Order of Continuity From Least to Greatest Complexity for the Excretory System is

Measures on Clans and on MV-Algebras

Barbieri Giuseppina , Hans Weber , in Handbook of Measure Theory, 2002

PROOF

(a)

Obviously, order continuity implies exhaustivity.

Let $a=\sup N\left(u\right)$ Then (L, u) is isomorphic to $\left(\left[0,a\right],{u|}_{\left[0,a\right]}\right)\times \left(\left[0,a\text{'}\right],{u|}_{\left[0,a\text{'}\right]}\right)$ u induces on $\left[0,a\right]$ the trivial uniformity and on $\left[0,a\text{'}\right]$ a Hausdorff uniformity. Therefore, by Theorem 4.2.3(b), $\left(\left[0,a\text{'}\right],{u|}_{\left[0,a\text{'}\right]}\right)$ is isomorphic to a product ${\displaystyle {\prod }_{\lambda \in \Lambda }\left(L_{\lambda },u_{\lambda }\right)}$ of metrizable

MV-algebras. Since u is order continuous and L is complete, u_λ is order continuous and L_λ; is complete (as a lattice) and therefore $\left(L_{\lambda },u_{\lambda }\right)$ is complete uniform space by Theorem 2.5. Since a product of complete uniform spaces is complete, it follows that (L, u) is complete.

(b)

Let (x_α ) be an increasing net. Then (x_α ) is a Cauchy net in (L, u) since u is exhaustive and therefore has a limit x in (L, u) since (L, u) is complete. By Proposition 4.1.7 $x=\sup x_{\alpha }$ . It follows that L is a complete lattice and u is order continuous.

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Geometric Continuity

Jörg Peters , in Handbook of Computer Aided Geometric Design, 2002

8.3.2 C ^k manifolds

Differential geometry has a well-established notion of continuity for a point set: to verify k th order continuity, we must find, for every point Q in the point set, an invertible C ^k map (chart) that maps an open surface-neighborhood of Q into an open set in R². If two surface-neighborhoods, with charts q ₁ and q ₂ respectively, overlap then q ₂○q ⁻¹ ₁ : R² → R² must be a C ^k function. This notion of continuity is not constructive: while it defines when a point set can be given the structure of a C ^k manifold, say a C ^k surface, it neither provides tools to build a C ^k surface nor a mechanism suitable for verification by computer.

However, geometric continuity and the continuity of manifolds are closely related: every point in the union of the patches of a G ^k free-form surface spline admits local parametrization by C ^k charts if the surface does not self-intersect: the union is an immersed C ^k surface with piecewise C ^k boundary. We face two types of obstacles in establishing this fact. First, the geometry maps should not have geometric singularities on their respective domains since these would prevent invertibility of the charts, and the spline should not self-intersect so that we can map a neighborhood of the point in R³ to the plane in a 1−1 fashion. Establishing regularity and non-self-intersection requires potentially expensive intersection testing (chapter 25 on Intersection Problems). The second apparent obstacle is that the patches that make up the surface are closed sets that join without overlap. Therefore the geometry maps cannot directly be used as charts. However, as illustrated in Figure 8.11, we can think of the charts as piecewise maps composed of n maps q _i = r ⁻¹ _i ○ g ⁻¹ _i that map open neighborhoods in R³ (grey oval) to open neighborhoods in R² (grey disk). The constructions [50],[78],[19] explicitly start by constructing the union of neighborhoods and connecting charts and then compose these with (rational) spline basis functions.

Figure 8.11. In the neighborhood of Q, a G ^k free-form surface spline with reparametrizations r = r i°r ⁻¹ _i can be viewed as a manifold with chart consisting of pieces q _i = r ⁻¹ _i ° g ⁻¹ _i , i = 1, …,n.

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Triangular norms and measures of fuzzy sets

Mirko Navara , in Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, 2005

13.6.1 Characterization of σ-order continuous charges on full tribes of constants

All σ-order continuous charges $\mu :\left[0,1\right]\to \mathrm{ℝ}$ on full tribes of constants are continuous functions satisfying (M1), (M2). As we shall see, this characterization extends to the general case rather directly.

We shall first characterize charges with respect to the Łukasiewicz t-norm 7   l. Due to Corollary 13.3.7, the assumption of σ-order continuity is unnecessary in this case.

13.6.1 Lemma

Each charge μ on the full tribe of constants ([0, 1], T _L ) is a multiple of the identity, i.e., it is of the form μ(a)   = a  · μ(1) for all a ∈ [0, 1].

Proof. Both T _L and S _L depend on the sum of their arguments only, hence

$T_{\mathbf{L}}\left(a,b\right)=T_{\mathbf{L}}\left(\frac{a+b}{2},\frac{a+b}{2}\right),$

and similarly for S _L Thus

$\begin{array}{lll}\mu \left(a\right)+\mu \left(b\right)& =& \mu \left(T_{\mathbf{L}}\left(a,b\right)\right)+\mu \left(S_{\mathbf{L}}\left(a,b\right)\right)\\ & =& \mu \left(T_{\mathbf{L}}\left(\frac{a+b}{2},\frac{a+b}{2}\right)\right)+\mu \left(S_{\mathbf{L}}\left(\frac{a+b}{2},\frac{a+b}{2}\right)\right)\\ & =& \mu \left(\frac{a+b}{2}\right)+\mu \left(\frac{a+b}{2}\right)\\ & =& 2\mu \left(\frac{a+b}{2}\right).\end{array}$

This, together with left-continuity, means that μ preserves convex combinations. The graph of μ is the set {(z, μ(z)) | z ∈ [0, 1]}. It is a convex subset of the plane, i.e., a linear segment, and, due to (M1), it contains the point (0, 0). This completes the proof. (The convexity of the graph of μ should not be confused with the convexity of μ as a function; in fact, being linear, μ is both convex and concave.)

Now we are ready to prove the characterization of σ-order continuous charges on full tribes of constants for all strict t-norms. This result will play a crucial role in the sequel. We divide it into two lemmas.

13.6.2 Lemma

Let T be an Archimedean nearly Frank t-norm and h_T a negation-preserving automorphism as in Definition 13.5.10. Then every multiple of h_T is a σ-order continuous charge on ([0, 1], T).

Proof. If h_T   =   id, T is a Frank t-norm, (M2) follows from (13.12) and the remaining properties are trivial. The general case then follows from Propositions 13.4.13 and 13.4.8.

13.6.3 Lemma

Let T be a strict t-norm. If μ is a non-zero σ-order continuous charge on ([0, 1], T), then T is nearly Frank and μ is a multiple of the negation-preserving automorphism h_T from Definition 13.5.10. Explicitly, μ(a)   = h_T (a)   · μ(1) for all a ∈ [0, 1],

Proof. Suppose that μ is a non-zero σ-order continuous charge on ([0, 1], T). Thus μ is a continuous function. In view of Lemma 13.5.3, μ attains its extreme values only at 0, 1 and μ(1)   ≠   0. The normalized charge h  = μ / μ(1) is positive and maps 1 to 1 and ]0, 1[ onto ]0, 1 [. We shall prove that h is the negation-preserving automorphism h_T , it satisfies (13.3) for some Frank t-norm T ^F _λ(T) and h_T is uniquely determined by T.

As a σ-order continuous charge, h is continuous. Suppose that it is not strictly increasing. Then there are a, b such that 0   < a  < b  <   1 and h(a)   ≥ h(b). We choose d ∈ argmin(h ↾ ]a, 1 [). This is possible because the values h(a),h(1) at the boundary points are not smaller than h(b). Then h(d) ≤   h(c) for all c ∈ [a, d]. We define

$d_i=\underset{j=1}{\overset{i}{\mathsf{S}}}d,i\in \mathrm{\mathbb{N}}.$

Then d_i ↗ 1 and h(d_i )   → h(1)   =   1. Starting from some index $n\in \mathrm{\mathbb{N}}$ , all i  ≥ n satisfy T(d, d_i ) ∈ [a, d], hence h(d)   ≤ h(T(d, d_i )). The subsequence (h(d_i ))^∞ _{i  = n} is non-increasing because (M2) implies

$h\left(d\right)+h\left(d_{i+1}\right)\le h\left(T\left(d,d_i\right)\right)+h\left(S\left(d,d_i\right)\right)=h\left(d\right)+h\left(d_i\right).$

This contradicts Lemma 13.5.3 which says that h(d_i )   < h(1)   =   lim _{i  →   ∞} h(d_i ). (Here we used strictness of T which ensures that d_i  ≠  1.) We proved that h : [0, 1]   →   [0, 1] is a strictly increasing bijection. According to (13.11), each a  →   [0, 1] satisfies

$h\left(a^{\prime }\right)=1-h\left(a\right)=h\left(a\right)\text{'},$

so h commutes with the standard negation. We proved that h is a negation-preserving automorphism.

Let us define a t-norm T* by (13.3). Its dual t-conorm is

$\begin{array}{lll}S*\left(a,b\right)& =& T*\left(a^{\prime },b^{\prime }\right)\text{'}\\ & =& h\left(T\left(h^{-1}\left(a^{\prime }\right),h^{-1}\left(b^{\prime }\right)\right)\right)\text{'}\\ & =& h\left(T\left(h^{-1}\left(a\right)\text{'},h^{-1}\left(b\right)\text{'}\right)\text{'}\right)\\ & =& h\left(S\left(h^{-1}\left(a\right),h^{-1}\left(b\right)\right)\right).\end{array}$

Using (M2) for T,

$\begin{array}{lll}T*\left(a,b\right)+S*\left(a,b\right)& =& h\left(T\left(h^{-1}\left(b\right)\right)\right)+h\left(S\left(h^{-1}\left(a\right),h^{-1}\left(b\right)\right)\right)\\ & =& h\left(h^{-1}\left(a\right)\right)+h\left(h^{-1}\left(b\right)\right)\\ & =& a+b.\end{array}$

By Theorem 13.5.4, T* is a Frank t-norm, T is nearly Frank and h satisfies the conditions for h_T from Proposition 13.4.13.

We shall complete the proof by proving that h_T in Proposition 13.4.13 is uniquely determined by T. Suppose that there are two negation-preserving automorphisms h ₁,h ₂ satisfying (13.3) for strict Frank t-norms $T_{{\lambda }_1}^{\mathbf{F}},T_{{\lambda }_2}^{\mathbf{F}}$ , respectively. According to Lemma 13.6.2, h ₁, h ₂ are charges on ([0, 1], T). Then also any linear combination of h ₁, h ₂, in particular h ₁  − h ₂, is a charge on ([0, 1], T). Flowever, we already proved that charges on ([0, 1], T) are multiples of negation-preserving automorphisms. As h ₁ – h ₂ vanishes at 1, it must be zero everywhere and h ₁ – h ₂. Thus the nearly Frank t-norm T determines the corresponding negation-preserving automorphism h_T uniquely and the charge μ up to a multiple.

As a by-product of the latter proof, we obtained (cf. [53]):

13.6.1 Proposition

If T is an Archimedean nearly Frank t-norm, then the negation-preserving automorphism h_T and the Frank t-norm T ^F _λ(1) in Definition 13.5.10 are unique.

Proof. For T strict see the proof of Lemma 13.6.3. Otherwise, T is nilpotent, the Frank t-norm T ^F _λ(1) from Definition 13.5.10 is also nilpotent, thus it is the Łukasiewicz t-norm. The uniqueness of h_T can be proved analogously also in this case.

Here we see the prominent role of Frank (more exactly, nearly Frank) t-norms: no other strict t-norms admit non-zero σ-order continuous charges on full tribes of constants. Each σ-order continuous charge on it is also a charge on ([0, 1], T) for some nearly Łukasiewicz t-norm T (which is obtained from (13.3) with T _L and the same negation-preserving automorphism h_T ).

13.6.2 Corollary

Let T be a continuous Archimedean t-norm. Then each σ-order continuous charge on ([0, 1], T) is uniquely determined by its value at 1.

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A mesh-free vibration analysis of strain gradient nano-beams

Lifeng Wang , ... K.M. Liew , in Engineering Analysis with Boundary Elements, 2017

5 Concluding remarks

The vibration of nano-beams is studied on the basis of strain gradient mesh-free method. A mesh-free scheme is built, in which the higher-order gradient of strain is directly approximated with the nodal components due to the higher-order continuity of the shape function. The MLS approximation is used to construct the shape function and its second- and third-order derivatives. The natural frequencies and vibration modes with different boundary conditions and lengths are obtained by the mesh-free analysis. The results by the mesh-free method coincide with the theoretical results very well in analyzing the simply-supported SWCNT which can be modeled as a strain gradient beam. Both the mesh-free and theoretical results indicated that the differences of natural frequency between that given by the strain gradient elastic beam and classical beam become obviously for the higher mode order and shorter length carbon nanotube.

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Recursive moving least squares

Hamid Mehrabi , Behzad Voosoghi , in Engineering Analysis with Boundary Elements, 2015

2.1 Moving least squares (MLS)

The origin of meshfree methods could be traced back to a few decades (1960s, introduced by Shepard [16]), but it was not until after the early 1990s when substantial and significant advances were made in this field. This method is considered as the means of generating a smooth surface interpolating among various specified point values. The procedure was later extended for the same purpose by Lancaster and Salkauskas [2] and [17]. The MLS is a common procedure for scattered data approximation that applies the local polynomial fitting in the least squares sense. It takes advantage of simple calculation, high precision and smoothness [18]. The advantages of the MLS approximation is to obtain the shape function in meshless methods with higher order continuity and consistency by employing the basis functions with lower order continuity and choosing a suitable weight function [19].

The MLS approximation is not the only means for building the regression model, but is fundamental for most of the current meshless methods (see Belytschko and Krongauz [20] and Fries and Matthies [21] for more details). The idea of MLS is estimating the desired parameters based on seeking a limited number of nearest point in the support domain of a nodal point rather than exploring the global solution. In this way the solution is obtained more rapidly and there is no need to high degree polynomial and the artificial unwanted oscillations in the unconstrained regions are avoided [22]. Meshless methods are typically designed to be local, i.e. evaluation of a function only involves values from nearby samples. These neighborhoods can be efficiently explored using search data structures, like k–d trees or quad-tree [23] in order to determine the nearest field points around the nodal points.

For describing the theoretical aspects of MLS method assume that $u_i(\mathit{x})$ is the function of the field variable defined in the domain $\Omega$ . The approximation of $u_i(\mathit{x})$ at point $\mathit{x}$ is denoted by $u_i^h(\mathit{x})$ . MLS approximation is written in the field function as [20,24] and [22]:

(1) $u_i^h\left(\mathit{x}\right)=\sum _{j=1}^m{a}_{ij}\left(\mathit{x}\right)p_j\left(\mathit{x}\right)=P\left(\mathit{x}\right)a_i\left(\mathit{x}\right)$

where $m$ is the number of terms of polynomial basis, $i$ is the $i\mathrm{th}$ coordinates component of the desired field function and $a_{ij}\left(\mathit{x}\right)$ is a vector of unknown coefficients which are the functions of $\mathit{x}$ presented as:

(2) $a_{ij}\left(\mathit{x}\right)=\left[\begin{array}{cc}\begin{array}{cc}{a}_{i1}\left(\mathit{x}\right)& a_{i2}\left(\mathit{x}\right)\end{array}& \begin{array}{cc}\cdots & a_{im}\left(\mathit{x}\right)\end{array}\end{array}\right]$

In Eq. (1), $P\left(\mathit{x}\right)$ are the monomial basis functions, which is quadratic in 2-D:

(3) $P\left(\mathit{x}\right)=\left(1,x,y,x^2,xy,y^2\right)$

For $m$ number of polynomial basis and $n$ number of field points, $P\left(\mathit{x}\right)$ will become:

(4) $P=\left[\begin{array}{ll}\begin{array}{cc}{p}_1\left(x_1\right)& p_2\left(x_1\right)\\ p_1\left(x_2\right)& p_2\left(x_2\right)\end{array}\hfill & \begin{array}{cc}\dots & p_m\left(x_1\right)\\ \dots & p_m\left(x_2\right)\end{array}\hfill \\ \begin{array}{cc}⋮& ⋮\\ p_1\left(x_n\right)& p_2\left(x_n\right)\end{array}\hfill & \begin{array}{cc}\ddots & ⋮\\ \dots & p_m\left(x_n\right)\end{array}\hfill \end{array}\right]$

The vector of unknown coefficients $a_{ij}\left(\mathit{x}\right)$ in Eq. (1) is determined by minimizing the quadratic form, Eq. (5), through a weighted least square (WLS) estimation as follows:

(5) $J_i={\left(a_{ij}\left(\mathit{x}\right)p_j\left({\mathit{x}}_I\right)-u_{iI}\right)}^T{W}_i\left(\mathit{x}\right)\left(a_{ij}\left(\mathit{x}\right)p_j\left({\mathit{x}}_I\right)-u_{iI}\right)$

where $W_i\left(\mathit{x}\right)$ is a weight function with compact support, $u_I$ is the function value of field points and ${\mathit{x}}_I$ is the component of the position of particle. Minimization of Eq. (4) yields a linear relation between $a_{ij}$ and $u_i$ which is written in standard form of batch least squares (BLS) as follows:

(6) $a_i\left(\mathit{x}\right)=N_i^{-1}{B}_i{u}_i$

where $N_i$ is called the moment matrix given by

(7) $N_i=P^T{W}_i\left(\mathit{x}\right)P$

and

(8) $B_i=P^T{W}_i\left(\mathit{x}\right)$

where

(9) $W\left(x\right)=\left[\begin{array}{cc}\begin{array}{cc}w\left(x-x_1\right)& 0\\ 0& w\left(x-x_2\right)\end{array}& \begin{array}{cc}\dots & 0\\ \dots & 0\end{array}\\ \begin{array}{cc}⋮& ⋮\\ 0& 0\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & w\left(x-x_n\right)\end{array}\end{array}\right]$

therfore, the approximation of $u_i^h\left(\mathit{x}\right)$ can be expressed as follows:

(10) $u_i^h\left(\mathit{x}\right)=a_{ij}\left(\mathit{x}\right)p_j\left(\mathit{x}\right)=P{a}_i=P{N}_i^{-1}{B}_i{u}_i$

Hence, by defining

(11) ${\mathrm{\Phi }}_i=P{N}_i^{-1}{B}_i$

(12) $u_i^h\left(\mathit{x}\right)={\mathrm{\Phi }}_i{u}_i$

is obtained: where ${\mathrm{\Phi }}_i$ is referred to as the shape function of approximation.

The support domain of a point $\mathit{x}$ determines the number of nodes that are used locally to approximate the function value and how does an increase in the radius of the support domain from R1 to R2 increase the numbers of field points, which leads to an increase in the degree of freedom of the estimation are illustrated in Fig. 2.

Fig. 2. Two support domains with different radius for a nodal particle.

The influence of the data points in traditional meshless methods is governed by a $W:\mathrm{\Omega }\times \mathrm{\Omega }\to ℝ$ , weight function with a compactly supported property which becomes smaller the further it gets from the nodal point [25]. Various functions are applied as weight function like compactly supported biquadratic B-spline [26], cubic spline [27], Hermitian interpolation [28], full Gaussian [29], truncated Gaussian function [28] and shifted Gaussian function [4]. In general, these weight functions are subsections of the radial basis functions (RBF) [30] and inverse distance weighting (IDW) functions [31] which are smoothing functions of distance among nodal and neighboring particles. These functions have a tendency towards zero out of the radius of influence domain.

It is worth noting that all kinds of the defined weight functions are usually used to ensure the compactness of the domain support. In this regards, the authors here are of the opinion that in surface interpolation and approximations all scattered points around the centered node should have equal weight, especially in scattered data points. It is noteworthy that fitting a surface to some scattered points is different from the estimations of Geoidal height and gravitational anomaly in Geodesy and Geophysics disciplines [32], where the amount of the aforementioned quantities are affected by the surrounding points.

In RMLS method, it is not mandatory to use the weight function for limiting the influences of the farther data points and size of the support domain; indeed, only the field points which have a positive and significant effect on the surface approximation and are not outliers are included in the recursive estimation procedure.

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