Propositions mathematical

It follows that 1/T is the integrating factor of SQ. Now since SQ is a function of two variables (in the simple case of a homogeneous fluid), and since the integrating factor of such a magni-" tude is usually also a function of the same two variables, we must regard the proposition that the integrating factor of SQ is a function of one variable only as expressing a physical, not a mathematical, truth. 

Most of the propositions in Chapters 2-4 are of independent value although, for the present book they are used only as part of the auxiliary mathematical apparatus. Some of them were known earlier and the rest were discovered and proven in recent years in connection with the rapid development of the theory of difference schemes. 

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

Proposition 1. All characters mentioned hitherto, with the exception of some mixed-types and some Rawls-types, are sensible. Furthermore, arbitrary intersections and unions of sensible types are also sensible. (For reasons of space, proofs of propositions have had to be omitted they are contained in a mathematical supplement, available on request from U. Krause). 

C ). (Some readers may already know that spin-1 spin states are described by vectors in C others might see Section 10.4.) We will use tensor products in Proposition 7.7, our mathematical description of the elementary states of the... 

Proposition 6.11 implies that irreducible representations are the identifiable basic building blocks of all finite-dimensional representations of compact groups. These results can be generalized to infinite-dimensional representations of compact groups. The main difficulty is not with the representation theory, but rather with linear operators on infinite-dimensional vector spaces. Readers interested in the mathematical details ( dense subspaces and so on) should consult a book on functional analysis, such as Reed and Simon [RS],... 

In this section we have verified mathematically what physicists have tested with long use. In spherically symmetric problems in L ( R ), the spherical harmonics of various degrees are the sensible building blocks they leave nothing out (Proposition 7.5) and they have no substitutes (Proposition 7.6). 

As far as experiments have been done, the state of a spin-1/2 particle is completely determined by its probabilities of exiting x-, y- and c-spin up from Stern-Gerlach machines oriented along the coordinate axes. This fact is consistent with the mathematical model for a qubit, as the following proposition shows. 

In fact, no measurement would show entanglement, as we shall prove once we have found a convenient mathematical description of entanglement in Proposition 11.1. 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

The propositional logic expressions can be expressed in an equivalent mathematical representation by associating a binary variable yi with each clause P. The clause P being true, corresponds to yi = 1, while the clause P, being false, corresponds to yt = 0. Note that (-iPj) is represented by (1 — 3/i). Examples of basic equivalence relations between propositions and linear constraints in the binary variables include the following (Williams (1988)) ... 

On the interior wall of the first circuit all the mathematical figures are conspicuously painted — figures more in number than Archimedes or Euclid discovered, marked symmetrically, and with the explanation of them neatly written and contained each in a little verse. There are definitions and propositions, etc. On the exterior convex wall is first an immense drawing of the whole earth, given at one view. Following upon this, there are tablets setting forth for every separate country the customs both public and private, the laws, the origins and the power of the inhabitants and the alphabets the different people use can be seen above that of the City of the Sun. 

The starting point for the mathematical description of diffusion in membranes is the proposition, solidly based in thermodynamics, that the driving forces of pressure, temperature, concentration, and electrical potential are interrelated and that the overall driving force producing movement of a permeant is the gradient in its chemical potential. Thus, the flux,. /,(g/cm2 s), of a component, i, is... 

For polysaccharide dispersions, SV is exceedingly small relative to Vi. Equations (3.11) and (3.12) are mathematical propositions that the exchangeable energy stored in a dispersed polysaccharide solute is equal to the energy absorbed from an external source and any increase in surface area of the solute is consequently a repository of +A . Conversely, aggregation and desorption correspond to a loss of energy, felt as heat in the latter occurrence ( —A ) when a dry polyaccharide powder is wetted (positive adsorption). 

THE PHILOSOPHY OF MATHEMATICS An Introductory Essay, Stephan Komer. Surveys the views of Plato, Aristotle, Leibniz 8c Kant concerning propositions and theories of applied and pure mathematics. Introduction. Two appendices. Index. 198pp. 54 x 84. 25048-2 Pa. 5.95... 

Proof. A term from logic and mathematics describing an argument from premise to conclusion using strictly logical principles. In mathematics, theorems or propositions are established by... 

Benzenoid (chemical) isomers are, in a strict sense, the benzenoid systems compatible with a formula C H, = (n s). The cardinality of C HS, viz. C HS = n, s is the number of isomers pertaining to the particular formula. The generation of benzenoid isomers (aufbau) is treated and some fundamental principles are formulated in this connection. Several propositions are proved for special classes of benzenoids defined in relation to the place of their formulas in the Dias periodic table (for benzenoid hydrocarbons). Constant-isomer series for benzenoids are treated in particular. They are represented by certain C HS formulas for which n s = In Sjl = n2 52 =. .., where (nk sk) pertains to the k times circumscribed C HS isomers. General formulations for the constant-isomer series are reported in two schemes referred to as the Harary-Harborth picture and the Balaban picture. It is demonstrated how the cardinality n s for a constant-isomer series can be split into two parts, and explicit mathematical formulas are given for one of these parts. Computational results are reported for many benzenoid isomers, especially for the constant-isomer series, both collected from literature and original supplements. Most of the new results account for the classifications according to the symmetry groups of the benzenoids and their A values (color excess). 

This book presents in a popular manner the elements of game theory—the mathematical study of conflict situations whose purpose is to work out recommendations for a rational behaviour of each of the participants of a conflict situation. Some methods for solving matrix games are given. There are but few proofs in the book, the basic propositions of the theory being illustrated by worked examples. Various conflict situations are considered. To read the book, it is sufficient to be familiar with the elements of probability theory and those of calculus. 

The hypothesis of Proposition 3.2 excludes the case Ai = A2. This is ordinarily not biologically important because the A, are computed from measured quantities it is unlikely that they would be exactly the same (or the same with respect to this environment). However, an interesting potential application is the case where the organisms are indeed the same (mutants of the same organism) except for their sensitivity to the antibiotic. Intuitively, if the organisms are the same except for sensitivity to the inhibitor, one expects the X population to lose the competition when the inhibitor is present. However, establishing this mathematically cannot be done directly from the comparison theorem as used before, since if A] = A2 then coexistence occurs with the chemostat equations used for comparison purposes. In order to use the comparison principle, one needs a better estimate than f(p) <1, p>0. This is the purpose of the following lemma. 

Greenwood (1989) stated that the word model should be reserved for well-constrained logical propositions, not necessarily mathematical, that have necessary and testable consequences, and avoid the use of the word if we are merely constructing a scenario of possibilities. A scientific model is a testable idea, hypothesis, theory, or combination of theories that provide new insight or a new interpretation of an existing problem. An additional quality often attributed to a model or theory is its ability to explain a large number of observations while maintaining simplicity (Occam s razor). The simplest model that explains the most observations is the one that will have the most appeal and applicability. 

The influence of Greek geometry on the mathematics communities of the world was profound for in Greek geometry was contained the ideals of deductive thinking with its definitions, corollaries, and theorems which could establish beyond any reasonable doubt the truth or falseness of propositions. For an estimated 22 centuries. Euclidean geometry held its weight. 

With the basic equivalent relations given in Table I (e.g., see Williams, 1988), one can systematically model an arbitrary propositional logic expression that is given in terms of OR, AND, IMPLICATION operators, as a set of linear equality and inequality constraints. One approach is to systematically convert the logical expression into its equivalent conjunctive normal form representation, which involves the application of pure logical operations (Raman and Gross-mann, 1991). The conjunctive normal form is a conjunction of clauses, gi gj A A gj. Hence, for the conjunctive normal form to be true, each clause g, must be true independent of the others. Also, since a clause g, is just a disjunction of literals, Pj V V " V t can be expressed in the linear mathematical form as the inequality. 

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