Spanning degree

However, a polynomial of degree w— 1 can always be fitted, however many such control points there are, and if w — 1 is less than or equal to the spanning degree, the limit curve will have polynomial pieces. 

If a given polynomial is to be reproduced, it must be within the span of the scheme, and it must also be interpolated. We can therefore see without any sophisticated argument that the degree of reproduction is just the lower of the spanning degree and the interpolation degree. 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

A gas turbine used in aircraft must be capable of handling a wide span of fuel and air flows because the thmst output, or pressure, covers the range from idle to full-powered takeoff. To accommodate this degree of flexibiUty in the combustor, fuel nozzles are usually designed with two streams (primary and secondary flow) or with alternate tows of nozzles that turn on only when secondary flow (or full thmst power) is needed. It is more difficult to vary the air streams to match the different fuel flows and, as a consequence, a combustor optimized for cmise conditions (most of the aircraft s operation) operates less efficiently at idle and full thmst. 

The entries in the table are arranged in order of increasing reaction coordinate or distance along the reaction path (the reaction coordinate is a composite variable spanning all of the degrees of freedom of the potential energy surface). The energy and optimized variable values are listed for each point (in this case, as Cartesian coordinates). The first and last entries correspond to the final points on each side of the reaction path. 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

If B— [bij] is an N xN matrix in which bu equals the degree of vertex i, bij = —1 if vertices i and j are adjacent and bij = 0 otherwise, then the number of spanning tree of G is equal to the determinant of any principal minor of B [hararybO]. The extremes occur for totally disconnected graphs that have no spanning trees and thus a complexity of zero, and for complete graphs of order N that contain the maximum possible number of distinct trees on N vertices. ... 

When we regard each of our spectra as a unique point in the n-dimensional absorbance space, we can say that the error in our data is isotropic. By this, we mean that the net effect of the errors in a given spectrum is to displace that spectrum some random distance in some random direction in the n-dimensional data space. As a result, when we find the eigenvectors for our data, each eigenvector will span its equivalent share of the error. But recall, we said that we must take degrees-of-ffeedom into account in order to understand what is meant by equivalent share. 

A wide range of degrees of freedom must be spanned, over which the approximated value changes appreciably. 

BCFs and BAFs measured before the steady state is reached have little value because they are dependent on the period of exposure of the organism to the chemical, and thus may greatly underestimate the degree of biomagnification that is possible. This statement should be qualified by the reservation that there may be situations in which the duration of exposure cannot be long enough for the steady state to be reached, for example, where the life span of an insect is very short. The principal processes of uptake and loss by different types of organisms are indicated in Table 4.1 (see also Box 4.2). 

The two strands, in which opposing bases are held together by hydrogen bonds, wind around a central axis in the form of a double helix. Double-stranded DNA exists in at least six forms (A-E and Z). The B form is usually found under physiologic conditions (low salt, high degree of hydration). A single turn of B-DNA about the axis of the molecule contains ten base pairs. The distance spanned by one turn of B-DNA is 3.4 nm. The width (helical diameter) of the double helix in B-DNA is 2 nm. 

One example of a structure (8) is the space of polynomials, where the ladder of subspaces corresponds to polynomials of increasing degree. As the index / of Sj increases, the subspaces become increasingly more complex where complexity is referred to the number of basis functions spanning each subspace. Since we seek the solution at the lowest index space, we express our bias toward simpler solutions. This is not, however, enough in guaranteeing smoothness for the approximating function. Additional restrictions will have to be imposed on the structure to accommodate better the notion of smoothness and that, in turn, depends on our ability to relate this intuitive requirement to mathematical descriptions. 

For this spectroscopic investigation 98 amidated pectin samples were provided by Copenhagen Pectin A/S (Hercules Inc.). The samples are spanning a degree of esterification between 20 and 55 per cent and a degree of amidation between 4 and 24 per cent (see Fig. 1). The powder samples were all measured as is without any form of pre-treatment such as drying and dilution. 

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