Bounding hyperplanes

An alternative method for computing S exists that relies on computing a basis for the null space of the stoichiometric matrix A. Vectors belonging to null (A ) are normal to S, and these vectors may be related to bounding hyperplane constraints that define S. This method is described in Feinberg (1987,2000a, 2000b). 

The central approach to constmction is to iterate over each comer of the current polytope P introduce new hyperplanes that eliminate unattainable space. Sharp corners of the polytope are slowly smoothed out by the introduction of additional bounding planes, and the level of accuracy obtained is hence a strong function of the number of unique hyperplanes that are introduced. Curvature of a region, such as that generated by a PFR manifold on the AR boundary, may be approximated by the use of many bounding hyperplanes. 

Figure 8.21 Graphical representation of the method of bounding hyperplanes. Hyperplanes are introduced at the comers of the polytope.

Figure 8.22 Rate vectors evaluated relative to a bounding hyperplane.

Three distinct classes of rate vectors, relative to the bounding hyperplane H(n, b), can be identified in Figure 8.22 ... 

S.2.2 The Algorithm The actual method of bounding hyperplanes may now be described. The following algorithm describes a general overview of the method ... 

AR constructions of systems involving five independent reactions have been investigated using the method of bounding hyperplanes (Abraham, 2005). Systems involving a larger number of independent reactions are difficult to construct due to the increased computational complexity at present, which is primarily due to the evaluation of points obtained from higher dimensional Cartesian products. 

Figure 8.23 Summary of construction results for the three-dimensional Van de Vusse kinetics, using the method of bounding hyperplanes.

Candidate AR construction via bounding hyperplanes (either via translation or rotation) may be extended to allow for constructions involving a parameter that may affect the direction of the rate vector. A system involving temperature-dependent kinetics is a well-known example of this. In Figure 8.26(b), the rotated hyperplanes method is used to compute an AR with temperature-dependent kinetics. At each point of evaluation, a temperature range between 300 and 1000 K is generated and rate vectors are checked for tangency with the hyperplane. This allows for... 

Figure 8.24 Computing candidate ARs using bounding hyperplanes, rotated about an edge.

Figure 8.25 Comparison of constructions between (a) bounding hyperplanes and (b) rotated hyperplanes.

Ming, D., Hildebrandt, D., Glasser, D., 2010. A revised method of attainable region construction utilizing rotated bounding hyperplanes. Ind. Eng. Chem. Res. 49, 10549-10557. 

We have seen that the output neuron in a binary-threshold perceptron without hidden layers can only specify on which side of a particular hyperplane the input lies. Its decision region consists simply of a half-plane bounded by a hyperplane. If one hidden layer is added, however, the neurons in the hidden layer effectively take an intersection (i.e. a Boolean AND operation) of the half-planes formed by the input neurons and can thus form arbitrary (possible unbounded) convex regions. ... 

Since (5.299) is solved in mixture-fraction space, the independent variables are bounded by hyperplanes defined by pairs of axes and the hyperplane defined by X = K, = 1. At the vertices (i.e., V = = (0, and e, (/el,..., AW)), where e, is the Cartesian unit vector for the /th axis), the conditional mean reaction-progress vector is null 121... 

There are very few examples of scalar-mixing cases for which an explicit form for (e, 0) can be found using the known constraints. One of these is multi-stream mixing of inert scalars with equal molecular diffusivity. Indeed, for bounded scalars that can be transformed to a mixture-fraction vector, a shape matrix can be generated by using the surface normal vector n( ) mentioned above for property (ii). For the mixture-fraction vector, the faces of the allowable region are hyperplanes, and the surface normal vectors are particularly simple. For example, a two-dimensional mixture-fraction vector has three surface normal vectors ... 

For the n-dimensional case, the region that is defined by the set of hyperplanes resulting from the linear constraints represents a convex set of all points which satisfy the constraints of the problem. If this is a bounded set, the enclosed space is a convex polyhedron, and, for the case of monotonically increasing or decreasing values of the objective function, the maximum or minimum value of the objective function will always be associated with a vertex... 

The unlabeled triangle is the simplex in E (2-simplex) and the unlabeled tetrahedron is the simplex in (3-simplex) evidently, whether enantiomorphous -simplexes can be partitioned into homochirality classes depends on the dimension of E". Recall that an /j-simplex is a convex hull of + 1 points that do not lie in any (n - l)-dimensional subspace and that are linearly independent that is, whenever one of the points is fked, the n vectors that link it to the other n points form a basis for an n-dimensional Euclidean space An n-simplex may be visualized as an n-dimensional polytope (a geometrical figure in E" bounded by lines, planes, or hyperplanes) that has n + vertices, n n + )/2 edges, and is bounded by n + 1 (u — l)-dimensional subspaces. It has been shown that the homochirality problem for the simplex in E is shared by all -sim-... 

Enantiomeric control is more difficult if the excited molecular potential energy surfaces do not possess an appropriate minimum at the o hyperplane configurations (see Figs. 1 and 2). In this case the method introduced in this section is not applicable. One may however be able to apply the laser distillation procedure by adding a molecule B to the initial L, D mixture to form weakly bound L — B and B - D, which are themselves right- and left-handed enantiomeric pairs [83]. The molecule B is chosen so that electronic excitation of B — D and L — B forms an excited species G, which has stationary rovibrational states that are either symmetric or antisymmetric with respect to reflection through t7>,. The species L - B and B - D now serve as the L and D enantiomers in the general scenario above, and the laser distillation procedure described above then applies. Further, the molecule B serves as a catalyst that may be removed from the final product by traditional chemical means. 

In Figure 6.5(a), a convex region is shown. A plane is introduced so that it meets the surface joined by extreme points ABCD. In this instance, points contained in the rectangular region bounded by ABCD are not considered as exposed points, for all points in this section are contained within the same hyperplane. (Points A, B, C, and D are all extreme points but they are not exposed points because all four points lie in the hyperplane and thus the hyperplane is not supported at a single point.)... 

A candidate AR construction method that utilizes hyperplanes to carve away unachievable space shall be discussed in Section 8.5.2. Linear constraints, such as non-negativity constraints on component concentrations and flow rates, may also be expressed in the form of a hyperplane equation. Hyperplanes therefore also arise in establishing bounds in state space. In Section 8.6, superstructure methods shall be described for the computation of candidate ARs. These methods, at their core, rely on the solution of a large... 

Hereafter, it is assumed that the reader is comfortable with the idea of the stoichiometric subspace S, which is discussed in Chapter 6. In this section, we shall describe how to numerically calculate the bounds of S (which is a convex polytope that is expressed as a collection of hyperplane constraints). 

Theorem 2.3 Let vectors x e belong to a sphere of radius R. Then the set of -margin separating hyperplanes has the VC dimension h bounded by the inequality... 

Adaptive-rejection-sampling for multiple parameters would be based on the idea of bounding the logarithm of a log-concave target by tangent hyperplanes. While this would work in theory, the boundaries of the envelope function would become extremely complicated. In practice, adaptive-rejection-sampling only works for a single dimension parameter. 

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