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1959: Principles of Invariance on Discrete Spaces

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1959: Principles of Invariance on Discrete Spaces Visibility Laboratory- University of California Scripps Institution of Oceanography San Diego 52, California PRINCIPLES OF INVARIANCE ON DISCRETE SPACES Rudolph W. Preisendorfer August 1959 Bureau of Ships Index Number NS 714-100 Contract NObs-72092 SIO REFERENCE 59-55 Approved: Approved for Distribution: A- Seibert n. Duntley, Director <---J RogejS/RevelleJ^evelle,, Director Vjsibility Laboratory Scripps Institution of Oceanography Principles of Invariance on Discrete Spaces Rudolph W. Preisendorfer Scripps Institution of Oceanography, University of California La Jolla, California IT WILL BE SHOWN THAT The discrete-space approach to radiative transfer theory leads to a derivation of the principles of invariance. The principles of invariance which once were without any links to the main body of radiative transfer theory are shown to be derivable by elementary means from the local interaction principle. This latter principle was introduced earlier ^ and shown to possess sufficient detail to allow a formulation and solution of the basic transfer equations on every discrete space. The local interaction principle thus serves as a unifying principle from which the entire discrete theory of radiative transfer can be constructed. By suitably designed convergence arguments these results may be carried over to the continuous theory. Thus, in the final analysis, the principle of local interaction can form the single point of departure for the construction of all of radiative transfer theory. The plan of the present note is as follows: First it is shown that the principle of local interaction, which describes how radiant SIO Ref. 59-55 - 2 - flux is scattered and absorbed by a single point of a discrete space Xn , can be rearranged so as to give the same description for an arbitrary subset of points of Xn • This is the s0 called divisi­ bility property of the principle. The principle of invariant imbedding is then an immediate consequence, and from this follows the derivation of the principles of invariance. The paper closes with a discussion of seven outstanding problems of the continuous theory which may be studied using the techniques of discrete-space theory. SIO Ref. 59-55 - 3 THE DIVISIBILITY PROPERTY OF THE LOCAL INTERACTION PRINCIPLE We begin with a brief explication of: The Notion of a Divisible Property Although the notion of a divisible property is definable in quite general spaces, we may and shall restrict our discussion to spaces with a finite number of "points". Suppose, then, that Xn consists of a finite number r. of points. (For concretcness, Xo may be envisioned as some well-defined bounded subset of euclidean three-space E3, such as a cube.) We now manufacture a now space from Xn as follows: Let R- be a given partition of Xn . That is, Pj. is a collection of pairwise disjoint subsets of Xn whose Thus union is Xn • > ^-f^fi *^ » —j *ft } is a partition of Xn if XP. has p-i elements of Xn , Xpx- ancl X Pj have no elements in common for each distinct AP- and Xp- > an<^ ^r,^'~ * We now consider /^ as a space whose elements or points are the H Xp- 1 *• -/}•• •-> • Thus, a point of Pf is a subset of the original set Xn . For reasons which are based on established topological terminology, we shall call Pt the quotient space generated by /,, . Now suppose that Xn , the original space, has some property P For example, P may be: "The number of elements of Xn within every 3 ~) sphere of radius h is less than H ". Or, again, / may be: "Lvery pair of points of Xn may be connected by a line which does not contain any other members of Xn ," and so on. Suppose now that the property P is meaningful for a quotient space /^ generated SIO Ref. 59-55 - 4 - by Xn- (By suitably defining the notion of "sphere", "line" and "intersect" in the preceding examples, the two specific properties cited above can be shown to be meaningful for PH .) Then, if f* holds for ft- when it holds for Xn we say that P is a divisible property.* Thus a divisible property on a space X/>is one which may be inherited by the quotient space of Xn . We now tum to the particular property of current interest. In particular, we will show that The Local Interaction Property is Divisible Suppose the principle of local interaction1 is applicable to a discrete space Xn • For brevity we say that Xn has the local inter­ action property: P . We shall now show that P is a divisible property. Let P^ ~lXpf , Afyf-.-j Xph f be a partition of a discrete space Xn We may then (using the conventions and notation of reference l) rearrange the analytic statement of the principle of local interaction: * Mathematical readers may verify that such properties as compactness and separability supply examples of divisible properties for topological spaces and their quotient spaces. SIO Rof. 59-55 " 5 " • , • n (1) which is clearly a system of P equations in the r) unknowns IN *j , into a new form which is governed by the given partition, as follows: First, we observe that we may, without any loss of generality, x DC renumber the points of Xn , if necessary, so that Xp(~ \ 'p '-•>- f't j } = r jC i = X Xn Xf.2 1^ + ' i •• ) *i+i i} 5 •" t ^i- '(• n~ph+-/ »- ••.> • -> - Second, for every pair of (not necessarily distinct) elements Xp- . Xp- of ^ we define the I X Pi Pj vector N^feJas follows: where K| is the least X -subscript occurring in Xp- > ^z. — ^i +"' * |<^=Kj 4-1 ^ and Ofj£, f Xp. , etc. Furthermore, we define the I * P<* P" vector: N-CPJP;}*^*,*, ,NK4i,,..., N/^i, ,..*, NKl£^ ,.-• ; N^.i^.] . (3) The vectors (2) and (3) are clearly generalized forms of the 1 * D vectors N^_ and N_ defined in reference 1. For by letting \J — \ , so that />^ - p. =» ,9 , (2) and (3) :reduce to N+. and N- respectively. SIO Rcf. 59-55 " 6 " Third, we define the matrix Z( PK.J £ / Pj) , which is a diagonal matrix of the form: P«P.*ZiPj (frlPJP;) o »(4) 2.(Px(xK jp.) I where, in turn, 2(pK | JCKJp ), X^ eX^ , is a pK * Pj matrix whose element in the U t^ row, and vih column is Fourth, we designate by N°(PJL) the row vector of the form: (5) where .X ^ is the first element of L , and A.x is the number of K \ 'A. source directions for DCK ; OCk is the last element of Xp 1 "x and «AX is the number of source directions for ^/^p . Clearly u \ N °L?x) has (2!/ ^ (elements, where the summation runs over the p index integers for X^ . As the fifth and final step, we define the matrix 2T °(Pt \f>), which is a /?. - (^ ^ j y px. pj matrix of the form: SIO Rcf. 59-55 7 - o IW- (6) o ^^KJPJ') P.- p where, in tum, S°CxK.|P;) , ^^fXp. is an ( 2/ ^-) * j matrix (A,: is the number of source directions accessible to OC^^ ) whose element in the <JLth row and Vth column is: S C^K^S -^"KV 3 ^v'? E^, y(. is a source direction at "XK^ £ X P</ J ^^ -*V ^ X p,- * With these formulations the principle of local interaction is readily cast into its appropriate form for the quotient space P^ : > N+(?,|P) ) (7) i= 1,...^ ; j - I, •--> *- The formal identity of (l) and (7) is evident under the following pairings of concepts within each parenthesis: (Xn;P*-J , ( ^j j Xp. ) ? (2°C*;3S;, j fij), Z°(PilPj)). It follows that property /^ is divisible. An immediate corollary of the divisibility of the local inter- action property is that the system (7) of Y equations in the SIO Ref. 59-55 - 8 - e f unknowns ft/ f ( P; \ Pj ) , /, j = I,...,/-, is uniquely and explicitly solvable using the techniques and proofs developed in reference 1. We now develop some further corollaries of a rather interesting kind, corollaries which show how the divisibility property CAN BE USED IN HIERARCHIES OF DISCRETE SPACES First of all, we shall understand here that a hierarchy H-<* ( Xn J of discrete spaces is a finite sequence,of at least two terms,of successively constructed quotient spaces, whose initial member is some given discrete space X,-( ' fy* ~j Xn , %-t > . .,-?£ } • Thus Ph^ is the quotient space derived from its immediate predecessor in the sequence. The discussion of the preceding section explicitly demonstrated the divisibility of the local interaction property for the simple However tne hierarchy 2^ ( Xn ) ~ [Xn , P^J' > arguments leading to the conclusion of the divisibility of P were independent of the explicit magnitudes /] and /; ; in fact n and /-f were arbitrary, fixed integers. It follows that the general demonstration of the divisibility of p is immediately applicable, mutatis mutandis, to the hierarchy /ph P. j . In this way we have shown that: if P holds for a given discrete space Xn , ^ holds for every member of a given hierarchy 9^-^ ( Xn ) constructed from Xn • The consequences of this conclusion are manifold: they range from the practical level in which numerical answers to transfer problems are of primary interest up to the conceptual level concerned with the basic principles of radiative transfer theory. We shall however SIO Ref. 59-55 - 9 - reserve the practical consequences for separate and detailed discussion in subsequent notes, and concentrate at present only on general theoretical conclusions. In particular, we now discuss with the aid of a specific example the basic idea behind the principles of invariance as seen from the hierarchy point of view in discrete- space theory. Imagine a relatively extensive lattice Xnof points in E3, which consists of points O,^,/) ; oc , y , and £ are integers, with — qqcj ^ oc^^ 999 > ~ q<?ss: ?- ^ O • Here n= (aoo^y.
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