Quotient set

In the chapter of Briiggemann and Carlsen some concepts introduced in the chapter of El-Basil are revitalized and explained in the context of the multivariate aspect. Basic concepts, like chain, anti-chain, hierarchies, levels, etc., as well as more sophisticated ones, like sensitivity studies, dimension theory, linear extensions and some basic elements of probability concepts are at the heart of this chapter. The difficult problem of equivalent objects, which lead to the items object sets vs. quotient sets are explained and exemplified. 

Equivalent objects in Hasse diagrams Different objects that have the same data with respect to a given set of attributes. Equality with respect to a given set of attributes defines an equivalence relation, 91 . Objects having the same values of all their attributes form disjoint subsets of A, the equivalence classes. An equivalence class with only one object is called a singleton and is called trivial. The equivalence classes can be considered as elements of a set, the quotient set El9E Usually the partial order is based on the quotient set and -if necessary- the equivalent elements are associated with that vertex, where a representative element out of the equivalence... 

A set that has an order relation is called a partially ordered set (poset). An order for a set, for example for the set E, is denoted by (E, <), the set E often being called the ground set (of objects). As the application of partial order, presented here, is based on attributes, just IB influences the partial order. Therefore, we often write (E, IB). If the quotient set is used, then we write (Em, IB). 

By scoring the data many equivalence classes (in fact 20) arise (vide infra). It is convenient to refer only to these classes by specifying a representative for each class Thus, besides the sensitivity study we apply the concept of quotient sets. With the equivalence relation 91 meaning equality in all five scores sFc, sCp, sCh, Smt and sGt, the following sediment samples appeared as equivalent, (Table 4) the quotient set being denoted as EhJ. ... 

Sometimes it is useful to refer to the quotient set, which is induced by the equivalence relation of equality, R (see for details Bruggemann and Bartel 1999). As usual we write E/R for the quotient set, and (E/R, IB) for the partially ordered quotient set. 

The application of weighting schemes as performed by METEOR is not the only way to get linear orders. Another possibility was found by (Winkler 1982) and worked out by Lerche and Sorensen (2003), Briigge-mann et al. (2004). The principle to get a linear order is first to find all order preserving maps of an empirical poset. Hereby a set of linear extensions, LE is found, where each element of this set is a single linear order preserving all <-relations of the empirical poset. The set LE can be very large. A very crude upper estimation of LT, the number of all linear extensions of an empirical poset is n , with n the number of all elements of the quotient set. The set LE of all linear extensions can be interpreted as probability space Let us assume that the rank of an object x, found for one specific linear extension, rk(x) has a certain value, Rk. Then the probability of x to get this value Rk is the number of linear extensions where rk(x) = Rk, L(rk(x) = Rk), divided by LT (see chapter by Briiggemann and Carl-sen, p. 86). We write... 

The basic relationships between solubility and pH can be derived for any given equilibrium model. The model refers to a set of equilibrium equations and the associated equilibrium quotients. In a saturated solution, three additional equations need to be considered, along with the ionization Eqs. (2a)-(2d), which describe the equilibria between the dissolved acid, base or ampholyte in solutions containing a suspension of the (usually crystaUine) solid form of the compounds ... 

Listed after the reactions are the corresponding equilibrium quotients. The law of mass action sets the concentration relations of the reactants and products in a reversible chemical reaction. The negative log (logarithm, base 10) of the quotients in Eqs. (3.1)—(3.4) yields the familiar Henderson-Hasselbalch equations, where p represents the operator -log ... 

Because the reaction quotient has a smaller value than the equilibrium constant, a net reaction will occur to the right. We now set up this solution as we have others, based on the balanced chemical equation. 

Finally, a measure of lack of fit using a PCs can be defined using the sum of the squared errors (SSE) from the test set, flSSETEST = Latest 2 (prediction sum of squares). Here, 2 stands for the sum of squared matrix elements. This measure can be related to the overall sum of squares of the data from the test set, SStest = -Xtest 2- The quotient of both measures is between 0 and 1. Subtraction from 1 gives a measure of the quality of fit or explained variance for a fixed number of a PCs ... 

The energy in variational calculations is obtained by minimizing the Rayleigh quotient. In the case of basis set (49), this quotient has the following form ... 

Here we just consider the set of closed points in the both hand side. Hence, the affine algebro-geometric quotient is considered as the set of closed GL (C)-orbits. This subject will be explained in Chapter 3 in more detail. 

The quotient space /r (0)/G is called a symplectic quotient (or Marsden-Weinstein reduction). It has a complex structure and natural Kahler metric (cf. Theorem 3.30) on points where G acts freely. On the other hand, the set of closed G -orbits is the affine algebro-geometric quotient and denoted by YjjG. In fact, it is known that the above identihcation intertwines the complex structures. 

It is easy to see a matrix has a closed orbit if and only if it is diagonalizable. Hence the set of closed orbits can be identified with the set of eigenvalues. On the other hand, a matrix B with [B, 5I] = 0 (i.e. a normal matrix) can be diagonalizable by a unitary matrix. Hence the quotient space is also identified with the set of eigenvalues. The identification can be seen directly in this example. 

Because bnite quotients are easier to handle than infinite quotients, it is useful to think of PZ7 (V) as a bnite quotient of the group SU V), the set of unitary transformabons from V to itself with determinant 1 (Debnibon 4.2). 

When X = C2, X can be identified with the set of GLJl(C)-orbits of (Hi, B2, i) where Bi, B2 are commuting n x n-matrices and i is a cyclic vector (Theorem 1.14). Many properties of (C2) are derived from this description. In Chapter 3, we shall regard the description as a geometric invariant theory quotient and a hyper-Kahler quotient. This description is very similar to the definition of quiver varieties which were studied in [62]. 

According to the principle of Le Chatelier. the reaction should go back to the left to partially offset the increase in dichromate, which appears on the right side of Reaction 6-7. We can verify this algebraically by setting up the reaction quotient, Q. which has the same form as the equilibrium constant. The only difference is that Q is evaluated with whatever concentrations happen to exist, even though the solution is not at equilibrium. When the system reaches equilibrium, Q = K. For Reaction 6-7. 

Given a scheme 5 and an open cover Uj) of 5, to give a locally free rank n quotient of Vv 0s is equivalent to give one such quotient over every open set Ut so that they patch together on the intersections Ut n Uj. Therefore 6 is a sheaf. 

For functions involving a combination of other elementary functions, we follow another set of rules if u and v represent functions fix) and g(x), respectively, then the rules for differentiating a sum, product or quotient can be expressed as ... 

---