数学最新文献

{"title":"Homogenisation Problems for Free Discontinuity Functionals with Bounded Cohesive Surface Terms","authors":"Gianni Dal Maso,&nbsp;Rodica Toader","doi":"10.1007/s00205-024-02053-0","DOIUrl":"10.1007/s00205-024-02053-0","url":null,"abstract":"<div><p>We study stochastic homogenisation problems for free discontinuity functionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimisation problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical study of magnesium dendrite microstructure under convection: Change of dendrite symmetry","authors":"Ang Zhang ,&nbsp;Minghang Yang ,&nbsp;Lang Qin ,&nbsp;Jing Cheng ,&nbsp;Yuchen Tang ,&nbsp;Jinglian Du ,&nbsp;Wenbo Yu ,&nbsp;Zhihua Dong ,&nbsp;Feng Liu ,&nbsp;Bin Jiang ,&nbsp;Fusheng Pan","doi":"10.1016/j.camwa.2024.10.038","DOIUrl":"10.1016/j.camwa.2024.10.038","url":null,"abstract":"<div><div>Besides diffusion and capillary, convection which is unavoidable under terrestrial condition has remarkable effects on the microstructure evolution during solidification. In this study, a phase-field lattice-Boltzmann model, accelerated by state-of-the-art parallel-adaptive mesh refinement algorithm, is solved to investigate the morphological evolution of the Mg-Gd dendrite under convection. The lengths of the dendrite primary arms are quantified to analyze the asymmetric dendrite patterns under convection. The effects of the multiple factors including the orientation angle, the flow intensity, and the undercooling are elucidated, and the relation between the length ratios and the three independent factors is established through multiple regression analysis. The upstream-downstream arm length difference and the included angle between the primary arms are characterized to illustrate the effect of convection on the evolution of the Mg-Gd dendrite. The 3D morphological selection, together with algorithm performance tests, is further discussed to elucidate the change of morphological symmetry under different growth conditions and to demonstrate the robustness of the numerical scheme. Deep understanding of the synergy between convection-induced solute transport and undercooling-driven growth, which largely determines the morphological selection, can assist guidance for the prediction and control of the magnesium alloy microstructures.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592656","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":"Upper bounds for the number of substructures in finite geometries from the container method","authors":"Sam Mattheus,&nbsp;Geertrui Van de Voorde","doi":"10.1016/j.jcta.2024.105968","DOIUrl":"10.1016/j.jcta.2024.105968","url":null,"abstract":"<div><div>We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and plane spreads in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, and caps in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The latter result extends work due to Roche-Newton and Warren <span><span>[21]</span></span> and Bhowmick and Roche-Newton <span><span>[6]</span></span>.</div><div>Finally, we investigate caps in <em>p</em>-random subsets of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and Roche-Newton and Warren <span><span>[6]</span></span>, <span><span>[21]</span></span>, and arcs in projective spaces by Chen, Liu, Nie and Zeng <span><span>[8]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593647","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":"Transverse Magnetic ENZ Resonators: Robustness and Optimal Shape Design","authors":"Robert V. Kohn,&nbsp;Raghavendra Venkatraman","doi":"10.1007/s00205-024-02023-6","DOIUrl":"10.1007/s00205-024-02023-6","url":null,"abstract":"<div><p>We study certain “geometric-invariant resonant cavities” introduced by Liberal, Mahmoud, and Engheta in a 2016 Nature Communications paper. They are cylindrical devices modeled using the transverse magnetic reduction of Maxwell’s equations, so the mathematics is two-dimensional. The cross-section consists of a dielectric inclusion surrounded by an “epsilon-near-zero” (ENZ) shell. When the shell has just the right area, its interaction with the inclusion produces a resonance. Mathematically, the resonance is a nontrivial solution of a 2D divergence-form Helmoltz equation <span>(nabla cdot left( varepsilon ^{-1}(x,omega ) nabla u right) + omega ^2 mu u = 0)</span>, where <span>(varepsilon (x,omega ))</span> is the (complex-valued) dielectric permittivity, <span>(omega )</span> is the frequency, <span>(mu )</span> is the magnetic permeability, and a homogeneous Neumann condition is imposed at the outer boundary of the shell. This is a nonlinear eigenvalue problem, since <span>(varepsilon )</span> depends on <span>(omega )</span>. Use of an ENZ material in the shell means that <span>(varepsilon (x,omega ))</span> is nearly zero there, so the PDE is rather singular. Working with a Lorentz model for the dispersion of the ENZ material, we put the discussion of Liberal et. al. on a sound foundation by proving the existence of the anticipated resonance when the loss parameter of the Lorentz model is sufficiently small. Our analysis is perturbative in character, using the implicit function theorem despite the apparently singular form of the PDE. While the existence of the resonance depends only on the area of the ENZ shell, its quality (that is, the rate at which the resonance decays) depends on the shape of the shell. It is therefore natural to consider an associated optimal design problem: what shape shell gives the slowest-decaying resonance? We prove that if the dielectric inclusion is a ball then the optimal shell is a concentric annulus. For an inclusion of any shape, we study a convex relaxation of the design problem using tools from convex duality. Finally, we discuss the conjecture that our relaxed problem amounts to considering homogenization-like limits of nearly optimal designs.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multisorted Boolean clones determined by binary relations up to minion homomorphisms","authors":"Libor Barto,&nbsp;Maryia Kapytka","doi":"10.1007/s00012-024-00878-0","DOIUrl":"10.1007/s00012-024-00878-0","url":null,"abstract":"<div><p>We describe the ordering of a class of clones by minion homomorphisms, also known as minor preserving maps or height 1 clone homomorphisms. The class consists of all clones on finite sets determined by binary relations whose projections to both coordinates have at most two elements. This class can be alternatively described up to minion homomorphisms as the class of multisorted Boolean clones determined by binary relations. We also introduce and apply the concept of a minion core which provides canonical representatives for equivalence classes of clones, more generally minions, on finite sets.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Chromatic Number of (P5, HVN)-free Graphs","authors":"Yian Xu","doi":"10.1007/s10255-024-1029-3","DOIUrl":"10.1007/s10255-024-1029-3","url":null,"abstract":"<div><p>Let <i>G</i> be a graph. We use <i>χ</i>(<i>G</i>) and <i>ω</i>(<i>G</i>) to denote the chromatic number and clique number of <i>G</i> respectively. A <i>P</i>₅ is a path on 5 vertices, and an HVN is a <i>K</i>₄ together with one more vertex which is adjacent to exactly two vertices of <i>K</i>₄. Combining with some known result, in this paper we show that if <i>G</i> is (<i>P</i>₅, <i>HVN</i>)-free, then <i>χ</i>(<i>G</i>) ≤ max{min{16, <i>ω</i>(<i>G</i>) + 3}, <i>ω</i>(<i>G</i>) + 1}. This upper bound is almost sharp.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniformity of Asymmetric Augmented Designs","authors":"Zhi-qing Wang,&nbsp;Xiang-yu Fang,&nbsp;Zu-jun Ou","doi":"10.1007/s10255-024-1135-2","DOIUrl":"10.1007/s10255-024-1135-2","url":null,"abstract":"<div><p>Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering. When some additional resources or information become available after the initial design of experiment is carried out, some additional runs and/or factors may be added in the follow-up stage. In this paper, the issue of the uniform row augmented designs and column augmented designs with mixed two-, three- and four-level is investigated. The uniformity of augmented designs is discussed under the wrap-around <i>L</i>₂-discrepancy. Some lower bounds of wrap-around <i>L</i>₂-discrepancy for the augmented designs are obtained, which can be used to assess uniformity of augmented design. Numerical results show that augmented designs have high efficiency, which have low discrepancy and close to the proposed lower bounds.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiple nontrivial solutions for a double phase system with concave-convex nonlinearities in subcritical and critical cases","authors":"Yizhe Feng,&nbsp;Zhanbing Bai","doi":"10.1007/s13324-024-00985-0","DOIUrl":"10.1007/s13324-024-00985-0","url":null,"abstract":"<div><p>In this article, we study the double phase elliptic system which contain with the parametric concave-convex nonlinearities and critical growth. The introduction of mixed critical terms brings some difficulties to the problem. For example, in proving that the solution is nontrivial, we need to do an additional series of studies on scalar equation. By introducing a new optimal constant <span>(S_{alpha ,beta })</span> in the double phase system, considering the different magnitude relationships of the exponential terms, and using the fibering method in form of the Nehari manifold and the Brezis-Lieb Lemma, the existence and multiplicity of solutions in subcritical and critical cases are obtained separately.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Local Existence of Solutions to the Fluid–Structure Interaction Problem with a Free Interface","authors":"Igor Kukavica,&nbsp;Linfeng Li,&nbsp;Amjad Tuffaha","doi":"10.1007/s00245-024-10195-6","DOIUrl":"10.1007/s00245-024-10195-6","url":null,"abstract":"<div><p>We address a system of equations modeling an incompressible fluid interacting with an elastic body. We prove the local existence when the initial velocity belongs to the space <span>(H^{1.5+epsilon })</span> and the initial structure velocity is in <span>(H^{1+epsilon })</span>, where <span>(epsilon in (0, 1/20))</span>.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10195-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An implementation of hp-FEM for the fractional Laplacian","authors":"Björn Bahr,&nbsp;Markus Faustmann,&nbsp;Jens Markus Melenk","doi":"10.1016/j.camwa.2024.10.005","DOIUrl":"10.1016/j.camwa.2024.10.005","url":null,"abstract":"<div><div>We consider the discretization of the 1<em>d</em>-integral Dirichlet fractional Laplacian by <em>hp</em>-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of <em>hp</em>-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>, where <em>N</em> is the problem size. Numerical examples illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for <em>hp</em>-finite element spaces based on shape regular meshes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}