Entity Finite elements

Numerieal finite element models of Astley et al. (1998) take aeeount of ehemieal eomposition, mierofibril angles and their dispersion within all the eell wall layers (not simply the S2) and eell geometry (shape, size, wall thiekness). Where adapted to realistie models of earlywood/latewood entities, these simulations reduee the disparity between stiffness as the MFA ehanges from 45 to 10° to a faetor of about 3 as opposed to about 5 in the original model (although that work was supported by experimental data) by Cave (1968). 

Dr. Ray Clough coins the term finite element—an entity that can model three-dimensional strain. 

Besides the Finite-Difference and Finite-Element mainstream, there has appeared a handful of alternative computational methods in Newtonian and non-Newtonian CFD that can be collectively characterized as particulate, in the sense that they involve discrete computational entities in addition to or in lieu of the standard continuum-mechanical discretization devices. In some of these methods (the first four and the last one in the following... 

We might, just accept it as a brute fact about the world that the series of elements was discrete. But if there were a finite number of properties, combinations of which generate the physical possibilities represented by the periodic table, then variation would necessarily be discrete rather than continuous. We can believe in the existence of these fundamental entities and properties without subscribing to any particular account of them (e.g. an account in terms of electronic configuration), such accounts at least show us the way in which chemical properties could be determined by more fundamental ones. The point is that, given the principle of recombination, unless those more fundamental properties exist, unactualized elements would not be physical possibilities (14). 

Finally, I would like to point out some specific points concerning Le Poidevin s analysis. Let me return to the question of the discrete manner in which the elements occur. This fact, it will be recalled, is taken by Le Poidevin to support a combinatorial argument whereby a finite number of fundamental entities or properties combine together to give a discrete set of composite elements. But what if we consider the combination of quarks (charge = 1/3), instead of protons (charge = 1) In the former case a finite number of quarks would also produce a discrete set of atoms of the elements, only the discreteness would involve increments of one-third instead of integral units. 

If I may, I would like to advert for a moment to the recent development of non-standard analysis and sketch how infinite and infinitesimal numbers can be presented. In this I follow a beautiful expository article of Ingleton (22) though, in my haste scarcely doing him, or Luxemburg on whom he leans, full justice. Consider all infinite sequences of real numbers X (x, x, ,xn, ) and let two such entities be equivalent if they differ only in a finite number of elements i.e., X E X if x =xf for all but a finite number of n. From now on we can consider the entity X to be the equivalence class and representable by any of its members just as the rational 1/2 is the class (1/2, 2/4, 3/6,..). We... 

Some facts are never stated upfront in chemistry textbooks. For example, the concept of the existence of indivisible atoms arose as a way of reconciling the paradox of having a finite entity that is composed of an infinite number of constituent elements (Zeno of Elea s paradox). Figure 2.2 shows a graphical representation of this paradox. If a frog is 1 m away from a tree and always hops half the distance between itself and the tree it will take theoretically an infinite number of hops to get there. 

Set-Based Representations Currently, the most prevalent entities in MSA are finite sets with ordered binary-valued elements, generally called (molecular or structural) fingerprints, bit (binary) vectors, or bit strings. Equation 15.4.1 provides an example of a molecular fingerprint... 

---