Closure, property

Valve dimension is flat width. Valve width can he made less than top width without affecting closure properties. 

The completeness relation for a multi-dimensional wave function is given by equation (3.32). However, this expression does not apply to the wave functions vs,A for a system of identical particles because vs,a are either symmetric or antisymmetric, whereas the right-hand side of equation (3.32) is neither. Accordingly, we derive here the appropriate expression for the completeness relation or, as it is often called, the closure property for vs,a-... 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

This property is known as the closure property of the group. The group is said to be closed under the given law of composition. 

Now the expectation (mean) value of any physical observable (A(t)) = Yv Ap(t) can be calculated using Eq. (22) for the auto-correlation case (/ = /). For instance, A can be one of the relaxation observables for a spin system. Thus, the relaxation rate can be written as a linear combination of irreducible spectral densities and the coefficients of expansion are obtained by evaluating the double commutators for a specific spin-lattice interaction X in the auto-correlation case. In working out Gm x) [e.g., Eq. (21)], one can use successive transformations from the PAS to the (X, Y, Z) frame, and the closure property of the rotation group to rewrite D2mG(Qp ) so as to include the effects of local segmental, molecular, and/or collective motions for molecules in LC. The calculated irreducible spectral densities contain, therefore, all the frequency and orientational information pertaining to the studied molecular system. 

By using the closure properties of the eigenstates one obtains easily ... 

A number of relationships exist between the elements of symmetry of a point group which are a consequence of the closure property of groups. They may be used to identify difficult-to-locate symmetry elements. 

Data validation Level 3 Risk assessment Site investigation Confirmation for site closure Property transfer UST site closure Groundwater monitoring... 

In practice, all combinations of frequent subgraphs and frequent subsequences may have a lot of infrequent subgraph-subsequence pairs, and so we can use the downward closure property on the product graph of the two enumeration trees, which can be clearly stated as follows ... 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

It is now easily verified that the general properties (2)—(5) of a Lie algebra are formally satisfied by this commutator multiplication, so that only the closure property (1) needs to be verified in each particular case. It should also be noted that commutators arise naturally in quantum mechanics. 

In theory, an infinite number of calculations for highly excited states is required to complete the expansion of the EP given by Eq. (24), since there are only a few occupied valence orbitals in neutral atoms. This difficulty also exists in the nonrelativistic case and is resolved by using the closure property of the projection operator with the assumption that radial parts of EPs are the same for all orbitals having higher angular momentum quantum numbers than are present in the core. The same approximation is applicable in the present... 

The family of all possible homeomorphisms of three-dimensional space is a group Evidently, any two such homeomorphisms applied consecutively correspond to one such homeomorphism (closure property). The unit element is the identity transformation. Each homeomorphism has an inverse, and the product of homeomorphisms is associative. 

Among the transformations in family G one finds all the symmetry operations, but also all reflections in curved mirrors, nonlinear stretchings, and all continuous distortions of the space. Evidently, all possible homeomorphisms of the 3D space form a group G. Any two such transformations applied consecutively correspond to one such transformation (closure property) the unit element is the... 

Accordingly, we derive here the appropriate expression for the completeness relation or, as it is often called, the closure property for... 

The closure property, which is used to derive (2.2), can also be used to express the higher moments of the strength function in closed form, for example,... 

A given string can be recognized by a TS or it can be rejected, in which case we have a "failure". The various types of failures are described and the relations between them are Investigated. Some closure properties and decidability results follow. 

The set of all rotations, R(Q) satisfy the postulates which define a group. For example, the product of two rotations R1R2 is also a rotation R3 — R1R2. This is just the closure property. Thus we can think of any rotation as being decomposable into any number of smaller rotations. Repeated use of Eqs (7.C.2) and (7.C.3) then gives the result... 

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