A spatial-multimedia-compartmental approach to modeling the partitioning and intermedia fluxes of particle-bound and volatile organics in the environment was developed with emphasis on a detailed description of intermedia transport processes associated with the gaseous, dissolved, and particle phases. Based on this approach a spatial-multimedia-compartmental model (COSMCM) of chemical transport and fate was developed. The COSMCM is composed of eight compartments, namely, air—gaseous phase, air—particulate phase, water, suspended solids (in water), biota (in water), sediment, soil, and vegetation. The COSMCM includes detailed modules of rain scavenging of gaseous and particle-bound chemicals, dry deposition, wind erosion and resuspension of soil, rain infiltration, surface runoff, and resuspension and deposition of sediment particles. In addition, the COSMCM accounts for the dependency of atmosphere/soil and atmosphere/water intermedia transport on particle size and the dynamic changes in the atmospheric particle size distribution. Test cases with benzo(a)pyrene and pyrene distributions in the Los Angeles region revealed that the current approach is of sufficient accuracy and flexibility for estimating pollutant fluxes and multimedia partitioning.

Consider mereology axiomatized as in Clay [1965]*. Sobociński has posed the question, “If the usual definition of class, DM1, is replaced by $$\left[ {Aa} \right]\therefore A\varepsilon {\text{Kl}}\left( a \right). \equiv :A\varepsilon A:\left[ B \right]:a \subset {\text{el}}\left( B \right). \equiv .A\varepsilon {\text{el}}\left( B \right),$$ is the resulting system equivalent to the original?”. This note gives a negative answer. Theses A12 and A13, together with the two trivial models which follow them, show where the resulting system is weaker than mereology.

INTRODUCTION.Mereology, it may be recalled, is Lesniewski's system consisting of:(1) A system of propositional logic, upon which is based (2) A system for characterizing the meaning of 'is', upon which is based(3) A system for characterizing the relation of 'part' to the 'whole'.The partial system of mereology consisting of just ( 1) is called protothetic.The partial system consisting of (1) and ( 2) is called ontology.Up to now, the models of mereology that have been constructed have given an interpretation for the terms 'part' and 'whole' of (3) but have left the term 'is' of (2) uninterpreted (see [3]).In this paper we give the first model for mereology in which 'is* is interpreted as well.In other words, based on ontology, we have a model of mereology that includes a model of ontology.(2) consists of a primitive semantical category (logical type) called the category of names, a proposition forming functor, ε (read is), of two name arguments, an axiom system 0.[Aa].\Aεa.^:[

In section 1 we prove that a certain characterization of class can be proved without the aid of auxiliary definitions. In section 2 we show that the main results in [l] still hold in the weakened system constructed by replacing the original definition of class by the characterization given in section 1. In what follows we assume that the reader is acquainted with the Ontological Preliminaries in [l].

In this note we prove that for inductive finiteness,[a]: Fin{α}.D. Fin{st(α)}.Sobociήski proved this previously under the added hypothesis, dscr{α}.Theorems quoted, but not stated in this note, refer to the Mereological Preliminaries in [1], We shall also need the following well-known definitions and properties concerning inductive finiteness.

In this note we show that in the standard axiom system for mereology which follows, the reflexive axiom, M2, is dependent on M3, DM, M4, and M5. Ml. [AB):A ε el(B) .3 . B ε B. M2. [A]:Aε A.D.A ε el(A). M3. [ABC]:A ε e\(B). B ε el(c). =>. Aεel(C). DM. [Aa].-.A ε KI(α).=:A εA:[ΰ]:Dε a .D .Z) ε el(A): [D]: Dεel(A).Ώ.[ 1 EF].E εa .F ε e\(D).Fε el(E).M4. [Λα]:A ε β.D.[ 3 5].5εKl(«) .M5. [A5to]:A ε Kl(α

This paper deals with a formal system introduced by Lesniewski called mereology, in which, as the name implies, the concept of "party of the whole" is primitive.This system studies the properties of the collective class.Mereology is based on ontology, a formal system in which "is" is the primitive term.Ontology in turn is based on protothetic or on propositional calculus and quantification theory.The collective class differs greatly from the distributive class.However, under the condition, "the a's are weakly discrete", which we introduce, the collective class of the a's and the distributive class of the a's become alike with respect to equinumerosity.We are thus able to prove the analogs of three important set-theoretic theorems under this condition.Two of these were previously known for the condition, "the a's are discrete", but the third is an entirely new theorem.We then prove that for a certain class of statements dealing primarily with equinumerosity, discrete and weakly discrete are inferentially equivalent. ONTOLOGICAL PRELIMINARIESOntology has the following very intuitive sole axiom. [Aa]:.

For the m-valued (2m and based on the functions f(p, q) = max(1,q—p+1) and g(p) = m—p+1, Evans and Schwartz have essentially2 proved in [1] that the addition of a constant function i, 1

As a consequence of theorem 1, p.p. 385-387, in [ l ] , Tarski has proved that two unary relations and four binary relations are definable by purely logical means, and that in general, "for every natural number n only a specifiable finite number of n-termed relations between individuals can be defined by purely logical means, and each of these relations can be expressed by means of identity and the concepts of the sentential calculus." It is the objective of this note to specify the above mentioned number and to exhibit the relations for n 3. Let n be a fixed natural number and x^ x2, . . . , xn n individuals. Dl. If R (xj, x2, . . . , xn) is an w-ary relation definable in terms of identity and the propositional calculus, it is called a modulus.