Semi-groups

Trotter H F 1959 On the product of semi-groups of operators Proc. Am. Math. Soc. 10 545-51... 

In a formal sense, a group must meet certain criteria. Collections of sets and operations that meet some of the required criteria my meet the lesser demands of semi-groups or groupoids. The groupoids demand only a set and an operation between them (10). Conditions (1) and (3) will sometimes not be met (14, 15). Hence, even though our theory is incomplete, the definition of a groupoid most closely applies to the present study. 

Other than for the classical case of the finite dimension, it is known that relation (21) is true for compact or analytic semi-groups. This covers the case of Newtonian flows, where the associated system of partial differential equations is parabolic. In this case, one also has the implication (S3) (Si) . (See [58] for instance.)... 

The proof of Theorem 5.3 consists in showing that the underlying semi-group is analytic (because of the degeneracy of the equation for a and t in (22)), and then in localizing the spectrum by the Routh-Hurwitz criterion. 

In addition to the continuous spectrum there is an infinite number of discrete eigenvalues which axe essentially lined up along a parallel to the imaginary axis— this is another illustration of the fact that the underlying semi-group is not analytic. 

M. Renardy, On the type of certain Co semi-groups. Comm. Part. Diff. Eq., 18 (1993) 1299-1307. 

In the theory of semi-groups, it is proved that the most general acceptable evolution equation has to be of the form (9), [Lindblad 1976 (b)], which is called the Lindblad form. 

Here, 1,1 is a one-parameter semi-group element and the state obeys the master equation given by its generator C,... 

Since it is not possible to reverse the direction of time so as to convert a later state into an earlier one, no inverse operation is possible in the chemical reaction system such systems are described by semi-groups rather than by groups. 

MeSeCH2CHjCH2)jSe IPdOjL] bidentate with one SeMie group not cootdioaled 60... 

Knopfinacher and Warlimont, following earlier work, use much more esoteric mathematical machinery (concerning semi-groups of polynomial-like structures) to develop methods to determine the asymptotic (large-A) form of isomer counts for A-atom alkanes. Polya " in fact developed asymptotic results, but Knopfinacher and Warlimont are interested in extensions, particularly to obtain asymptotics for the numbers of generalized isomers , viewed as sets of molecules such that different sets as a whole have the same elemental composition. Their main result is a theorem on the asymptotic number of ways in which a mixture of molecules may be realized when the total number of particular kind of atoms is fixed, and the asymptotic results for ordinary single-molecule isomers is already known. Examples of application to mixtures of alkanes, alkenes, substituted alkanes and achiral alkanes and alkanols are given. 

E. Hille and R. S. Phillips, Functional analysis and semi-groups. New York, 1957, Chapters XI-XII. 

E. Hille, Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publications, vol. 31, 1948. 

A more complete study of A shows that the Hille-Yosida theorem [8] applies and does indeed insure the existence of a semi-group of bounded operators T(t) providing the unique solution to the problem posed. The solution may be represented in the form of a Laplace integral. 

The successive operations represented by the arrows can be associated with a mathematical group, called the renormalization group (actually a semi-group because we cannot define a unique inverse to the operations). 

Summarizing the loss of information, i.e. restricting the full unitary time evolution to an isometry (weak convergence) before the conversion via a suitable Lyapunov converter to a contractive semi-group (strong convergence) is an objective process in contrast to the subjective preparation of any initial state involving various... 

Coleman, B. D., and Mizel, V. J., "Norms and Semi-Groups in the Field of Fading Memory," Arch. Rat. Mech. Anal., 23,... 

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