Contractible to a point

Consider an K-dimensional set X. Set X is simply connected if and only if every -dimensional (k < n) topological sphere Sk in set X is contractible to a point. 

Two of the methods proposed appear at first sight to be non-arbitrary. In fact, absolute accuracy has been claimed for one of these (25), a refined two-dimensional variational procedure in which all frequencies but one are used as constraints, while this last one is used as a criterion. In fact, it is not possible to fix two unknowns from one parameter, and the method seems to this reviewer an unusually ingenious and elaborate exercise in self-deception, in which the range of possible solutions of the independent parameter method appears to contract to a point. 

Informally, a subset A of X is said to be contractible to a point if a continuous contraction of the set A to a single point is possible without requiring any point of A to pass through points not in set X. 

Fig. 15. IMPS plots recorded at different potentials for p-InP in 1.0mol dm" H2SO4 [35]. Note that the diameters of the semicircles decrease as the band bending increases and recombination becomes less effective. In the saturation photocurrent region, the IMPS response contracts to a point.

The glass fabric reinforcement serves another function in that it lowers the overall coefficient of linear thermal contraction to a point where thermal stresses at the sealant-substrate interface are reduced to a minimum. 

We can imagine the coil tightening up as this point is approached from better conditions. This is not a shrinking to the vanishing point as suggested by the u = 0 criterion, but a contraction to the point where intramolecular exclusion effects are offset by shrinkage. 

On the progressive dehydration of a silica gel the gel rapidly contracts to a certain point. Van Bemmelen s first inversion point after which but little contraction takes place. On continued dehydration the clear gel suddenly becomes cloudy, then opaque, and finally loses its opacity again—Van Bemmelen s second point of inversion, when the water content has sunk to a ratio of one mol. of water to one of silica. 

The isotherm between those numbered 3 and 4 in the figure represents the transition between isotherms corresponding to Ihe gas phase only, and those including a horizontal portion corresponding to a liquid-gas equilibrium. In this isotherm the horizontal line has contracted to a single point of inflection C. This is ihe critical point characterized by the relations... 

The second type of transition curves is related to the appearance or disappearance of a closed line. A closed line may disappear either by contracting into a point or by joining with the rest of the curve. In the first case the intermediate type we seek will be the kinetic curve of the form in Fig. 8. In both the case of a single isolated point and that of a point of self-intersection of the curve, we have the following conditions2 [cf. the form of equation (6a)] ... 

Subrahmanyam, 1989). It can therefore be successfully used for constrained optimization of expensive black-box functions. Where it is less effective, however, is in the treatment of noisy functions. If the underlying noise-free function has similar values at the vertices, then the effect of the added noise may be sufficient to change the apparent ordering of the vertices, causing the optimization routine to make a bad decision such as unnecessary contraction (a process that should only happen when the simplex is close to the true minimum). The effect of such a contraction is to bring the vertices closer together, which means the values of the noise-free function at the vertices will be even closer and the distorting effect of noise even worse. The routine may therefore make a series of erroneous decisions that cause it to collapse to a point away from the true minimum. 

Exact calculations of the potential energy surfaces for complex molecular systems are impossible to carry out from a practical point of view. Such calculations involve the solution of the electronic Schrodinger equation for the system including electron correlation effects and full geometry optimization. However, an estimate of the 8 A lq> value can be obtained in a different way. One can carry out an ab initio calculation in the Hartree-Fock (HF) approximation by using a simple basis set, e.g., (7s, 3p/3s), contracted to a minimal basis set, STO-3G, or 3-21G, etc., with full geom-... 

To continue with Laplace s model As the rate of rotation increased to the point where the centrifugal force at the periphery exceeded gravitation, a ring of material separated from the main mass, eventually contracting towards a point. As the process continued, more planets condensed in the outer regions, while the inner region contracted to form the snn. The satelhtes were formed by condensation from the contracting planets. Excess material between the planets turned into comets and meteors. 

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