Academic literature on the topic 'Ε-Regularity'

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.

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.

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.

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.

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.

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.

Cette thèse est consacrée à l'étude de la relation entre les variables dans les équations des fluides micro-polaires. Ce système, basé sur les équations de Navier-Stokes, consiste en un couplage de deux variables: le champ de vitesse vec{u} et le champ de micro-rotation vec{w}. Notre objectif est de mieux comprendre comment l'information concernant une variable influence le comportement de l'autre. À cette fin, nous avons divisé cette thèse en quatre chapitres, où nous étudierons les propriétés de régularité locale des solutions faibles de type Leray, puis nous nous concentrerons sur la régularité et l'unicité des solutions faibles dans le cas stationnaire. Le premier chapitre présente une rapide déduction physique des équations micro-polaires, suivie de la construction des solutions faibles de type Leray. Dans le chapitre 2, nous commençons par prouver un gain d'intégrabilité pour les deux variables vec{u} et vec{w} lorsque la vitesse appartient à certains espaces de Morrey. Ce résultat souligne un effet de domination de la vitesse. Nous montrons ensuite que cet effet peut également être observé dans le cadre de la théorie de Caffarelli-Kohn-Nirenberg, i.e., sous une hypothèse de petitesse supplémentaire uniquement sur le gradient de la vitesse, nous pouvons démontrer que la solution devient Hölder continue. Pour cela, nous introduisons la notion de solution partiellement adaptée, qui est fondamentale dans ce travail et représente l'une des principales nouveautés. Dans la dernière section de ce chapitre, nous obtenons des résultats similaires dans le contexte du critère de Serrin. Dans le chapitre 3, nous nous concentrons sur le comportement de la norme L^3 de la vitesse vec{u} autour des possibles points où la régularité peut être perdue. Plus précisément, nous établissons un critère d'explosion pour la norme L^3 de la vitesse et améliorons ce résultat en présentant un phénomène de concentration. Nous vérifions également que le cas limite L^infty_t L^3_x du critère de Serrin reste valable pour les équations des fluides micro-polaires. Enfin, le problème de l'existence et de l'unicité des équations stationnaires des fluides micro-polaires est abordé dans le chapitre 4. En effet, nous prouvons l'existence de solutions faibles (vec{u}, vec{w}) dans l'espace d'énergie naturel dot{H}^1(mathbb{R}^3) imes H^1(mathbb{R}^3). De plus, en utilisant la relation entre les variables, nous déduisons que ces solutions sont régulières. Il convient de noter que la solution triviale peut ne pas être unique, et pour surmonter cette difficulté, nous développons un théorème de type Liouville. Ainsi, nous démontrons qu'en imposant une décroissance plus forte à l'infini uniquement sur vec{u}, nous pouvons en déduire l'unicité de la solution triviale (vec{u},vec{w})=(0,0)
This thesis is devoted to the study of the relationship between the variables in the micropolar fluids equations. This system, which is based on the Navier-Stokes equations, consists in a coupling of two variables: the velocity field vec{u} and the microrotation field vec{w}. Our aim is to provide a better understanding of how information about one variable influences the behavior of the other. To this end, we have divided this thesis into four chapters, where we will study the local regularity properties of Leray-type weak solutions, and later we will focus on the regularity and uniqueness of weak solutions for the stationary case. The first chapter presents a brief physical derivation of the micropolar equations followed by the construction of the Leray-type weak solutions. In Chapter 2, we begin by proving a gain of integrability for both variables vec{u} and vec{w} whenever the velocity belongs to certain Morrey spaces. This result highlights an effect of domination by the velocity. We then show that this effect can also be observed within the framework of the Caffarelli-Kohn-Nirenberg theory, i.e., under an additional smallness hypothesis only on the gradient of the velocity, we can demonstrate that the solution becomes Hölder continuous. For this, we introduce the notion of a partial suitable solution, which is fundamental in this work and represents one of the main novelties. In the last section of this chapter, we derive similar results in the context of the Serrin criterion. In Chapter 3, we focus on the behavior of the L^3-norm of the velocity vec{u} near possible points where regularity may get lost. More precisely, we establish a blow-up criterion for the L^3 norm of the velocity and we improve this result by presenting a concentration phenomenon. We also verify that the limit point L^infty_t L^3_x of the Serrin criterion remains valid for the micropolar fluids equations. Finally, the problem of existence and uniqueness for the stationary micropolar fluids equations is addressed in Chapter 4. Indeed, we prove the existence of weak solutions (vec{u}, vec{w}) in the natural energy space dot{H}^1(mathbb{R}^3) imes H^1(mathbb{R}^3). Moreover, by using the relationship between the variables, we deduce that these solutions are regular. It is worth noting that the trivial solution may not be unique, and to overcome this difficulty, we develop a Liouville-type theorem. Hence, we demonstrate that by imposing stronger decay at infinity only on vec{u}, we can infer the uniqueness of the trivial solution (vec{u},vec{w})=(0,0)

In the paper, three dimensional unsteady CFD numerical method is established to simulate load rejection transient of Francis pump-turbine in pump mode. First, pumped storage power station’s whole flow system geometric model including diversion tunnel, surge tank, penstock, pump turbine unit and tailrace tunnel has been built and subdivided with prism and tetrahedral mesh. Then, three dimensional unsteady CFD simulation begins from the original steady pump condition, with the Realizable k–ε turbulent model. Through numerical calculation, the variation regularity of unit rotate speed, flow rate, torque, axial thrust and static pressure of measuring points with time are revealed during the load rejection transient. By comparison with experimental data, the changing regularity of transient dynamic parameters can be verified and their differences in the detail also can be reflected. Simultaneously, the phenomenon of water hammer has been captured by recording time-varying static pressure of measuring points. Furthermore, flow configuration in the passage undergoes very complex and unsteady change during load rejection transient.

In order to reveal the characteristics of pressure fluctuation in centrifugal pump, based on finite volume method, RNG k–ε turbulence model and sliding mesh, the three-dimensional unsteady incompressible viscous flow in a low-specific-speed centrifugal pump is simulated numerically at different flow rates, wherein SIMPLE arithmetic is used to couple pressure and velocity. The calculation region consists of straight suction chamber, impeller and spiral casing. The results show that pressure wave presents periodic sine or cosine regularity in spiral casing, while the characteristic doesn’t appear in suction chamber. In suction chamber, the dominant frequency of pressure fluctuation is equal to rotational frequency of impeller. And in spiral casing, the dominant frequency of pressure fluctuation is equal to the product of rotational frequency of impeller and blade numbers. The dominant frequency of pressure fluctuation for each detection point is constant at any operation conditions. With the augment of operation flow rate, local average pressure in suction chamber will gradually increase, while local average pressure in spiral casing will gradually decrease. The pressure fluctuation at tongue will be more violent as flow rate increases. The pressure fluctuation in spiral volute will gradually decline along rotational direction of impeller.

Helically coiled tubes are widely used in many industrial applications such as the steam generator in the high-temperature gas-cooled reactor which is recognized as one of the new generation advanced reactors. The thermophysical properties of fluids exhibit drastic and fast changes in the pseudocritical region so that the flow and heat transfer characteristics of supercritical pressure fluids are greatly different from those at the subcritical pressure. The paper presents results of numerical investigation on turbulent heat transfer of supercritical CO2 in a helically coiled tube with a tube diameter of 9 mm, a coil diameter of 283 mm and a coil pitch of 32 mm under the constant wall heat flux. Both the RNG k-ε model with enhanced wall function and the SST k-ω model were applied in the simulations, and the results showed that the SST k-ω model agreed better with the experimental results in the literature. Effects of buoyancy and flow acceleration were evaluated. Details of developing heat transfer characteristics at three specific cross sections were analyzed. The heat transfer regularity and mechanism presented in this work can be useful for the design and development of more economic and safer design of the supercritical steam generator.