Domain of a function

It is the custom in certain branches of engineering science to seek out the most singular point in the domain of a function. Such a point serves as an organizing center in its neighborhood all possible forms of behavior flourish and may be apprehended coherently. Though it smack a little of the vulgar solecism very unique, I venture to commend... 

This similarity in spectral properties implies that haemocyanins should also have catalytic activity. From the available body of experimental data, it is clear that the distinction between the two major functions — oxygen transport and enzymatic activity — is determined by the presence or absence of a protein domain covering the active site. In the case of tyrosinase and catechol oxidase, inactive pro-enzyme forms are activated by removal of an amino acid which blocks the entrance channel to the active site (indicated by the black bar in Figure 14.7). Haemocyanins behave as silent inactive enzymes but can be activated in the same way if the blocking amino acid is removed. In arthropods, like crabs, this is located in the N-terminal domain of a subunit whereas in molluscs, like octopus, it is in the C-terminal domain of a functional unit. 

A function y = f(x) is a rule that assigns a nnique value of y for each of the given values of x in the definition of a domain of /, and x is said to be an independent variable and y the dependent variable. There may exist more than one value of x for a given value of y. The set of values of x is called the domain of a function, and the corresponding set of y is called the range of a function. 

The domain of a functional is a space that holds all possible forms of the associated functions or vectors with some common specification. For example, the domain of the functional I in Equation (2.1) is the space holding all forms of f x) with the common specification that each form be continuous for a < X [Pg.25]

The aforementioned anal3dical procedure furthermore requires a single-valued measure for the size of the function or vector in the specified domain of a functional. That measure is termed the norm. We use it to quantify the difference between two functions, or equivalently, the change in the function from some reference form. 

If the domain of a function is (a subset of) one dimensional teal number space R, it is said to be a functional . The potential functions, such as the internal energy, tire atypical functional. 

In this section we consider the boundary value problem for model equations of a thermoelastic plate with a vertical crack (see Khludnev, 1996d). The unknown functions in the mathematical model under consideration are such quantities as the temperature 9 and the horizontal and vertical displacements W = (w, w ), w of the mid-surface points of the plate. We use the so-called coupled model of thermoelasticity, which implies in particular that we need to solve simultaneously the equations that describe heat conduction and the deformation of the plate. The presence of the crack leads to the fact that the domain of a solution has a nonsmooth boundary. As before, the main feature of the problem as a whole is the existence of a constraint in the form of an inequality imposed on the crack faces. This constraint provides a mutual nonpenetration of the crack faces ... 

Here [ ] is the jump of a function across the crack faces and v is the normal to the surface describing the shape of the crack. Thus, we have to find a solution to the model equations of a thermoelastic plate in a domain with nonsmooth boundary and boundary conditions of the inequality type. 

The fundamental unit of tertiary structure is the domain. A domain is defined as a polypeptide chain or a part of a polypeptide chain that can fold independently into a stable tertiary structure. Domains are also units of function. Often, the different domains of a protein are associated with different functions. For example, in the lambda repressor protein, discussed in Chapter 8, one domain at the N-terminus of the polypeptide chain binds DNA, while a second domain at the C-terminus contains a site necessary for the dimerization of two polypeptide chains to form the dimeric repressor molecule. 

A random variable is a real-valued function defined over the sample space S of a random experiment (Note that tliis application of probability tlieorem to plant and equipment failures, i.e., accidents, requires tliat the failure occurs randomly, i.e., by chance). The domain of tlie function is S, and tlie real numbers associated witli tlie various possible outcomes of the... 

Figure 3 The radius distribution function between the centers of the P4VP domains of (a) the block copolymer BCl, and (b) the microsphere MCI [24].

The absolute maximum (or minimum) of f(x) at x = a exists if f(x) < f(a) (or f(x) > f(a)) for all x in the domain of the function and need not be a relative maximum or minimum. If a function is defined and continuous on a closed interval, it tvill altvays have an absolute minimum and an absolute maximum, and they tvill be found either at a relative minimum and a relative maximum or at the endpoints of the interval. 

Fig. 9.6 The length of liquid domain as a function of Pcl-Reprinted from Peles et al. (2001) with permission...

In principle, the reactivity of a functional group should not be altered when it is attached to a polymer ( 1). However, special effects may be encountered when a reagent is attracted to a polymer or repelled from it, when the polymer-bound reactive group is activated or inhibited by a neighboring group or when the local polarity of the polymer domain differs from that of the bulk solvent. A review of studies of such effects... 

Figure 1 Schematic representation of the 13C (or 15N) spin-lattice relaxation times (7"i), spin-spin relaxation (T2), and H spin-lattice relaxation time in the rotating frame (Tlp) for the liquid-like and solid-like domains, as a function of the correlation times of local motions. 13C (or 15N) NMR signals from the solid-like domains undergoing incoherent fluctuation motions with the correlation times of 10 4-10 5 s (indicated by the grey colour) could be lost due to failure of attempted peak-narrowing due to interference of frequency with proton decoupling or magic angle spinning.

Fields of Application. In SAXS a calibration to absolute intensity is required if extrapolated or integrated numerical values must be compared on an absolute scale. Examples are the determination of density fluctuations or the density difference between matrix and domains as a function of materials composition. 

The cap-binding protein eIF4E promotes folding of a functional domain of yeast translation initiation factor eIF4Gl.J. Biol. Chem. 274, 21297—21304. 

All fungal prion proteins have a so-called prion domain and a functional domain. The prion domain is a region of the polypeptide chain that is necessary and sufficient for prion formation and maintenance (Fig. 1 Wickner et al., 2002). For Ure2p and Sup35p, the functional domain is responsible for the cellular activity of the normal form of the protein. 

Just as a vector is projected as components on orthogonal axes, a given function defined on a given domain can be projected onto an orthogonal set of functions. The Fourier series decomposition of a function /(x) defined over the interval [ —X, X] is a convenient example... 

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