Entire function

The mathematical behavior of the critical point is characterized by the two minima, the maximum, and the two inflection points all merging into a common point so that the entire function displays the smooth features seen outside the PQ region. 

A function/(x) starts with a number, x, performs mathematical operations, and produces another number, /. It transforms one number into another. A functional starts with a function, performs mathematical operations, and produces a number. It transforms an entire function into a single number. The simplest and most common example of a functional is a definite integral. The goal in Example 6.5 was to maximize the integral... 

We do not need to expand the entire function into partial fractions. The functions Gb G2, etc., are better viewed as simply first and at the most second order functions. In frequency response analysis, we make the s = jto substitution and further write the function in terms of magnitude and phase angle as ... 

Complement A term originally used to refer to the heat-labile factor in serum that causes immune cytolysis, the lysis of antibody-coated cells, and now referring to the entire functionally related system comprising at least 20 distinct serum proteins that is the effector not only of immune cytolysis but also of other biologic functions. Complement activation occurs by two different sequences, the classic and alternative pathways. The proteins of the classic pathway are termed components of complement and are designated by the symbols... 

The mechanisms of inhibition by peroxide decomposers, metal deactivators, and ultraviolet absorbers are fairly well understood in general terms, although many details of the individual reactions remain to be elucidated. Classifying a preventive antioxidant into one of the three categories above will only rarely describe its entire function. The dual behavior of dialkyl dithiophosphates in the liquid phase has been mentioned. Many other phosphorus- and sulfur-containing antioxidants commonly classified as peroxide decomposers can also act as chain breakers. Similarly, the structure of many metal deactivators and ultraviolet absorbers indicates that they must also have some chain-breaking activity. 

For a degenerate system we cannot always determine beforehand the zero-order or model functions. The mixing of the degenerate eigenfunctions in the zero-order depends on the perturbation. Therefore, generally speaking, we can only confine the model function to a certain subspace of the entire functional space. Let us call this subspace the model (or P)... 

The Mittag-Leffler function [44-46] can be viewed as a natural generalization of the exponential function. Within fractional dynamics, it replaces the traditional exponential relaxation patterns of moments, modes, or of the Kramers survival. It is an entire function that decays completely monotoni-cally for 0 < a < 1. It is the exact relaxation function for the underlying multiscale process, and it leads to the Cole-Cole behavior for the complex... 

The endothelial cells assume their entire function in the neointima. In all one-year animals (N = 5) no thrombosis was recognized. 

Figure 1.24. Schlegel diagrams and individual hemispheres of two constitutional isomers resulting from sixfold [3 + 2] cycloaddition of 1,1,3,3-tetramethylazomethine ylide to C o- The two isomers can be considered as made up of two isoconstitutional, C3-symmetric hemispheres the combination of homochiral moieties leads to the D3-symmetric regioisomer and the combination of heterochiral moieties affords the 7h-symmetric regioisomer. It should be noted that the front hemisphere is viewed from the convex side (extra-cage view) and the rear hemisphere from the concave side (intra-cage view). The indicated symmetry elements refer to the entire functionalized carbon sphere resulting from translational superposition of the represented hemispheres (the C3-axis is also valid for the individual hemispheres).

For many paramagnetic substances it is found empirically that c v - Ai/T2 where Ax is independent of H0 and T. Furthermore, for such materials Curies Law M — A2Ha/T holds, in which A2 is also independent of H0 and T. Note that the magnetic moment is an entire function of H0/T. On introducing these relations into (5.8.23) one finds... 

Much as Eq. (10) can be expanded in inverse powers of (X —X ) to yield the simple form, Eq. (17) can be expanded in invert powers of (x — Xo), the constants Oo, i>o, and Xo defined in terms of a,-, bi, and X<, and higher order terms neglected. In the light of the theory and the approximations from which this equation is developed, the constants bo and Xo are principally, though not entirely, functions of the helical skeleton alone and accordingly should be relatively insensitive to environmental factors such as solvent changes and the nature of side chains. The constant ao, on the other hand, represents both the intrinsic residue rotations, which would be present irrespective of the helix, and interactions within the helix. Since intrinsic residue rotations are known to vary with environment, ao should likewise be expected to vary. 

For the normal electronic state of the molecule, since Pauli s principle must be obeyed, the entire function must be antisymmetric in the two electrons, i,e. when the electrons are transposed, the sign of the wave function must change. Let us consider the symmetry of each of the above five wave functions with reference to the two electrons. The electron orbital motion function is (see Chapter 3),... 

This expression is of intrinsic interest for many classes of materials the magnetic moment is found to be an entire function of 7fo/T, in which case, as is readily verified, (du /dHo)T,v = 0. Then u and U do not depend on Ho, in which case the heat capacity is a function of T and of V alone. 

Analytic functions are also called regular and holotnorphic. If f z) is analytic for all finite values of z, it is an entire function. If /(z) is not analytic at a point zq, the point Zo is a singular point. 

Here we find a perturbing potential at the position of the other monomer (caused by its nuclei and electrons), so now functions with cusps are perfect for the physical situation the monomer is in, allowing it to recover the full BSSE. Formally, one may say that the monomer can still use the entire function space of the ghost, and only a physical perturbation (the Pauli principle) forces it away from the ghost occupieds, into a function space orthogonal to these. So in both dimer and monomer calculations the extra orbitals are adapted to the physical requirements of the system. Using only the virtual orbitals of the ghost in the monomer counterpoise calculation introduces restrictions on the orbital space which are only justified in the dimer calculation. 

---