Smooth Compactness Research Articles

Enhancing Resistance to Corrosion and Fouling Using Epoxy Coatings with Superhydrophobic Cells

AbstractAs maritime transport becomes increasingly important, metal corrosion results in significant economic losses. Superhydrophobic materials have significant potential for corrosion prevention. However, poor compatibility between superhydrophobic materials and resin reduces the long‐term corrosion resistance. Interfacial strengthening cells (IS‐Cells) are designed to maintain superhydrophobicity and retain active hydroxyl groups participating in resin‐curing. IS‐Cells also contribute to achieving the maze effect. A novel anti‐corrosion IS coating is fabricated by compatible IS‐Cells and epoxy. IS coatings feature smooth surfaces and vertical compactness, protecting against liquid and gaseous corrosive medium. Strong diffusion resistance and extended diffusion distance ensure long‐term corrosion resistance of IS coatings. Approximately 85 µm of IS coatings remain corrosion‐free after 112 days of immersion in the NaCl solution. The low‐frequency impedance, self‐corrosion potential, and self‐corrosion current density are 1.68 × 109 Ω cm2, −0.0708 V, and 5.38 × 10−11 A cm−2, respectively. IS coatings can also withstand 2400 h of neutral salt spray testing without corrosion in the artificial scratches. Due to the electrostatic repulsion between the hydrophobic C–F chains onto IS‐Cells and protein molecules, IS coatings maintain anti‐fouling properties after dry‐wet immersion tests in the natural marine environment for 90 days. IS coatings show great potential for marine anti‐corrosion applications.

Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

Abstract In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.

Smooth Compactness for Spaces of Asymptotically Conical Self-Expanders of Mean Curvature Flow

AbstractWe show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all 2D self-expanders of a fixed topological type and, in all dimensions, for the set of self-expanders of low entropy and for the set of mean convex self-expanders with strictly mean convex asymptotic cones. From this we deduce that the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones is proper for these classes.

On the Closure of Smooth Compactly Supported Functions in Weighted Hölder Spaces

The closure of the set of smooth compactly supported functions in a weighted Holder space on ℝn, n ≥ 1, with a weight controlling the behavior at the point at infinity is described. As an application, a solvability criterion for operator equations generated by de Rham differentials both in these spaces and on the closure of the set of smooth compactly supported functions in them for n ≥ 2 is obtained.

Quaternion-Valued Smooth Compactly Supported Orthogonal Wavelets with Symmetry

We construct two novel quaternion-valued smooth compactly supported symmetric orthogonal wavelet (QSCSW) filters of length greater than existing ones. In order to obtain their filter coefficients, we propose an optimization-based method for solving a specific kind of multivariate quadratic equations. This method provides a new idea for solving multivariate quadratic equations and could be applied to construct much longer QSCSW filters.

On the Closure of Smooth Compactly Supported Functions in Weighted Hölder Spaces

В работе получено описание замыкания гладких финитных функций в весовом пространстве Гeльдера на пространстве $\mathbb R^n$ для $n\ge 1$ с весом, контролирующим поведение в бесконечно удаленной точке. В качестве применения мы получаем критерий разрешимости операторных уравнений, порожденных дифференциалами де Рама как в этих пространствах, так и на замыкании гладких финитных функций в нем при $n\ge 2$. Библиография: 14 названий.

Curvature estimates for stable free boundary minimal hypersurfaces

Abstract In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

Erratum to: Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

The line between the displayed formulas (16) and (17) was copied incorrectly from [41, Theorem 1].

Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Let ( M n + 1 , g , e − f d μ ) (M^{n+1},g,e^{-f}d\mu ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 2\leq n\leq 6 and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded f f -minimal hypersurfaces in M M with uniform upper bounds on f f -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in R n + 1 \mathbb {R}^{n+1} with 2 ≤ n ≤ 6 2\leq n\leq 6 . We also prove some estimates on the f f -index of f f -minimal hypersurfaces.

Free boundary minimal surfaces in the unit ball with low cohomogeneity

We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers ( m , n ) (m,n) such that m , n > 1 m, n >1 and m + n ≥ 8 m+n\geq 8 , we construct a free boundary minimal surface Σ m , n ⊂ B m + n \Sigma _{m, n} \subset B^{m+n} (1) invariant under O ( m ) × O ( n ) O(m)\times O(n) . When m + n > 8 m+n>8 , an instability of the resulting equation allows us to find an infinite family { Σ m , n , k } k ∈ N \{\Sigma _{m,n, k}\}_{k\in \mathbb {N}} of such surfaces. In particular, { Σ 2 , 2 , k } k ∈ N \{\Sigma _{2, 2, k}\}_{k\in \mathbb {N}} is a family of solid tori which converges to the cone over the Clifford torus as k k goes to infinity. These examples indicate that a smooth compactness theorem for free boundary minimal surfaces due to Fraser and Li does not generally extend to higher dimensions. For each n ≥ 3 n\geq 3 , we prove there is a unique nonplanar S O ( n ) SO(n) -invariant free boundary minimal surface (a “catenoid”) Σ n ⊂ B n ( 1 ) \Sigma _n \subset B^n(1) . These surfaces generalize the “critical catenoid” in B 3 ( 1 ) B^3(1) studied by Fraser and Schoen.

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$| \det A|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.

Stability and compactness for complete 𝑓-minimal surfaces

Let ( M , g ¯ , e − f d μ ) (M,\overline {g}, e^{-f}d\mu ) be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that in M M there is no complete two-sided L f L_f -stable immersed f f -minimal hypersurface with finite weighted volume. Further, if M M is a 3 3 -manifold, we prove a smooth compactness theorem for the space of complete embedded f f -minimal surfaces in M M with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in R 3 \mathbb {R}^3 by Colding-Minicozzi.

Closure, Interior and Compactness in Ordinary Smooth Topological Spaces

It presents the concepts of ordinary smooth interior and ordinary smooth closure of an ordinary subset and their structural properties. It also introduces the notion of ordinary smooth (open) preserving mapping and addresses some their properties. In addition, it develops the notions of ordinary smooth compactness, ordinary smooth almost compactness, and ordinary near compactness and discusses them in the general framework of ordinary smooth topological spaces.

Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

Some Bivariate Smooth Compactly Supported Tight Framelets with Three Generators

For any dilation matrix with integer entries and , we construct a family of smooth compactly supported tight wavelet frames with three generators in . Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.

Smooth Compactly Supported Solutions of Some Underdetermined Elliptic PDE, with Gluing Applications

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the initial elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel of P and also solutions vanishing in a chosen relatively compact open region. We apply the result for natural geometric and physics contexts such as divergence free fields or TT-tensors.

Smooth compactness of self-shrinkers

We prove a smooth compactness theorem for the space of embedded self-shrinkers in \mathbb{R}^3 . Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow.

Some Smooth Compactly Supported Tight Framelets

For any dilation matrix with integer entries, we construct a family of smooth compactly supported tight wavelet frames in $L^2(\mathbb{R}^d)$, $d\ge 1$. Estimates for the degrees of smoothness of these framelets are given. Our construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler.

Relative Smooth Topological Spaces

In 1992, Ramadan introduced the concept of a smooth topological space and relativeness between smooth topological space and fuzzy topological space in Chang's (1968) view points. In this paper we give a new definition of smooth topological space. This definition can be considered as a generalization of the smooth topological space which was given by Ramadan. Some general properties such as relative smooth continuity and relative smooth compactness are studied.

WEAK SMOOTH α-STRUCTURE OF SMOOTH TOPOLOGICAL SPACES

In [3] and [6] the concepts of smooth closure, smooth interior, smooth <TEX>${\alpha}-closure$</TEX> and smooth <TEX>${\alpha}-interior$</TEX> of a fuzzy set were introduced and some of their properties were obtained. In this paper, we introduce the concepts of several types of weak smooth compactness and weak smooth <TEX>${\alpha}-compactness$</TEX> in terms of these concepts introduced in [3] and [61 and investigate some of their properties.