Least element

In order to discriminate between the individual approaches one needs measurements of the energy spectra for individual elements, or at least elemental groups, the arrival directions of CRs, and their mass composition. In this lecture recent experimental results are reviewed and their implications on the contemporary understanding of the origin of the knee is discussed. 

If there is only one minimal element, then this is also called a least element. In Fig. 5, there is no least element, however in Fig. 4 element A is a least element. 

If the Hasse diagram, supplied (if necessary) by a greatest and least element can be drawn in the plane without crossing... 

Fig. 16. Hasse diagram of a poset with dimension 3. The Hasse diagram on the left side follows not the convention explained earlier The Hasse diagram on the right side is supplied by a greatest G and least element L .

Least element order Quadratic (Babuska-Brezzi) Linear ... 

A lattice is a set, a partial order on the elements of this set, a least element i., and a greatest element T. 

The relation unification algorithm). The lub operator computes the msg of two terms (by an anti-unification algorithm). The msg of two terms thus always exists, and is unique up to variable renaming. 

When attempting the measurements the system should be zeroed with the aid of the reference element, which should remain close to the studied element for at least 8 hours. 

The total change d.S can be detennined, as has been seen, by driving the subsystem a back to its initial state, but the separation into dj.S and dj S is sometimes ambiguous. Any statistical mechanical interpretation of the second law requires that, at least for any volume element of macroscopic size, dj.S > 0. However, the total... 

Table Bl.5.1 Independent non-vanishing elements of the nonlinear susceptibility, for an interface in the Ay-plane for various syimnetry classes. When mirror planes are present, at least one of them is perpendicular to they-axis. For SFIG, elements related by the pennutation of the last two elements are omitted. For SFG, these elements are generally distinct any syimnetry constraints are indicated in parentheses. The temis enclosed in parentheses are antisymmetric elements present only for SFG. (After [71])...

The sensitivity in an NMR experiment is directly proportional to the number of spins, making quantification of the amount of a particular element present straightforward, at least for spin / = inuclei. Furtliemiore, the... 

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

The modem era of biochemistry and molecular biology has been shaped not least by the isolation and characterization of individual molecules. Recently, however, more and more polyfunctional macromolecular complexes are being discovered, including nonrandomly codistributed membrane-bound proteins [41], These are made up of several individual proteins, which can assemble spontaneously, possibly in the presence of a lipid membrane or an element of the cytoskeleton [42] which are themselves supramolecular complexes. Some of these complexes, e.g. snail haemocyanin [4o], are merely assembled from a very large number of identical subunits vimses are much larger and more elaborate and we are still some way from understanding the processes controlling the assembly of the wonderfully intricate and beautiful stmctures responsible for the iridescent colours of butterflies and moths [44]. 

In this appendix, we discuss the case where two components of Xm, namely, x p and XMg (p and q are Cartesian coordinates) are singular in the sense that at least one element in each of them is singular at the point B p = a,q = b) located on the plane formed by p and q. We shall show that in such a case the adiabatic-to-diabatic transformation matrix may become multivalued. 

Let us first define the information content per object. A (discrete) system can be split into classes of equivalence, whose number can vary from 1 to n, where n is the number of the elements (objects) in the system. No element can belong simultaneously to more than one class. Therefore, the information content (JQ of the system is additive, at least class-wise. This means that the information content of the system is the sum of the information contents of the classes. The IC of a class can be given by Eq. (2), where h is the number of elements in the ith class. (Recall, also, that log (x) = -log (l/x)). 

Structural keys describe the chemical composition and structural motifs of molecules represented as a Boolean array. If a certain structural feature is present in a molecule or a substructure, a particular bit is set to 1 (true), otherwise to 0 (false). A bit in this array may encode a particular functional group (such as a carboxylic acid or an amidelinkage), a structural element (e.g., a substituted cyclohexane), or at least n occurrences of a particular element (e.g., a carbon atom). Alternatively, the structural key can be defined as an array of integers where the elements of this array contain the frequency of a specific feature in the molecule. 

II energy values where the energy is between Ef j and E and where the value of the opriate array element H( ) is greater than some minimum value (e.g. 20). In other is, that particular energy level must have been visited at least twenty times during iimulation. The following two parameters are now calculated ... 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

The basic procedure for the derivation of a least squares finite element scheme is described in Chapter 2, Section 2.4. Using this procedure the working equations of the least-squares finite element scheme for an incompressible flow are derived as follows ... 

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