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Zeitschriftenartikel zum Thema „Ε-Regularity“

Autor: Grafiati

Veröffentlicht am 21. Dezember 2024

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1

FOX, JACOB, LÁSZLÓ MIKLÓS LOVÁSZ und YUFEI ZHAO. „On Regularity Lemmas and their Algorithmic Applications“. Combinatorics, Probability and Computing 26, Nr. 4 (28.03.2017): 481–505. http://dx.doi.org/10.1017/s0963548317000049.

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Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

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CONLON, DAVID, JACOB FOX und BENNY SUDAKOV. „Hereditary quasirandomness without regularity“. Mathematical Proceedings of the Cambridge Philosophical Society 164, Nr. 3 (26.01.2017): 385–99. http://dx.doi.org/10.1017/s0305004116001055.

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AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

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3

Chen, Shibing, und Alessio Figalli. „Boundary ε-regularity in optimal transportation“. Advances in Mathematics 273 (März 2015): 540–67. http://dx.doi.org/10.1016/j.aim.2014.12.032.

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4

Gerke, Stefanie, Yoshiharu Kohayakawa, Vojtěch Rödl und Angelika Steger. „Small subsets inherit sparse ε-regularity“. Journal of Combinatorial Theory, Series B 97, Nr. 1 (Januar 2007): 34–56. http://dx.doi.org/10.1016/j.jctb.2006.03.004.

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5

Zhang, Yanjun, und Qiaozhen Ma. „Asymptotic Behavior for a Class of Nonclassical Parabolic Equations“. ISRN Applied Mathematics 2013 (01.09.2013): 1–14. http://dx.doi.org/10.1155/2013/204270.

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This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations ut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.

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Hasselblatt, Boris. „Regularity of the Anosov splitting and of horospheric foliations“. Ergodic Theory and Dynamical Systems 14, Nr. 4 (Dezember 1994): 645–66. http://dx.doi.org/10.1017/s0143385700008105.

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Abstract‘Bunching’ conditions on an Anosov system guarantee the regularity of the Anosov splitting up toC2−ε. Open dense sets of symplectic Anosov systems and geodesic flows do not have Anosov splitting exceeding the asserted regularity. This is the first local construction of low-regularity examples.

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HOSSEINI, KAAVE, SHACHAR LOVETT, GUY MOSHKOVITZ und ASAF SHAPIRA. „An improved lower bound for arithmetic regularity“. Mathematical Proceedings of the Cambridge Philosophical Society 161, Nr. 2 (11.03.2016): 193–97. http://dx.doi.org/10.1017/s030500411600013x.

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AbstractThe arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemerédi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → [0, 1], there exists a subgroup H ⩽ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for 1 − ε fraction of the cosets, the nontrivial Fourier coefficients are bounded by ε, then Green shows that |G/H| is bounded by a tower of twos of height 1/ε3. He also gives an example showing that a tower of height Ω(log 1/ε) is necessary. Here, we give an improved example, showing that a tower of height Ω(1/ε) is necessary.

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Chen, Jianyi, Zhitao Zhang, Guijuan Chang und Jing Zhao. „Periodic Solutions to Klein–Gordon Systems with Linear Couplings“. Advanced Nonlinear Studies 21, Nr. 3 (17.07.2021): 633–60. http://dx.doi.org/10.1515/ans-2021-2138.

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Abstract In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u , v u,v satisfy the Dirichlet boundary conditions on spatial interval [ 0 , π ] [0,\pi] , b > 0 b>0 and f , g f,g are 2 ⁢ π 2\pi -periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) with time period 2 ⁢ π 2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε → 0 \varepsilon\to 0 , ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

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Miura, Tatsuya, und Felix Otto. „Sharp boundary ε-regularity of optimal transport maps“. Advances in Mathematics 381 (April 2021): 107603. http://dx.doi.org/10.1016/j.aim.2021.107603.

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Han, Xiaoli, und Jun Sun. „An ε-regularity theorem for the mean curvature flow“. Journal of Geometry and Physics 62, Nr. 12 (Dezember 2012): 2329–36. http://dx.doi.org/10.1016/j.geomphys.2012.07.009.

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11

Colton, David, und Rainer Kress. „Time harmonic electromagnetic waves in an inhomogeneous medium“. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 116, Nr. 3-4 (1990): 279–93. http://dx.doi.org/10.1017/s0308210500031516.

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SynopsisWe consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support, i.e. the permittivity ε = ε(x) and the conductivity σ = σ(x) are functions of x ∊ ℝ3. Existence, uniqueness and regularity results are established for the direct scattering problem. Then, based on existence and uniqueness results for the exterior and interior impedance boundary value problem, a method is presented for solving the inverse scattering problem.

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EBMEYER, C. „STEADY FLOW OF FLUIDS WITH SHEAR-DEPENDENT VISCOSITY UNDER MIXED BOUNDARY VALUE CONDITIONS IN POLYHEDRAL DOMAINS“. Mathematical Models and Methods in Applied Sciences 10, Nr. 05 (Juli 2000): 629–50. http://dx.doi.org/10.1142/s0218202500000343.

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In this paper the system of partial differential equations [Formula: see text] is studied, where e is the symmetrized gradient of u, and T has p-structure for some p<2 (e.g. div T is the p-Laplacian and p<2). Mixed boundary value conditions on a three-dimensional polyhedral domain are considered. Ws,p-regularity (s=3/2-ε) of the velocity u and Wr,p′-regularity of the pressure π are proven.

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Zhuge, Jinping. „Regularity theory of elliptic systems in ε-scale flat domains“. Advances in Mathematics 379 (März 2021): 107566. http://dx.doi.org/10.1016/j.aim.2021.107566.

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14

Schikorra, Armin. „$$\varepsilon $$ ε -regularity for systems involving non-local, antisymmetric operators“. Calculus of Variations and Partial Differential Equations 54, Nr. 4 (20.08.2015): 3531–70. http://dx.doi.org/10.1007/s00526-015-0913-3.

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15

Huang, Shaosai. „ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons“. International Mathematics Research Notices 2020, Nr. 5 (18.04.2018): 1511–74. http://dx.doi.org/10.1093/imrn/rny069.

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Abstract A closed four-dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^{2}$-norm of the curvature. In this paper, we localize this fact in the case of gradient shrinking Ricci solitons by proving an $\varepsilon $-regularity theorem, thus confirming a conjecture of Cheeger–Tian [20]. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four-dimensional gradient shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

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FOX, JACOB, und BENNY SUDAKOV. „Decompositions into Subgraphs of Small Diameter“. Combinatorics, Probability and Computing 19, Nr. 5-6 (09.06.2010): 753–74. http://dx.doi.org/10.1017/s0963548310000040.

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We investigate decompositions of a graph into a small number of low-diameter subgraphs. Let P(n, ε, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ ⋅⋅⋅ ∪ Ek such that |E0| ≤ εn2, and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi's regularity lemma, Polcyn and Ruciński showed that P(n, ε, 4) is bounded above by a constant depending only on ε. This shows that every dense graph can be partitioned into a small number of ‘small worlds’ provided that a few edges can be ignored. Improving on their result, we determine P(n, ε, d) within an absolute constant factor, showing that P(n, ε, 2) = Θ(n) is unbounded for ε < 1/4, P(n, ε, 3) = Θ(1/ε2) for ε > n−1/2 and P(n, ε, 4) = Θ(1/ε) for ε > n−1. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low-diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, Rödl, Ruciński and Szemerédi.

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Fischer, Julian, und Stefan Neukamm. „Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems“. Archive for Rational Mechanics and Analysis 242, Nr. 1 (30.06.2021): 343–452. http://dx.doi.org/10.1007/s00205-021-01686-9.

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AbstractWe derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $$\mathbb {R}^d$$ R d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $$\varepsilon >0$$ ε > 0 , we establish homogenization error estimates of the order $$\varepsilon $$ ε in case $$d\geqq 3$$ d ≧ 3 , and of the order $$\varepsilon |\log \varepsilon |^{1/2}$$ ε | log ε | 1 / 2 in case $$d=2$$ d = 2 . Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $$\varepsilon ^\delta $$ ε δ . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $$(L/\varepsilon )^{-d/2}$$ ( L / ε ) - d / 2 for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) $$C^{1,\alpha }$$ C 1 , α regularity theory is available.

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18

Elshamy, M. „Randomly perturbed vibrations“. Journal of Applied Probability 32, Nr. 02 (Juni 1995): 417–28. http://dx.doi.org/10.1017/s0021900200102876.

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Let u ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u ε(t, x) from u 0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u ε(t, x).

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Matache, A. M., und J. M. Melenk. „Two-Scale Regularity for Homogenization Problems with Nonsmooth Fine Scale Geometry“. Mathematical Models and Methods in Applied Sciences 13, Nr. 07 (Juli 2003): 1053–80. http://dx.doi.org/10.1142/s0218202503002817.

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Elliptic problems on unbounded domains with periodic coefficients and geometries are analyzed and two-scale regularity results for the solution are given. These are based on a detailed analysis in weighted Sobolev spaces of the so-called unit-cell problem in which the critical parameters (the period ε, the wave number t, and the differentiation order) enter explicitly.

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Amosov, Andrey, und Nikita Krymov. „On a Nonlinear Initial—Boundary Value Problem with Venttsel Type Boundary Conditions Arizing in Homogenization of Complex Heat Transfer Problems“. Mathematics 10, Nr. 11 (31.05.2022): 1890. http://dx.doi.org/10.3390/math10111890.

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We consider a non-standard nonlinear singularly perturbed 2D initial-boundary value problem with Venttsel type boundary conditions, arising in homogenization of radiative-conductive heat transfer problems. We establish existence, uniqueness and regularity of a weak solution v. We obtained estimates for the derivatives Dtv, D12v, D22v, D1D2v with a qualified order in the small parameter ε.

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Tuan, H. D. „On solution sets of nonconvex Darboux problems and applications to optimal control with endpoint constraints“. Journal of the Australian Mathematical Society. Series B. Applied Mathematics 37, Nr. 3 (Januar 1996): 354–91. http://dx.doi.org/10.1017/s0334270000010729.

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AbstractWe prove a continuous version of a relaxation theorem for the nonconvex Darboux problem xlt ε F(t, τ, x, xt, xτ). This result allows us to use Warga's open mapping theorem for deriving necessary conditions in the form of a maximum principle for optimization problems with endpoint constraints. Neither constraint qualification nor regularity assumption is supposed.

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Uskov, Vladimir I. „Boundary layer phenomenon in a first-order algebraic-differential equation“. Russian Universities Reports. Mathematics, Nr. 144 (2023): 436–46. http://dx.doi.org/10.20310/2686-9667-2023-28-144-436-446.

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The Cauchy problem for the first-order algebraic differential equation is considered A du/dt=(B+εC+ε^2 D)u(t,ε), u(t_0,ε)=u^0 (ε)∈E_1, where A,B,C,D are closed linear operators acting from a Banach space E_1 to a Banach space E_2 with domains everywhere dense in E_1, u^0 is a holomorphic function at the point ε=0, ε is a small parameter, t∈[t_0; t_max]. Such equations describe, in particular, the processes of filtration and moisture transfer, transverse vibrations of plates, vibrations in DNA molecules, phenomena in electromechanical systems, etc. The operator A is the Fredholm operator with zero index. The aim of the work is to study the boundary layer phenomenon caused by the presence of a small parameter. The necessary information and statements are given. A bifurcation equation is obtained. Two cases are considered: a) boundary layer functions of one type, b) boundary layer functions of two types. Newton’s diagram is used to solve the bifurcation equation. In both, the conditions under which boundary layer phenomenon arises are obtained — these are the conditions for the regularity of degeneracy. Case a) is illustrated by an example of the Cauchy problem with certain operator coefficients acting in the space R^4.

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Bourgeois, Laurent, und Lucas Chesnel. „On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates“. ESAIM: Mathematical Modelling and Numerical Analysis 54, Nr. 2 (18.02.2020): 493–529. http://dx.doi.org/10.1051/m2an/2019073.

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We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

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Zhang, Hui. „A Global Regularity Result for the 2D Generalized Magneto-Micropolar Equations“. Journal of Function Spaces 2022 (24.02.2022): 1–6. http://dx.doi.org/10.1155/2022/1501851.

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In this paper, we proved the global (in time) regularity for smooth solution to the 2D generalized magneto-micropolar equations with zero viscosity. When there is no kinematic viscosity in the momentum equation, it is difficult to examine the bounds on the any derivatives of the velocity J ε u L 2 . In order to overcome the main obstacle, we find a new unknown quantity which is by combining the vorticity and the microrotation angular velocity; the structure of the system including the combined quantity obeys a Beale–Kato–Majda criterion. Moreover, the maximal regularity of parabolic equations together with the classic commutator estimates allows us to derive the H s estimates for solutions of the system.

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Hai, Dang Dinh, und Dang Dinh Ang. „Regularisation of Abel's integral equation“. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 107, Nr. 1-2 (1987): 165–68. http://dx.doi.org/10.1017/s0308210500029425.

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SynopsisConsider the Abel integral operatorwhere 0 < α < 1. Suppose u is in H1(0, 1) of H1-norm ≦E, and f is an element of L2(0, 1) such that ∥Au – f∥L−2 < ε. We give a regularised approximate solution uβ(f) of the equationwhich satisfiesand can be computed simply by performing some integrations. The preceding error estimate can be sharpened by strengthening regularity conditions on u

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Nesvidomin, Andrii, Serhii Pylypaka, Tetiana Volina, Irina Rybenko und Alla Rebrii. „Analytical connection between the Frenet trihedron of a direct curve and the Darboux trihedron of the same curve on the surface“. Technology audit and production reserves 4, Nr. 2(78) (27.08.2024): 54–59. http://dx.doi.org/10.15587/2706-5448.2024.310524.

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Frenet and Darboux trihedrons are the objects of research. At the current point of the direction curve of the Frenet trihedron, three mutually perpendicular unit orthogonal vectors can be uniquely constructed. The orthogonal vector of the tangent is directed along the tangent to the curve at the current point. The orthogonal vector of the main normal is located in the plane, which is formed by three points of the curve on different sides from the current one when they are maximally close to the current point. It is directed to the center of the curvature of the curve. The orthogonal vector of the binormal is perpendicular to the two previous orthogonal vectors and has a direction according to the rule of the right coordinate system. Thus, the movement of the Frenet trihedron along the base curve, as a solid body, is determined. The Darboux trihedron is also a right-hand coordinate system that moves along the base curve lying on the surface. Its orthogonal vector of the tangent is directed identically to the Frenet trihedron, and other orthogonal vectors in pairs form a certain angle ε with the orthogonal vectors of the Frenet trihedron. This is because one of the orthogonal vectors of the Darboux trihedron is normal to the surface and forms a certain angle ε with the binormal. Accordingly, the third orthogonal vector of the Darboux trihedron forms an angle ε with the orthogonal vector of the normal of the Frenet trihedron. This orthogonal vector and orthogonal vector of the tangent form the tangent plane to the surface at the current point of the curve, and the corresponding orthogonal vectors of the tangent and the normal of the Frenet trihedron form the tangent plane of the curve at the same point. Thus, when the Frenet and Darboux trihedrons move along a curve with combined vertices, there is a rotation around the common orthogonal vector point of the tangent at an angle ε between the osculating plane of the Frenet trihedron and the tangent plane to the surface of the Darboux trihedron. These trihedrons coincide in a separate case (for a flat curve) (ε=0). The connection between Frenet and Darboux trihedrons – finding the expression for the angle ε, is considered in the article. The inverse problem – the determination of the movement of the Darboux trihedron at a given regularity of the change of the angle ε, is also considered. A partial case is considered and it is shown that for a flat base curve at ε=const, the set of positions of the orthogonal vector of normal forms a developable surface of the same angle of inclination of the generators. In addition, the inverse problem of finding the regularity of the change of the angle ε between the corresponding orthogonal vectors of the trihedrons allows constructing a ruled surface for the gravitational descent of the load, conventionally assumed to be a particle. At the same time, the balance of forces in the projections on the orthogonal vectors of the trihedron in the common normal plane of the trajectory is considered. This balance depends on the angle ε.

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Arutyunov, Aram V., Dmitry Yu Karamzin und Fernando Lobo Pereira. „Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints“. Axioms 11, Nr. 2 (20.01.2022): 40. http://dx.doi.org/10.3390/axioms11020040.

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This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an ε-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained.

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Cai, Yongyong, und Yan Wang. „A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime“. ESAIM: Mathematical Modelling and Numerical Analysis 52, Nr. 2 (März 2018): 543–66. http://dx.doi.org/10.1051/m2an/2018015.

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A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε2) and O (1) in time and space, respectively. In the nonrelativistic regime,i.e., 0 < ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds inε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as hm0+τ2/ε2andhm0 + τ2 + ε2, where his the mesh size, τis the time step and m0depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ2) in the regimes when either ε = O (1) or 0 < ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.

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MUNTEAN, A., und T. L. VAN NOORDEN. „Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity“. European Journal of Applied Mathematics 24, Nr. 5 (02.04.2013): 657–77. http://dx.doi.org/10.1017/s0956792513000090.

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We prove an upper bound for the convergence rate of the homogenization limit ε → 0 for a linear transmission problem for a advection–diffusion(–reaction) system posed in areas with low and high diffusivity, where ε is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity.Eur. J. Appl. Math.22, 493–516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.

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Favini, Angelo, Ciprian G. Gal, Gisèle Ruiz Goldstein, Jerome A. Goldstein und Silvia Romanelli. „The non-autonomous wave equation with general Wentzell boundary conditions“. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, Nr. 2 (April 2005): 317–29. http://dx.doi.org/10.1017/s0308210500003905.

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We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε > 0 in [0, 1] × [0, + ∞) and βj(t) > 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

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Chrysafinos, Konstantinos. „Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation“. ESAIM: Mathematical Modelling and Numerical Analysis 53, Nr. 2 (März 2019): 551–83. http://dx.doi.org/10.1051/m2an/2018071.

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Fully-discrete approximations of the Allen–Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/ε) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique.

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MÁRMOL, MACARENA GÓMEZ, und FRANCISCO ORTEGÓN GALLEGO. „EXISTENCE OF SOLUTION TO NONLINEAR ELLIPTIC SYSTEMS ARISING IN TURBULENCE MODELLING“. Mathematical Models and Methods in Applied Sciences 10, Nr. 02 (März 2000): 247–60. http://dx.doi.org/10.1142/s021820250000015x.

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We study some nonlinear elliptic systems governing the steady-state of a two-equation turbulence model that has been derived from the so-called k–ε model. Two kinds of problems are considered: in the first one, we drop out transport terms and we deduce the existence of a solution for [Formula: see text]; in the second one we take into account all transport terms; in this case, the existence result holds for N=2 or 3. Positivity and [Formula: see text]-regularity of the scalar quantities are also shown here.

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33

Elshamy, M. „Randomly perturbed vibrations“. Journal of Applied Probability 32, Nr. 2 (Juni 1995): 417–28. http://dx.doi.org/10.2307/3215297.

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Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

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Pratelli, Aldo, und Giorgio Saracco. „The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets“. Advanced Nonlinear Studies 20, Nr. 3 (01.08.2020): 539–55. http://dx.doi.org/10.1515/ans-2020-2074.

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AbstractWe prove the validity of the {\varepsilon-\varepsilon^{\beta}} property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

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Ricco, Laura, Saverio Russo, Giustino Orefice und Fernando Riva. „Anionic Poly(ε-caprolactam): Relationships among Conditions of Synthesis, Chain Regularity, Reticular Order, and Polymorphism“. Macromolecules 32, Nr. 23 (November 1999): 7726–31. http://dx.doi.org/10.1021/ma9909004.

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Zhang, Jie, und Wenjun Liu. „The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain“. Nonlinearity 37, Nr. 7 (12.06.2024): 075024. http://dx.doi.org/10.1088/1361-6544/ad5131.

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Abstract In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter ε ∈ ( 0 , ∞ ) , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are μ = O ( 1 ) and ν = O ( ε α ) , and the orders of magnetic diffusion coefficients κ and σ are κ = O ( 1 ) and σ = O ( ε α ) , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For H 1 -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For H 1 -initial data with additional regularity ( ∂ z A ~ 0 , ∂ z B ~ 0 ) ∈ L p ( Ω ) ( 2 < p < ∞ ) , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal. 272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For H 2 -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order O ( ε γ / 2 ) , where γ = min { 2 , α − 2 } with α ∈ ( 2 , ∞ ) . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order O ( ε ) .

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Harfash, Akil, und Ghufran A. Al-Musawi. „Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential“. Gulf Journal of Mathematics 16, Nr. 1 (25.02.2024): 9–35. http://dx.doi.org/10.56947/gjom.v16i1.1368.

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We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

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Haghighi, Hassan, und Mohammad Mosakhani. „Containment Problem for Quasi Star Configurations of Points in ℙ2“. Algebra Colloquium 25, Nr. 04 (Dezember 2018): 661–70. http://dx.doi.org/10.1142/s1005386718000469.

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In this paper, the containment problem for the defining ideal of a special type of zero-dimensional subscheme of ℙ2, the so-called quasi star configuration, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number [Formula: see text], we construct an infinite family of homogeneous radical ideals of points in 𝕂[ℙ2] such that their resurgences lie in the interval [2−ε, 2). Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that all of these ordinary powers have a linear resolution.

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Zhang, Yu Hui, und Xian Min Zhang. „The Study of Large Vibration Compaction on Soil-Stone Mixture Poisson′s Ratio with Wave Velocity“. Advanced Materials Research 446-449 (Januar 2012): 1497–501. http://dx.doi.org/10.4028/www.scientific.net/amr.446-449.1497.

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The equipment of large vibration compaction was designed and adopted to simulate the changes of mechanical properties of mixture, by large laboratory vibration compaction test under the condition of different water content and different soil-stone proportion with sandy soil, silt soil and low liquid-limit clay. The deformation capability in each direction of foundation mixture in different compactness was studied by analyzing poisson′s ratio. The change law of poisson′s ratio, shear-wave velocity, longitudinal wave velocity of mixture in different compaction degree were revealed. Model of correlation between poisson′s ratio (μ) and ratio of the S to P wave velocity (ε) was proposed. The model was verified by extra test. The results show that the relationship between μ and ε is not accord with theoretical formula because of special particle shape and distribution of the soil-stone mixture. The change regularity of mixture poisson′s ratio in different conditions can be reflected well by the model. The new method of field test of poisson′s ratio in soil-stone foundation with ratio of the S to P wave velocity is proposed. The conclusion is valuable in poisson′s ratio study of soil-stone mixture.

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Fernández-Real, Xavier, und Joaquim Serra. „Regularity of minimal surfaces with lower-dimensional obstacles“. Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, Nr. 767 (01.10.2020): 37–75. http://dx.doi.org/10.1515/crelle-2019-0035.

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AbstractWe study the Plateau problem with a lower-dimensional obstacle in {\mathbb{R}^{n}}. Intuitively, in {\mathbb{R}^{3}} this corresponds to a soap film (spanning a given contour) that is pushed from below by a “vertical” 2D half-space (or some smooth deformation of it). We establish almost optimal {C^{1,\frac{1}{2}-}} estimates for the solutions near points on the free boundary of the contact set, in any dimension {n\geq 2}. The {C^{1,\frac{1}{2}-}} estimates follow from an ε-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi’s improvement of flatness. To prove it, we follow Savin’s small perturbations method. A nontrivial difficulty in using Savin’s approach for minimal surfaces with thin obstacles is that near a typical contact point the solution consists of two smooth surfaces that intersect transversally, and hence it is not very flat at small scales. Via a new “dichotomy approach” based on barrier arguments we are able to overcome this difficulty and prove the desired result.

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Yan, Ya Guang. „Influence of Vented Tunnel Hood on Aerodynamic Characteristics“. Applied Mechanics and Materials 256-259 (Dezember 2012): 1340–43. http://dx.doi.org/10.4028/www.scientific.net/amm.256-259.1340.

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On the basis of the three-dimensional ,unsteady-state , viscous N-S Equation and k-ε two equation turbulence model , the simulation calculation is made with the finite volume method on the aerodynamic effect of a high-speed train passing a tunnel.The numerical computation simulates the pressure change in the tunnel,analysises regularity of the the initial compression wave and the pressure gradient wave,compares the mitigation effect of the two hoods on the pressure gradient.The following conclusions have been drawn from the research: The amplitude of the initial compression wave is not significantly reduced with hood,however,the amplitude of the pressure gradient is significantly reduced with hood.And the mitigation effect of the vented hood is better than the unvented one.

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Herrmann, L., und C. Schwab. „Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients“. ESAIM: Mathematical Modelling and Numerical Analysis 53, Nr. 5 (06.08.2019): 1507–52. http://dx.doi.org/10.1051/m2an/2019016.

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We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

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Briane, Marc. „Homogenization of the Torsion Problem and the Neumann Problem in Nonregular Periodically Perforated Domains“. Mathematical Models and Methods in Applied Sciences 07, Nr. 06 (September 1997): 847–70. http://dx.doi.org/10.1142/s0218202597000438.

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This paper is devoted to the homogenization of the torsion problem (or stiff problem) and the Neumann problem (or soft problem) for second-order elliptic but not necessarily symmetric linear operators set in a bounded open subset Ω of ℝN. More precisely, we study the asymptotic behavior of the equations [Formula: see text] in Ω with ε → 0, where Sε is a closed subset of Ω, which represents the set of the inclusions for the stiff problem or the holes for the soft one, and Ωε = Ω \ Sε. The stiff problem corresponds to δ → + ∞ and ν = 0, the soft one to δ → 0 and ν = 1. We prove a homogenization result in the periodic case without assuming any regularity on the set Sε and thus generalizing the result of Cioranescu and Saint Jean Paulin.7

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Sugiyama, Yoshie. „ε-Regularity theorem and its application to the blow-up solutions of Keller–Segel systems in higher dimensions“. Journal of Mathematical Analysis and Applications 364, Nr. 1 (April 2010): 51–70. http://dx.doi.org/10.1016/j.jmaa.2009.11.019.

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Pinto, Martin. „A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation“. International Journal of Applied Mathematics and Computer Science 17, Nr. 3 (01.10.2007): 351–59. http://dx.doi.org/10.2478/v10006-007-0029-9.

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A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson EquationThis article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1 + 1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be saidoptimalin the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge inL∞towards the exact ones as ε and δttend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.

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LUCE, ROBERT, CÉCILE POUTOUS und JEAN-MARIE THOMAS. „WEAKENED CONDITIONS OF ADMISSIBILITY OF SURFACE FORCES APPLIED TO LINEARLY ELASTIC MEMBRANE SHELLS“. Analysis and Applications 06, Nr. 03 (Juli 2008): 247–67. http://dx.doi.org/10.1142/s0219530508001158.

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We consider a family of linearly elastic shells of the first kind (as defined in [2]), also known as non inhibited pure bending shells [7]. This family is indexed by the half-thickness ε. When ε approaches zero, the averages across the thickness of the shell of the covariant components of the displacement of the points of the shell converge strongly towards the solution of a "2D generalized membrane shell problem" provided the applied forces satisfy admissibility conditions [1,3]. The identification of the admissible applied forces usually requires delicate analysis. In the first part of this paper, we simplify the general admissibility conditions when applied forces h are surface forces only, and obtain conditions that no longer depend on ε [5]: find hαβ = hαβ in L2(ω) such that for all η = (ηi) in V(ω), ∫ω hi ηi dω = ∫ω hαβγαβ(η)dω where ω is a domain of ℝ2, θ is in [Formula: see text] and [Formula: see text] is the middle surface of the shells, where (γαβ (η)) is the linearized strain tensor of S and V(ω) = {η ∈ H1(ω), η = 0 on γ0}, the shells being clamped along Γ0 = θ(γ0). In the second part, since the simplified admissibility formulation does not allow to conclude directly to the existence of hαβ, we seek sufficient conditions on h for hαβ to exist in L2(ω). In order to get them, we impose more regularity to hαβ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE's with hαβ as unknowns. The existence of solutions depends both on the geometry of the shell and on the choice of h. We carry through the study of four representative geometries of shells and identify in each case a special admissibility functional space for h.

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Du, Yihong, Zongming Guo und Kelei Wang. „Monotonicity formula and $$\varepsilon $$ ε -regularity of stable solutions to supercritical problems and applications to finite Morse index solutions“. Calculus of Variations and Partial Differential Equations 50, Nr. 3-4 (25.06.2013): 615–38. http://dx.doi.org/10.1007/s00526-013-0649-x.

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Bir, Bikram, Deepjyoti Goswami und Amiya K. Pani. „Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data“. Computational Methods in Applied Mathematics 22, Nr. 2 (12.02.2022): 297–325. http://dx.doi.org/10.1515/cmam-2022-0012.

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Abstract In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 {\mathbf{H}_{0}^{1}} . New regularity results which are valid uniformly in time as t → ∞ {t\to\infty} and in the penalty parameter ε as ε → 0 {\varepsilon\to 0} are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 {\mathbf{L}^{2}} as well in 𝐇 1 {\mathbf{H}^{1}} -norms and for the penalized pressure in L 2 {L^{2}} -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.

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Yan, Ya Guang, Qing Shan Yang und Jian Zhang. „Aerodynamic Comfort Analysis of High-Speed Trains Passing each other at the same Speed through a Tunnel“. Applied Mechanics and Materials 90-93 (September 2011): 2147–51. http://dx.doi.org/10.4028/www.scientific.net/amm.90-93.2147.

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On the basis of the three-dimensional,compressible,unsteady-state,viscous N-S Equation and k-ε turbulence model,the simulation calculation is made with the finite volume method on the aerodynamic effect of two high-speed trains passing each other at the same speed through a tunnel,simulating the pressure change,analysising regularity of the pressure,researching comfort in the train.The following conclusions have been drawn from research:The maximum of transient pressure is concerned with the meeting point,when the two high-speed trains meet in tunnel.When the length,the speed and the sealing factor of the train are same,the transient pressure is maximum with the trains meeting at the midpoint of tunnel.When the tunnel length,the speed of the train,the sealing factor and the point of meeting are same,the transient pressure of the longer train is maximum.When the sealing factor is greater than 15s,the standard of permissible pressure (1250Pa\3s) can be satisfied with all circumstances.

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Antonopoulou, Dimitra C. „Higher moments for the stochastic Cahn–Hilliard equation with multiplicative Fourier noise“. Nonlinearity 36, Nr. 2 (03.01.2023): 1053–81. http://dx.doi.org/10.1088/1361-6544/acadc9.

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Abstract We consider in dimensions d = 1 , 2 , 3 the ɛ-dependent stochastic Cahn–Hilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order , γ > 0. The initial condition is non-layered and independent from ɛ. Under general assumptions on the noise diffusion σ, we prove moment estimates in H 1 (and in L ∞ when d = 1). Higher H 2 regularity p-moment estimates are derived when σ is bounded, yielding as well space Hölder and L ∞ bounds for d = 2 , 3 , and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter ɛ. As in the deterministic case, in H 1, H 2, the bounds admit a negative polynomial order in ɛ. Finally, assuming layered initial data of initial energy uniformly bounded in ɛ, as proposed by Chen (1996 J. Differ. Geom. 44 262–311), we use our H 1 2d-moment estimate and prove the stochastic solution’s convergence to ± 1 as ε → 0 a.s. when the noise diffusion has a linear growth.

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