试验和三维表面高度测量，功率谱密度（PSD）随时间变化的行为分为两个阶段，即初始非稳定阶段和长期稳定阶段。通过对初始阶段磨损橡胶表面的详细光谱分析，得出了介于自亲和表面模型和 K - 相关表面模型之间的混合 PSD 函数，该函数由一个变量（赫斯特指数）组成，该变量因基底自亲和而饱和。在格林伍德-威廉姆森粗糙接触力学理论的支持下，混合 PSD 函数的体积估计值被用于评估稳定阶段磨损碎片的体积率，并得到了实验验证。这些发现不仅改进了根据磨损表面的初始测量结果对软材料磨损的预测，还有助于阐明磨损的多尺度约束机制。
{"title":"Spectral wear modelling of rubber friction on a hard substrate with large surface roughness","authors":"H. Tanaka, S. Yanagihara, K. Shiomi, T. Kuroda, Y. Oku","doi":"10.1098/rspa.2023.0587","DOIUrl":"https://doi.org/10.1098/rspa.2023.0587","url":null,"abstract":"Soft-hard matter friction is a long-standing tribology problem that remains unclarified, requiring engineers to empirically predict the wear life. To clarify this issue, this study examines the transient running-in regime of rubber friction on a hard rough substrate and models the temporal wear progression using the spectrum curves of surface roughness for both materials. Performing a series of friction tests and three-dimensional surface-height measurements, the time-dependent behaviours of the power spectral densities (PSDs) are divided into two phases, namely the initial non-steady and long-term steady phases. The detailed spectral analyses of worn rubber surfaces in the initial phase lead to a blended PSD function between self-affine and <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>K</mml:mi> </mml:math> </jats:inline-formula> -correlation surface models, consisting of one variable (the Hurst exponent) that is saturated by the substrate self-affinity. Supported by the Greenwood–Williamson theory concerning rough contact mechanics, the volumetric estimate with the blended PSD function is used to assess the volume rate of wear debris in the steady phase, which is validated experimentally. These findings not only improve the wear predictions of soft materials from the initial measurements of worn surfaces but also help clarify the constrained multiscale mechanism of wear.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"47 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The electrosteric interaction energy for a charged hydrogel and hard plane, and between two charged hydrogels is derived in the Debye–Hückel approximation. This is combined with a van der Waals potential that explicitly addresses the Hamaker constant for the solvent-mediated hydrogel interactions. Then, in the Derjaguin approximation, DLVO-type interaction potentials are provided for hydrogel and hard/rigid spheres, accounting for elastic deformation that accompanies adhesion. As examples, this furnishes the energy for cohesion of soft polyelectrolyte microspheres, and provides a quantitative interpretation for the adhesion of rigid latex spheres to a soft deformable hydrogel, as reported by Sato et al. (Sato et al. 2017 Sci. Rep.7 , 1–10 ( doi:10.1038/s41598-017-06257-1 )). The theory demonstrates that weak van der Waals attraction of hydrogels is readily balanced by electrosteric interactions, e.g. making colloidal hydrogel dispersions less stable than their rigid-particulate counterparts.
根据 Debye-Hückel 近似法推导出了带电水凝胶与硬平面以及两个带电水凝胶之间的静电相互作用能。这与范德华势结合在一起，范德华势明确解决了溶剂介导的水凝胶相互作用的哈马克常数问题。然后，在德雅金近似中，为水凝胶和硬/刚性球提供了 DLVO 型相互作用势，并考虑到了伴随粘附而来的弹性变形。例如，这提供了软聚电解质微球的内聚能，并为硬质乳胶球粘附到软质可变形水凝胶提供了定量解释，如 Sato 等人的报告（Sato et al.该理论表明，水凝胶的弱范德华吸引力很容易被静电相互作用所平衡，例如，使胶体水凝胶分散体的稳定性低于其刚性颗粒对应物。
{"title":"Electrosteric, van der Waals and elastic interaction of polyelectrolyte hydrogels","authors":"Reghan J. Hill","doi":"10.1098/rspa.2023.0541","DOIUrl":"https://doi.org/10.1098/rspa.2023.0541","url":null,"abstract":"The electrosteric interaction energy for a charged hydrogel and hard plane, and between two charged hydrogels is derived in the Debye–Hückel approximation. This is combined with a van der Waals potential that explicitly addresses the Hamaker constant for the solvent-mediated hydrogel interactions. Then, in the Derjaguin approximation, DLVO-type interaction potentials are provided for hydrogel and hard/rigid spheres, accounting for elastic deformation that accompanies adhesion. As examples, this furnishes the energy for cohesion of soft polyelectrolyte microspheres, and provides a quantitative interpretation for the adhesion of rigid latex spheres to a soft deformable hydrogel, as reported by Sato <jats:italic>et al.</jats:italic> (Sato <jats:italic>et al.</jats:italic> 2017 <jats:italic>Sci. Rep.</jats:italic> <jats:bold>7</jats:bold> , 1–10 ( <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"uri\" xlink:href=\"http://dx.doi.org/doi:10.1038/s41598-017-06257-1\">doi:10.1038/s41598-017-06257-1</jats:ext-link> )). The theory demonstrates that weak van der Waals attraction of hydrogels is readily balanced by electrosteric interactions, e.g. making colloidal hydrogel dispersions less stable than their rigid-particulate counterparts.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"23 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a specific class of matrices that participate in factorization problems that turn out to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang–Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order N=2 matrices of this specific class are equivalent to the homogeneous normalization map . From order N=3 matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang–Baxter maps.
我们提出了一类特殊的矩阵，它们参与的因式分解问题等价于恒定和缠绕（非恒定）五边形、反五边形或杨-巴克斯特映射，用非交换变量表示。具体而言，我们证明了这一特定类别的 N = 2 阶矩阵的因式分解等价于同质归一化映射。从 N = 3 阶矩阵中，我们得到了同质归一化映射的扩展，以及新颖的缠绕五边形、反五边形和杨-巴克斯特映射。
{"title":"Matrix factorizations and pentagon maps","authors":"Pavlos Kassotakis","doi":"10.1098/rspa.2023.0276","DOIUrl":"https://doi.org/10.1098/rspa.2023.0276","url":null,"abstract":"We propose a specific class of matrices that participate in factorization problems that turn out to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang–Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> </jats:inline-formula> matrices of this specific class are equivalent to the <jats:italic>homogeneous normalization map</jats:italic> . From order <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:math> </jats:inline-formula> matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang–Baxter maps.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"221 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time t evolves from large negative to large positive, the triangle lump reverses itself along the y -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the y -direction when time t evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.
本文研究的是 Davey-Stewartson (DS) II 方程。基于 Kadomtsev-Petviashvili (KP) 简化法和舒尔多项式理论，我们构建了一般高阶块体解。通过提取真解的舒尔多项式的前阶渐近值，得出了由基本块体及其位置组成的预测解。当选择解的索引为不同的正整数组合时，大时间的预测解反映了真解的两类块状模式。第一类是三角形的块状模式，由 Yablonskii-Vorob'ev 多项式的根结构分析描述。当时间 t 由大负值变为大正值时，三角形凸块沿 y 方向反转。第二类块状模式包括外部区域的非三角形，它由弗伦斯基-赫米特多项式的非零根结构分析描述，以及内部区域的可能三角形，它由雅布隆斯基-沃罗布夫多项式的根结构分析描述。此外，当时间 t 从大负值变为大正值时，外部区域的非三角形块状图案会旋转一个角度，而内部区域的可能三角形块状图案则会沿 y 方向反转。这些结果加深了我们对高阶凸块时间演化机制的理解。
{"title":"Prediction of general high-order lump solutions in the Davey–Stewartson II equation","authors":"Xue-Wei Yan, Haie Long, Yong Chen","doi":"10.1098/rspa.2023.0455","DOIUrl":"https://doi.org/10.1098/rspa.2023.0455","url":null,"abstract":"Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive, the triangle lump reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction when time <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"263 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties.
{"title":"Effective statistical control strategies for complex turbulent dynamical systems","authors":"Jeffrey Covington, Di Qi, Nan Chen","doi":"10.1098/rspa.2023.0546","DOIUrl":"https://doi.org/10.1098/rspa.2023.0546","url":null,"abstract":"Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"67 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135515145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Honeycomb materials frequently encounter hypergravity conditions in both aerospace and biological contexts during phases such as launch, reentry or under centrifugal motion. The significant body force engendered by hypergravity induces alterations in the microstructure of honeycomb materials, which in turn, influences their macroscopic mechanical behaviour. Leveraging the stiffness of the beam element as a pivotal variable, we successfully derived the equivalent moduli of the honeycomb material under hypergravity conditions. We further proposed the concept of a ‘hypergravity factor’, elucidating that the density of the base material, the dimensions of honeycomb cells and the magnitude of the hypergravity contribute to amplifying hypergravity effects. The results, numerically validated through finite-element simulations, could be reduced to the case that neglects body force. The critical buckling load of the honeycomb material under hypergravity can be assessed by setting the derived moduli to zero. In the presence of hypergravity, a honeycomb material undergoes a transition into a gradient material along the hypergravity direction, thereby exacerbating anisotropy. This phenomenon is theoretically expected to occur in virtually all porous materials. The analytical framework adopted, which employs beam stiffness as an intermediary variable, facilitates the extension of these results to honeycomb materials which encompass beam elements with functional gradients or varying cross-sectional morphologies.
{"title":"Equivalent in-plane elastic moduli of honeycomb materials under hypergravity conditions","authors":"Lei Wang, Guannan Wang, Chaofeng Lü","doi":"10.1098/rspa.2023.0638","DOIUrl":"https://doi.org/10.1098/rspa.2023.0638","url":null,"abstract":"Honeycomb materials frequently encounter hypergravity conditions in both aerospace and biological contexts during phases such as launch, reentry or under centrifugal motion. The significant body force engendered by hypergravity induces alterations in the microstructure of honeycomb materials, which in turn, influences their macroscopic mechanical behaviour. Leveraging the stiffness of the beam element as a pivotal variable, we successfully derived the equivalent moduli of the honeycomb material under hypergravity conditions. We further proposed the concept of a ‘hypergravity factor’, elucidating that the density of the base material, the dimensions of honeycomb cells and the magnitude of the hypergravity contribute to amplifying hypergravity effects. The results, numerically validated through finite-element simulations, could be reduced to the case that neglects body force. The critical buckling load of the honeycomb material under hypergravity can be assessed by setting the derived moduli to zero. In the presence of hypergravity, a honeycomb material undergoes a transition into a gradient material along the hypergravity direction, thereby exacerbating anisotropy. This phenomenon is theoretically expected to occur in virtually all porous materials. The analytical framework adopted, which employs beam stiffness as an intermediary variable, facilitates the extension of these results to honeycomb materials which encompass beam elements with functional gradients or varying cross-sectional morphologies.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"39 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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