Homotopic path

Of course, if the critical point of the catchment region is an element of the level set, then within the homotopy classes of paths passing through the catchment region the maximal local symmetry is g(K(X,i)). Nevertheless, within the family of homotopic paths, the paths with this maximal locd symmetry form a subset of measure zero. 

In Mi and M2, the closed path is homotopic to a null path. In Mi, this cycle is the boundary of a face, while in M2, the closed path is the boundary of all faces put together. More generally, a plane graph and a finite plane graph minus a face are simply connected. But the closed path in M3 is not homotopic to a null path. Actually, this closed path is a generator of the fundamental group Jti(Mf) — Z. 

Take a subclass Pi(C(A,iXC(A, i )) of class Pi, define by the following condition Pi(C(A,i), C(A, i )) is the family of all paths from class Pi which start at the catchment region C(A,i), end at the catchment region C(A, i ), and are homotopic to one another (continuously deformable into one another) while preserving these properties. Evidently the above conditions correspond to an equivalence relation among paths, and, consequently, Pi(C(A,i), C(A, i )) is an equivalence class. Such equivalence classes Pi(C(A,i), C(A, i )) represent formal reaction mechanisms defined in terms of the shape of Density Domains (D-shape). 

Consumption of homotopic sites (or subsites) is aselective because the distinct paths at such sites are isoenergetic at every point along the reaction coordinate, and the homotopic sites react at exactly the same rate. For example, the pig-liver esterase catalyzed hydrolysis of enantiopure dimethyl trfl s-l,2-cyclopentanedicarboxylate would be an aselective process, the two methoxycarbonyl groups are homotopic (C2 axis) and would react at equal rates. 

It is useful to generalize the model development procedure just described, adopting the idea of homotopy methods that have become important in the design of algorithms to solve various types of optimization problems. The basic idea is the construction of a continuous path connecting a difficult problem that we wish to solve with a simpler problem that we can solve easily. By following the path in sufficiently small steps from the simple problem, we ultimately obtain an approximate solution to the more complicated problem of interest. More specifically, two continuous functions f X Y and g X Y, axe said to be homotopic if there exists a continuous function 77 X x [0,1] "K such that 77(2 , 0) = f x) and 77(a , 1) = g x) [18, ch. 11]. The idea behind homotopy methods in minimization, for example, is to find a homotopy function 77(i, A) such that H x, 0) = f x) defines an easy minimization problem and H x,l) = g x) defines the minimization problem we would like to solve. This approach is useful in cases where we can construct a sequence of intermediate values satisfying... 

A loop path within a level set F(A) is a path starting and ending at the same nuclear configuration K. Some loop paths of F(A) can be deformed continuously into one another within the level set F(A), while some others cannot. Such continuous deformations are called homotopies. Those loop paths which can be deformed into one another are said to be homotopically equivalent and to belong to the same homotopy class. 

Homotopy theory is the theory of continuous deformations. The topological representation of reaction mechanisms by homotopy equivalence classes of reaction paths is based on the actual chemical equivalence of all those reaction paths that lead from some fixed reactant to some fixed product, and are "not too different" from one another. This "chemical" condition, combined with an energy constraint, corresponds to a precise topological condition two reaction paths are regarded as "not too different" if they can be continuously deformed into each other below some fixed energy value A, that is, within a level set F(A). Such paths are homotopically equivalent at energy bound A. This leads to a classification of all reaction paths a collection of all paths that are deformable into one another within F(A) is a homotopy equivalence class of paths at energy bound A. 

Consider two paths, qi and q2, within a level set F(A). These paths qjand q2 are homotopic relative to their endpoints (or, using a simpler but somewhat imprecise terminology, homotopic), if they have common origin as well as common extremity and if they can be continuously d ormed into one another within F(A) while keeping their endpoints fixed ... 

In general, a family of paths which are homotopic to one another form a homotopy equivalence class, or in short, a homotopy class. (In fact, a precise statement requires a reference to the preservation of endpoints these classes are homotopy classes relative to endpoints.) The homotopy class of all paths homotopic to some path p is denoted by [p]. 

Homotopical equivalence within each homotopy class implies that this product, if it exists, is unique and does not depend on the choice of reaction paths Pi, P2 P, representing equivalence classes [pj], [p2] e IT. 

The unit element [Kq] for these homotopy classes is defined as the homotopy equivalence class that contains the constant path p(u) = Kq. Since all these loop paths, when multiplied by their inverse paths, generate a loop path that is homo-topically contractible to Kq, therefore, these path-products are all homotopically equivalent to the constant path p(u) = Ko, so they must all belong to the same, and unique, homotopy equivalence class. 

However, in all instances, the paths p = (pi p2> ps and pB = pi (p2 Ps) are homotopically equivalent. Consequently, for the products of loop homotopy equivalence classes, the associativity condition applies ... 

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