Adaptive functions

Bronson F. and Coquelin A. (1980). The modulation of reproduction by priming pheromones in house mice speculations on adaptive function. In Chemical Signals in Vertebrates 2 (Miiller-Schwarze D. and Silverstein R.M., eds.). Plenum, New York, pp. 243-266. 

Helicobacter pylori is a gram-negative microaerophilic rod that has a number of adaptive functions allowing it to live within the acidic environment of the stomach. It is an S-shaped... 

Among insectivores, at least five genera of hedgehogs anoint themselves with toad toxins. Although the exact adaptive function of this behavior is not known, it has been observed that the anointed spines hurt humans more than clean ones (Brodie, 1977). 

Dominy NJ, Davoust E, Minekus M (2004) Adaptive function of soil consumption an in vitro study modeling the human stomach and small intestine. J Exp Biol 207(2) 319-324... 

Axis V, to reflect the highest level of adaptational functioning in the previous year information should be provided, when possible, in all relevant areas. 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

Prove that if the irreducible representation r, is one dimensional, then the function g of (9.67) is a symmetry-adapted function that transforms according to r,. Start by writing the basis function / as some linear combination of symmetry-adapted functions g, apply Or to this equation then multiply by x (R) and sum over R. 

Because the group is of infinite order, (9.67) is cumbersome to use, and it is simplest to find the symmetry-adapted functions by inspection. Each symmetry operation either leaves lja and sb alone or transforms them into each other, and the following functions are easily seen to transform... 

These operators are examples of projection operators they project out the symmetry-adapted functions from the basis AOs.) Application of PAt to l o gives... 

Note, however, that since we now work with only the trace of the matrix, we have no information about off-diagonal elements of the irrep matrices and hence no way to construct shift operators. The business of establishing symmetry-adapted functions therefore involves somewhat more triad and error than the approach detailed above. Character projection necessarily yields a function that transforms according to the desired irrep (or zero, of course), but application of character projection to different functions will be required to obtain a set of basis functions for a degenerate irrep, and the resulting basis functions need not be symmetry adapted for the full symmetry species (irrep and row) obtained above. 

In a quantum chemical calculation on a molecule we may wish to classify the symmetries spanned by our atomic orbitals, and perhaps to symmetry-adapt them. Since simple arguments can usually give us a qualitative MO description of the molecule, we will also be interested to classify the symmetries of the possible MOs. The formal methods required to accomplish these tasks were given in Chapters 1 and 2. That is, by determining the (generally reducible) representation spanned by the atomic basis functions and reducing it, we can identify which atomic basis functions contribute to which symmetries. A similar procedure can be followed for localized molecular orbitals, for example. Finally, if we wish to obtain explicit symmetry-adapted functions, we can apply projection and shift operators. 

As was discussed in Chapter 2, the need to have full matrix representations available to obtain basis functions adapted to symmetry species is something of a handicap. Although character projection itself is not adequate for this task, Hurley has shown how the use of a sequence of character projectors for a chain of subgroups of the full point group can generate fully symmetry-adapted functions. Further discussion of this approach is beyond the scope of the present course, but interested readers may care to refer to the originad literature [6]. 

Note that the average over k on the RHS of Eq. 4.5 means that the integral itself is independent of the value of the row index i. Thus the matrix of O in a basis of symmetry-adapted functions is diagonal on symmetry species, and the diagonal blocks corresponding to rows of the same irrep are the same. 

Viewing the relationship between culture and psychopathology within an evolutionary framework, Hammer and Zubin (1966) also suggest some possible culturally adaptive functions of schizophrenia. They point out that the characteristics of the mentally ill occur in most people, and symptoms such as anxiety, guilt, shame, and depression are often harnessed to socially useful purposes. Similarly, Bowers and Freedman (1966) and Ludwig (1966) emphasize the healing function of schizophrenic states and consider them a major avenue of new knowledge and creative experience. 

The development of the types of skeletons that characterize Tommotian faunas constituted a major evolutionary event. Although skeletons are known to support soft tissue and to facilitate locomotion, such adaptive functions cannot explain why so many different kinds of skeletons developed suddenly in the early part of Tommotian time. It has been suggested that a chemical change within the oceans triggered the production of these skeletons, but this hypothesis does not explain why some skeletons were composed of calcium carbonate and others of calcium phosphate, two compounds with quite different chemical properties. The rapid evolution of various kinds of external skeletons is probably in part attributable to the fact that animals... 

Chapter 2 - The aim of this chapter is to examine the role of noradrenaline in interpersonal functioning. Healthy interpersonal functioning is important for the development of relationships in both work and personal situations. Many psychiatric disorders including depression are associated with poor interpersonal functioning and less social activity but improvement in interpersonal functioning can be independent of symptom resolution. Noradrenaline may be involved in the adaptive function of human social behaviours. 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

In the basis of these orthonormal symmetry-adapted functions, matrix elements of the invariant Hamiltonian are given for two different configurations A and B by... 

Symmetry-adapted functions Table 4.1. a/x table for functions 2S (abc)... 

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