Family of functions

The Slater-type orbitals are a family of functions that give us an economical way of approximating various atomic orbitals (which, for atoms other than hydrogen, we don t know anyway) in a single relatively simple form. For the general case, STOs are written... 

A family of wavelets is a family of functions with all its members derived from the translations (e.g., in time) and dilations of a single, mother function. If iffit) is the mother wavelet, then all the members of the family are given by... 

In brief, wavelets are a family of functions of constant shape that are localized in both time and frequency. A family of discrete dyadic wavelets is represented as... 

The trigonometric functions developed in the previous seefidtiare lef pd to as circidar functions, as they are related to the circle shown in Fig, 11, somewhat less familiar family of functions, the hyperbolic funefens, c also be derived from the exponential. They are analogous to the circular iom considered above and can be defined bv the relations... 

DEFINITION The family of primitive recursive functions is the least family of functions closed under functional composition and primitive recursion and containing the base functions ... 

Given the simplistic nature of this relationship, we forbear to plot the function, although we note that there are again a family of functions, corresponding to the various values of k. 

Nathan, A., Zalipsky, S., Ertel, S.I., Agathos, S.N., Yarmush, M.L., and Kohn, J. (1993) Copolymers of lysine and polyethylene glycol A new family of functionalized drug carriers. Bioconjugate Chem. 4, 54-62. 

There is no such thing as a typical cell. They come in a variety of sizes, shapes, and structures, each suited to a particular function or family of functions. Cells are isolated from their environment by a cell membrane that regulates what gets in and what does not. At present, we need to know just two things about cells. 

To use what is termed universal kriging, it is assumed that Z(2 ) is an intrinsic random function of order k. But the problem of identifying the drift and the semi-variogram when they are both unknown is still present. However, Matheron (11) defined a family of functions called the generalized covariance, K(h). and the variance of the generalized increment of order k can be defined in terms of K(h ). That is. 

HCTH Hamprecht. Cohen. Tozer. and Handy GGA exchange-coiTelation functional based on a reoptimization/extension of empirical parameters in B97 and a removal of HF exchange. Now a family of functionals with optimizations over different numbers of test-set molecules, typically denoted HCTH/w where n is the number of molecules in the test set, e.g., HCTH/93, HCTH/120, HCTH/147, and HCTH/407. Hamprecht, F. A., Cohen, A. J., Tozer, D. J., and Handy, N. C. 1998. 7. Chem. Phys., 109, 6264. (Most recent refinement, Boese, A. D., Martin, J. M. L., and Handy, N. C. 2003. 7. Chem. Phys., 119, 3005.)... 

This can be considered as the simplest member of a family of functions in which... 

Figure 6.6 Integration of the function f(x) =18x2 (right) yields a family of functions given by the indefinite integral F(x) = 6x3+C (left), where C can take any value. Differentiation of F(x) yields the original function, f(x)...

The spectrin superfamily is a group of cytoskeletal proteins that have been found to perform a variety of cellular functions. The role of each protein and their interactions within the cellular environment stem from the specific domains found within each protein and the manner in which they are organized. Each of the family members is formed from discrete modular domains that have the ability to interact or modulate specific interactions or impart physical abilities on the protein relevant to its function. The particular members of this protein family have been shown to be evolutionary related. a-Actinin is believed to be the ancestor of the whole group and, indeed, sequence and phylogenetic analysis has found this to be the case. It is astounding that from a simple precursor containing few domains such a family of functionally diverse proteins can be... 

Valenta, R., Duchene, M., Ebner, C. et al. 1992. Profilins constitute a novel family of functional plant pan-allergens. J Exp Med 175 377-385. 

An efficient synthesis of l-aryl-4,6-diamino-l,2-dihydro-l,3,5-triazines from an acid-catalyzed reaction between the corresponding arylbiguanidine and carbonyl compounds has been reported <02SC2089>. A new family of functionalized terpyridine-like ligands containing a central 1,3,5-triazine, (8a) and (8b) has been synthesized. These compounds exhibit room-temperature luminiscence as their Ru(II) complexes (9a), (9b) and (9c) <02CC1356>. 

In this connection there may be an objection that if an expression is given, e.g., for time as a function of the degree of advancement, and this contains a number of constants, one can always choose the constants so as to make the theoretically predicted curve fit the experimental curve. This would be possible if we could choose at will certain families of functions and if we were free to dispose of many constants. But the functions appearing in the equations cannot be chosen arbitrarily since they follow as logical consequences of the assumed mechanism, and the number of constants is restricted by the finite number of partial reactions in the sequence. Only if the range of the degree of advancement is small and if the precaution to make experiments with different initial concentrations is not taken, may it be possible to obtain agreement between the same set of experiments and different mechanisms, but not otherwise. 

With minor modifications, the fuzzy electron density membership function formalism of molecular families can also be applied to a family of functional groups within a molecule. Consider a molecule X and some electron density threshold a within the functional group range of density. Consider the functional groups appearing as separate density domains... 

For this we return to thinking of A as in some sense functions on Spec A. The open set Spec A Z(f) is canonically homeomorphic to the spectrum of the localized ring <4[/ 1] = Af, so it is reasonable to consider Af as the functions on that open set. Intuitively we are just allowing rational functions on the set where the denominator does not vanish. One can show that these functions have a reasonable local-determination property a function on a large open set U = (J U, is precisely determined by a family of functions /, on Ua in which/, and ff agree on Ua n Ufi. This says we have a sheaf of functions (see (15.6)). 

Consider a macromolecule M of some nuclear configuration K. For an electron density threshold a, the family of functional groups... 

A solution that involves one or more arbitrary constants represents a family of functions that satisfy the differential equation and is called a general solution of that equation. Not surprisingly, a differential equation may have more than one general solution. A general solution is usually referred to as the general solution or the complete solution if every solution of the equation can be obtained from it as a special case. A solution that can be obtained from a general solution by assigning particular values to the arbitrary constants is called a specific solution. 

A function/of the essential variables, is said to be of finite co-dimension if a small perturbation of/gives rise to only a finite number of topological types. It is in this case that a map / of co-rank k can be embedded in a family of deformations (/, x R), parametrized by q variables which form the control space a R. By definition, q, the dimension of the control space, is the co-dimension of the singularity represented by /. The family of functions (/, / X R R) thus defined, describes the universal unfolding of /, in which the function/itself is the member associated with the origin of control space. In our applications the control space is a subset of the nuclear configuration space. 

Mevalonate diphosphate decarboxylase (MVP EC4.1.1.33) catalyzes the conversion of mevalonate diphosphate to isopentenyl diphosphate, a key building block for a large family of functionally important terpenoids. Fig. (6). This reaction is the third step in the biosynthesis of steroids and terpenoids from the mevalonate pathway, and the last well characterized step in the mevalonate pathway for the biosynthesis of isopentenyl pyrophophaste, the isoprenoids precursor [296-298]. Some reports showed that MVP is located predominantly in the cytosolic fraction and its expression is independent of peroxisome proliferation [299-300]. 

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